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Chem Rev. 2009 May 13; 109(5): 1793–1899. Published online 2009 March 5. doi: 10.1021/cr030440j. | PMCID: PMC2682929 |
Modeling Kinetics of Subcellular Disposition of Chemicals Stefan Balaz Department of Pharmaceutical Sciences, College of Pharmacy, North Dakota State University, Fargo, North Dakota 58105 Received March 24, 2006. This is an open-access article distributed under the ACS AuthorChoice Terms & Conditions. Any use of this article, must conform to the terms of that license which are available at http://pubs.acs.org. The effects of low-molecular-weight (<1000 g/mol) chemicals on biosystems ranging from membranes to organisms are of interest in medicinal chemistry, chemical biology, pharmacokinetics, pesticide science, environmental toxicology, and several other areas of science and technology. Most effects are difficult to predict, because of their heavy dependence on the chemical structures. The effects in homologous series in biosystems containing at least one membrane, however, are often (6174 cases documented) 1,2 a smooth function of the physicochemical properties of chemicals, most frequently represented by the reference partition coefficients (4223 cases), 1,2 but also by electronic parameters and steric characteristics. The structure-nonspecific effects described by nonlinear dependencies on the reference partition coefficients (553 cases) 1,2 are mostly related to the passive distribution of chemicals in the aqueous and less-polar phases of biosystems rather than to the structure-specific interactions with the receptors, enzymes, and transporters. The disposition-related homologous series data provide initial outlines of the relationships between structure and disposition, equivalent to the cross sections along one variable through complex dependencies. Can the relationships be extended to cover broader classes of chemicals or eventually the entire chemical universe? The research area that is striving to understand the principles of distribution of chemicals at the subcellular level of biosystems, in terms of the structures and properties of chemicals, can be appropriately called structure-based subcellular pharmacokinetics (SBSP). Theoretical fundamentals, status, applications to experimental data, and probable future development trends of SBSP are reviewed here. Numerous reviews are available on different aspects of distribution and how it is related to the properties of the chemicals, 3–25 and the subject is covered, to some extent, in several monographs on quantitative structure−activity relationships (QSARs), 26–32 drug design, 33–41 pesticide design, 42,43 and medicinal chemistry. 44–50 This treatise introduces several new concepts and presents a systematic, critical, integrative, and chemically oriented view. To foster readability, a unified nomenclature is used throughout the text, and the details of the model development are deferred to section . These details can be skipped without a major impact on the understanding of the remaining content. For a quick, semi-quantitative overview of the disposition of chemicals in biosystems and its relation to structure of chemicals, the reader is referred to the following psrts of the paper. In section , relevant physiology is concisely summarized and prevailing transport routes in individual organs are identified. Section relates the trans-bilayer transport rates to the interactions of chemicals in the bilayer regions, which can be characterized using surrogate solvent systems, as described in section . These results are used in section to obtain coarse estimates of equilibration times for crossing a single bilayer on a gross time scale, in regard to its dependence on the structure of the chemicals. 1.1. Subcellular Pharmacokinetics The main goal of structure-based subcellular pharmacokinetics (SBSP) is a model-based description of the kinetics of the distribution of chemicals, in terms of the properties of both chemicals and biosystems. For this purpose, the conceptual kinetic models are constructed, comprising physically distinct subcellular compartments such as membranes or their regions, and the extracellular and intracellular aqueous phases. The considered processes include transport and accumulation in a set of the aqueous phases and membranes, as well as protein binding, metabolism, hydrolysis, and other reactions of chemicals with body constituents. The resulting differential equations are solved either numerically (see section ) or explicitly (see sections and ), depending on the complexity of the model. To simplify the solutions, the complexity of mathematical description of the subcellular models is often reduced in the process of solving the pertinent differential equations, using the experimentally verified time hierarchy of the processes that determine the disposition of chemicals. Individual rate and equilibrium parameters are related to the computed characteristics or experimentally determined physicochemical properties by the extra-thermodynamic linear free-energy relationships (LFERs). 51,52 The properties of chemicals characterize their behavior in water (hydration, ionization, spontaneous hydrolysis rate) or in macroscopic surrogate systems imitating parts or processes in the biosystem, which are either difficult to analyze directly, or for which generalizations are sought (reference partitioning systems (section ), protein binding (section ), in vitro enzymatic reactions (section ), and surrogate chemical reactions (section )). Like most other models of biological processes, the SBSP models belong to the category of a posteriori semi-empirical models, 53,54 because the modeled system is not known in sufficient detail to allow for the formulation of a priori theoretical models. The unknown properties, which do not change under given experimental conditions and are difficult to measure, are usually collected in adjustable coefficients, which are then optimized by regression analysis to provide the best agreement between the model and experiment. To formulate tractable models, decisive features governing the behavior of the system must be identified and captured in the description. For a meaningful optimization of the regression coefficients in the SBSP models, a proper balance between the number of adjustable coefficients and the information content of available experimental data is required. In the following, every attempt will be made to follow these principles in the descriptions of chemicals’ disposition. Individual steps in the SBSP model construction are analyzed in section . The functional form, in which the time is incorporated in the SBSP models, has been repeatedly proven by classical pharmacokinetics. Therefore, a proper application of the SBSP concepts to the chemicals’ disposition in organisms should result in expressions that provide similar time dependencies as the equations of classical pharmacokinetics (sections and ). As the ultimate outcome of the SBSP modeling, the kinetics of disposition is expressed as a nonlinear disposition function of properties, with adjustable coefficients containing the biological and chemical attributes, which do not vary under given experimental conditions. The properties are represented by the reference partition coefficients, acidity, reactivity, and other parameters, which may also include three-dimensional (3D) structures of chemicals and structure-related characteristics. The disposition function is calibrated using experimental data of, typically, several dozens of compounds, and then used for the prediction of distribution of untested chemicals under identical experimental conditions. The SBSP models have a better predictive ability than empirical models, especially outside the ranges of tested properties( 55) (section ). Finally yet importantly, the conceptual nature of the SBSP models facilitates their flexible extensions accommodating some changes in experimental protocols (e.g., section ). SBSP has its roots in the distribution-based quantitative structure−activity relationships (QSAR or QSTAR, if the exposure time, T, is included), as defined by the pioneering works of Hansch and Fujita,( 56) Stehle and Higuchi,( 57) Penniston,( 58) McFarland,( 59) Yalkowsky and Flynn, 60,61 Leo,( 62) Martin,( 63) Kubinyi,( 64) Dearden,( 65) van de Waterbeemd,( 66) Cooper,( 67) Seydel,( 68) Schaper,( 69) Franke,( 70) Berner,( 71) Amidon,( 72) Anderson and Xiang,( 73) Benet,( 74) and others. SBSP is also related to classical pharmacokinetics,( 75) with which it shares the representation of biosystems as a set of compartments. However, the compartment size has been reduced to subcellular dimensions, with the SBSP compartments representing the membranes or their regions, and extracellular and intracellular aqueous phases. The nanoscale dimensions ensure fast diffusion of chemicals inside the less-structured subcellular compartments (section ). Fast diffusion makes the SBSP models suitable for modeling the effects of chemicals by facilitating a coherent description of their concentrations in the immediate surroundings of the receptors, as known in chemical kinetics. The rate/equilibrium constants, characterizing elementary steps in the subcellular distribution, are related to physicochemical properties of chemicals via the extra-thermodynamic LFERs, as known in the QSAR research and physical organic chemistry. 51,52SBSP relies on other disciplines for methodological support: principles of dynamic modeling( 53) are used to simplify the models; computational chemistry provides simulation techniques and molecular characteristics; chemical kinetics supplies some modeling schemes and approximations; physical organic chemistry lends LFER; biophysical methods are used to study transport; and biochemical approaches deliver the binding data for the interactions of chemicals with body constituents. The interplay between SBSP and other disciplines is outlined in Figure . A special effort has been made to compare and integrate, where feasible, the approaches to the analysis of the same phenomenon in different areas. Experimental techniques providing data for the calibration of the SBSP models can be classified as indirect and direct, depending on whether the phase, where the concentration of chemicals is determined, has been separated in the course of the experiment from the rest of the biosystem or not, respectively. The indirect techniques include measurement of the chemical’s uptake into membranes,( 76) vesicles,( 77) subcellular organelles,( 78) cells,( 79) tissues,( 80) and organs.( 81) The direct approaches, suitable for simpler biosystems, are represented mainly by spectroscopic techniques: ultraviolet and visible (UV Vis) difference spectroscopy, 82–85 circular dichroism, 84–86 fluorescence lifetime measurements,( 87) fluorescence quenching techniques, 88–95 fluorescence anisotropy,( 96) Fourier-transform infrared (FT-IR) spectroscopy( 84) combined with attenuated total reflection (ATR)( 97) and high pressure,( 98) electron paramagnetic resonance (EPR) with phospholipids containing spin labels in different positions, 99–104 and various nuclear magnetic resonance (NMR) techniques, 84,87,90,99,105–129 including the measurement of nuclear Overhauser enhancement spectroscopy (NOESY) 130,131 cross-relaxation rates.( 127) Other direct techniques include neutron diffraction, 132–138 small-angle X-ray diffraction (XRD),( 139) autoradiography,( 140) and fluorescence microscopy. 141,142The time courses of the concentrations of chemicals in individual cells and their organelles are difficult to measure precisely, and the intracellular data for the calibration of the SBSP models are quite scarce. For this purpose, the uptake and release kinetics of chemicals, monitored in the extracellular media, are often used. This is a rigorous approach, because the extracellular and intracellular concentrations are variables of the same model and any of them can be, in principle, used for the model calibration. Although this statement is not valid for all types of compartment models, it does hold for the SBSP models. Another option is to use a biological effect, which has been assumed or shown to be (i) an immediate consequence of the chemical’s presence in a biological phase,( 143) and (ii) proportional to the chemical’s concentration,( 144) as a measure of the time course of the chemical’s concentration in the relevant biological phase. According to the dependence on the detailed molecular structures of chemicals, the processes in which chemicals participate in biosystems can be classified in two groups: (i) disposition of chemicals in biosystems, except protein-mediated transport and enzymatic biotransformation; and (ii) interactions of chemicals with specific macromolecules, which are responsible for biological effects, mediated transport, and biotransformations of the compounds. The processes in the first group can be fully characterized by properties of chemicals in aqueous solutions (ionization, hydrolysis) and surrogate systems (e.g., partitioning, reactivity, redox potential). The processes in the second group depend predominantly on the molecular structure of compounds, given as the 3D constitution of atoms, and a spatial distribution of the structure-related properties. These processes can be examined in vitro, using the relevant macromolecule(s). The first-group processes governing disposition of chemicals, except enzymatic biotransformation and protein-mediated transport, are considered property-related or conformation-averaged processes, and the second-group interactions with the fate- and effect-determining macromolecules are deemed structure-specific or conformation-dependent processes. The property-related processes are easier to model and the models can be generalized for all biosystems. The conformation-dependent processes convey structural specificity to the disposition of chemicals and must be modeled individually for each biosystem. The overall description of disposition for diverse chemicals requires that the results for conformation-dependent processes are embedded in the models for property-dependent processes. With the improvement of experimental techniques, multiple binding modes are detected with increased frequencies in some binding processes.( 145) The processes still retain a significant level of structural specificity and require specific treatment in QSAR modeling.( 146) The methods for descriptions of both process types have been developed separately. SBSP conceptually combines modeling of both the property-related processes and the structure-specific processes, as illustrated in eq 7 in section . Therefore, SBSP represents a suitable framework for integrated description of the fates of chemicals in biosystems. One of the outcomes is the cell-QSAR approach, which extends, by incorporating the disposition function, conceptual 3D-QSAR techniques for the application to cell-level assay data. 1.2. Disposition in Organisms To elicit an effect in a biosystem, a chemical must reach the site of action, frequently located far away from the points of entry, which, for higher organisms, are typically the gastrointestinal and pulmonary tracts, skin, mucosa, and cornea. We will first focus on the physiologic aspects, and later, in section , cover the mechanisms of the transport processes. In higher organisms, the first step is absorption, which brings the molecules from the point of entry or the site of administration into the bloodstream or the lymphatic system. Although the lymph circulation is much slower than that of the blood,( 147) very lipophilic chemicals may use the former route for distribution, piggy-backing on the transport systems of lipophilic nutrients. 148–150 These compounds may also exhibit slow absorption into the blood and lymph capillaries in the villi, and enter the lymph via endocytosis in the Peyer’s patches in ileum, especially in lipid-based formulations. 151–160Absorption mostly proceeds by the transcellular route, with the molecules crossing epithelia and other tissues by passive and active transport through the bilayers and cell interiors. An exception is the absorption through the skin, where the only cells in the external stratum corneum layer are keratinized corneoytes.( 161) Corneocytes are embedded in a lamellar phase of lipids,( 162) with the composition differing from that of other membranes. Absorption through stratum corneum is assumed to proceed mainly by passive lateral diffusion( 163) through the lipid phase.( 164) Could a similar passage through a cell layer by lateral diffusion in the bilayers,( 165) without entering cell interiors, be accomplished by amphiphilic molecules in other tissues? This does not seem to be a widespread mechanism. Tight junctions would need to be bypassed by the flip-flop to the cytoplasmic side of the bilayer, followed by the return to the outside leaflet before the release into medium on the opposite site of the cell layer. Moreover, for many amphiphilic compounds, the flip-flop is a slow process (section ). Once in the blood or lymph, chemicals quickly reach distant body parts by convection and diffusion, although circulation of the lymph is slower. The details of vasculature have been long known in great detail, starting with the description of capillaries by Malpighi in 1661, the capillary walls by Schwann in 1839, and the cellular nature of the capillary walls by von Recklinghausen in 1862, as well as the introduction of the term “endothelium” by His in 1865.( 166) The distribution of chemicals from the biological fluids into tissues involves (i) paracellular transport into interstitial fluid directly mixing with the content of blood and lymph capillaries and (ii) transcellular transport bringing the molecules of chemicals into the cells. In the hierarchy of the vasculature, we shall focus on the capillaries, because (i) their walls are composed of a single layer of endothelial cells and have openings in some organs, in contrast to larger blood vessels, which have continuous and more-complex structures, and are more difficult to permeate; and (ii) they represent the majority of ~300 m 2 of the vascular endothelium surface and of ~19  000 km of overall length of blood vessels in a standard 70-kg man.( 167) The walls of blood capillaries in individual organs differ in porosity (Table 1). Continuous capillary walls with endothelial cells connected by tight junctions and supported by a complete basement membrane are encountered in places requiring a barrier protection, as in brain (but not choroid plexus), lungs, intestine, placenta, and testis. Continuous endothelia are still capable of paracellular transport of small molecules, with diameters of <3 nm, such as urea and ions.( 168) The endothelial junctions are regulated by various signaling mechanisms and can operatively respond to physiological conditions.( 168) The pores in capillary walls are associated with specific functions. Renal glomerulus, serving for plasma filtration in the kidney, exhibits 30−60 nm openings between the endothelial cells, which are supported by a complete basement membrane. The most porous capillaries allow the transcellular passage of macromolecules and even some cells, thanks to the broken basement membrane and the openings with the diameters reaching 100−300 nm in the bone marrow, spleen, and sinusoids of the liver, 169–171 and up to 1.2 μm in tumors.( 172) For a point of reference, the average capillary diameter is only slightly larger than the average diameter of erythrocytes (7.5 μm),( 173) and the minimum capillary diameter is ~4 μm.( 167) Lymph capillaries are also porous.( 167) The exchange of water between the blood and interstitium through the leaky blood capillaries attains 70% of the volume per minute( 174) and is somewhat slower for the lymph capillaries, because of the difference in hydrostatic pressures. Because all pores are substantially larger than small organic molecules, the transport through the leaky capillary walls into the interstitial space proceeds mainly through the paracellular route on the time scale of minutes. This type of transport has the specificity determined only by the interactions of chemicals with cells that affect the free concentrations and, thus, the overall transport rates. In contrast, the transcellular route that the chemicals must use to cross the nonporous capillaries exhibits widely differing transport rates and mechanisms, which depend on the properties of the chemicals (section ). | Table 1Perfusion Characteristics of Some Organs in a Standard 70-kg Human167,175,176 |
Chemicals can enter the tissues cells directly from the capillary walls, from the interstitial fluid or from other tissue cells. Individual organs differ in density of vasculature and, consequently, in the number of cells, which do not maintain a direct contact with the capillaries or primary interstitial fluid directly mixing with the blood. 175,176 According to the decreasing capillary densities, the tissues can be classified into four categories: (i) lungs; (ii) kidney; (iii) brain, liver, intestines, and heart; and (iv) muscles, connective tissues, fat, and bones (Table 1). The average thickness of the cell layer served by a capillary increases in this order. The basal blood flows for individual organs are also summarized in Table 1. The basal blood flow can be increased 5−10 times in most organs during physical exercise and other conditions. The exceptions are brain and kidneys, where the blood flow is buffered, and the basal flow cannot increase more than twice.( 167) These data determine relative equilibration times for individual tissues, which are primarily given by the number of crossed bilayers, including, in each cell, cellular, and organelle membranes, as well as endoplasmic reticulum. For a single bilayer, the equilibration times vary between less than a second and several days, depending on the structure of the transported chemical (section ). The growing number of crossed bilayers increasingly magnifies these differences. Chemicals can be eliminated by processes that do or do not change their molecular structure, i.e., by metabolism or excretion, respectively. Metabolism and excretion processes collectively represent elimination. The prevailing portion of metabolism in higher organisms happens in the liver, where the majority of metabolizing enzymes, predominantly cytochromes P450 (CYPs, section ), are located. However, CYPs and other metabolizing enzymes are also expressed in the intestine and some tissues. Many chemicals can also participate in spontaneous reactions with body constituents (section ). The excretion of chemicals and their metabolites proceeds mainly in the kidneys, although excretion into feces, bile, saliva, and sweat, as well as transpiration into air, may contribute. From more than a liter of the blood passing through the kidneys every minute, ~100 mL of plasma are filtered at the glomerulus of nephrons, through the 30−60 nm pores, which are normally too narrow to allow the passage of proteins and protein-bound chemicals. Active, protein-mediated secretion and reabsorption of chemicals proceed in the proximal tubule (sections and ), and passive reabsorption happens along the entire tubule. Water is also reabsorbed, reducing the volume of the filtrate by a factor of ~100. The time courses of concentrations of chemicals in individual parts of higher organisms are determined by absorption, distribution, metabolism, and excretion (ADME). We will refer to these processes collectively as disposition, although, in classical pharmacokinetics, the term disposition denotes only the last three processes.( 75) In vitro cellular systems (e.g., microbial populations) do not possess specialized structures participating in absorption and excretion, and the processes that the chemicals undergo can be classified as distribution (or transport) and metabolism. During absorption, distribution, and elimination, the majority of chemicals seem to enter and leave the majority of the cells through phospholipid bilayers of the membranes by passive diffusion, without the help of protein carriers, unless the chemicals are analogs of physiologic molecules with active or passive carrier-mediated transport. The baseline diffusional trans-bilayer transport rates of many chemicals may be increased by influx transporters in the intestine, liver, and kidney( 177) (section ) or decreased by the efflux pumps and transporters (section ), which are expressed in organs with excretory functions such as the kidney and liver; in sheets or tissues with barrier functions such as the intestine and capillaries of brain, testis, and placenta; and also in tumors.( 178) The list of chemicals undergoing facilitated or active transport is steadily growing, because this area is a subject of extensive research, because of the obvious importance in drug delivery.( 179) It is assumed that practically each drug is a substrate of a transporter somewhere in the body but the phenomenon may not always be clinically relevant.( 180) If the trans-bilayer transport is fast, its further enhancement has no significant impact on the overall process. Understanding of both passive and mediated transport mechanisms is necessary for prediction of distribution of chemicals. The generalizing conclusions about the key roles of transporters for the entry of all drugs into all cell types( 181) are not based on sound arguments. The transport mechanisms utilizing protein carriers can be described in terms of structure in a similar way as other structure-specific processes, using receptor-based or ligand-based approaches (section ). After crossing the cell membrane, intracellular distribution of chemicals proceeds chiefly by passive diffusion through the bilayers of organelles and of endoplasmic reticulum, although several transporter proteins are expressed in intracellular membranes and may affect the transport rates. Some hydrophobic chemicals, planar aromatic hydrocarbons, and similar compounds( 182) enter the nucleus in a complex with the aryl hydrocarbon receptor, 183,184 but the passive diffusion across the nuclear envelope cannot be excluded. In summary, passive trans-bilayer diffusion is a ubiquitous and most important process affecting the distribution of chemicals in biosystems. Because of frequent references to this process, we will use the term “transport” to denote the flux and the term “accumulation” to characterize the pseudo-equilibrium retention of chemicals in bilayers. The SBSP models are essentially common mass-action based models for the interactions of chemicals with body constituents, which are placed within the framework of intracellular and extracellular compartments separated by membranes. The term “ baseline disposition” can be used to denote the disposition that is not affected by protein-mediated processes, such as active or passive mediated transport or enzyme-catalyzed metabolism. The baseline disposition is always occurring in any biosystem and is implicitly assumed if no specific knowledge about the concentration and kinetics of transporters or metabolizing enzymes is available. The carrier-mediated influx or efflux become important if they significantly change the baseline disposition. Understanding the transport and its interplay with other property-related processes (ionization, hydrolysis) is a crucial requirement for the development of the relationships between the baseline disposition kinetics and structure or properties of chemicals. The interactions of chemicals with biosystems will be analyzed at the elementary level of individual processes in section , and for the biosystems of increasing complexity in section . 2. Individual Steps vs. Properties To describe chemical disposition via explicit or numerical solutions of appropriate differential equations, the rate and equilibrium parameters for individual processes must be expressed as functions of physicochemical properties of chemicals and biosystems. In sections − , we will focus on the structural dependencies of the interactions with lipids, protein carriers, inert proteins, and enzymes, which are involved in disposition of chemicals. 2.1. Interactions with Lipids Lipids of mammalian cells can be classified as bilayer-forming lipids and other lipids. The first category is represented by phospholipids, cholesterol, sphingomyelins, and other species,( 185) whereas the second category consists mainly of triglycerides and their derivatives. Triglycerides are contained in lipoproteins and lipid droplets, which are present in practically all cells, and accumulate to a large extent in adipocytes (section ).( 186) From the viewpoint of chemicals’ disposition in biosystems, transport through phospholipid bilayers and accumulation in phospholipid bilayers are among the most important processes. Chemicals usually interact with lipid aggregates, not with individual lipid molecules. Therefore, the interactions are mostly of structure-nonspecific nature, resemble interactions with phases or solvents, and can be analyzed using the data obtained in macroscopic surrogate systems (section ). These facts permit generalization of the rules for the interactions of chemicals with phospholipids to the level of predictions of the kinetics and equilibria from the structure of chemicals. 2.1.1. Bilayer Structure and Properties History of the research on bilayer structure and properties is tightly interwoven with structural studies of biological membranes. The timeline showing the major events in the history of the bilayer structural studies is given in Figure , with the details and references summarized in the next section. The key findings about the protein interactions with the bilayer, if not related to the bilayer structure, are only mentioned in the text. The bilayer structure is analyzed in more detail in section . 2.1.1.1. History of Membrane Structure The concept of the cell membrane originated in the second half of the 19th century from the work of botanists, who recognized that the protoplasm of plant cells was separated from the cell walls by an osmotic barrier. 187–189 The determination of surface tension was established as the measurement of the force needed to pull a metal plate out of the interface.( 190) The predecessor of the Langmuir trough was invented and used to measure the dynamic surface tension.( 191) Numerous observations of lipid-soluble compounds partitioning into the membrane led to the hypothesis about the lipid character of the membrane.( 192) The measurements of the area to which oils were able to spread at the air/water interface resulted in the proposal for the monolayer structure of fatty acids having the headgroups immersed in water and chains protruding into air.( 193) The notion of a double lipid layer came from the observations showing that the lipid content of erythrocytes from several animals was just sufficient to cover the cell surface by two monolayers.( 194) The resulting structural model of the membrane as a bilayer sandwiched between two protein layers holding the lipids in place( 195) was later modified, on the basis of electron-microscopic observations, to a self-standing bilayer with adsorbed polypeptide and possibly polysaccharide layers,( 196) and finally to the fluid mosaic model with the bilayer representing a support for peripheral and integral proteins.( 197) The geometry of phospholipid molecules was considered a factor in formation of the maximal bilayer curvature.( 198) The X-ray( 199) and calorimetric( 200) observations indicated the presence of the gel and liquid-ordered lipid phases in the membranes. The X-ray structural studies clarified the orientation of hydrocarbon chains.( 201) The hydrophobic effect, attributed to entropy-driven changes in water structure,( 202) which was initially studied in connection with colloid behavior, 203–205 waterproofing of textiles,( 206) and protein folding, 207,208 was recognized as the driving force in the formation of the bilayer.( 209) The structure of the phospholipid bilayer in the gel phase was studied by neutron diffraction. 210–212 The headgroup dipoles change orientations in response to the overall charge of the bilayer.( 213) Integral proteins were shown to interact with the bilayer core via hydrophobic residues.( 214) In a bilayer composed of a lipid mixture, the lipids with the lengths of the fatty acid chains matching the dimensions of the hydrophobic exteriors of integral proteins concentrate around the proteins, to minimize hydrophobic mismatch.( 215) Combined X-ray and neutron diffraction analyses provided a detailed picture of distribution of individual phospholipid fragments along the normal of a liquid-ordered phosphatidylcholine (PC) bilayer. 216–220 High-resolution X-ray diffraction was used to determine the key structural parameters of the fully hydrated PC bilayer.( 221) Coexistence of the gel and liquid-ordered phases was directly observed by atomic force microscopy.( 222) These observations contributed to the current view of biological membranes as dynamic structures, where peripheral and integral proteins interact with the bilayer containing different phospholipids and other lipid species in liquid-ordered and gel states. 2.1.1.2. Membrane Composition Phospholipid bilayers form the basis of endomembranes, including cell membrane, endoplasmic reticulum, Golgi bodies, vesicles, and nuclear envelope, which participate in several vital processes in eukaryotic cells, such as attachment, intracellular compartmentalization, energy transduction, signaling, protein sorting, and other secretory and endocytic processes. The protein and lipid contents are roughly equal on the weight basis, with significant deviations for specialized membranes. For instance, myelin, the multilayered insulating sheath of neural axons contains 80% lipid, and mitochondrial membranes, the locale of respiration, oxidative phosphorylation, and intense metabolism, contain 80% protein. In the simplest case of inert proteins, the significant protein content decreases the area available for passive trans-bilayer transport and modifies the structural arrangement of surrounding bilayer phospholipids. The irregularities may increase the transport rates, as described for the perturbed bilayer structures at the transition temperature (sections and ). Mammalian bilayers are composed of phospholipids, sphingomyelin, glycolipids, cholesterol and other sterols, and many other non-bilayer-forming lipid species.( 185) Phospholipids, mainly PC, less phosphatidylserine, phosphatidylethanolamine, and phosphatidylinositides, account for 40%−80% of the bilayer weight. Cholesterol, the preponderant mammalian sterol, is equimolar with phospholipids in hepatocytes, erythrocytes, and myelin; it represents 20% w/w of stratum corneum( 223) and is less abundant in other cells and tissues. The lipid composition of individual membranes varies widely, even between the membranes of the same cell,( 224) and affects the permeation rates. The tubular cells of barrier epithelia in the kidney nephrons, manifesting low membrane permeability, contain the outer bilayer leaflets composed of a specific mixture of PC, sphingomyelin, glycosphingolipids, and cholesterol. The omission of any component dramatically increases permeability.( 225) Caveolae, invaginated membrane structures with high levels of cholesterol, sphingomyelin, and caveolin (section ), exhibit high permeabilities, thanks to a high degree of unsaturation of fatty acid chains.( 226) 2.1.1.3. Phospholipid Distribution in Membranes Phospholipids in membrane bilayers do not form homogeneous mixtures. Lipid distribution may differ, under physiological conditions, for different cells or different membranes of the same cell, for the two monolayers forming a bilayer, and for a monolayer. For different cell membranes, sphingolipids and sterols form gradients along the secretory pathways, with the highest abundance in the plasma membranes and the lowest level in endoplasmic reticulum. In a bilayer, lipids display asymmetric trans-bilayer arrangements maintained by energy-dependent transporters: sphingomyelin, glycolipids, and PC concentrate in the outer leaflet of the normal cell membrane, whereas phosphatidylserine and other phospholipids prefer the inner monolayer. 227,228 These differences are particularly pronounced in renal and intestinal epithelial cells, which exhibit low permeabilities to carbon dioxide and ammonia.( 229) Perturbation of this asymmetry, especially the appearance of phosphatidylserine at the cellular surface, is a hallmark of apoptosis, which enables an orderly removal of the affected cell by phagocytes.( 230) In any of the two monolayers of the bilayer, some mixtures of lipids are immiscible to various degrees and may form lateral domains, occasionally termed lipid rafts, especially in the surroundings of proteins. 215,231–237 Formation of a domain in a monolayer affects the phospholipid distribution in the opposite monolayer. 238–242 The domains participate in many of the membrane-related vital processes mentioned in the introductory paragraph of section , 243–252 as well as in the transport 253,254 and accumulation 255–258 of chemicals. The perturbed bilayer structures at the raft perimeters may affect the transport rates, because increased permeabilities were observed for coexisting phases at transition temperatures (see the next section). 2.1.1.4. Phase Behavior of Bilayers Phospholipid molecules in aggregates exhibit collective behavior. The main determinant of this behavior, for the given composition and normal pressure, is temperature. At lower temperatures, the bilayer is in the gel phase, with the fatty acid chains tightly packed and ordered, and mostly assuming extended trans conformations.( 259) When the temperature increases above the main transition temperature, the bilayer enters the liquid-ordered phase, with less-regular conformations of fatty acid chains. Among additional chain-ordered phases, the ripple phase of PC and phosphatidylglycerols, exhibiting periodic ripples several hundred angstroms apart,( 260) is most frequently studied. The profile of individual ripples may be symmetrical or asymmetrical, with the two shoulders differing in the thickness: one side has the same thickness as the gel phase and the rest is thinner, possibly because of a different tilt or the melted state of the chains.( 261) The formation of ripples, consisting of gel and liquid-ordered regions in individual monolayers, which were periodically arranged for geometrical and topological reasons, was modeled by Monte Carlo (MC) simulations( 260) and other computational methods (section ). In PC, the ripple phase occurs just below the gel-to-liquid transition chain-melting temperature that marks the coexistence of the gel and fluid phases.( 261) In biological membranes, the main chain-melting transition occurs within the range of 20−60 °C. The heat capacity, as well as the volume and area compressibility, reach their peak values during this transition.( 262) The heat-capacity changes seem to be proportional to the volume changes,( 263) as well as to the relaxation times, which range from seconds to approximately a minute.( 264) Usually, the transition temperature is affected less by the headgroup composition than by that of the tails: it increases with the length and saturation of the fatty acid chains. The transition temperature exhibits a decreasing sigmoidal dependence on hydration: the upper limit is observed for dehydrated bilayers, whereas the lower limit is achieved at the saturation of water-binding capacity of the lipids. The gel-liquid transition occurs in a narrow temperature interval in pure phospholipid bilayers, especially for multi-bilayer liposomes.( 264) Additives usually broaden the transition temperature interval. In the gel phase, cholesterol decreases the fatty acid chain ordering and reduces the main transition temperature.( 265) The gel−liquid phase change results in an abrupt increase in bilayer volume (~4%),( 266) area (~25%),( 267) enthalpy (20−40 kJ/mol),( 268) and the headgroup hydration( 269) (approximately three-fold in PC).( 270) The thickness of the bilayer decreases upon the chain-melting transition,( 271) as reflected in the disproportionate changes of the volume and area. Above the transition temperature, further temperature elevation results in an increased abundance of folded conformations, leading to a continuous reduction of phospholipid surface density and bilayer thickness( 272) and a concomitant increase in headgroup hydration.( 270) Within a range of ~20°, these changes are dependent on temperature in an approximately linear way.( 271) The relationship between the phospholipid surface area and the average number of water molecules intercalated in the headgroup region is almost linear.( 273) Cholesterol increases the fatty acid chain order in the liquid-ordered phase, as opposed to its effect in the gel phase.( 265) The increased order, thanks to cholesterol or diminished unsaturation, is associated with a decrease in the water concentration in the core.( 274) Hydration of the headgroups is not closely correlated with the chain order.( 274) 2.1.1.5. Bilayer Structure at Atom Level The dynamic bilayer structure is characterized by average attributes. The most important structural characteristic is the surface area per lipid molecule that is related to the bilayer thickness via the specific volume. To describe the thickness, one can use (i) the steric thickness, characterizing interbilayer distances in multilayer preparations; (ii) the electron density thickness, obtained from electron density profiles as the head-head separation; (iii) the hydrophobic thickness, which is most important for hydrophobic matching of protein structures to bilayers; and (iv) the Luzzati’s thickness, obtained by the gravimetric X-ray method,( 275) as amended by inclusion of the compression to eliminate hydration defects.( 276) XRD determines the electron density profiles along the bilayer normal. 199,201,275 Neutron diffraction provides the mean positions of the deuterium atoms in bilayers containing selectively deuterated lipids. 219,277,278 The experimental data have been available earlier for the more-ordered gel phase 210–212 than for the liquid phase. A combination of the XRD and neutron diffraction data provided a detailed picture of distribution of individual phospholipid atoms along the normal of a liquid-ordered PC bilayer with comparatively low hydration. 216–220 High-resolution XRD and a computational reduction of the fluctuations enabled determination of the key structural parameters of the fully hydrated PC bilayer, including the area and volume per lipid.( 221) The status of the fatty acid chains can be monitored in the bilayers containing selectively deuterated lipids. 2H NMR results allow for the calculation of the order parameters, isomerization times, and trans/gauche populations of the fatty acid chains. 279,280 The latter attribute can also be studied by Raman spectroscopy( 281) and IR spectroscopy: the CD 2 bands differ for trans and gauche orientations of the adjacent C−C bonds. 259,2822H-NMR spectroscopy was used to study hydration of the headgroups,( 283) and, in combination with 31P-NMR, to characterize conformations and orientations of the headgroups.( 284) Monitoring of the phosphate and carbonyl bands by FT-IR allowed quantification of hydration and interactions with cations.( 285) For oriented bilayers, 2H-NMR was used to determine the overall tilt of the phospholipid molecules.( 286) The bilayers formed by the most abundant mammalian phospholipids, PCs, are studied most frequently. Individual parts of the PC molecules exhibit the following approximate time-averaged orientations, with respect to the bilayer plane. The dipoles formed by the zwitterionic phosphocholine groups align at angles starting at ~0° and increasing( 287) up to ~50°, when the surface charge becomes more positive.( 213) A similar movement of the dipoles, from the in-plane orientation to an orientation more parallel to the bilayer plane normal, was observed in the PC monolayer at the air/water interface upon compression from the liquid-ordered phase to the gel phase.( 269) The glycerol backbones are almost perpendicular or slightly slanted, to improve the access of sn-1 carbonyl to water molecules,( 288) although the sn-2 carbonyl remains better hydrated.( 289) The fatty acid chains are tilted to compensate for the larger surface area of the headgroup, compared to the relevant cross section of the extended acyl chains. The charged phosphocholine groups form a regular network of electrostatic interactions.( 290) The thickness of the bilayer increases( 291) and the area per phospholipid decreases with prolongation of the acyl chains. 272,273Despite intense thermal motion, 220,221 the subregions of (i) low headgroup density, (ii) high headgroup density, (iii) high tail density, and (iv) low tail density were discerned in the structure of a hydrated PC bilayer patch that was simulated by molecular dynamics (MD).( 292) The borders of the subregions cannot be precisely expressed in terms of the phospholipid structural elements because of their thermal fluctuations. The ranges of thermal motions and the subregion borders of a fluid PC bilayer are outlined in Figure . Subregion 1 (perturbed water) is represented by a water layer, whose structure is affected by interactions with trimethylammonium and phosphate groups, which temporarily protrude into this subregion. Subregion 1 starts where the orientational ordering of water by the headgroups can be observed and ends where the densities of water and lipid become comparable. In this direction, the overall mass density is gradually increasing along the normal to the bilayer plane. Significant lipid movement in the direction of the bilayer normal creates a rough bilayer/water interface that averages in time to a smooth density profile. Just above the gel−liquid transition temperature, the primary hydration shell consists of 12−16 water molecules per the PC headgroup. 293,294 The thickness of this subregion is ~1 nm in PC and is affected by the surface charge density.( 213) Sometimes, this subregion is considered to be part of the bulk water phase.( 295) Subregion 2 (interphase) extends over 0.8 nm and includes, most of the time, the hydrated headgroup atoms up to the ester groups and the first two or three tail methylenes. The distance between the phosphate groups and the headgroups/core interface is 0.41−0.54 nm.( 273) This subregion has the highest density and the lowest free volume fraction. Just above the gel−fluid transition temperature, practically all water molecules are engaged in hydration shells of the headgroup atoms. The extent of hydration and the phospholipid surface area increase with temperature as described previously. 270–273Subregion 3 (soft polymer) starts close to the carbonyl groups, where the water density drops to 1% of bulk water density. It contains the first 6−8 methylene segments of the fatty acid chains, with the thickness equal to ~0.7 nm. The chains are tightly packed, with restricted atom movement, so that the subregion behaves like a viscous liquid. The chain order parameters do not vary much for individual methylene segments. The exception is the second sn-2 methylene, which seems to be less ordered than the other methylenes due to two prevalent conformations.( 265) The mass density in subregion 3 is lower than that in subregion 2. The decrease in density continues in subregion 4 (alkane),( 296) which starts where the mass density becomes equal to the density of hexadecane. Subregion 4 represents the center of the hydrocarbon core and is formed by the remaining portions of the fatty acid chains. This subregion resembles a liquid alkane with low viscosity.( 292) Thanks to the diffusive hopping transport, 292,297,298 some water is present in the bilayer core formed by subregions 3 and 4; 299,300 however, the actual concentrations may be too low to measure by some techniques. 218,301Phosphatidylethanolamines (PEs), the most abundant bacterial phospholipids, differ from PC mainly in the hydration of the headgroups. The amine hydrogens participate in hydrogen bonds with water molecules and phosphates of other PE molecules. These interactions, along with the smaller headgroup size, result in a significant decrease in the surface area per lipid molecule.( 302) The surface charge of the bilayer is dependent on the ion composition of the aqueous phase, and on adsorption of charged chemicals, in addition to the headgroup composition. A difference in the monovalent salt concentrations on the two sides of the bilayer can induce significant asymmetry in structure, dynamics, and electrostatic properties of the bilayer.( 303) Surface charge density affects the conformation of the zwitterionic phosphate−nitrogen dipoles( 213) and the partitioning of charged solutes. 304–306 The surface charge at the shear plane of the bilayer, ~2 Å from the surface,( 307) is characterized by the zeta potential,( 308) which is related to electrophoretic mobility. This attribute was used to determine the liposome/water partition coefficients of charged and neutral solutes by capillary electrophoresis with liposomes (section ) acting as a pseudo-stationary phase.( 309) The electrostatics of the surface are also affected by the dipole potential 310,311 which originates from the alignment of electric fields of the charged groups, carbonyls of the ester groups,( 213) and the first layer of hydrating water molecules.( 312) The influence of the dipole potential on electrostatic properties of the headgroup region is smaller than that of surface charge but not negligible. 2.1.2. Phospholipid Systems Interactions of chemicals with phospholipids represent critical factors for the kinetics of both disposition and effects of chemicals. Accumulation in biological membranes is important primarily for understanding pseudo-equilibrium pharmacokinetic parameters such as the volume of distribution and the tissue/blood partition coefficients, as well as for elucidation of the biological effects of chemicals. The trans-bilayer transport is a key process in absorption and distribution. Direct measurements on biological membranes are obstructed by interactions of chemicals with proteins and other membrane components, complex membrane architecture, and variability of membrane composition. To avoid these complications, self-aggregating phospholipid systems are frequently used instead of biological membranes to study the interactions of chemicals with phospholipids. The bilayer/water partition coefficients are expected to be good descriptors of equilibrium partitioning in tissues and organisms, because the bilayers are the parts of biological membranes where many compounds accumulate. Published studies indicate successful descriptions for a limited series of compounds. 313–315 A broader application to more-diverse compounds may be limited by the variable phospholipids composition of biological membranes. An important goal of the SBSP modeling is the prediction of the bilayer interactions from structure of chemicals. The bilayer partitioning is a composite process that generally includes interactions with the headgroups, interface, and core. Consequently, the overall bilayer/water partition coefficients are more difficult to scale in the extrathermodynamic relationships (e.g. eq 1 in section ) than the elementary core/water, interface/water, and headgroups/water partition coefficients. Deconvolution of the liposome partitioning into core/water and interface/water contributions was attempted for limited datasets. 316,317An inherent drawback common to all bilayer systems is the potentially long incubation time needed to achieve the equilibrium for amphiphilic compounds: the flip-flop between the two interfaces of the bilayer can significantly slow the partitioning process,( 20) as well as lead to experimental errors, because of the termination of the experiment before reaching the equilibrium. Transport is also slow for compounds with extreme lipophilicities (section );( 20) however, the errors caused by a premature experiment termination are less frequent and less significant. For very hydrophilic chemicals, there is no fast step providing the illusion of reaching the equilibrium. Very lipophilic chemicals may have biphasic kinetics but the slow second phase is the partitioning into the small volume of internal water, which is less extensive and causes comparatively small errors. For the monolayer systems, the risk of a slow kinetics is limited to hydrophilic chemicals, which may exhibit a slow entry into the core. Depending on the preparation, the support used, the lipid composition, the lipid concentration, and the temperature, the phospholipid aggregates can have nanoscale dimensions and spherical, discoid, or oblate ellipsoid shapes (see Figure ), or macroscopic spherical or planar geometries. The aggregates include free and gel-trapped liposomes and other bilayer-based corpuscles, supported bilayers, and monolayers at solid, liquid, and gas interfaces with water. 2.1.2.1. Liposomes and Other Bilayer Microsystems Liposomes are spherical corpuscles consisting of the aqueous central phase surrounded by one (Figure A) or more phospholipid bilayers, which are separated by thin layers of water. They were first described by Bangham et al. in 1965,( 318) although the suspensions of phospholipids, stained by histological dyes, have been microscopically observed much earlier.( 192) Free (Figure A), immobilized, and supported liposomes (Figure B) have a long history of being used as surrogate systems for membranes. In these studies, unilamellar vesicles are preferred to multilamellar liposomes because of better-defined and simpler structures. Based on the size, liposomes are classified into three categories: the minimum-size unilamellar vesicles (diameters of 15−30 nm), large unilamellar vesicles (up to several hundred nanometers), and giant unilamellar vesicles (on the order of micrometers). The enthalpy and entropy contributions to binding may differ in small and large unilamellar vesicles,( 319) although the area per lipid, chain ordering, and dynamics seem to be similar.( 320) The enthalpy−-entropy compensation may lower the differences in the overall free energies.( 319) Giant unilamellar vesicles are generated from specific lipid mixtures by various techniques including detergent dialysis from aqueous mixtures of organic solvents 321,322 and chaotropic solutions,( 323) freezing-thawing,( 324) gentle hydration, 325,326 rapid solvent evaporation,( 327) and electroformation on electrodes.( 328) The vesicles are sufficiently large to be observed by optical microscopes and manipulated by micropipettes.( 329) They are well-suited for the fluorescence monitoring of lateral inhomogeneities also called lipid rafts or domains, 224,330,331 examination of mechanical( 332) and electrical( 333) properties, and patch-clamp recording of incorporated ion channels.( 334) Limitations such as a complicated kinetics with the difficult-to-predict duration of transport for some compounds,( 20) a tedious separation, 335,336 a laborious spectroscopic data analysis that is due to the light scattering if the separation step is omitted, 337–339 and low stability of some preparations( 340) restrict the use of liposomes in routine experiments with increased throughput. The direct approaches, avoiding the separation of liposomes, are limited to fluorescent and UV-Vis-absorbing compounds( 82) or ionized compounds. 341,342Immobilization of liposomes by embedding in agarose gel beads,( 343) adsorption on alkylated surfaces,( 344) or by biotin−avidin interactions alleviate the problem of liposome separation but introduce the risk of the matrix effects.( 345) Similar issues may arise with the liposomes that were noncovalently attached to a sensor chip and had the kinetics and equilibria of interactions with drugs analyzed using the surface plasmon resonance technology that measures the changes in liposome mass caused by drug binding. 346–349 The examination of matrix effects significantly increase the workload, especially if it is rigorously performed for each studied compound. Planar bilayers of nanoscale dimensions and circular shape, reminiscent of some plasma lipoproteins (section ), can be stabilized if their edges are covered by either membrane scaffold proteins 350,351 (nanodisks; see Figure D) or by phospholipids with shorter fatty acids and detergents 352–354 (bicelles; see Figure F). They are mostly used in imaging,( 355) NMR studies,( 356) and stabilization( 357) of membrane proteins and less in the studies of chemicals’ accumulation, because the stabilizing proteins, lipids, or detergents could interfere with the solute binding. The lipid−protein interactions in nanodisks were analyzed using MD simulations( 358) and mesoscale modeling.( 359) The self-assembly of lipoprotein nanodisks and their disassembly caused by cholate addition were studied by mesoscale modeling and small-angle X-ray scattering. 360,361To improve stability and separation properties of liposomes, microspherical supported phospholipid bilayer systems (Figure B) have been developed. The most commonly used solid supports are silica 362,363 (Transil beads( 364)), mica,( 365) glass 366,367 (supported spherical vesicles( 368)), and polydimethylacrylamide hydrogel (Lipobeads).( 369) The attachment of phospholipids is generally based on either chemisorption on the contact monolayer 366,370,371 or adhesion of the bilayer due to preferred planar shape of the bilayer formed by certain type of phospholipids.( 365) Polymerized liposomes containing rafts of nonpolymerized phospholipids exhibit concentration-dependent blue−red color shift upon interaction with chemicals.( 372) The effective concentration ranges seem to be specific for three elementary mechanisms: interaction with the surface, bilayer penetration, and no interaction. Polymerized liposomes undergo color transitions in response to external stimuli such as changes of pH or temperature, or binding of chemicals even if no phospholipids are incorporated.( 373) If the importance of phospholipids in the response to binding is confirmed, the system could be used for a fast screening of chemicals for interactions with phospholipids. 2.1.2.2. Planar Bilayer Systems The use of planar membranes for elucidation of some aspects of membrane transport dates back to the 1920s. The first membranes’ only attribute shared with the biological membrane was the presence of pores. The gelatine membranes that had been hardened in formaldehyde( 374) and contained copper ferrocyanide, 375,376 as well as collodion membranes( 377) exhibited selective permeability, based on the size of the permeants. The membranes with the structure of phospholipid bilayers were produced four decades later. A macroscopic planar bilayer can be generated at an aperture in a thin Teflon barrier separating the donor and acceptor aqueous compartments. 378−380 A drop of the solution of phospholipids in an organic solvent is transferred on the aperture that is ~1 mm in diameter and is immersed in an aqueous phase. The bilayer forms spontaneously, creating a black spot in the microscope view of the rainbow-colored droplet. Based on this phenomenon, the system is called the black lipid membrane (BLM). Asymmetrical bilayers were created using two different air/water monolayers separated by a barrier with an aperture, upon lowering the barrier across the interface.( 381) A similar principle was used for the formation of the bilayer in a microfluidic chip.( 382) Polycarbonate filters with the apertures in the micrometer range were used as a support for the formation of the PC/cholesterol bilayers and provided enhanced mechanical stability. 383–385 Accessibility of the apparatus to analytical, electrical,( 386) optical, and mechanical measurements makes BLM an excellent tool for detailed studies of transport, 387–390 bilayer electroporation,( 391) reconstitution of membrane proteins( 392) or channels, 393–395 and other bilayer-related phenomena. The bilayer stacks of skin lipids—cholesterol, ceramides, and free fatty acids—can be reconstituted in vitro.( 396) The multilamellar sheets mimic the real situation better than liposomes 397,398 and can be used for permeability measurements.( 399) Macroscopic solvent-free planar bilayers were formed from the corresponding monolayers or by bilayer adsorption between two porous cellulose sheets,( 400) on patch-clamp pipette opening,( 401) quartz,( 402) glass, mica,( 362) silicon,( 403) agar gel,( 404) organosulfate- and organophosphate-modified gold surfaces,( 405) metal (stainless steel,( 406) silver,( 407) platinum( 408)) and metal-coated (gold,( 370) platinum( 408)) or polymer-coated( 409) glass plates, and porous supports. 410–412The throughput of the experiments with planar bilayers was significantly increased by the introduction of the Parallel Artificial Membrane Permeation Assay (PAMPA). 413–415 A multiwell microtiter plate is sandwiched with the multiwell filter plate of the same format. The porous hydrophobic filter is impregnated with a mixture of PC (2%−10%) in alkane solvents. The opposite well pairs, separated by the filter, represent the aqueous donor and acceptor compartments. Individual wells can be stirred to regulate the thickness of the stagnant water layer so that it imitates that in the intestine, and to accelerate the procedure for lipophilic compounds.( 416) Other support materials were examined, 417,418 and the composition of the impregnating mixture was optimized to reproduce permeation through the Caco-2 cell monolayer,( 419) intestine, 420,421 blood-brain barrier,( 422) and skin.( 423) The corrections for the effects of the filter porosity( 424) and the unstirred water layers were introduced to provide more-realistic permeabilities.( 425) Later, the PAMPA approach was modified to measure the permeabilities and the partition coefficients of chemicals in hexadecane( 424) and 1-octanol.( 426) The PAMPA permeabilities of a limited set of compounds depend in a bilinear way on the reference partition coefficients, and linearly on the indicator variables for some ionizable groups.( 427) The permeability parameters were modeled in terms of hydrogen-bond donor and acceptor abilities, refraction, and polarizability using the solvatochromic LFER approach.( 428) In view of the extensive industrial use, it is interesting that the structure of the impregnated PAMPA filters has not been investigated in detail, although the formation of multiple bilayers in the filter pores was anticipated. 429,430 Some assurance about the bilayer formation comes from the following facts. The phospholipid-alkane mixtures used for the impregnation of the filters have similar composition as those used to prepare BLM. 378−380 The phospholipids in the filters are capable of fusion with phospholipid vesicles. 431–433 A multiple-bilayer structure was inferred from electrical resistance measurements on a hydrophobic filter impregnated with a synthetic lipid analog. 434–436 The formation of a single PC/cholesterol bilayer in the pores of a polycarbonate filter was inferred from the current−voltage relationships and the response to the polyene antibiotics amphotericin-B( 383) that forms a trans-bilayer channel in association with sterols. A better defined bilayer structure in the filter pores could definitely improve the reproducibility and interlaboratory variation of the PAMPA measurements, as well as interpretation of the results. A nonsolvent version of the impregnation, with liposomes filling the pores and forming a layer on the filter, was recently reported. 437–439 2.1.2.3. Monolayers at the Air/Water Interface Lipid monolayers have been widely used to study the behavior of lipid bilayers, with the underlying assumption that two weakly coupled monolayers, having appropriate surface pressure, 440,441 imitate the bilayer’s adsorption and partitioning behavior. From the viewpoint of the interactions with chemicals, monolayer systems differ from the bilayer, inter alia, by the absence of the opposite headgroup region that leads to a faster achievement of the equilibrium due to the removal of the potentially slowest step, the flip-flop of amphiphilic molecules between individual bilayer leaflets (section ). Phospholipid monolayers have been deposited on the interfaces between water and other phases, which can exist in the gas, liquid, or solid state. Each system has unique advantages and disadvantages. However, the monolayers with liquid support or solid support formed by hydrocarbon chains have the cores, which have the appropriate structure and sufficient size to solvate the permeants in a similar way as the bilayer core. The Langmuir trough with a movable barrier that can regulate the surface pressure in the monolayers 191,193,442 at the air/water or argon/water( 441) interfaces is a classical tool of surface chemistry. Its invention( 191) and use greatly contributed to the elucidation of monolayer and bilayer structures. Adsorbed phospholipids have the headgroups solvated in the aqueous phase. Their fatty acid chains are facing the air,( 193) where they may exhibit a higher ordering of the end methylenes than that in bilayers.( 443) The typical trough experiments monitor surface pressure, i.e., the tendesurface pressurency to keep the current volume, surface area, and shape, in the relation to the overall surface area. The dependence of surface pressure on the overall surface area leads to a straightforward determination of the phospholipid surface area and its change upon binding of the chemicals.( 444) The entire dose-response curve can be measured for one monolayer using the continuous exchange of the chemical solution in the subphase for the stock chemical solution under the conditions of rapid mixing.( 441) The surface pressure versus area measurement, combined with other techniques, provides a basis for the inference about the approximate positions of interacting chemicals in the monolayer. 445,446 Surfactants, amphiphilic compounds with clearly separated polar and nonpolar parts, reduce the surface pressure at the air/water and lipid/water interfaces. The lateral dimensions of the monolayer allow the use of a broad palette of techniques, ranging from electrochemical methods such as the surface potential measurements,( 445) to analytical methods that examine the composition of the aqueous phase, and surface imaging. The last category includes fluorescence microscopy examining surface domains in phospholipid monolayers doped with fluorescent phospholipid analogs 447–449 or lateral diffusion of phospholipids after photobleaching,( 450) as well as specular neutron reflection,( 451) ellipsometry, FT-IR reflectance,( 452) X-ray reflectance,( 453) and grazing incidence XRD( 454) determining some geometry aspects of the molecules of phospholipids or interacting chemicals.( 455) Binding of radioactive compounds to phospholipid monolayers can be easily tracked using a detector positioned closely above the interface. The method was widely applied to monitoring the adsorption of Ca 2+ ions( 456) and proteins. 457,458There are two concerns with the phospholipid monolayer at the air/water interface as a mimic for phospholipid bilayers and membranes. First, only the aqueous subphase can be sampled for the change in the concentrations of the studied chemical so the experiments do not clearly locate the chemical in the headgroup or core regions of the monolayer. Second, the hydrocarbon core is too thin to effectively solvate larger hydrophobic molecules. The hydrocarbon layer can be made thicker by adding hydrocarbons that float on top of the fatty acid chains. This phenomenon, however, was only observed for hexane. Longer hydrocarbons (12−16 carbons), which would be preferred because of lower volatility, intercalate between the fatty acid chains and reduce their tilt angle so that the chains become practically perpendicular to the interface. 453,454 2.1.2.4. Monolayers at the Solvent/Water Interface The hydrocarbon-doped monolayers described in the previous paragraph represent a transition between air/water and alkane/water interfaces. While a microscopic alkane layer at the air/water interface in the former setup is a prerequisite for surface imaging, thicker layers in the latter arrangement enable one to use conventional analytical and electrochemical techniques to monitor concentration changes in the nonpolar phase.( 459) The performed experiments focused on the behavior of phospholipids at the interfaces between water solutions and 2,2,4-trimethylpentane,( 460) heptane,( 461) and other alkanes.( 462) If the alkanes are of comparable lengths with the fatty acyl chains, they intercalate into the monolayer, which then assumes a different state than that of the bilayer.( 462) The first systems of this nature were created at the beginning of the last century. The globules, formed by mixing the solutions of PC in various water-immiscible solvents with water, 192,463 were ~0.1 mm in diameter and could be observed microscopically, after staining with histological dyes. The structure of the monolayer was not known at that time. If sufficiently stable and uniform, the globules could become a valuable tool for studying the interactions of chemicals with the phospholipids, thanks to the absence of the second interface, which eliminates the potentially slow flip-flop step (section ). 2.1.2.5. Monolayers on Solid Supports The phospholipid monolayers on solid supports, which can be easily adapted for routine experiments with an increased throughput, are more relevant to pre-clinical drug candidate screening and other high-throughput applications than other monolayer setups. The monolayers can be bound to the support by covalent or noncovalent forces. Phospholipids or their derivatives, which are covalently immobilized to the surfaces of porous silica microspheres, served as a column material in high-pressure liquid chromatography. They are called immobilized artificial membranes (IAMs) and the structure is sketched in Figure E (the monolayer can also form inside the pores).( 464) The retention factors were used as descriptors for the prediction of trans-bilayer transport of chemicals. 465–469 The liposome and IAM binding were correlated for some compounds( 470) and not correlated for other compounds. 471,472 A systematically higher partitioning to IAMs than to liposomes was measured for several other series of compounds. 343,473 The IAMs were not recommended for the characterization of ionized chemicals, because the matrix charges seemed to be insufficiently shielded.( 474) In the molecular dynamics simulations,( 475) no structural differences were observed between the PC headgroup in IAMs and the liquid-ordered bilayer. However, the 31P-NMR studies( 476) indicated that the surface density of IAM headgroups is significantly lower than that in liquid-ordered membranes. The decrease is ~30% for ether-linked PC and ~40% for ester-linked PC.( 477) The hydrocarbon core of IAMs is characterized, in comparison with the bilayer core, by smaller thickness,( 470) higher packing density,( 475) and, consequently, lower rotational freedom of the fatty acid chains. Moreover, the ends of the phospholipid fatty acid chains in IAMs are covalently bound to porous silica beads, which may exhibit extreme surface curvatures, especially in the pores.( 478) At these spots, disturbances in the continuous bilayer may exist, which are not seen in the averaging NMR experiments. Diffusion of chemicals in the pores may complicate the measurement of the transport rates, if attempted. Monolayers adsorbed on alkylated surfaces are expected to resemble the bilayer more closely than IAMs. Adsorbed monolayers can be prepared by several techniques. A multistep immersion process has been used to create layers of dimyristoylphosphatidylcholine (DMPC) on the surface of silica plates with covalently attached octadecyl chains.( 479) The uneven coverage of the hydrophobic surface and the formation of multilamellar islands make this technique unsuitable for the exact bilayer partitioning studies where a well-defined system is needed. The Langmuir−Blodgett technique( 480) produced a phospholipid monolayer on the surface of gold-coated plates with covalently attached octadecyl chains.( 481) The obtained monolayer was uniform and stable in both dry and rehydrated forms. Alkylated glass plates were used to obtain a phospholipid monolayer by vesicle fusion.( 482) The solid lipid nanoparticles with the triglyceride core covered with adsorbed phospholipids and other surfactants have been obtained by industry-scale processes, such as emulsification of melted triglycerides and homogenizaton. 483–491 The slow diffusion of compounds in the solid core that makes the suspension a useful drug delivery vehicle, is a drawback for mechanistic studies of interactions of chemicals with phospholipids, along with the presence of surfactants, irregular coating, and nonspherical shapes.( 488) We have developed a monolayer system that overcomes most of the aforementioned problems. The system consists of uniform nonporous octadecylated silica (ODS) particles (1.5 μm in diameter) that are coated with phospholipids (Figure C).( 492) The coating is based on a spontaneous self-assembly of phospholipid molecules that is expected to lead to formation of a monolayer with the packing density comparable to that in liposomes. Formation of a continuous monolayer in a similar setting has also been demonstrated for planar surfaces.( 481) The presence of a single interface eliminates the possibility of the time-consuming flip-flop process and leads to a significantly faster binding and simpler kinetics, compared to that observed in free or immobilized liposomes. The smooth surface of the nonporous particles is a prerequisite for formation of a defect-free monolayer. The hydrocarbon core, formed by phospholipid acyls and the octadecyl chains attached to the microspheres, is expected to be more similar to the bilayer core in thickness and solvation properties than the core in IAMs. The used ODS particles provide a significantly larger specific surface area than that of planar systems. Easy separation of the DMPC-coated ODS-particles from the medium is a prerequisite for a plate-friendly assay that is suitable for routine testing with higher throughput than using liposomes or IAMs. If an estimate of the overall bilayer partitioning is of interest, the monolayer partition coefficients are first deconvoluted into the core and headgroup contributions, and the headgroup contribution is counted twice.( 492) 2.1.3. Surrogate Solvent Systems Individual regions of phospholipid bilayers are difficult to access experimentally, especially in biological membranes. Various organic solvents, imitating the bilayers differing in lipid composition, packing, and hydration levels, have been examined as membrane or bilayer surrogates, with the objective of obtaining thermodynamic or kinetic information on the partitioning process. 52,56,73,272,389,493–508 Curiously, air was also used as the surrogate phase imitating the nonpolar hydrocarbon core of the bilayer. The measurements of the surface pressure of chemicals at the air/water interface have been miniaturized and adapted for a high throughput.( 509) A parameter approximating the bilayer/water partition coefficient was derived using the experimental data for the air/water partition coefficient and the surface area. Partitioning equilibria of solutes in bilayer regions result from the interplay of several factors, which include the interactions of solutes with phospholipids and water molecules, the energy for creation of the cavities for solute molecules, 510–512 electrostatic interactions with the membrane and dipole potentials,( 513) and entropic consequences of partitioning in individual regions. 514–516 Estimates for the first factor have traditionally been obtained using organic solvents as surrogates of the bilayer or one of its regions. 2.1.3.1. Collander Equation The partition coefficients ( Px) of a set of compounds in a solvent system that is capable of similar interactions with the studied compounds as the reference system, are related to the reference partition coefficient P via the Collander equation 52,517 as
where A x (A represents accumulation) and β x are empirical adjustable coefficients, which are specific for system x. The coefficients A x and β x are obtained by the fit to experimental data. The approach has been widely used in the design of bioactive compounds,( 518) computational chemistry,( 519) protein folding,( 520) and other areas.( 31) At the biological levels, from cells to organisms, the empirical coefficients also account for the variability in the composition of bilayers. 2.1.3.2. 1−Octanol as the Reference Solvent After early experiments with loosely defined oils 192,493–497,521 and other solvents, 52,498,517,522,523 the studies of Hansch and Fujita,( 56) as well as those of Leo et al.,( 524) have established 1-octanol as a widely used bilayer-mimicking solvent. The authors, and many followers in the areas of QSAR and the design of bioactive compounds, intuitively considered hydrated 1-octanol to be a hydrocarbon core surrogate, despite its comparatively high equilibrium water content at room temperature (2.18 mol/L).( 525) There is some water in the bilayer core, because of trans-bilayer water transport. 292,297,298 The equilibrium water concentrations in the core increase as the fatty acyl chains become more disordered,( 274) but they are generally low 218,301 (section ). Contrary to the core-surrogate hypothesis, studies in peptide binding to liposomes( 526) (section ) and in transport through BLM( 507) (section ) assumed that 1-octanol mimics the interfacial region of the bilayer. Interestingly, both views can be correct for different compound sets: XRD analyses,( 527) spectroscopic experiments,( 528) and MD simulations 529,530 of the structure of liquid, water-saturated 1-octanol revealed fluctuating inverted micellar aggregates encompassing polar and nonpolar regions, which could imitate hydrated headgroup regions and cores, respectively. Inverted micelles consist of the centers comprising hydrated hydroxyl groups,( 531) which are surrounded by outward-extending alkyl chains. 527,529 The centers are thin and prolonged in neat 1-octanol and become longer and more spherical in hydrated 1-octanol.( 530) As the criteria for selection of a proper bilayer surrogate, early studies used the uptake of chemicals into cells and transport through biological sheets, which were affected by protein binding and other factors. These problems are avoided when using phospholipid liposomes, which are more suitable for evaluation of the surrogate phases. Liposome partitioning is affected by several factors,( 185) including surface charge density, especially for ionized solutes, 304–306 and phospholipid surface density, which, in turn, is dependent on temperature, 272,504 fatty acid composition,( 532) and cholesterol content 272,504 (section ). Ideally, all data used for comparisons should be measured under identical conditions that imitate the studied biosystem, so that the variation can be taken into account by the empirical regression coefficients in eq 1. Simple chemicals affect various bilayer properties at higher concentrations: surface charge, 304,533 thickness,( 534) fluidity in individual regions, 397,398 and transition temperatures, to name just a few. Chemicals in biosystems act at low concentrations, except for the first bilayers coming into contact with the dose (section ). To mimic the reality for the majority of chemical−phospholipid interactions, the liposome partitioning data should be measured at low concentrations of chemicals or, if feasible, extrapolated to zero concentrations. The correlations of the partition coefficients of chemicals in liposomes and in the 1-octanol/water system become nonlinear for large molecules, presumably because of the energy needed to create a cavity( 535) in the structured bilayer subregions 2 and 3 (section ). The correlations displayed separate trends for neutral, negatively charged, and positively charged molecules, with the last category exhibiting the largest scatter.( 536) The relationships broke down even for a homologous series exhibiting different preferred bilayer locales for individual members.( 537) If more-diverse chemicals were considered, the relationship seemed to hold, with a broader scatter and within certain size and lipophilicity limits, 535,538 for neutral molecules( 539) but not for charged molecules. 342,540 The differences are more significant for acids, as opposed to bases, because the former show more-pronounced pH dependences of the bilayer/water partition coefficients ( PB/W) than the latter. This phenomenon is caused by the interactions with the headgroups of neutral and acidic phospholipids, which lead to a rearrangement in the headgroups. 541,542 The former profiles are still rather shallow in comparison with those for the 1-octanol/water partition coefficients ( PO/W). The differences of the PB/W values for different pH of the aqueous phases are dependent on the ionization character of the chemical: they are comparatively small for bases and can reach a magnitude100-fold larger for acids.( 543) The PO/W values of both acids and bases can differ by 3−4 orders of magnitudes in the systems with differing pH values of the aqueous phase. A careful comparison of the data measured above the gel-to-liquid transition temperature with the experimental or calculated PO/W values( 544) showed that, even for neutral chemicals, different lines can be discerned for polar compounds with or without hydrogen-bond donor ability( 538) and nonpolar compounds,( 532) as shown in Figure . On the bilogarithmic scale, the lines have the slopes of 0.63 for nonpolar compounds, 0.82 for polar compounds without hydrogen-bond donor groups, and 1.27 for the group consisting of hydrogen-bond donors and acceptors (phenols and alcohols). However, the entire dataset exhibits a reasonable linear relationship with the squared correlation coefficient r2 = 0.950, as noticed previously.( 539) In summary, the PO/W describes partitioning in the phospholipid bilayer very well for a homologous set of compounds,( 537) and acceptably for diverse neutral molecules (Figure ), within certain size limits. In both cases, the relative affinities of the chemicals for the headgroups and core must not significantly vary among the studied chemicals. PO/W is a perfect reference coefficient in the cases when overall partitioning is targeted, e.g., for the description of pseudo-equilibrium disposition (section ). Other surrogate solvent systems are needed if the distribution between bilayer regions is of interest: the hydroxyl group of 1-octanol can hardly emulate the interaction capabilities of phosphate, ammonium, and carbonyl functionalities of the phospholipid headgroups. In addition to the main advantage that 1-octanol resembles some membranes as assessed through the application of eq 1, other desirable properties of this solvent include chemical stability, easy availability and purification, UV-Vis transparency (which is important for the spectrophotometric determination of many chemicals), and a low volatility.( 518) 2.1.3.3. Other Solvent Systems The studies with BLM (section ) showed that the selectivity of the bilayer to the partitioning of the hydrogen-bonding groups lies somewhere between those of 1-octanol and pure hydrocarbons,( 389) but it is significantly closer to the latter. 545,546 Additional support for the use of hydrocarbons as core surrogates 500,501 comes from the experimental observation of similar molecular packing of the fatty acid methylene and methyl groups in subregion 4 of the bilayer and in bulk liquid alkanes( 219) (section ). The hydrocarbon cores seem to have similar solvation properties in all bilayers, although fine differences were noted. 507,508 Most likely, a single reference phase, e.g. n-hexadecane, can be used, with the differences being taken into account by applying the Collander equation (eq 1). Whereas alkanes approximate solvation properties of the hydrocarbon core, is there a phase that emulates most interaction capabilities of the headgroups? Four solvents were used to model the amphiprotic, hydrogen-bond acceptor and donor, and hydrophobic properties of membranes: 1-octanol, propylene glycol dipelargonate (PGDP), chloroform, and alkanes, respectively.( 503) The structure of PGDP contains some fragments found in phospholipids; however, no substructure imitates the charged functional groups, and an equivalent of the hydroxy group of PGDP is not found in PC. Other tested solvents include isopropylmyristate( 547) and N-butyl-acetate.( 548) The ethylene glycol−heptane system 549–551 represents a rare attempt to imitate separately the headgroup and core regions of the bilayer. None of the solvents used so far emulates the charged functional groups of phospholipids. A straightforward approach to inclusion of all functional groups of a phospholipid into the surrogate system is represented by the direct use of acylated headgroups. We have found( 552) that the PC headgroup with truncated fatty acid chains, diacetylphosphatidylcholine (DAcPC), when dissolved in water in a molar ratio of 1:14, resembling the conditions in the hydration shell of the PC headgroups in the bilayer slightly above the transition temperature, 265,553 forms a homogeneous and only slightly viscous solution. Albeit isotropic, the hydrated DAcPC imitates the type of solvation that solutes experience in the PC headgroup region, including the lack of free water molecules. The methyl groups in acetyls mimic the first two or three methylenes of the sn-2 fatty acid chain, which thermally fluctuate at the level of sn-1 carbonyls in a PC bilayer and can be considered a part of the headgroup region. The DAcPC molecules do not measurably partition from the concentrated aqueous solution (1.96 mol/L) into n-hexadecane (C 16) in the two-phase system. For a set of 16 compounds, the C 16/DAcPC partition coefficients were measured and used to estimate the prevalent location of the compounds in the bilayer. The difference log PC16/W − log PO/W (formally equal to log PC16/O) is frequently considered to be an indicator of the hydrogen-bonding ability, 549,550,554,555 a correlate for permeability,( 556) and a parameter characterizing the preferential location of solutes in the bilayer. The fraction of compounds present in the core, as estimated from the C 16/DAcPC partition coefficients, is plotted against log PC16/O in Figure . To connect the compounds with known locale in the headgroups or in the core, the data were fitted( 557) with the Boltzmann sigmoidal function:( 552) Standard deviations of the two optimized coefficients are <20% of the coefficient values, and the squared correlation coefficient is r2 = 0.992. The locales estimated using the C 16/DAcPC partitioning (Figure ) are in perfect agreement with the known locales for all compounds where this information is available. However, eq 2 can be used to obtain approximate estimates in the absence of C 16/DAcPC data. The solvation energies of a solute in C 16 and in hydrated DAcPC phase can be rationalized in terms of two contributions: the energies needed to form cavities in both phases, and the interactions of the solute molecules with the solvent molecules lining the cavities.( 548) The cavity formation energy is maximal in bulk water and decreases in the headgroup region, to a minimum in the core. 513,558 The cavities in the isotropic DAcPC solution can probably be formed easier than those in the headgroups, because the headgroup anchoring effect of the high-density tail subregion 3 (section ) is missing in DAcPC. Based on these facts, we expect that the main contributions to the solvation energies in DAcPC and C 16 will come from the solute−solvent interactions. Thus partitioning in the C 16/DAcPC system is expected to provide only information about one of the factors contributing to the location in the bilayer, namely, the energy of interactions with the phospholipids. The cavity formation energy, the energy of electrostatic interactions of the permeants with the membrane and dipole potentials, and entropic factors are not included. The C 16/DAcPC data can help factorize the contributions to solvation energies in individual bilayer regions. Lipophilic chemicals also partition into plasma lipoproteins and the adipose tissue (section ). Triacylglycerols (TAGs), representing ~90% of the adipocyte’s mass( 176) and forming the lipoprotein cores, lack structural similarity to common surrogate molecules: alkanes miss the hydrogen-bond acceptor abilities of the TAG ester carbonyls, 1-octanol has a surplus hydrogen-bond donor function, and DAcPC features additional charged groups. Olive oil has been advocated as the best surrogate;( 559) however, a chemically defined phase would provide more-reproducible results. Long-chain TAGs with saturated chains are solid at room temperature. The melting temperatures decrease with shortening the chains, e.g., trioctanoylglycerol melts at 6 °C. This clear liquid boils at 233 °C and can be distilled easily. Trioctanoylglycerol seems to be suitable for partitioning studies at room temperature, although no experiments have been performed yet. Triolein (trioleoylglycerol) 192,560–563 represents another option. Surrogate phases were also used to mimic the binding sites of proteins, using solvents that represent the fragments of amino acid side chains: benzene, toluene, pyrrole, 1-propane-thiol, and methyl ethyl sulfide.( 564) The applicability of the approach to protein interactions is limited, because the protein binding sites are much more heterogeneous than the bilayer regions. The protein phases, which could represent homogeneous environments, are not large enough to completely solvate the interacting drug molecules, as observed in the macroscopic surrogate phases. 2.1.3.4. Partitioning and Interfacial Transfer Experimental techniques for monitoring transport of chemicals through the planar 387,389,390,506,565–567 or spherical bilayers 73,397,507,546,568–585 focus on overall permeability. The elementary steps have not been studied, with the sole exception of the flip-flop rates for amphiphilic compounds. 586,587 Computational simulations of chemical distribution in hydrated bilayer patches( 588) (section ) do not analyze this phenomenon either, because no experimental data are available for comparison. Despite this lack of interest, the rate of interfacial transfer is an important characteristic for all chemicals, which accumulate in the bilayer regions. Some information about the process can be obtained from the data on interfacial transfer between water-immiscible organic solvents and water (see below). Structure and dynamics of liquid−liquid interfaces is a matter of intense experimental research relying mainly on voltammetry,( 589) vibrational sum frequency spectroscopy, 590–596 and X-ray reflectance, 597–603 as well as on computational simulations. 558,604–607 The interface structures are rather different, even for similar organic solvents. For lower alkanes( 597) and carbon tetrachloride systems, where the nonpolar phase has no hydrated groups interacting with water, the interface is only a few angstroms wide and exhibits molecular sharpness, with a slight mutual permeation of both solvents due to thermally excited capillary waves. Interfaces of higher alkane systems are wider than predicted from the capillary-wave theory, and the deviations increase with the chain length. 608,609 Dichloromethane also exhibits a wide, more diffuse interface in contact with water.( 610) The interface is broader for the solvents with well-hydrated functional groups, including alcohols 602,611,612 and ketones,( 600) as well as for other solvents containing amphiphilic compounds, such as surfactants 590,613 and phospholipids,( 614) which adsorb at the interface. Physical resemblance of the interfaces between bilayer/water and organic solvent/water systems is closer for the solvents with functionalized molecules than for alkanes.( 558) Computational simulation of partitioning at the interfaces suggested interesting behavior of several polar solutes, such as alcohols, anesthetics,( 615) monofluoromethane, difluoromethane, and trifluoromethane (but not methane and tetrafluoromethane),( 513) as well as trifluoroethane (but not hexafluoroethane)( 616) at the hexane/water and glycerol-1-monooleate/water interfaces. The solutes exhibited accumulation at the interface, although molecular structures of at least some of them did not show a clear separation of polar and nonpolar fragments as known for surfactants. This fact indicates that interactions of functional groups and substructures with interfaces may be difficult to extrapolate from hydration energies. Despite the dissimilarities of interfaces, solvent/water systems have been used to monitor the solute transfer, in the hope that relative trends can be discerned, which would also hold for the bilayer/water interface. The experimental arrangement for the measurement of the transfer rate parameters from an organic phase to water and backwards is simple.( 498) Both immiscible phases sit one above the other in a temperature-controlled jar and are stirred at an optimal rate that secures sufficiently rapid equilibration in the bulks. Under the stirring conditions, the interface is usually not planar and adopts a slightly conical shape, but the interfacial area can be approximately calculated. The studied chemical is added to the phase in which it has a higher solubility, and the kinetics of its partitioning is monitored, preferably in both phases. From the time courses of the concentrations, the rate parameters of transfer, li and lo (for the direction from water to the organic phase and backwards, respectively), are determined. This so called slow-stirring method precisely determines the partition coefficient P = li/ lo, because the achievement of the equilibrium is verified in the course of the experiment, and the risk of the mutual droplet cross-contamination of the phases is completely eliminated,( 617) as opposed to the more frequently used shake-flask approach. The slow-stirring method was also used to measure the P values of unstable compounds with the rate of decomposition comparable to the rate of transfer.( 618) Arrangements of multiple phases have been used in the water−1-octanol−water system 619,620 and in the system consisting of a series of 11 alternating aqueous and 1-octanol phases.( 621) A moving-drop technique was developed and used to measure the transfer rates of short-chain alcohols and tritiated water across the 1-octanol/water interface.( 622) Electrochemical techniques are useful for the monitoring of the interfacial transfer in solvent/water systems. 589,623–630For the QSARs of the compounds, which do not interact with the headgroups, the relationships between P and the rate parameters li and lo are of special interest. Experimental observations( 620) were quantified as( 631)
where A t and B t (the subscript “t” denotes the transfer) are empirical coefficients, which are dependent on the organic phase, the geometry of the apparatus, and the stirring rate, but are independent of the molecular structure of the tested chemicals. The rate parameter li for the transfer from water to the organic phase initially increases with the partition coefficient ( P = li/ lo) and levels off for high P values. The rate parameter for the reverse direction ( lo) is constant for lower P values and drops for high P values. The observation, summarized in eqs 3, was later extended to chemicals that do not belong to a homologous series, may ionize or form ion pairs.( 66) The plateaus in the dependencies were assumed to originate as a consequence of the diffusional control of partitioning in the diffusion layers. In a steady-state treatment of the interfacial transport process,( 66) the empirical coefficients in eqs 3 were expressed using the diffusion coefficient D and the effective thickness h of the corresponding diffusion layer as A t = D org/ horg and B t = D orghaq/(D aqhorg). The apparent independence on the molecular structure of the coefficient A t is rather puzzling as the diffusion coefficients in the apolar solvents are known to vary with the molar volume.( 632) A small dependence of the permeability coefficient on the molar volume was observed( 633) in experiments with rotating diffusion chamber,( 634) which, however, monitor transport through planar systems (section ) that is more complex than the simple interfacial transfer. Another view of the empirical coefficients A t and B t was provided by considering convective diffusion caused by the laminar flow in the vicinity of the interface.( 635) Practical impact of these observations on the distribution of chemicals in biosystems remains to be assessed. A further matter of dispute is the interfacial resistance to transfer of chemicals through the phase interface, which is sometimes found to be an effective barrier to the chemical transfer 634,636 and, in other cases, turns out to be negligible. 637,638 This phenomenon probably is dependent on the ability of transported chemicals to bind to individual bilayer regions (see sections and ), as shown in MD simulations of trans-bilayer transport.( 588) 2.1.4. Accumulation and Transport Historically, transport and accumulation were first measured using cells (section ) and biological sheets (section ). Most organic molecules with molecular weights up to 1000 g/mol are expected to pass the bilayer through the bilayers. The measurements on spontaneously aggregating phospholipid systems, void of the effects of other membrane components, started after liposomes (section ) and BLM (section ) were discovered. Typically, the measurements in phospholipid systems use high chemical-to-lipid ratios, corresponding to molar fractions of 0.1 and more. Such high concentrations of chemicals are relevant only for the bilayers, which come into a direct contact with certain types of local doses of chemicals. Upon absorption from the gastrointestinal and pulmonary tracts or through the skin, and subsequent distribution, however, the chemicals become significantly diluted. A typical ballpark value for the drug dosing is 1 mg/kg, because only ~10% of the drug candidates show some potency in the 0.1 mg/kg range.( 639) For the lipid weight fractions in lean human tissues,( 176) ranging from 0.01 to 0.1, there are 10 4−10 5 times more lipids than drug molecules under pseudo-equilibrium conditions (section ). Even if all molecules would accumulate in the lipid phases, the chemical-to-lipid ratios are much lower than in the experiments with phospholipid aggregates. The measured data usually must be extrapolated to zero concentrations, to become relevant for the majority of observed chemical-bilayer interactions in human body. From this viewpoint, the experimental methods that “see” the molecules of chemicals reflect the real conditions better than the methods that examine lipids. While the used biological systems such as cells and biological sheets contain several bilayers, the data were frequently processed by the models consisting of a single bilayer. In several studies of transport mechanisms, the results from different test systems were used interchangeably, although the number of crossed membranes varied significantly. The outcomes of these studies will be sorted out and presented separately for different test systems: bilayers in sections − , and membrane vesicles, subcellular organelles, cells, cell monolayers, biological sheets, tissues, organs, and organisms in sections − , respectively. 2.1.4.1. Binding in Bilayer Regions The headgroup and core regions of the phospholipid bilayer differ in the types of interactions with the transported molecules of chemicals, and in the ability to shield electrostatic charges. Both regions are sufficiently voluminous to fully solvate the molecules of chemicals.( 220) The headgroup region can be subdivided into perturbed water and interphase (subregions 1 and 2), and the core region consists of soft polymer and alkane (subregions 3 and 4), as described in section . Chemicals can accumulate in the headgroup subregions 1 and 2 ( cephalophiles), subregions 2−3 ( amphiphiles), and core subregions 3−4 ( lipophiles). Taking into account that the solvation properties of the headgroup region differ from those of bulk water, the existence of the headgroups/water interface is justified. Published examples of individual interaction modes are summarized in Charts 1− 3, with Tables 2− 4 providing more details. The headgroup region is well-hydrated but the water molecules are tightly bound to the phospholipids headgroups. Consequently, the solvation properties of the headgroup region differ from those of bulk water. Therefore, we prefer to use the term “cephalophilicity” to denote the tendency to accumulate in the headgroup region, instead of “hydrophilicity”, as occasionally seen. 640–643 | Table 2Cephalophiles: Compounds Experimentally Shown To Accumulate in the Headgroup Region of the Phosphatidylcholine Bilayer |
| Table 4Lipophiles: Compounds Experimentally Shown To Accumulate in the Core of Phosphatidylcholine Bilayer |
Structural determinants of binding in individual bilayer regions are quite subtle, and the overall outcome is dependent on a fine balance of individual structural elements of the molecule, including their spatial arrangement. Benzene derivatives, for instance, accumulate in the core ( 42 in Table 4 and Chart 3), 644–646 unless a hydrogen-bonding substituent is available to anchor them in the headgroup region as seen for benzyl alcohol ( 2 in Table 2 and Chart 1).( 647) The partitioning in the hydrocarbon core prevails, however, if the hydrogen-bonding ability of the substituent is obstructed by other groups as in 2,6-di-C(CH 3) 3-phenols ( 42 in Table 4 and Chart 3).( 648) Several experimental techniques, summarized in Tables 2− 4, have been used to characterize—often only qualitatively or semi-quantitatively—preferred locations of compounds in the phospholipid bilayer. This information is important for quantitative assessment of the transport rates, because analyses of published trans-bilayer transport data (section ) indicate that the transport rates are dependent on the tendencies of chemicals to interact with the hydrocarbon core and the headgroup region. The two tendencies of chemicals are lipophilicity and cephalophilicity, respectively. Amphiphilicity denotes a mixed scenario, when a molecule interacts with both headgroup and hydrocarbon regions. Smaller amphiphilic molecules bind in subregions 2 and 3; larger molecules reach into subregions 1 and 4. Our preliminary data indicate that some chemicals could also accumulate at the interface between the bulk water and the headgroups, pointing to the possibility that there may be two types of amphiphilicity. Lipophilicity is usually parametrized by a reference partition coefficient (section ), which is defined as the ratio of activities (or concentrations, if they are sufficiently low) of the solute in a surrogate phase and the aqueous phase under equilibrium conditions. Because lipophilicity is given by the relative levels of accumulation in the hydrocarbon core of the bilayer and in bulk water, the core/water partition coefficient is the most suitable parameter for characterization of lipophilicity in the context of bilayer transport and accumulation. Lipophilicity generally is defined vaguely, because there are different nonpolar phases in the body: bilayer cores, lipid droplets, and hydrophobic lipoprotein interiors. To avoid having different lipophilicities reflecting composition variability of biological nonpolar phases, we prefer to define lipophilicity as the core/water partition coefficient for a saturated PC, which should be properly parametrized by the hexadecane (C 16)/water partition coefficient (section ), according to eq 1 in section . Fatty acid chains are hydrocarbons, which only differ in the length, number, and positions of double bonds, and sometimes in branching. This variability should not lead to pronounced differences in the interactions of the chains with chemicals. Therefore, the Collander equation (eq 1) can be used to correlate the core/water partition coefficients for the cores of varying composition with the C 16/water partition coefficient. Other reference partition coefficients may be more suitable for lipid droplets and lipoprotein cores (section ). All partition coefficients are additive-constitutive properties, which can be, in principle, partitioned among individual fragments of the molecule. 501,518,649–652 This procedure allows for a fast structure-based prediction of all partition coefficients. Cephalophilicity denotes the tendency of a chemical to bind in the headgroup region, forming the interactions in subregion 1 and, predominantly, in subregion 2. Experimentally, cephalophilicity can be quantified as the headgroups/core or headgroups/water partition coefficients, which are quite tedious to measure directly in the bilayers. In the absence of fast and convenient microtechniques, which would be capable of a routine monitoring of the concentrations in the bilayer regions, solvation energies in the systems imitating the bilayer environments have been used to provide initial estimates. As mentioned in section , we have found that diacetylphosphatidylcholine (DAcPC) forms a homogeneous solution with acceptable viscosity, when mixed with water in a similar ratio as in the fully hydrated bilayer headgroup region. From this solution, the DAcPC molecules practically do not partition into nonpolar solvents. The two-phase system n-hexadecane/DAcPC aqueous solution was used to precisely determine the differences in solvation energies of organic chemicals, which can be used to estimate the preferred bilayer location (section )( 552) and the transport rates (section ). Amphiphilicity corresponds to the ability of a molecule to have, in at least one conformation, lipophilic and cephalophilic parts placed in two subspaces separated by a plane, so that they are able to interact with the headgroup region and the hydrocarbon core across the headgroups/core interface. Amphiphilicity has been roughly expressed as asymmetry of the spatial distribution of lipophilicity in the molecule. For a given conformation, amphiphilicity was characterized by the lipophilic moment( 653) and similar parameters, 316,654–656 or by an empirical function describing position-dependent solvation energies.( 640) All these approaches utilize the hydration characteristics of chemicals in bulk water instead of true cephalophilic characteristics. These characteristics may not be adequate because the water molecules, although abundant in the headgroup region, are mostly confined forming strong hydrogen bonds in the hydration shells and behave differently than those in bulk water. Cephalophilic fragment contributions must be developed, in a similar way as known for lipophilic contributions.( 518) 2.1.4.2. Interactions with Plasma Lipoproteins and Lipid Droplets A common feature of plasma lipoproteins and intracellular lipid droplets is the presence of a core that is composed mainly of triacylglycerols (TAGs). TAGs have different solvation properties than phospholipids. Specific surrogate phases are needed for a precise estimation of the partitioning of chemicals into TAGs. Plasma lipoproteins are spherical or discoidal particles, which are composed of apolipoproteins, a phospholipid−cholesterol monolayer coat, and the core that mainly contains TAGs and esterified cholesterol. They are classified into four groups, based on the particle density, decreasing from 1.21 g/mL to <0.95 g/mL, as follows: high-density lipoproteins, low-density lipoproteins, very-low-density lipoproteins, and chylomicrons. In this order, the diameter of the particles rises from 35 Å to 6 μm, the TAG content increases from 2 % to 88 % d.w., and the protein, cholesterol, and phospholipid contents decrease.( 657) The self-assembly of lipoprotein nanodisks and their disassembly caused by cholate addition were analyzed by mesoscale modeling and small-angle X-ray scattering. 360,361 The physiological function of plasma lipoproteins is the transport of, respectively, cholesterol from tissues to the liver, cholesterol to tissues, endogenous TAGs, and exogenous TAGs and cholesterol. Interactions with lipoproteins can significantly contribute to intestinal absorption of lipophilic chemicals via lymph. 148–150 The potential to be absorbed by this pathway was assessed using the binding to chylomicrons( 658) and predicted using the partial least squares (PLS) regression analysis of numerous VolSurf descriptors.( 659) Lipid droplets are ubiquitous cellular structures, composed mainly of TAGs. They are highly concentrated in adipocytes, which contain ~90% (w/w) TAGs. Partitioning into adipose tissues heavily affects the distribution of lipophilic compounds. Synthetic TAGs with saturated fatty acids melt in the range of 60−80 °C, and the melting temperatures rise with chain prolongation. Solid TAGs mixed with emulsifiers form platelets with discrete molecular layers. 660,661 Liquid TAGs under physiological conditions are not completely isotropic,( 662) and loose remnants of the layered organization from the solid state can be recognized.( 663) The aggregates of the ester groups are more diffuse than the headgroup regions observed for phospholipid bilayers (section ), bind much less water than the phospholipid headgroups, and do not form distinct phases. Consequently, the solvation behavior of TAGs can probably be emulated by one solvent, although the solubilization process can be complex in some cases.( 664) Vegetable oil has been proposed as a TAG surrogate,( 665) but the dependence of the composition on the source may be a problem. The partition coefficients of chemicals have been measured in the triolein (trioleoylglycerol)-water system, 192,562,563 and used for structure-based predictions( 561) and in correlations with bioconcentration in fish.( 666) 2.1.4.3. Transport versus Bilayer Fluidity The bilayers exhibit generally higher permeabilities in the liquid-ordered phase than in the gel phase.( 667) Several barrier epithelia such as gastric mucosa, and those in the renal collecting duct and urinary bladder, maintain high solute gradients between body compartments and the blood. Low fluidity of their apical membranes has been postulated as the mechanism underlying the low permeabilities. 225,668–670 Regulation by the lipid composition of the bilayer fluidity at the body temperature has been considered a major mechanism for controlling bilayer permeability. 225,226,670,671 Expulsion from the bilayer into bulk water of di- tert-butyl nitroxide,( 672) perylene,( 673) propranolol ( 34 in Table 3 and Chart 2), timolol,( 674) benzene, hexane,( 272) and Bay K 8644, which is a 1,4-dihydropyridine derivative ( 27 in Table 3 and Chart 2),( 675) was observed after cooling liquid-ordered bilayers to the gel state. Increasing the surface density of the liquid-ordered phospholipids by addition of cholesterol, cooling in the temperature range above the transition temperature, or extending the fatty acid chains resulted in decreased partitioning of benzene,( 272) hexane,( 504) non-ionized acetic acid,( 505) and polychlorinated biphenyls.( 532) Increased surface density also reduced permeability of acetic acid through the bilayer composed of DMPC and cholesterol.( 505) | Table 3Amphiphiles: Compounds Experimentally Shown To Accumulate at the Interface between the Headgroup and Core Regions of the Phosphatidylcholine Bilayer |
The temperature dependence of the bilayer transport is not monotonously sigmoidal. At the gel-to-liquid transition temperature, the inhomogeneous bilayer structure results in the peak values of the transport rates, as observed for water,( 676) Na + cations, 673,677,678 glucose, methylphosphoethanolamine,( 580) and trimeprazine,( 572) as well as in a faster insertion of an amphiphilic hydroxycoumarin alkylammonium fluorophore.( 256) These phenomena have been observed in the bilayers composed of pure phospholipids, where the transition occurs in a narrow temperature interval, as opposed to the bilayers consisting of mixed phospholipids and membranes. In membranes, the interactions of phospholipids with integral proteins may cause distortions in the bilayer structure. These irregularities could have a similar enhancing effect on the transport rates as observed during gel-to-liquid transitions. 2.1.4.4. Steady-State Transport Most experimental data on the steady-state trans-bilayer transport were obtained using the BLM technique, utilizing a small-area bilayer separating two aqueous compartments (section ). Many models used to describe the BLM data were previously derived for the cellular and sheet systems (sections and , respectively), represented by the oversimplified one-membrane or one-compartment systems. Before looking at detailed descriptions, let us make a few general remarks on the steady-state disposition of chemicals in biosystems. A flow-through transport system reaches the steady state, when a constant flux passes through each cross section of the system, and the concentrations of the transported chemical in all parts of the system become time-invariant. For this situation to happen, the concentration in the entry compartment—and, consequently, the entry rate—must be constant for a sufficiently long time. Although membrane transport studies are mostly performed in closed systems, the data are often processed using the assumption of the steady-state conditions in the membrane, because the concentrations of the permeant in the bilayer regions are assumed to be much smaller than those in the donor aqueous compartment. Consequently, the respective gradients are close to zero and considered to be approximately constant. The constant flux through the bilayer is described by the permeability coefficient, which is quantified as the slope of the approximately linear increasing part of the concentration−time curve in the acceptor compartment. The shape of this curve documents the adherence of the situation to the steady-state conditions. The presence or absence of an initial lag phase indicates that the permeant at least temporarily does or does not accumulate in one or more bilayer regions. The leveling off at the end of the transport experiment reflects the achievement of the equilibrium, when the chemical potentials in both aqueous compartments become equal. For the non-ionized solute not interacting with the bilayer, the equilibrium concentrations are easy to calculate using the initial concentration in the donor compartment and the volumes of the two compartments. For the opposite case, membrane accumulation can be determined from the difference between the experimental equilibrium concentration and the hypothetical concentration for the situation without membrane accumulation. In summary, for chemicals interacting with the bilayer, the permeability coefficient alone does not suffice to describe and predict the overall distribution of chemicals, and additional parameters are needed to account for the possible lag phase and the final phase of the distribution kinetics. Despite this shortcoming, the permeability coefficient is frequently used as a characteristic of the interactions of chemicals with bilayers in pharmaceutics and in membrane biology. As shown several decades ago for the cellular transport, 521,679,680 the permeability coefficient for the steady-state flux through the bilayer was linearly related, on the bilogarithmic scale, to the surrogate partition coefficient (section ) of the transported chemicals forming a homologous series. 297,668,681–685 In terms of elementary steps, this dependence was explained by the solubility-diffusion mechanism, which assumes a fast dissolution of the permeant into the entry region of the bilayer, slow diffusion through the interior, and another fast dissolution step at the distal bilayer/water interface. Within the framework of this mechanism, the permeability coefficient is proportional to the product of the partition coefficient and the diffusion coefficient, with both quantities mostly measured in surrogate systems. The linear log-log relationships between trans-bilayer permeability and the partition coefficients for more-diverse chemicals were augmented by molecular weight as another linear term, because the diffusion coefficient is inversely related to molecular weight. This relation was obtained based on a structural model of the bilayer as a homogeneous, hydrophobic slab. The model has been made more realistic by the addition of two resistances at the interfaces,( 499) and significantly improved by inclusion of scaling factors accounting for the effect of chain ordering( 506) and the permeant size and shape.( 73) 2.1.4.5. Specific Mechanism for Small Molecules Water,( 298) urea, formic acid,( 387) formamide,( 297) oxygen,( 686) and nitric oxide( 687) are frequently assumed to permeate the bilayer using a mechanism that is specific for small molecules. 292,688 Details of this mechanism may include mobile chain kinks taking the permeants along,( 689) hopping of the permeant molecules between the kinks( 690) or between free volumes created by thermal fluctuations in the headgroup region and in the core, 677,691 and collective density fluctuations (phonons).( 692) Longer, transient, and water-filled pores( 693) were implicated in trans-bilayer permeation of monovalent cations including protons and potassium.( 684) Molecular simulations indicated that the mechanisms involves active participation of the ions such as protons, Na + cations, and hydroxyls.( 694) The main support for the involvement of transient pores comes from the fact that the measured transport rates are higher than predicted from the corresponding permeation models. Imperfections in these models, demonstrated in sections − , indicate that this criterion may not sufficiently support the claim that the special mechanisms are the main transport route, at least for some small molecules and in some membranes. A rare experimental opportunity to examine the trans-bilayer transport mechanism is based on the fact that the pores produced by thermal fluctuations occur less frequently for longer phospholipid molecules,( 695) resulting in an exponentially decreasing dependence of the permeability coefficient on the bilayer thickness.( 696) Experiments with the bilayers made of PC with unsaturated fatty acids of varying lengths showed that water, urea, glycerol,( 684) and halide anions( 685) do not use the short-lived pores as the dominant transport mechanism. Protons and Na + cations do utilize this mechanism as the main route, but only for thinner membranes. Apparently, more experiments are needed to reconcile the contradictory views. The thermal fluctuations occur with differing frequencies in individual bilayer regions. Therefore, it is possible that some molecules will cross the bilayer using a combined mechanism consisting of classical diffusion in the more-dense subregions 2 and 3 and a pore-hopping mechanism in subregion 4. The question of the transport mechanism is fundamental for the phenomenological description of transport kinetics. For a pore-hopping mechanism, the maximum concentration of a chemical in a bilayer region would be limited by the pore density, whereas no such limits are expected for diffusing chemicals. 2.1.4.6. Kinetics of Transport The overall kinetics of transport and accumulation is described completely and exhaustively by mechanistic models with several rate constants, characterizing the transfers between individual bilayer regions. The use of transient kinetics prevails in the areas of drug design and membrane biophysics. The observed trans-bilayer transport rates of diverse chemicals range from seconds to days, which are needed to achieve the partitioning equilibrium across a single liquid-ordered bilayer. The transient kinetics of transport provides a more-complete picture of the overall process, including accumulation in individual regions of the bilayer that is important for the description of the interactions of chemicals with the transport proteins, metabolizing enzymes, and the receptors. The first kinetic model of transport in a multicellular system assumed slow transfers through both bilayer interfaces, which can include subregions 1−3 (section ), and fast diffusion in the core, where the chemical could accumulate.( 58) This mechanism is in accordance with the bilayer density profile (section ), and accurately describes the trans-bilayer transport of some chemicals, as summarized below. The resulting peak-shaped dependencies of the transport rates on the partition coefficients were frequently observed at the cellular level.( 1) The model was extended by considering the different solvation abilities of the headgroup regions and the core in some cases. 667,697 The prediction of trans-bilayer transport rates remains elusive, despite tremendous efforts to develop descriptors of compounds that would correlate with this property.( 469) Here, we present an analysis of the transport data and properties of the permeants that could help in resolving the prediction problem. Tendency to associate with the headgroups, interface, and core, as reflected in cephalophilicity, amphiphilicity, and lipophilicity of chemicals (section ), leads to the compartmentalization of the bilayer for transport models. The observed situations are summarized in Figure . Cephalophilicity is not included in this scheme because, at this point, we do not completely understand structural determinants of this property and cannot tell whether a permeant accumulates in the headgroups, as compared to the bulk water or not. Lipophilicity is precisely characterized by the reference partition coefficient and, therefore, three levels of lipophilicity are distinguished in Figure . Amphiphilicity is considered in a semi-quantitative way, with high magnitudes expected for compounds with separated polar and nonpolar parts, and low magnitudes otherwise. The interface regions, where the amphiphiles partition, are drawn in Figure as the volumes comprising parts of subregions 2 and 3, i.e., the high-density headgroup and core regions (section ), which are sometimes called the diffusion layers.( 698) The compartmentalization rules are simple. The aqueous phases are always regarded as compartments. The bilayer can be represented by maximally five compartments: two headgroup regions, two interfaces, and the core. The actual number of compartments is given by the permeant properties. The compartments, in which the permeant exhibits measurable accumulation, are included in the scheme. Although the headgroups are shown in Figure , we do not yet know which types of chemicals prefer to partition there. Therefore, in the following paragraphs, we will only deal with the amphiphilicity and lipophilicity of chemicals. Let us first consider non-amphiphilic chemicals. If hydrophilic, they do not interact significantly with any part of the bilayer, which then becomes just an inert barrier between the two aqueous compartments (row 1). If lipophilic, they accumulate in the hydrocarbon core of the membrane, which then represents a compartment (rows 2 and 3). Amphiphilic chemicals tend to interact with the headgroup/core interface, forming one additional compartment on each side of the bilayer. For hydrophilic amphiphiles, the interfaces are the only two compartments of the bilayer. The third bilayer compartment, corresponding to the hydrocarbon core, is added for chemicals that are both amphiphilic and lipophilic. The experimentally observed rates of trans-bilayer transport of chemicals vary in a wide range. The results of published transport studies are summarized in a common framework in Figure . Diffusion in bulk water, as well as in the low-density headgroup and core subregions 1 and 4, is much faster than the steps, which involve crossing of the more-structuralized interface subregions 2 and 3 (sections and ).( 699) The four schemes represent known mechanisms that have been used to describe the trans-bilayer transport. The first row, which is valid for non-amphiphilic hydrophilic chemicals, represents the inert barrier concept.( 700) A single parameter, the permeability coefficient PC is sufficient for a complete description of the process. A typical duration of the transport process is minutes to hours, 569,701–703 because molecules of the bilayer-inert chemicals enter the bilayer with difficulty. Because the membrane accumulation is negligible and the intracellular aqueous phases are usually much smaller than their extracellular counterparts, the transport of these chemicals is ideally measured as their release from preloaded vesicles. The scheme in the first row is also suitable for description of kink-based and pore-based mechanisms for the permeants with a low bilayer accumulation. The second and third rows describe the situation of the non-amphiphilic chemicals with a higher lipophilicity, which can accumulate in the bilayer core( 56) but do not interact with the interfaces. Two rate parameters, li and lo—for entering and leaving the core, respectively—are needed for a complete description in this case. Both rate parameters and, consequently, the overall transport rate are dependent on lipophilicity( 631) (see eq 3 in section ). The fastest transport, with the equilibrium reached in the time interval ranging from milliseconds to several seconds, 82,577,704 is observed for chemicals with intermediate lipophilicity. Very hydrophilic chemicals (row 1) and very lipophilic chemicals (row 3) cross the bilayer at much slower rates. 56,143,705 This mechanism has been most extensively used in simulations of transport in biosystems. 16,58,706 The agreement between phenomenological simulations and the experiment (sections and ) supports the adequacy of the used description, at least for this mechanism. The fourth row applies to chemicals, which are both hydrophilic and amphiphilic. The adsorption and desorption rates, characterized by the parameters la and ld, respectively, are usually high (on the order of milliseconds( 707) or seconds( 708)). The flip-flop between the two interfaces of the bilayer, characterized by the rate parameter lf, can take minutes, 578,584 hours,( 709) or days, 710–712 depending on the strength of the interaction between the chemical and the headgroup region. This point is illustrated by the fact that the flip-flop rates are much higher for the non-ionized species of amphiphilic molecules compared with the ionized species.( 713) The fifth and sixth rows in Figure describe the interactions with the interfaces, combined with the accumulation in the core, which are typical for amphiphilic and lipophilic chemicals. Two rate parameters ( ldc and lac) for the desorption/adsorption between the interface and the core were added. The desorption from the interface into water was accomplished within a fraction of a second.( 714) This mechanism has been described for ionizable chemicals, with the desorption from the interface into the core caused by charge neutralization. 107–110,429The conclusions summarized in Figure can be extended by including cephalophilicity. The four laminae, two phases (the headgroups and the core) and two interfaces (between the bulk water and the headgroups and between the headgroups and core) play important roles in the overall bilayer permeability. Our hypothesis is that an intermediate strength of interactions of a chemical with all four laminae is necessary for its fast trans-bilayer transport. Any deviation of the interaction strengths from the intermediate values, even for a single lamina, leads to a slowdown in the overall transport. The hypothesis is based on simple and straightforward logic: weak interactions with a lamina prevent a compound from entering it, and strong interactions keep the compound inside the lamina. This behavior has been previously postulated for the bilayer core by Hansch and Fujita,( 56) and used as a theoretical basis for an interpretation of the concave activity−lipophilicity profiles. For the frequent case of non-amphiphilic chemicals, a cell or a higher biosystem can be represented by a catenary set of alternating aqueous and membrane compartments, as it has been proposed for pure transport of non-amphiphilic compounds.( 58) All presented models reviewed below in sections and apply to chemicals, which do not differ in cephalophilicity and amphiphilicity. The assumption of the instantaneous diffusion in bulk water and the core allows the use of the conventional mass balance description of the kinetics and equilibria of all processes affecting the fates of chemicals within the compartments (part 10.1.1). 2.1.5. Computational Approaches Simulation of different types of the bilayer behavior is complicated by widely differing length and time scales of the studied system: microscopic thickness contrasts with almost macroscopic surface area, and the rates of the bilayer processes including the transport (section ) vary in a broad range. Another complicating factor is the rich phase behavior of phospholipid aggregates, which may adopt the form of micellar, lamellar, hexagonal, and cubic phases. The time scales of individual motions in the bilayer( 715) increase in the following order: fluctuations of bonds, angles, and dihedrals within the same conformation (femtoseconds for covalent bonds to non-hydrogen-bonded hydrogens, picoseconds for other covalent bonds); trans−gauche isomerizations of dihedrals (tens of picoseconds in the core to nanoseconds in the headgroups); lipid rotation along the long axis and “wobbling” (nanoseconds);( 716) lipid position changes in the monolayer due to lateral diffusion (nanoseconds to microseconds);( 265) collective undulatory motions of lipids (on the order of 0.1−10 ms, and sped up, with a smaller amplitude, by protein binding),( 717) relaxation during transitions, 264,718 trans-bilayer transport of drugs and other small molecules lacking strong interactions with bilayer regions as summarized in section (seconds to minutes); and, finally, the spontaneous flip-flop of some lipids and lipid derivatives between monolayers of the same bilayer, 711,719–721 and transport of chemicals with extreme lipophilicity and amphiphilicity, as described in section (minutes to days). The major events and their time scales are summarized in Figure . Atomistic MD simulations belong to the most informative computational approaches for studying bilayers. The maximum duration of published force-field-based MD studies of hydrated patches of phospholipids, currently a hundred( 722) or more nanoseconds,( 723) is by a wide margin too short to cover many bilayer-related phenomena with significant time-scale (Figure ) and length-scale differences. In addition, the slower phenomena usually require the use of a larger phospholipid ensemble than those currently utilized.( 724) Modified approaches are needed for the situations where atomistic simulations cannot provide answers. A broad palette of available computational approaches can be classified based on (i) the used lipid molecule representations—simplified particles or all-atom molecules; (ii) solvation—approximated by a continuum or explicit solvent molecules; (iii) interactions between molecules—varying levels of approximation; (iv) spatial arrangement of the molecular representations—one-dimensional to study, e.g., some aspects of the ripple phase, two-dimensional to study surface phenomena, or three-dimensional to study complete structures; (v) movement of molecules—free, confined to a lattice, or none, i.e. the use of average equilibrium positions; and (vi) the used simulation method − in the order of increasing complexity: energy minimization, Self-Consistent Mean-Field Theory (SCMFT), MC simulations to study mainly equilibria, and Brownian dynamics (BD) and MD to study both kinetics and equilibria. Selection of the approach is dependent on the structures and/or processes to be modeled. The approaches are frequently combined and the multiscale methods are of special promise. 725–7352.1.5.1. Bilayer Formation and Rearrangements Collective phenomena requiring the use of large systems and long time scales are usually simulated using approximate molecular representations of lipids and other participating molecules. The mesoscale approaches can be classified as faster lattice-based models 677,736–741 and more-detailed off-lattice models.( 742) The lateral rearrangements are frequently studied in two dimensions: on-lattice MC models were used to simulate the ripple phase formation 743–745 and de-mixing of lipids differing in chain flexibility caused by cholesterol 746,747 (also analyzed by SCMFT( 748)) or membrane proteins. 234,236,746,749 The off-lattice models were utilized to analyze complex phenomena proceeding at long time scales such as lipoprotein nanodisk 359–361,750 and micelle 742,751 formation, bilayer formation, 752–754 dissolution and rearrangements( 755) (also analyzed by SCMFT( 756)), mechanical properties, 757–759 pore formation,( 752) vesicle shape fluctuations 760–763 (also analyzed by SCMFT( 764)), vesicle tubulation upon interaction with a protein,( 765) vesicle fusion, 766,767 lipid interactions with transmembrane proteins,( 733) lipid−cholesterol interactions, 241,768–770 lipid lateral diffusion, 241,768,769,771 and lipid demixing. 241,768,769,772 These approaches, also known as coarse-grained simulations, use the representations of small groups of atoms by beads or particles with specified interaction potentials. The particle consistency can be either hard( 742) or penetrable and interacting via soft potentials such as in dissipative particle dynamics (DPD). 755,766 Coarse-grained models with particles corresponding to individual phospholipid substructures, which interact through potentials resembling atomistic force fields, 359–361,750,752–754,773–779 provide more quantitative results. Mesoscale modeling methods have become more available after the release of specialized software suites, such as Mesodyn( 780) and DPD.( 781) 2.1.5.2. Bilayer Structure and Dynamics Early approaches to characterization of bilayer structure focused on the acyl chains using an SCMFT approach 782–785 and BD simulations of a single lipid molecule 786,787 with a continuous version of the mean-field potential.( 782) The force-field energy minimization of the monolayer and bilayer models 788,789 was used to analyze the intrinsic dipole potential, 790–792 acyl chain packing, 793–797 and the mode of insertion of drugs 641–643 and proteins.( 798) More-extensive sampling of the simulated systems can be obtained using either the probabilistic MC methods or the deterministic MD methods. The explicit time dependencies resulting from the integration of equations of motion for all atoms in a system made MD the preferred technique in most applications. In MC approaches, time-dependent properties can be, at best, obtained indirectly. For some situations, especially for the analysis of the equilibrium characteristics of processes, which are slow at the time scale covered by the MD simulations, MC can be the method of choice.( 799) The first MC studies of the bilayer structure in three dimensions focused on the chains that were anchored at the interface, with the surface density corresponding to that of the respective phospholipid bilayer. The dihedrals were initially restricted to three states( 746) or to a three-dimensional diamond lattice,( 800) to allow the simulations to finish in a reasonable time. The order of chains was studied, both for neat chains and under the influence of cholesterol,( 801) a simple cylindrical protein model,( 749) and gramicidin.( 802) United-atom phospholipid models, confined to the diamond lattice and used in fast MC simulations of bilayer structure,( 803) reproduced all states of the bilayer along the temperature scale, from the crystal phase through liquid-crystal and quasi-hexagonal phases to an isotropic liquid. United-atom representation of phospholipid molecules was used in the MD simulations combined with: (i) an MC procedure to analyze the distribution and shapes of the free volumes( 804) and (ii) the mean-field potential BD approach to study the chain properties.( 805) An early full-atom MC simulation of the bilayer patch of 36 phospholipid molecules( 806) analyzed the chain-order parameters and hydration differences between the gel and liquid-ordered phases. The used Metropolis procedure did not approach the recommended thousand moves per a degree of freedom.( 807) To enhance the sampling, several techniques were proposed to replace the random moves in the studied system, including configurational bias( 808) and extension-based rotation sampling.( 799) The techniques enabled full-atom studies that analyzed orientation of water molecules,( 809) provided structural information on pure bilayers and on the effects of cholesterol, 810–812 and simulated voids in the bilayers of pure saturated( 813) and unsaturated phospholipids,( 814) as well as phospholipid−cholesterol mixtures.( 815) The analysis of voids is relevant for the hypothetical transport mechanism of small molecules, as described in section . Availability of the general-purpose molecular simulation suites such as CHARMM,( 816) AMBER,( 817) GROMOS,( 818) OPLS,( 819) and later NAMD 722,820,821 promoted the use of MD methods. Two monolayers separated by a pool of water molecules were used to analyze the repulsive hydration force, 822–824 as well as the structure of hydrating water, 823–826 the headgroup region, 823,824,826,827 and the core region. 826–828 Structure and dynamics of hydrating water were also analyzed in monolayer systems. 829,830For the bilayer, the initial coarse-grained models utilizing particles, 804,831 hexadecane chains confined on one end,( 832) and other surfactants, with either continuum solvent approximation 833–835 or with explicit solvent, 513,616,836–838 were soon replaced by atomistic models of fully hydrated phospholipid molecules. 839–843 Several technical issues, including the refinement of force fields, 838,844–853 the choice of simulated ensembles maintaining the invariability of selected traits (pressure, box dimensions, surface area per lipid), 846,854–858 the choice of periodic boundary conditions,( 859) and effective calculation of electrostatic 840,860–862 and van der Waals interactions( 842) had to be resolved before the force-field MD simulations could be deployed for the analyses of structural and dynamical issues in bilayers. Regarding the use of surface tension, 553,863,864 it seemed that this need was diminished for the simulations of larger systems.( 865) The equilibration times were significantly shortened by augmenting MD simulations with carefully chosen MC steps. 866–868The MD simulations of bilayer patches were mostly applied to liquid-ordered phases but the behavior of the gel phase was also analyzed.( 869) Among individual phospholipids, PC as the most abundant mammalian phospholipid was the most frequently studied molecule. Behaviors of other zwitterionic phospholipids, phosphatidylethanolamine 302,826 and sphingomyelin,( 870) and anionic phosphatidylserine 871–875 and phosphatidylglycerol,( 443) were also analyzed. As admixtures, cholesterol 810,876–884,884–886 and sphingomyelin 887–889 were studied most frequently; the former because of its abundance and physiological significance, and the latter because of the role as a component of the canonical mixture forming the lipid rafts.( 890) The most popular tails were myristoyl and palmitoyl chains but MD simulations for lauroyl chains, 825,826 stearoyl chains,( 891) branched chains( 892) as components of archaeal and thermostable bacterial membranes, unsaturated oleoyl chains, 294,849,875,893,894 (also analyzed by the molecular-level SCMFT( 895)), di-unsaturated linoleoyl chains,( 896) and poly-unsaturated docosahexaenoyl chains 891,897,898 were performed. The simulated bilayer patch usually contained 64 lipids 842,854,891,899–901 or 72 lipids; 716,856,857,897,902 however, in some cases, larger systems 294,553,903–905 up to 1600 lipid molecules,( 870) were analyzed. The size of the patch is important for lateral lipid diffusion and does not affect internal dynamics of the lipid molecules.( 906) The duration of simulation extended up to several hundred nanoseconds.( 723) The studied aspects included fluidity of the fatty acyl chains, 839,842,907–911 structure of the four bilayer subregions differing in density (section ), 292,912 structure of the headgroup region,( 913) hydration, 294,302,902,914–916 free volume distribution,( 912) lateral diffusion of phospholipids,( 917) and the bilayer structure changes under the influence of ions, 872–875,918–921 cholesterol and other sterols, 876,878,880–883,885,886 and acyl chain unsaturation. 849,886,896,897,922–924 Extended MD simulations were used to analyze slower phenomena, such as undulatory and thickness fluctuations;( 904) the evolution of hydrophobic defects;( 925) the development of lipid domains, 888,889,926 the hexagonal phase,( 927) and the ripple phase;( 928) the pore opening by mechanical 929–931 and electrostatic forces; 929,930 the formation of the bilayer( 905) and a vesicle( 932) from diluted phospholipid solutions; and bilayer fusion. 933,934The MD simulations involve details that are frequently beyond the reach of current analytical and imaging techniques, so that the match of simulations with experiments can only be tested in a limited number of cases. Special protocols have been developed to compare the MD results with some NMR data, 716,723,856,935,936 IR data,( 911) and with the results of diffraction experiments.( 937) As common in the modeling research, plausibility of the simulations is judged by their ability to imitate known phenomena. The situation that is selected for testing the suitability of an approach should imitate, as closely as possible, the studied structure, event, or effect. Although atomistic simulations are able to reproduce several bilayer-related phenomena, they are under continuous development. For instance, the two most frequently used MD simulation packages, GROMOS and CHARMM, in then-current phospholipid parameterization, were not able to completely reproduce experimental bilayer structure from diffraction experiments.( 937) The largest deviations were seen in the distribution of terminal methyl groups of acyl chains.( 937) These analyses led to the development of constraints ensuring the agreement of the observed and simulated bilayer structure,( 938) as well as to further developments in the phospholipid force fields.( 853) The limitations of the used approach with respect to the simulated event should always be kept in mind to avoid over-interpretation of obtained results. In particular, the simulations are based on numerous parameters, including, for the most detailed MD approaches, partial charges,( 844) the force-field parameters for van der Waals interactions, and the equilibrium values and force constants for bonds, angles, and dihedrals. Many of these parameters are obtained from either experimental or quantum mechanical results for simpler compounds. The simplified forms of the common potential functions currently do not include polarizability of atoms and induced electrostatic effects, although a vigorous development is noticeable in this area. 939–948 The most common force fields can only be used for systems exhibiting weak interactions and no covalent bond changes. Even for such systems, there is no guarantee that the simulations will provide quantitatively correct results. For instance, free binding energies of ligand−protein interactions are best-described by the methods that recalibrate the simulation results by optimization of additional coefficients. 949,950 This approach could work also for the interactions of chemicals with bilayers but the experimental data required for such recalibration are scarce. 2.1.5.3. Equilibrium Partitioning of Chemicals The interactions of chemicals with bilayers are affected by fine structural details. Mesoscale models do not work well in this context,( 753) and atomistic simulations are the methods of choice. The overall partitioning is often the only experimentally measured quantity, and the intra-bilayer distribution of the studied compounds remains largely unknown. The compounds with known bilayer locations are listed in section , in Tables 2− 4 and Charts 1− 3. The importance of partitioning is occasionally underestimated because the concentrations of compounds in the bilayer are believed to be too low to play significant roles. As summarized in section , some chemicals accumulate in the bilayer regions in much-higher concentrations than in the aqueous phases. Although the volume of all bilayers in the cell is 2−3 orders of magnitude smaller than the cell volume, the amount in the bilayer is higher than that in water for all chemicals with the bilayer/water partition coefficient being >1000. The transfers between the bilayer subregions (section ) and between the headgroup region and surrounding water proceed at widely different time scales. The fast processes can be described as pseudo-equilibria in the kinetic models of slow processes. Under such circumstances, the equilibrium partitioning in the bilayer regions is of interest, even for kinetic transport studies. Early approaches to partitioning of chemicals used a continuum representation of the bilayer, most frequently in the form of a hydrophobic slab wetted by two water phases. Conformational analysis, utilizing different dielectric constants for the aqueous and nonpolar phases, and atomic transfer energies between the two phases were used to estimate the overall energetics. Selected conformations were optimized by energy minimization. 951–953 The initial insertion of chemicals and peptides was guided by the hydrophobic and hydrophilic centers of the molecules that were determined using the atomic energies of transfer between water and a nonpolar solvent.( 641) The binding mode was refined by a force-field geometry optimization of a complex with several phospholipids.( 642) The approach was extended to membrane proteins,( 954) utilizing a combination with hydrophobicity profiles,( 955) molecular lipophilicity potential, 956–958 and an MC procedure with a continuous lipophilicity function. 959–962A continuum approach with a smooth function describing the bilayer properties was devised for geometry optimization and MD simulation of bilayer interactions with solutes( 640) and for insertion of proteins into bilayer.( 963) The location of comparatively complex chemicals in the bilayer was studied by the molecular-level SCMFT models. 964,965 United-atom phospholipid models, with the movements restricted to a diamond lattice, were deployed in MC simulations of peptide insertion into bilayers. 966,967 The partitioning of six noble gases was analyzed using MD simulations and the particle insertion method( 968) for the united-atom representations of phospholipids, with the simplified headgroups tethered to a mobile plane by a harmonic potential.( 969) Solute−lipid interaction energies increased more rapidly with the molecular size than the reversible work for cavity creation, leading to a more-favorable partitioning of larger noble gas molecules in the bilayer core. The atomistic models with frozen bond length and angles were used to obtain free-energy profiles for small molecules such as H 2O, O 2, CO, CO 2, NO, NH 3, chloroform, and formamide, using the particle insertion( 968) MC simulations in the bilayers of pure DMPC( 970) and phospholipid−cholesterol mixtures.( 971) This approach utilizes pre-existent voids in the bilayer for the placement of the solute molecules. In this way, it counteracts the nonphysical assumption of the particle-insertion method that the lipid environment remains unperturbed by the inserted solute molecules. The use of voids is relevant because small molecules may utilize them for trans-bilayer transport, which is faster than predicted from the permeation models (section ). The approach can be guided by detailed analyses of the size distribution of the voids in individual bilayer regions. 813,815 The method could possibly be extended to larger molecules using the idea of a gradual growth of the solute molecules. 972,973Estimates of partitioning from the MD-simulated trajectories have been attempted by monitoring the time that the solute molecules spent in different regions 616,969 or probability distribution of molecules.( 973) Taking into account that the experimental values of the partition coefficients of chemicals span a dozen orders of magnitude, these direct approaches seem to be limited to the chemicals with comparable affinities for water and the bilayer. Partitioning between water and a given region of the bilayer is quantitatively characterized by the difference in chemical potentials in these two phases representing the reversible work to bring the solute from bulk water to a given position in the bilayer. To sample the interactions in different bilayer regions in reasonable time, the permeants can be dragged through the bilayer. This approach was used in the steered MD simulations of oxygen diffusion in a monolayer of hexadecane and cholesterol molecules, with one end constrained by a harmonic potential to imitate the headgroup attachment.( 832) The techniques for reconstruction of the free-energy profiles from these experiments are being developed. 974–976 The placement of the permeants’ molecules at various positions in the bilayer at the start of the simulation and subsequent monitoring of the trajectories, or confinement of the permeants to certain depths have also been used.( 616) For the majority of studied chemicals, MD simulations provided mostly qualitative results because of the complications arising from the size and interactions of permeants with the bilayer. A possible solution to the size problem is a gradual grow of the permeants’ molecules in the bilayer. 972,973The first atomistic MD simulations of non-physiological molecules 514,977,978 in the fully hydrated DMPC bilayer patch used the mean displacement method to calculate the diffusion coefficients. The role of the void size in transport of small molecules was illustrated for benzene on one hand, and adamantane and a nifedipine derivative (skeleton 27 in Chart 2) on the other hand.( 979) Benzene exhibits faster diffusion in subregion 4 of the bilayer, where the voids are concentrated. Curiously, the two larger molecules did not differ in the diffusion rates in individual bilayer subregions. Benzene was mainly found in subregion 4 at 310 K, and in subregion 3 at 340 K (the subregions are specified in section ). A preference of rigid aromatic molecules for subregion 3 was also observed in other simulations (see below). This phenomenon was explained by a lower entropy loss for intercalation in the ordered subregion 3, compared to the loss of chain mobility that would be elicited by partitioning in subregion 4. A fluorescent probe, 1,6-diphenyl-1,3,5-hexatriene (DPH, 33 with R = H in Table 3 and Chart 2) in a dipalmitoylphosphatidylcholine (DPPC) bilayer patch was studied in 250-ps( 980) and 50-ns simulations.( 515) The latter study showed that DPH assumes the position with the mass center in subregion 3, and the long axis approximately parallel with the acyl chains. This orientation affects chain packing and lateral diffusion of DPPC molecules.( 981) Fluorescence anisotropy( 96) indicated a wide bimodal orientational distribution along the molecular axis that was pronounced for angles that are parallel and perpendicular to the bilayer normal. A fraction of DPH associated with headgroups has also been detected.( 982) As shown by 2H NMR experiments, the order parameter of deuterated DPH decreases with temperature and increases with the cholesterol content.( 111) MD simulations placed pyrene into subregions 3 and 4 (section ) of the 1-palmitoyl-2-oleoylphosphatidylcholine (POPC) bilayer.( 983) Pyrene molecules adopted, presumably for entropic reasons, the orientation parallel with the chains, similar to that of DPH, as shown by solid-state 2H NMR of oriented bilayers and extended MD simulations.( 516) Numerous MD simulations focused on finding the preferred locale and the effects of chemicals on the bilayer. Dihydropyridine calcium channel antagonists nifedipine and lacidipine (skeleton 27 in Table 3 and Chart 2) were shown to interact with subregions 2 and 3 (section ) of a DMPC bilayer patch.( 984) The changes in the dioleoylphosphatidylcholine (DOPC) bilayer caused by trichloroethane( 985) were monitored. The MD simulations of benzyl alcohol ( 2 in Table 2 and Chart 1) in a DMPC bilayer patch( 986) placed this molecule in subregion 3, in contrast to experimental data indicating the headgroup subregion 2 as the main locale.( 647) In the POPC bilayer, indole and N-methylindole ( 10 in Table 2 and Chart 1) were located in subregion 2, using the interaction energies, or in subregion 3, based on the free energies.( 718) Experimental data indicate that subregion 2 is the preferred locale. 123,124 This study was undertaken to help explain frequent observations of the placement of tryptophan in trans-membrane helices in the headgroup region,( 987) although the partitioning studies using surrogate solvents indicate Trp’s lipophilicity, which would indicate partitioning in subregions 3 and 4. The authors assumed that rigidity of Trp prevents its partitioning into the core for entropic reasons, and the quadrupole potential facilitates electrostatic interactions in the headgroup region. Anesthetics exhibit interesting interactions with the bilayer. Halothane ( 3 in Table 2 and Chart 1) was found mainly in subregion 3 of the DPPC bilayer, 973,988 in accordance with the 2H and NOE NMR studies,( 119) and contrary to the FT-IR ATR spectroscopy results( 97) and the 19F NMR experiments in combination with the use of paramagnetic shifting agents.( 106) Hexafluoroethane accumulated in subregions 3 and 4, with a slight preference for the latter.( 989) An anesthetic bicyclo[4.1.0]heptan-3-one derivative, KP-23, was found to associate mainly with the interface of the POPC bilayer. The protonated form preferred the headgroup region and the neutral form intercalated deeper and protruded into the hydrocarbon region.( 990) Two pools of β-carotene were identified in the POPC bilayer patch, both having the cyclohexene rings localized in the headgroup regions, and differing in the chain conformations, with bent and stretched geometries.( 991) Salicylate in the DPPC bilayer assumed the position typical for amphiphilic compounds, led to a decreased area per lipid and a higher chain order, and affected the electrical properties of the bilayer.( 992) Pentachlorophenol mostly diffused into the positions in subregions 2 and 3, with the hydroxy group interacting with the water and lipid oxygens, regardless the starting position in bulk water or in the core of the 1-palmitoyl-2-oleoylphosphatidylethanolamine and POPC bilayers.( 993) Large anions were found to intercalate deeper than small anions into the POPC bilayer patch, below the level of phosphate groups,( 921) presumably as a consequence of the easier shedding of the less-structured hydration shell, and the influence of the dipole potential. This observation is in accordance with the experiments showing that chaotropic, water-structure-perturbing anions affect the conformations of the PC headgroups( 994) and reduce the dipole potential.( 995) Both phenomena behave as the Hofmeister effects,( 996) i.e., exhibit the magnitudes proportional to the surface charge density of anions. A detailed MD study of partitioning of n-hexane into the DOPC bilayer at three temperatures provided the profiles of free energy, enthalpy, entropy, and heat capacity. The prevalent partitioning into the core was entropy-driven, while the partitioning into the dense alkyl chain subregion is dominated by a favorable enthalpy change.( 997) To study the effects of membrane potential, the standard setup with one bilayer and two aqueous phases in the unit cell was replaced by three aqueous phases and two bilayers, allowing a different salt composition in the central aqueous phase. 998,999 2.1.5.4. Transport of Chemicals The trans-bilayer transport of chemicals is most frequently characterized experimentally by the permeability coefficients, which are valid for the steady-state situation in the bilayer. The steady state can only be achieved if the permeant concentration in the entry compartment remains constant over the time interval when permeability is determined. This condition is not routinely satisfied in many experimental permeability measurements and may affect the accuracy of the determined permeability coefficient values. Under physiological conditions, the need for an invariant entry concentration of the permeant is satisfied by continuous dosing (e.g., intravenous drug infusions and the exposure of aqueous fauna and flora to pollutants in large basins with balanced input and output). However, other transport processes are more completely characterized by individual transport rate constants as described in section . Trans-bilayer transport of any permeant is too slow (section ) to be simulated completely from the relevant starting state on one side of the bilayer to a steady state or equilibrium. Shorter transfers between individual bilayer regions have a better chance to be sufficiently sampled in simulations. The permeability estimates have been made by the application of the solubility-diffusion theory, combining estimates of free energy and diffusion coefficients at given depths of the bilayer. 292,513,1000The first full-atom MD simulations of transport of water( 292) provided a detailed view of the PC bilayer structure. The diffusion coefficient profile was obtained by the mean-squared-displacement method in the headgroup regions, and by the force correlation method for the molecules confined at a given depth in the chain subregions. The diffusion was found almost isotropic except subregion 3, where the transversal movement was faster. The free-energy profile of water molecules in the bilayer was calculated directly from water concentrations in the headgroup subregions 1 and 2, from the average force acting on confined molecules in subregion 3, and by the particle insertion method( 968) in subregion 4. The diffusion and partitioning data were used to calculate the permeabilities, which reached their maximum magnitudes in the center of the bilayer and were minimal in subregion 3. Similar studies were performed for oxygen and ammonia,( 1001) which are assumed to exhibit the same void-based transport mechanism as water. The highest permeabilities were observed in the core subregion 4, and depended greatly on the polarity of the molecules. Oxygen diffusion was actually faster in the bilayer than in bulk water. Proton transport properties were inferred from the formation of transient water pores( 1002) and from direct simulations.( 1003) In another study of small permeants, diffusion of nitric oxide was found to be isotropic.( 1004) The transport of small molecules was shown to be associated with chain dynamics: the rate significantly decreased for diphytanoylphosphatidylcholine, because of a denser packing of branched acyl chains,( 892) compared to the straight-chain phospholipids. The permeability coefficients of eight small organic compounds (acetamide, non-ionized acetic acid, benzene, ethane, methanol, methyl acetate, methylamine, and water) were systematically analyzed using the depth-dependent free energies of partitioning and diffusion coefficients on the basis of the solubility-diffusion theory.( 1005) The approach allows identification of the rate-limiting steps and has a potential to provide a better correlation with experimental data thanks to the use of the relevant, depth-specific partition coefficients instead of the experimental partition coefficients, which are based on space-averaged concentrations in the heterogeneous bilayers. The rate-limiting steps or resistances were identified in subregions 1 and 2 for ethane, methylacetate, and benzene; subregions 3 and 4 for water; and subregion 4 for methanol, methylamide, non-ionized acetic acid, and acetamide.( 1005) The calculated permeability coefficients were about one order of magnitude larger than the experimental values but the relative trends were correctly reproduced. Interestingly, the partition coefficients of the small studied compound series in individual bilayer subregions 3 and 4 correlated well with both hexadecane/water and 1-octanol/water partition coefficients.( 588) The diffusion coefficients decreased with the molecular weight and the cross-sectional area of the permeants. Inclusion of larger drug molecules (β-blockers alprenolol, atenolol, and pindolol, with the skeleton similar to propranolol: 34 in Table 3 and Chart 2)( 1006) provided the opportunity to examine other aspects of the transport process. When crossing the bilayer, the drug molecules tended to keep their long axis parallel with the bilayer normal, and deviated from this orientation only when it was advantageous for the formation of hydrogen bonds. The polar vectors of permeants were more ordered than in liquid water. These tendencies were most pronounced in the dense bilayer subregions 2 and 3. The conformer populations in individual subregions differed: the molecules were more packed near the interface, to form hydrogen bonds, and more elongated in the core. 2.2. Vesicle-Mediated Transport Absorption, distribution, and excretion of some chemicals can be affected by mechanisms that evolved in specialized cells for transport of physiological molecules and macromolecules, to enable the cells to perform key functions of living organisms. The transport of chemicals can be mediated by vesicles pinching off the membranes and delivering the enclosed cargo into or across the cell, or by membrane proteins transferring molecules across the membranes (section ). The rates of trans-bilayer transport of chemicals vary in a broad range (section ). The mediated mechanisms play meaningful roles only in the situations when they significantly contribute to the actual trans-bilayer transport rates, regardless of the direction. While the mediated transport can affect the trans-bilayer transport rates in the specialized cells, the interactions with the bilayers (section ) determine the transport rates for crossing other membranes. The overall absorption and distribution of chemicals are determined by both passive diffusion and mediated transport, and cannot be understood without a proper consideration of both contributions. Currently, we do not know the detailed transport mechanisms for many chemicals,but this picture will gradually change, because of intense research in the area of mediated transport. Formation of the vesicles can be spontaneous ( pinocytosis) or initialized by the binding of macromolecules or small molecules to the receptors in cell membranes. Pinocytosis was first studied for amoeba in the 1950s.( 1007) The concept of receptor-based endocytosis was elaborated about two decades later for the intracellular entry of proteins( 1008) and low-density lipoproteins. 1009,1010 The receptors are encountered in two distinct domains of cell membranes, for which the mechanisms and the cargo differ. The receptors for both small molecules and macromolecules are located in caveolae and the respective mechanism is termed potocytosis (section ) .The exclusively macromolecular receptors that are found in clathrin-coated pits elicit receptor-mediated endocytosis (section ). Other vesicle-mediated transport mechanisms are used for secretion of proteins or signaling molecules from the cells ( exocytosis) and will not be described here. This mechanism can also play a role in drug efflux via shedding of vesicles and leading to multiple drug resistance.( 1011) Vesiculo-vacuolar organelles are clusters of vesicles and vacuoles in endothelial cells of lymphatic vessels and small veins in normal tissue and tumors, which can interconnect and create irregular trans-cellular channels.( 1012) Their openings at the levels of the cell membrane are either unobstructed or covered by a glycoprotein diaphragm. The channels can serve for transport of chemicals and macromolecules.( 1013) 2.2.1. Potocytosis Caveolae are 50−100 nm wide invaginations of the cell membrane that are rich in cholesterol, and coated, on the cytoplasmic side, by caveolins and other proteins.( 1014) The most frequently observed invaginations have the flask form, but tubules and grapelike shapes have also been observed.( 1015) The flask-shaped invaginations can remain anchored to the cell membrane or detach and form separate vesicles. The caveolae are flattened by cholesterol depletion, and the exposure to polyene, cholesterol-binding antibiotics, such as filipin and nystatin. The caveolae are found in many cell types, most frequently in capillary endothelium, Type I pneumocytes, adipocytes, as well as in skeletal, cardiac, and smooth muscles. 1016,1017 Potocytosis was discovered as a mechanism for pre-concentration of nutrients and signaling molecules, enabling their faster carrier-mediated transport. 1018,1019 The intracellular vesicle, which forms upon binding of the ligand to the receptor, encapsulates the ligand molecules in a higher concentration than that which is in the cell surroundings. The vesicle remains attached near the cellular membrane, and the concentrated molecules are transported into the cell interior by a carrier-mediated mechanism. The term potocytosis has been extended to encompass all types of vesicular transport, which involve caveolae.( 1020) Chemicals, macromolecules, and viruses may be transported by one of four possible mechanisms. The original pre-concentration mechanism, with the carrier-mediated release of small molecules from the surface-anchored vesicles into cytoplasm, was observed for folate, retinol, iron, and calcium. Cholesterol and some proteins are delivered by endocytosis to the endoplasmic reticulum or other organelles, and the unloaded vesicles return to the surface. Transcytosis, the transfer of ligands through the epithelial cell, with relocation of the caveolae to the opposite surface, was documented for plasma proteins including albumin and transthyretin. Lastly, the formation of vesicles functioning as depots for temporary sequestration, without an immediate intracellular release, were observed for acetylcholine and several proteins.( 1020) 2.2.2. Receptor-Mediated Endocytosis Clathrin-coated pits are cell membrane invaginations containing several receptors. Clathrin is a large coat protein that associates with the membrane surface. Upon binding of macromolecular ligands, the invaginations transform into vesicles, which detach from the cell membrane, with the help of dynamin and other proteins. Protons are pumped into the vesicles, causing the ligands to dissociate from the receptors, and the coat protein to separate from the bilayer. The bilayer vesicles then associate with endosomes.( 1015) Endocytosis can proceed at a significant speed; some cultured mammalian cells can internalize 50% of their surface per hour.( 1021) The process is inhibited by ammonium chloride, chlorpromazine,( 1015) and monodansylcadaverine.( 1022) Endocytosis was described in detail in the Peyer’s patches in the mucose of ileum, 1022–1028 in renal tubules, 1029–1031 and for the cases involving the receptors for low-density lipoprotein, transferrin,( 1032) and scavenger receptors in the blood-brain barrier( 1033) and liver.( 1034) 2.2.3. Structure-Permeability Considerations The vesicle-mediated transport has been considered as a route for targeting of drugs to specific cells. 1034–1041 Chemicals can participate in the vesicle-mediated transport via direct binding to the receptors in potocytosis, binding to the internalized membrane, or piggy-backing on the internalized macromolecules. These interactions can affect the rate of both potocytosis and receptor-mediated endocytosis. Currently, no QSARs are available for the vesicle-mediated transport. It can be expected that the respective QSARs will be rather complex. In addition to the direct interactions with the receptors, the accumulation of chemicals in the internalized part of the membrane, and the binding of chemicals to the internalized macromolecules must be included. 2.3. Carrier-Mediated Transport Some chemicals are transported in specialized tissues via carrier systems that have evolved for physiological reasons. The transporters are expressed at the highest levels in absorptive and excretory tissues such as intestinal epithelia, renal tubules, and liver;( 1042) in the blood-tissue barriers in the brain (also at the surfaces facing cerebrospinal fluid),( 1043) testis,( 1044) ovaries,( 1045) and placenta; 1046,1047 as well as in tumors. 1048,1049 In the past decade, hundreds of transporter proteins have been cloned and characterized,( 1050) and this number is expected to grow. The mechanisms of carrier-mediated transport include (i) facilitated transport along the concentration gradient; (ii) facilitated influx of cations and efflux of anions due to the inside-negative membrane potential, which can lead to some pre-concentration; and (iii) active transport that proceeds against the concentration gradient on the account of the energy of coupled processes, most frequently adenosine triphosphate (ATP) hydrolysis and the co-transport of Na +, H +, and HCO 3−.( 1051) Physiological ions, such as Na +, K +, Ca 2+, HCO 3−, and Cl −, are much smaller than the majority of organic molecules. Consequently, ion channels are not known to be directly involved in the transport of chemicals, except for their roles in co-transport. The channels can be blocked by chemicals and are targeted by several drugs.( 1052) The transporters are integral membrane proteins that are difficult to crystallize due to the structural changes, which can be elicited by the extraction from the natural membrane environment. Therefore, detailed structural information is scarcer than for other protein types, despite significant progress made in recent years, especially in elucidation of the structure of the efflux pumps (section ). Chemicals utilizing the protein-mediated mechanisms must resemble the native substrates to the extent that is dependent on the specificity of the carrier proteins. Nutrient transporters seem to have a higher specificity than other transporter proteins, which are usually rather promiscuous. In contrast to facilitated transport, 1053–1055 active transporters seem to be strictly unidirectional and mediating either cellular influx or efflux.( 1051) This feature leads to the typical polarity of epithelia forming the monolayers of tightly packed cells separating the bloodstream from non-tissue body compartments. The apical side of most epithelia, frequently rich in glycolipids and having villi, is oriented toward the nontissue compartments, such as the lumen of the gastrointestinal tract, air in the lungs, the glomerular filtrate in the tubule of nephrons, bile in the liver canaliculi, or urine in the urinary bladder. The basolateral side, facing the bloodstream, is separated from the apical side by tight junctions, which prevent both the paracellular transport of chemicals and the lateral diffusion of transporters.( 1056) To ensure a flux through an epithelial layer, the influx and efflux transporters are co-expressed( 1057) and properly arranged on the apical and basolateral sides of the cells. Combinations of active and passive processes (influx and efflux, respectively, for some anions;( 1050) efflux and influx, respectively, for Type II cations,( 1058) both described in section ) were observed in the tubular epithelium of the nephron. In the intestinal enterocytes, some transporters (GLUT and SGLT; see section ) are expressed in both apical and basolateral cell membranes. Most transporters are encoded by the solute carrier (SLC) family genes. The exceptions are the efflux pumps that belong to the ATP-binding cassette (ABC) family. The used nomenclature is recommended by the Gene Nomenclature Committee of the Human Genome Organisation.( 1059) By convention, the gene symbols are italicized and the respective protein symbols are not. Human genes and gene products are shown in upper case, while rodent genes and gene products are shown in lower case. Additional specifications and numbers denoting subfamilies and individual members complete the characterization of the transporters. All transporter subfamilies include several members. Although the roles of subfamily members frequently differ, the structural similarity of transporters within families complicates structure−activity analyses at the cellular and higher levels. Several physiological processes are of significance in the context of transport of chemicals: (i) active transport of nutrients to the bloodstream, with absorption from the intestine and reabsorption from the tubular filtrate as typical examples; (ii) active renal and biliary secretion; and (iii) active transport of nutrients from the blood and the backward transport of metabolites and signaling molecules through the barrier tissues in the brain, testis, ovaries, and placenta. Although the participating transporters are not exclusively expressed in the aforementioned tissues and perform significant roles in other tissues as well, it is convenient to review the carrier-mediated transport along the themes of individual physiological processes, because the processes determine the overall disposition of chemicals in the body. This approach also leads to an intuitive recognition of the tissues with prevalent expression of individual transporters. To maintain the focus on the transport in individual cells, the influx and efflux carriers are covered separately, despite the fact that there is some overlap between the two groups. 2.3.1. Influx Transporters Protein carrier systems have evolved in all organisms for the uptake of sugars, 1060–1062 amino acids, dipeptides and tripeptides, 1063,1064 nucleosides,( 1065) lipids,( 1062) vitamins, 1062,1066 and hormones in higher organisms.( 1067) This route has been considered a possible mechanism for the increase in bioavailability of the drugs bearing, at least in some parts, some similarity to the nutrient molecules such as antiviral nucleosides( 1065) and peptidomimetic drugs including β-lactam antibiotics and inhibitors of angiotensin converting enzyme, renin, and thrombin.( 1068) 2.3.1.1. Intestinal Absorption The apical brush-border membrane of the enterocytes in the small intestine contains a plethora of carriers for glucose, amino acids, peptides, nucleosides and other nutrients. Usually, the transporters from the same families, and, for glucose and amino acids, sometimes the same transporters, are expressed at both sides of epithelial cells, supporting enhanced specificity of nutrient transport processes. The carriers for the universal mammalian cell fuel, d-glucose, belong to the most intensively studied nutrient transporters. They include 1069,1070 facilitative glucose transporters (GLUT) belonging to the SLC2A (facilitated glucose transporter) family,( 1059) and concentrative sodium glucose co-transporters (SGLT) from the SLC5A (sodium/glucose co-transporter) family.( 1059) Glucose transport in individual tissues is regulated, in addition to other factors, by specific distribution of transporters. For instance, GLUT1 is found in the erythrocytes, placenta, brain, kidney, and colon; GLUT2 in the liver; GLUT3 in the brain; insulin-sensitive GLUT4 in the adipose tissue and the skeletal and cardiac muscles; GLUT5 in the small intestine, and others in the kidneys. Selectivity of the GLUT carriers is high: GLUT1 in erythrocytes transports d-deoxyglucose and 3- O-methylglucose, but not l-glucose.( 1071) Insulin elicits recruitment of intracellular GLUT4 to the cytoplasmic membrane, resulting in an increase in the d-glucose transport. 1072,1073The results of protein cleavage studies, immunologic epitope mapping experiments, chemical modifications, and glycosylation-scanning and cystein-scanning mutagenesis allowed for the construction of a working structural model of GLUTs that is based on 12 transmembrane segments. 1061,1074 Similar steps are being made toward the elucidation of the SGLT structure.( 1075) These results represent an important foundation for a rational development of GLUT and SGLT effectors promoting absorption and inhibiting reabsorption of glucose, as potential anti-diabetic drugs. 1076–1079Amino acids are absorbed via a complex system of amino acid transporters.( 1080) The peptide transporters, the low-affinity PEPT1 and high-affinity PEPT2 carriers, which are members of the SLC15A (oligopeptide transporter) family,( 1059) perform H +-coupled influx of dipeptides and tripeptides across apical membranes in the intestine and renal tubules.( 1081) PEPT1 has a broader substrate specificity and was examined as a means to enhance intestinal absorption of ester prodrugs.( 1082) The influx of nucleosides into the enterocytes( 1080) is facilitated by equilibrative nucleoside transporters ENT1 and ENT2, belonging to the SLC29A (nucleoside transporter) family,( 1059) and by other carriers.( 1065) Specific transporters are available for phosphate, monocarboxylates, fatty acids, bile acids, and vitamins. Some transporters participating in renal secretion and reabsorption (OATPs and OCTNs; see the next section) were also found in the apical membrane of the enterocytes.( 1080) 2.3.1.2. Renal Excretion Renal excretion consists of glomerular filtration, tubular secretion, and reabsorption, which proceed in the nephrons of the kidneys. While filtration and some reabsorption steps are passive processes, transporters are involved in secretion and reabsorption of some physiologic molecules and chemicals. Active renal secretion was discovered ~80 years ago,( 1083) using an anionic substrate, phenolsulfonphthalein, and proved for organic cations two decades later.( 1084) Active secretion can be extremely fast: p-aminohippuric acid is almost completely secreted in one pass of the blood through the kidney.( 1085) Some of the transporters involved in renal excretion are also expressed in the intestine, hepatocytes,( 1086) brain capillaries,( 1087) and the choroid plexus.( 1088) The involved protein carriers include( 1089) organic anion transporters (OATs) and organic cation transporters (OCTs and OCTN, where N stands for novel), all belonging to the SLC22A family,( 1059) as well as organic anion transporting peptides (OATPs), which form the SLCO (organic anion transporter) family.( 1059) The carriers have 12 transmembrane domains, with the N- and C-terminals located in the cytoplasm. 1058,1090,1091 From the mechanistic viewpoint, they mostly act as exchangers: OATs exchange organic anions for dicarboxylates,( 1091) OATPs exchange organic anions for bicarbonate, glutathione, or glutathione-S-conjugates,( 1092) and OCTs and OCTN1 exchange organic cations for protons.( 1093) In contrast, OCTN2 supports Na + co-transport and electrogenic-facilitated diffusion.( 1058) The substrates are mostly amphiphilic organic anions and cations, usually having distinct hydrophobic parts. Notably, amphiphilic compounds frequently exhibit slow passive trans-bilayer transport (section ), so the transporters significantly affect the overall transport rates. The influx transporters are involved in the first step of active and facilitated tubular secretion, the uptake from the bloodstream through the basolateral cell membrane into the renal proximal tubule cells.( 1094) The systems have evolved for numerous organic anions and cations as a means to clean the organisms of endogenous metabolites, toxins, and hormone and neurotransmitter metabolites, and to maintain homeostasis.( 1050) Concentrations of chemicals and their metabolites are also affected by the secretion process. Influx of organic anions into the tubular cells proceeds through a series of three coupled processes. The Na +/K +-ATPase uses the energy released by the ATP hydrolysis to support the Na + efflux that drives the α-ketoglutarate (α-KG) influx via the Li +-blocked Na +-dicarboxylate co-transporter 3 (NaDC3), member 3 of the SLC13A (sodium-dependent dicarboxylate transporter) family.( 1059) Organic anions are exchanged for α-KG by the probenecid-blocked dicarboxylate/organic anion exchangers OAT1,( 1095) OAT3, and possibly OAT2.( 1050) Organic cations are classified as Type I (smaller, monovalent cations, e.g. tetraethylammonium) and Type II (bulkier, often polyvalent cations, e.g. daunomycin). Type I cations enter the tubular cells via facilitated diffusion mediated by OCT1, OCT2, and OCT3, while Type II cations probably use passive transport.( 1058) Many endogenous compounds (e.g., glucose, electrolytes, and amino acids) and some drugs are subject to active reabsorption, which occurs predominantly in the proximal tubule. Glucose transport in the proximal tubules is mediated by several transporters, e.g., GLUT1 and SGLT2. 1078,1094 OATs are involved in reabsorption of anionic substances.( 1096) Peptides and peptidomimetics are reabsorbed via PEPT1 and PEPT2 transporters.( 1097) 2.3.1.3. Biliary Excretion Metabolic and excretion functions of the liver are facilitated by a comparatively dense network of capillaries, with each capillary serving several hepatocyte layers (Table 1, section 1.2), and by the direct contact of practically each hepatocyte with a bile canaliculus. The fenestrated liver capillaries (sinusoids) allow the entry of plasma proteins with bound ligands into the interstitial space (the space of Disse), where they can come into a direct contact with the basolateral membrane of hepatocytes.( 171) Biliary excretion plays an important physiological role in the bile formation, by the excretion of synthesized bile salts from hepatocytes into the bile, and by the concentrative transport of bile salts from the blood into the bile, contributing to the bile salts conservation, in a process called enterohepatic cycling. After synthesis from cholesterol in hepatocytes, bile salts are excreted into the bile, in which they enter the digestive tract, and help in emulsification of lipid substances. Bile salts are absorbed in the small intestine, and retrieved from the bloodstream in hepatocytes. The influx through the basolateral (sinusoidal) hepatocyte membrane is mediated by OAT1 and OATPs, 1098–1102 which are described in section . The excretion from hepatocytes into bile is mediated by specific biliary transporters and multidrug resistance related proteins (MRPs), which are described in section . The bile salt excretion mechanism can be used by some chemicals to get from the bloodstream into bile and then to the intestinal content. If the chemicals exhibit slow intestinal absorption, they will be eliminated in feces. The chemicals undergoing both intestinal absorption and biliary excretion can get trapped in enterohepatic cycling.( 1103) Some data suggest that albumin-bound compounds are preferentially excreted into bile and metabolized. 1100,1104–1106 The mechanism of this interesting phenomenon could include albumin endocytosis through caveolae (section ) or clathrin-coated regions (section ) of the basolateral membrane of hepatocytes.( 1107) The significance of the endocytotic contribution to biliary excretion of chemicals remains to be assessed. Biliary excretion is dependent on the structure of chemicals,( 1108) although no structure-dependent QSAR have been formulated so far. In a limited series of compounds, lipophilicity was seen as the rate-determining factor. 1109,1110 An atypical convex parabolic dependences of the excretion rate on lipophilicity were observed for a series of nitroimidazoles.( 1111) 2.3.1.4. Blood-Brain Barrier Permeation The blood-brain barrier, represented by the brain capillary endothelium with tight junctions and low endocytotic activity, contains glucose, amino acid,( 1112) and nucleoside( 1033) transporters for the nutrient uptake. Secretion of organic anions through the brain capillary endothelial cells and the choroids plexus epithelial cells is mediated by OAT3 (section ) at the brain side, and by the efflux via MRP1 (section ) at the basolateral membrane.( 1091) Thanks to therapeutic importance of the blood-brain barrier permeation, numerous attempts have been made to correlate structure of chemicals with distribution into the brain. The models are mostly based on the assumption of passive diffusion, although the transporters can contribute to permeation of some chemicals. 1033,1087,1113 In smaller datasets, lipophilicity, 1114,1115 hydrogen-bonding ability, 550,1116–1118 polarizability, size and shape, 1119–1125 cross-sectional area, 656,1126 polar surface area, 1127–1129 and solvation free energies( 1130) were found as the rate- or extent-determining properties. For larger datasets, numerous topological,( 1131) electrotopological,( 1132) and 3-D descriptors based on the VolSurf 1133,1134 and GRID( 1135) approaches were used, and processed by the PLS and principal component analysis (PCA)( 1136) techniques. Simple rules were formulated for the CNS-active compounds: the number of O and N atoms should be less than five and smaller than the log P value.( 1118) 2.3.1.5. QSAR Approaches In the absence of structural data on the influx transporters, qualitative pharmacophore mapping methods( 1088) and quantitative 3D-QSAR techniques( 1137) were used to infer the binding site models of human OATP1B1 and rodent Oatp1a5, respectively. These approaches provide a spatial arrangement of electrostatic and steric regions that represent a putative binding site. Assignment of the site model to a specific state of the protein is complicated by large-scale protein motions, which are anticipated for a functioning transporter. Among several possible interpretations, the simplest one would be that the protein is in the resting unliganded state, to which the ligands bind quickly before the molecular machinery is set into motion. A descriptor-based QSAR and a pharmacophore model were derived for human OCT1.( 1138) The binding to human dipeptide/tripeptide transporter PEPT1, determined in the Caco-2 monolayers (section ), was analyzed( 1139) using the Volsurf descriptors. 655,1140–1142 2.3.2. Efflux Transporters Extrusion of physiologic molecules and chemicals from cells has mainly been studied for (i) renal and biliary excretion of physiological molecules, and (ii) cellular efflux of drugs, especially in association with the multidrug resistance. The two research streams have discovered two groups of transporters, which are frequently co-expressed in tissues and have overlapping functions and specificities, but differ in structures and mechanisms. The latter attributes represent the basis for naming the carriers. Some names, which were initially assigned based on the location and function (e.g., canalicular multispecific organic anion transporter, CMOAT), have been changed to reflect the structure and mechanism (CMOAT is now multidrug resistance protein 2—MRP2, which is described in section ). The excretion-associated transporters are usually coupled with compatible influx transporters in the opposite cellular membrane, while the proteins involved in multidrug resistance may lack this attribute. During intestinal absorption of nutrients, the efflux from enterocytes into the bloodstream is often performed by the same transporters as the influx (GLUT, SGLT), reflecting the higher specificity of the nutrient transporters, compared to other transporters. 2.3.2.1. Tubular Secretion Excretion of organic ions from the renal proximal tubule cells into the urine through the apical membrane proceeds via facilitated diffusion that is sensitive to membrane potential, and via active exchange transport. 1050,1058 The anion exchange is performed by the urate transporter (URAT1, SLC22A12),( 1059) member 12 of the SLC22A (organic anion/cation transporter) family, OAT4, and possibly OAT5, 1050,1094 as well as by OATP1,( 1089) and multidrug resistance associated proteins, such as MRP2, 1089,1143 MRP4,( 1144) and others,( 1145) which are described in section . Anion transporters with apical location, such as MRP2, MRP4, PEPT2, OATP1, and some OATs, contain the PDZ domains, consisting of 70−90 amino acid sequences.( 1089) The Type I organic cations (section ) are excreted into urine by OCTN1 and OCTN2, while the Type II cations (section ) utilize ATP-dependent P-glycoprotein (PGP or MDR1), which is described in section .( 1058) If the chemicals, which are actively transported into the tubular cells, exhibit slow efflux, a tubular injury may evolve as a result of the increase of intracellular concentrations.( 1094) 2.3.2.2. Biliary Excretion Efflux through the apical (canalicular) membrane of hepatocytes is performed 1100–1102 by MRPs, characterized in section , which are also expressed in other organs, and specific biliary transporters such as Na +-taurocholate co-transporting polypeptide (NTCP, SLC10A1),( 1059) member 1 of the SLC10 (sodium/bile acid cotransporter) family, and the bile salt export pump (BSEP, ABCB11),( 1059) member 11 of the ATP-binding cassette family, sub-family B MDR/TAP. 2.3.2.3. Efflux Pumps The ATP-binding cassette (ABC) transporters form a large, ancient family of active transporters that are widely expressed across species. In bacteria, the ABC transporters mediate both inward and outward transport, and they represent up to 4% of all proteins in some proteomes. Eukaryotes express several dozen ABC transporters each, all of them acting as efflux pumps.( 1146) They rank among the most extensively studied transporters, because of their impact on drug disposition, especially in the connection with the resistance of bacteria 1147,1148 and tumors 1149–1154 to drug treatments. The range of affected compounds is wide and includes lipids, bile salts, drugs, and toxic and environmental agents, which are transported against the concentration gradient across the cell or organelle( 1155) membranes, using ATP hydrolysis as the driving force.( 1044) Human ABC transporters (49 genes) are divided into seven subfamilies: ABC1 (subfamily A, 12 members), MDR/TAP (subfamily B, 11 members), MRP/CFTR (subfamily C, 13 members), ALD (subfamily D, 4 members), OABP (subfamily E, 1 member), GCN20 (subfamily F, 3 members), and White (subfamily G, 5 members).( 1156) Subfamilies B and C seem to be most relevant for the transport of chemicals.( 1157) Human ABC transporters typically consist of 1200−1700 amino acids. Polypeptides shorter than 800 amino acids, which are frequently found in subfamilies B and D−G, are mostly half-transporters, which require dimerization to function. Some members of subfamily A contain 2100−2600 amino acids, and the predicted member 13 is even supposed to have more than 5000 amino acids.( 1156) The ABC transporters are integral membrane proteins that are folded into several trans-membrane domains (TMDs), each composed of fewer than 10, and most commonly 6 trans-membrane helices. They frequently contain extracellular N-glycosylation branches that do not seem to be associated with transport function.( 1158) The minimal functional unit of an ABC transporter involved in the efflux of chemicals consists of two TMDs and two nucleotide binding domains (NBDs or ABCs), 1159,1160 as illustrated in Figure . The TMDs contain one or more substrate binding sites. The TMDs exhibit some sequence similarity only if they mediate transport in the same direction.( 1161) The NBD sequences are much more conserved than those of TMDs, because each contains the conserved Walker A and B motifs, which are shared by numerous ATP-binding proteins,( 1162) and an ABC signature motif (ALSGGQ) that is also called the C motif. The bound nucleotides (Figure ) are sandwiched between the C motif and the Walker A motif from the opposite NBD. 1161,1163,1164 The NBDs act as molecular motors, initiating conformational changes of TMDs.( 1160) The same motor seems to be connected to differing translocation channels. Therefore, a common mechanism may be shared by the majority of ABC transporters.( 1161) Large conformational changes, expected in both the TMDs and NBDs during the transport process, were confirmed by diverse conformations that were observed in experimentally determined structures of transporters. Structural information for eukaryotic ABC transporters is limited to low-resolution, electron cryo-microscopy structures of PGP 1165–1167 and MRP1 (see below).( 1168) For bacterial transporters, structures of NBDs 1169,1170 as well as those of complete transporters are available. The latter category includes multidrug transporter Sav1866 from Staphylococcus aureus( 1171, 1172) (Protein Data Bank (PDB)( 1173) files 2ONJ, shown in Figure , and 2HYD); molybdate/tungstate transporter ModBC-A from Archaeoglobus fulgidus( 1174) ( 2ONK), vitamin B12 permease BtuCD from Escherichia coli( 1163) ( 1L7V), putative metal-chelate importer HI147Q/71 from Haemophilus influenzae( 1175) ( 2NQ2); multidrug transporter MsbA from E. coli ( 3B5W), Vibrio cholerae ( 3B5X), Salmonella typhimurium ( 3B5Y, 3B5Z, 3B60), and multidrug transporter EmrE from E. coli( 1176) ( 3B5W, 3B61, 3B62). The MsbA 1177–1179 and EmrE 1180,1181 structures were retracted after publication of the Sav1866 structure( 1171) and recalculated from original data to correct the handedness and topology. This structural insight, as well as the results of the uptake, cross-linking, labeling, and mutational experiments, and conformational changes of the NBD domains studied by molecular dynamics simulations,( 1182) represent the basis for mechanistic hypotheses. The majority of data in the last four categories are available for PGP, so most mechanisms are being developed for this transporter. Various mechanisms for the extrusion step have been recently proposed. 1160,1182,1183 Flipping of the amphiphilic substrate, i.e., the change of the inward-facing orientation of the hydrophilic part to the opposite orientation, is a key step in the extrusion process. A synthesis of experimental and structural data delineates the following features of the mechanism, which starts with free PGP in basal conformation, i.e., with the disengaged NBD domains: (i) the substrate binds from the inner leaflet to a high-affinity binding site and increases the affinity of the NBDs for ATP; (ii) binding of ATP brings the NBDs to a closed position and causes the “power stroke”, which is a movement of the TMDs leading to the expulsion and flip of the substrate through the cavity between TMDs to a low-affinity binding site at the level of the outer leaflet; and (iii) ATP hydrolysis and the release of ADP and phosphate are required to bring the ABC transporter to the basal state with open NBDs. 1159,1184The most frequently studied efflux pump is PGP, also known as multiple drug resistance protein 1 (MDR1, ABCB1), which is member 1 of the subfamily B (MDR/TAP) of ATP-binding cassette family.( 1059) PGP was discovered as the cause of the resistance of normal( 1185) and cancer cells 1149–1153 to several cytotoxic agents. The 170-kDa single polypeptide, consisting of 1280 amino acids, is the most studied ABC transporter, because of its early discovery and medicinal importance. The expression in apical membranes of the intestine, 1186–1188 hepatocytes, and renal proximal tubular cells underlines the actions of PGP in reducing absorption and enhancing renal and biliary excretion of chemicals. PGP can also be found in the capillary cells of barrier systems, especially in the brain, testes, and ovaries,( 1045) as well as in a variety of stem cells( 1189) and tumors. ( 1149– 1153) Its localization in the cytoplasmic and intracellular membranes leads to the efflux from the cells and changes in the intracellular distribution of chemicals.( 1190) The activity of PGP, 1191–1193 and probably all transporter proteins, is heavily dependent on the lipid environment. The substrate binding sites seem to be in contact with the inner leaflet of the apical cytoplasmic membrane, 1194–1196 requiring special considerations for the description of transport kinetics.( 1197) The transport kinetics of rhodamine and Hoechst 33342 was explained by a model containing two distinct binding sites, termed R- and H-sites according to the increased affinity for the respective substrate.( 1198) The sites exhibited positive cooperativity, i.e., binding of one substrate stimulated the efflux of the other substrate, and the efflux was maximal when both sites were modified. Many but not all tested substrates could be classified as either H- or R-substrates. The H-substrates stimulate the transport of rhodamine and inhibit the transport of Hoechst 33342, while the R-substrates exhibit an inverse pattern.( 1158) In addition to the induction by many chemicals, PGP is up-regulated in some in vitro models of tissue regeneration, and in cells treated by growth factors, as recently reviewed.( 1189) The PGP expression prevents apoptosis via a direct interference with the caspase-dependent pathway. 1199,1200 These facts have been put together in a hypothesis stating that PGP protects the stem cells from apoptosis during natural stress conditions, aside from the effects of chemicals.( 1189) Tumors seem to contain a subpopulation of cells with the stem cell characteristics, which are capable of initiating new tumor growth.( 1201) Through PGP induction, some chemotherapies could actually enrich the stem cell subpopulation in tumors and promote their resistance.( 1189) These facts may help analyze the failures of clinical trials with the PGP inhibitors.( 1202) Extensive photoaffinity labeling studies on binding of different substrates to PGP and single-point mutagenesis analyses, as recently reviewed,( 1045) delineate the associations between structure and function. The PGP ability to bind a variety of structures was hypothesized to result from the presence of a large binding site with a set of hydrophobic and hydrogen-bonding points, which can anchor substrate molecules in different binding modes.( 1203) The pregnane X receptor, exhibiting a broad substrate specificity, is involved in transcriptional regulation of PGP and multiple CYPs.( 1204) A broad substrate specificity of PGP overlaps remarkably, although not completely, with that of CYP3A4 (described in section ). This fact, along with the co-localization of the two proteins, indicates the existence of a concerted defense mechanism against exogeneous compounds. 1157,1205–1207Multidrug-resistance-related proteins (MRPs), the subfamily C (CFTR/MRP)( 1059) of the ABC family, have a potential to confer resistance to chemotherapy. Clinically, this role has been proven( 1044) for MRP1 ( ABCC1( 1059)). MRPs act as efflux pumps for glutathione, glucuronate, and sulfate conjugates of many drugs and chemicals.( 1208) The efflux caused by several MRPs was inhibited by plant flavonoids.( 1209) MRP1 is expressed in all tissues except the liver, and it is localized at the basolateral pole of the cells. MRP2 is found mainly in the apical membranes in the liver, intestine, and kidney.( 1044) MRPs 1−3 contain ~1530 amino acids( 1156) in three TMDs and two NBDs. The N-terminus protrudes, in contrast to other ABC transporters, outside the cell, and the C-terminus is located in the cytoplasm. 1158,1210 MRPs 4 and 5 are shorter; they contain 1325 and 1437 amino acids, respectively, which are organized in two TMDs and two NBDs.( 1158) For MRP1, the low-resolution structural data were obtained by electron microscopy of negatively stained particles and 2D crystals of the protein in a lipid bilayer.( 1211) The X-ray structure of MRP1 NBD shows a nonproductive conformation that may be associated with low ATPase activity.( 1212) Breast cancer resistance protein (BCRP, ABCG2), member 2 of subfamily G of the ABC family,( 1059) is a half-transporter that contains 655 amino acids in only one six-helix TMD, and one NBD.( 1044) BCRP needs to dimerize for function.( 1213) Its substrate spectrum is similar to those of MRPs, but is more restricted. In addition to the breast tissues, BCRP is found in the liver, intestine, placenta, and also in stem cells.( 1213) Its expression is promoted by hypoxia;( 1214)therefore, it is found in venous and capillary endothelia, but not in arteries.( 1215) 2.3.2.4. QSAR Approaches Medical significance and comparatively early discovery have made PGP a prominent target of the studies between structure of chemicals and their efflux. The algorithms for prediction of efflux transport of chemicals by PGP and other transporters, and the effects of chemicals on this process were recently reviewed. 1216,1217Homology between bacterial efflux transporters and PGP( 1149) was used to obtain structural models of PGP. Kinetic parameters of the interactions between PGP and 34 drugs were correlated with the partition coefficients and surface areas.( 1218) Structural requirements for binding of chemicals to PGP, such as amphiphilicity and certain volumes, were generalized based on a structural comparison of known PGP substrates.( 1219) More-detailed information about structural requirements for binding and, indirectly, the geometry of the binding site was obtained 1220,1221 by classification analysis,( 1222) pharmacophore studies, 1203,1223,1224 pseudo-receptor modeling,( 1225) GRIND descriptor studies,( 1226) and 3D-QSAR analyses, 1227–1232 which were combined with the homology models, based on the available bacterial ABC transporter structures. 1233,1234 Diverse PGP substrates and modulators were shown to be predominantly localized at the headgroup/core interface.( 1235) In addition to binding affinity, the interface concentration is an important determinant of the overall efflux rate and must be considered when making quantitative predictions. In this context, the knowledge of solvation energies in individual headgroup regions (section ) may help in a quantitative understanding of the efflux by PGP and other transporters. The QSAR studies for MRPs and BCRP were performed less frequently and generally are less sophisticated than those for PGP. In many cases, the conclusions are limited to the importance of individual substructures for efflux or its inhibition in MRP 1236–1242 and BCRP.( 1243) The MRP1 and MRP2 efflux characteristics were correlated with fragment occurrences( 1244) and topological indices.( 1245) A spatial pharmacophore was generated for MRP1( 1246) and MRP2,( 1247) and compared with that for PGP. Ligand-based 3D-QSAR studies were performed for MRP2.( 1248) 2.4. Binding to Body Constituents Along with lipids, proteins are the most probable candidates for binding of chemicals, because of their significant physiological concentrations and the ability to participate in practically all types of intramolecular and intermolecular interactions. Strong (covalent) and weak (noncovalent) interactions differ in the magnitude of the liberated interaction energy (more or less than 40 kJ/mol, respectively), and the distance between the most closely located atoms of the molecules of the chemical and the cell constituent (more or less than 2 Å, respectively).( 1249) The weak, noncovalent interactions include hydrogen bonding, electrostatic interactions, dispersion (van der Waals) interaction, and the charge-transfer interaction. The weak complexes are created quickly and spontaneously when two molecules come into contact, while the covalent bond formation requires certain activation energy to overcome the energy barrier. These facts are closely related to the kinetics of the processes: while a weak interaction is completed within milliseconds,( 1250) formation of a strong covalent bond often requires minutes, hours, or days. Protein binding is usually reversible and equilibrium is reached, as a rule, within milliseconds, unless covalent bonds are formed or large protein motions are elicited. Proteins can be classified into four types, according to the consequences of the binding event: receptors, metabolizing enzymes, protein carriers, and the remainder, which are sometimes called silent receptors. The receptor binding elicits bioactivity, the binding to enzymes is a prerequisite for metabolism, and the binding to protein carriers facilitates some transport steps. In these three structure-specific events (see section ), the binding of chemicals usually exhibits high affinity, limited capacity, and a tendency toward one major orientation and conformation (binding mode) of the bound molecules of chemicals. The majority of proteins belong to the category of silent receptors. They frequently contain several binding sites within each molecule. From the pharmacokinetic viewpoint, the silent receptors serve as a depot: they bind free molecules of chemicals, decrease their concentration that is accessible to other processes, and liberate the bound molecules when the concentration has dropped. The binding to silent receptors is typically linked with low affinity, high capacity, and multiple binding modes. For the formulation of the QSAR models for large sets of diverse compounds eliciting the effects at the cell to organism levels, binding to the most abundant proteins must be analyzed and correlated with structures. The preponderant human proteins are albumin in plasma, collagen in interstitial space, and G-actin in cells. Analyses of a broader set of proteins would lead to more-precise predictions. 2.4.1. Serum Albumin Albumin is present in human plasma in concentrations of ~0.6 mM or 40 g/L.( 1251) The binding of sulfonamides to human serum albumin (HSA) and the consequent drug inactivation were observed in the early 1940s.( 1252) At the end of that decade, the first comprehensive review on drug interactions with plasma proteins was published.( 1253) The binding to HSA is usually determined after protein separation by dialysis, ultracentrifugation, or ultrafiltration.( 1254) For compounds absorbing or emitting UV-Vis light, direct methods avoiding the separation of HSA are available.( 1255) HSA, immobilized in a way that maintains accessibility of the binding sites, was used to determine the binding of chemicals by liquid chromatography, 1256–1267 capillary electrophoresis,( 1268) and surface plasmon resonance analysis.( 1269) These methods are reproducible and fast, especially the last technique in combination with microfluidics.( 1270) However, there is a concern regarding the artifacts, which may originate from the inability of the immobilized HSA to undergo large-scale movements, possibly induced in free HSA molecules by the binding event. 1271–1273 While the studies usually document that the HSA binding affinities for the studied set are comparable with those determined by other techniques, the role of large-scale HSA movements may become important for the next analyzed ligand. The PAMPA setup (section ) with HSA present in the acceptor compartment allows the binding affinity determinations at an intermediate throughput, and without the potential for aforementioned complications.( 1274) HSA is composed of three homologous α-helical domains, differing in structural and functional characteristics. Each domain contains subdomains A and B, which share common structural elements. HSA undergoes three pH-dependent structural transitions: two in the acidic range and one in the slightly alkaline range.( 1275) Among the up to 11 binding sites for fatty acids,( 1276) there are at least 3 high-affinity sites for small molecules: the primary hemin site in domain 1,( 1277) the primary warfarin site (drug site I) in domain 2A, and the primary diazepam site (drug site II) in domain 3A.( 1278) Displacement of fluorescent markers by studied chemicals can be used to distinguish between associations with some sites: warfarin and dansyl-sulfonamide mark the 2A site, and dansyl-sarcosine points to the 3A site.( 1279) Structural details of the binding sites were clarified after the 3D structures of albumin 1280–1284 and its complexes with small molecules 1276,1285–1290 had become available. One of the X-ray structures of HSA, with binding cavities and ligands 1287,1290 (structures in Chart 4) bound in sites I and II visualized using the Sybyl module SiteID,( 1291) is shown in Figure . Interestingly, the drugs prefer the drug sites I and II, although each ligand also has at least one secondary site in other domains.( 1277) Binding of a fatty acid changes the polarity and volume of the drug site I.( 1290) These facts, along with the pH-dependent transitions, underline the difficulty of the HSA binding predictions under physiologic conditions. Binding in multiple sites was detected by diffusion NMR and transferred nuclear Overhauser effect (NOE).( 1272) The NOE spectroscopy was combined with conformational analysis to identify the binding modes of some site I ligands.( 1292) Restricted MD simulations were used to analyze the ionization states, water bridges, and functional roles of lysines in the 2A binding site.( 1293) The binding of transition metal ions to the N-terminal of HSA was examined by both classical mechanics and quantum mechanics.( 1294) The development of prediction approaches to the HSA binding affinities follows the general trend in the area of QSAR modeling: qualitative identification of binding-enhancing structural features( 1295) was developed into the dependencies, within homologous series, on physicochemical properties, such as lipophilicity 1263,1266,1296,1297 and ionization;( 1254) followed by the fragment-based approaches for diverse chemicals; 1278,1298–1300 and the PLS or neural networks models with numerous global 1263,1301–1303 and 3D variables. 1304,1305 The most advanced step, the estimation of free energies of binding to HSA based on the atom-level simulations using the flexible 3D structure of albumin 1276,1280–1290 is yet to appear. A reliable prediction of HSA binding is important for estimation of the overall fate of chemicals because, at the high HSA concentration in plasma, even the modest affinities result in a significant decrease of the free concentrations of chemicals. This task is complicated by two factors: (i) the low association constants represent a challenge for the prediction methods; and (ii) the low-affinity binding to the abundant protein results in an easy replacement of the bound chemical by another, only slightly better ligand, with subsequent significant effects on the disposition of both compounds. 2.4.2. Extracellular Matrix The basement membranes, which are parts of the extracellular matrix (ECM), are formed by Type IV collagen in association with laminin, entactin, and heparan sulfate proteoglycans.( 1306) ECM is the most abundant protein mixture in the extracellular space. Recently, we described a simple method for monitoring of the binding of chemicals to ECM.( 1307) The simple dependencies of the association constants on lipophilicity hold only for rather small homologous series.( 1307) To obtain predictions of ECM binding for large and diverse series, the 3D-QSAR methods( 1308) that capture the averaged shapes and properties of the binding sites have been applied.( 1309) 2.4.3. Intracellular Proteins G-actin represents 10%−20% of intracellular protein of mammalian cells. Analytical techniques to be used for the determination of the association constants are similar to those utilized for albumin (section ). The G-actin association constants can be analyzed by receptor-based QSAR methods, using the available X-ray structure.( 1310) There seem to be no QSAR studies available yet for the binding of chemicals to G-actin. 2.4.4. Binding versus Structure and Properties The QSAR models for binding to albumin and other proteins are summarized in the preceding sections. There are some unique aspects of the QSAR models of nonspecific protein binding that deserve attention. The specificity of binding to nonreceptor proteins is much lower than that of the association with the receptors. As a consequence, multiple binding modes of a compound can be encountered much more often than in the case of the receptors. The formulation of QSARs for this situation requires special treatment, because the observed association constant is the sum of the association constants for individual modes. Based on this premise, we have been developing a ligand-based multimode 3D-QSAR procedure to model unknown receptor sites for several years.( 146) The multimode approach is also useful for the receptors with known structures, because the standard MD simulations do not sample the configurations too different from the starting situations, and the use of several starting conformation improves the sampling.( 1311) Among different weak interactions of ligands in the binding sites, hydrophobic interactions exhibit the lowest structural specificity. The reason is that the substructures forming hydrophobic interactions, i.e., nonpolar amino acid residues and alkyl or aromatic ligand parts are quite flexible. Consequently, within a homologous series, the association constants K quantifying binding to silent receptors frequently are dependent on lipophilicity, characterized by the reference partition coefficient P,( 1312) as
Here, B and β are empirical coefficients, which are described in more detail in eq 43 in section . Equation 4 is also valid for the reciprocal values of the Michaelis−Menten constants (eq 45) in some cases.( 1312) The empirical coefficients B (for binding) and β are specific for the chemicals and the protein. For the binding of various series of chemicals to the same protein, the Collander exponent β often remains identical for all series, and the linear coefficient B varies to reflect the involvement of other than hydrophobic interactions participating in the binding of the parent molecule.( 1313) It is not clear if this fact might play a role in the QSAR models of complex biosystems, where many different proteins function as silent receptors, and the coefficients B and β in eq 4 are their averaged values. Depending on the diversity of the binding sites, the protein binding to a mixture of proteins can still exhibit specificity, as we recently showed for ECM.( 1309) In such cases, the 3D-QSAR techniques must be applied, which must be modified to describe the averaging process for binding to multiple proteins. The observed association constant, shown in eq 41 in section , must be described by the expression containing the elementary association constants in the right-most term of eq 39. The reason is that the standard 3D-QSAR equations can only be applied to the elementary association constants for the binding to one site. The proper equation for derivation of a conceptual 3D-QSAR study is obtained from eq 41 by back-substitution from eqs 40 and 39. The resulting 3D-QSAR equation is nonlinear with regard to the optimized coefficients, and it contains the fractions of individual binding sites and the fractions of individual species. While the former fractions are usually unknown and must be fitted as the optimized coefficients, the latter fractions may be estimated using the known dissociation constants and ion-pairing association constants of chemicals in aqueous media. Unbound fractions are used in the practical pharmacokinetic models for individual drugs to adjust the dosing regimens. The typical ranges of this parameter are tabulated for individual drugs in drug monographs. However, the unbound fraction is not a true drug characteristic, because it is dependent on the protein concentration. Consequently, a direct use of this parameter in the QSAR models 1304,1314,1315 is not warranted, unless the data are measured under the same conditions. Rather, the unbound fraction values should be recalculated to the association constants, which are related to the free energy of binding. 2.5. Metabolism Covalent reactions, along with excretion, represent the route for elimination of chemicals from biosystems. The reactions are either spontaneous or enzymatically catalyzed. Metabolism in organisms proceeds predominantly in the hepatocytes( 1316) but other cell types also contribute. Metabolism can affect the outcomes of in vitro experiments in preparations of subcellular organelles and cell suspensions, as well as in isolated tissues, organs, and organisms. 2.5.1. Enzymatic Reactions Generally, a distinction is made between two phases of biotransformation. Phase-I reactions are mostly catalyzed by cytochromes P450 and include hydrolytic, oxidative, and reductive processes.( 1317) The products of the Phase-I reactions or the unchanged molecules are conjugated with hydrophilic endogenous molecules (e.g., amino acids, glutathione, glucuronide, sulfates) in the Phase-II reactions. 2.5.1.1. Cytochromes P450 The largest fraction of metabolism, up to 90% of the Phase-I metabolism of current drugs,( 1318) is performed by cytochromes P450 (CYPs). The 40−50 kD enzymes with a single heme group( 1319) catalyze the hydroxylation of numerous substrates, increasing their aqueous solubility and accelerating renal excretion.( 1319) Originally discovered in mammalian liver microsomal preparations (section )( 1320) and named after the distinct UV-Vis absorption band at 450 nm, 1321,1322 CYP mono-oxygenases have subsequently been found in almost all organisms. The counter-examples are rare and include, e.g., Escherichia coli and Salmonella typhimurium.( 1323) In the superfamily of more than 1000 CYPs, 57 CYPs are expressed in humans.( 1324) Human CYPs metabolize more than a thousand known substrates, and ~15 CYPs are involved in the degradation of xenobiotics.( 1323) The CYP isoforms 3A4 and 2D6 are responsible for the metabolism of 50% and 30%, respectively, of current clinically used drugs.( 1325) The levels of some CYPs in humans are highly regulated, whereas others (e.g., CYP2D6) exhibit considerable differences,( 1326) leading to significant individual variation in pharmacokinetics of drugs. CYP3A4 is upregulated by the nuclear hormone receptor PXR (pregnane X receptor). Both proteins bind a similar broad range of lipophilic chemicals( 1327) and represent the sensor and executioner of the axis of defense against xenobiotics.( 1328) The CYP isoforms 1A1, 1B1, and 1A2 are regulated by the aryl hydrocarbon receptor. 1329,1330 Monitoring the actual CYP isoform levels in individual patients is one of the first components of modern approaches to personalized medicine. In mammals, CYPs are located mainly in the liver; however, they can also be encountered in the intestine and tumors. While CYPs of eukaryotes are mainly anchored in the endoplasmic reticulum, some are also found in mitochondrial membranes. 1331,1332 All CYPs in the endoplasmic reticulum receive electrons from a single flavoprotein: NADPH−P450 reductase. The mitochondrial CYPs use an electron transport chain with the iron−sulfur protein adrenodoxin and the flavoprotein adrenodoxin reductase.( 1323) The electron transfer proteins bind on the side of the CYP that is proximal to the heme. 1333,1334 Although the structures of individual components have been determined, only structural models of the complex are available.( 1335) CYPs can be induced by various chemicals and this ability, in limited series of compounds, increases with lipophilicity.( 1336) The studies of CYP interactions with model bilayers 355,1337 are important for understanding the CYP mechanism and the prediction of CYP metabolism. Bacterial CYPs are soluble; therefore, they were crystallized and analyzed using XRD much earlier 1338–1350 than rabbit CYP2C5, 1333,1333,1351,1352 and human CYP3A4 1353,1354 (Figure ) and CYP2C9. 1355,1356 The structures show that CYPs have large, flexible binding sites, which can accommodate more than one substrate molecule, 1357,1358 and allow for mutual influences of chemicals, which can act as substrates, activators, or inhibitors. The size of the binding cavity can explain the majority of complicating factors of CYP-mediated metabolism, which include broad specificity of individual subtypes,( 1359) atypical allosteric kinetics, 1360–1362 activation, substrate inhibition,( 1362) stereoselectivity,( 1362) regioselectivity, 1363–1368 cooperativity, 1369,1370 and formation of multiple metabolites of some chemicals. 1369,1371–1373The active site of CYPs is buried deep in the protein next to the heme co-factor that plays an important role in activating the molecular oxygen.( 1374) In the absence of substrate, the Fe 3+ ion of the heme is hexacoordinated with four heme-pyrrole nitrogens in a planar arrangement, the sulfur of a cysteine below the heme, and a water molecule on the opposite side.( 1339) When ligated to water, the Fe 3+ ion prefers the hexacoordinated, low-spin state ( S = 1/ 2), in which the five 3d electrons are maximally paired. The spin state remains low when the inhibitors are bound, forming a coordination bond with the iron. The substrates interacting with the binding pocket above the heme and expulsing the aqua ligand without a substantial interaction with the iron, which then becomes pentacoordinated, induce a transition to the high-spin state ( S = 5/ 2), when the five 3d electrons are maximally unpaired.( 1375) The spin changes are reflected in the UV-Vis spectra and frequently are used for a convenient measurement of binding affinities of inhibitors and substrates. The absorbances at 390 and 420 nm have been linked to the low-spin iron and the high-spin, pentacoordinated iron, respectively, using a range of quantum-chemical techniques. 1375–1377The high-spin Fe 3+ ion moves out of the heme plane and becomes more prone to reduction to the Fe 2+ ion by the electron transfer from NAD(P)H. Dioxygen binds to the Fe 2+ ion, where it is stabilized by a network of hydrogen-bonded water molecules and acidic amino acid residues,( 1378) and cleaved, following the second reduction. The resulting iron-linked O atom is a potent oxidant. The substrate is bound in the way ensuring proximity of the pertinent substrate atom to the iron-linked O atom for regioselective and stereoselective hydroxylation. The overall stoichiometry of the hydroxylation reaction includes the chemical, dioxygen, two protons, and two electrons on the substrate side, and the hydroxylated metabolite and water on the product side. The efficiency of the process can be diminished by reduction of the intermediateds to hydrogen peroxide or water. The futile reactions are facilitated by the presence of water in the active site. The elementary steps of CYP-catalyzed metabolism have been studied in great detail,( 1379) and only the most important steps have been summarized above. The CYP active site is often completely isolated from the medium. The protein must undergo dynamic motions to allow substrates to enter and products to leave the active site. These movements are accompanied by complementary water diffusion, for which a water channel was postulated. 1340,1380 This route can also be used by protons.( 1334) The replacement of water by a substrate shifts the redox potential toward the Fe 2+ form. 1381,1382 Three routes for substrates and products were identified, based on Random Expulsion( 1383) and Steered MD simulations,( 1384) and an analysis of the thermal motion factors in the X-ray structures.( 1385) The preference for a route is dependent on properties of the transported chemical and varies among CYPs.( 1386) For mammalian CYPs, two routes were suggested: a one-way route from the bilayer for membrane-bound substrates, associated with a one-way route into the aqueous surroundings for the products; and a two-way route connecting the active site with the aqueous medium for more water-soluble substrates and products.( 1387) CYPs are rather flexible, and the X-ray structures with different geometries of the channels (closed, open just for water molecules, and open for larger substrates( 1343)) were observed.( 1334) Several steps in hydroxylation of camphor by CYPcam, revealing structures of the intermediates, conformational movements of the amino acid residues, and changes in the water network, were captured using cryocrystallography.( 1388) Reactive intermediates produced by CYPs may bind to proteins or react with water and other low-molecular-weight constituents. The protein adducts have long been implicated in organ toxicity, especially in the liver. 1389–1393 Although some nontoxic compounds also form protein adducts, the potential for toxicity leads to a close monitoring of the adduct formation in the development of bioactive compounds. 1394,1395 The approaches include binding of labeled compounds to microsomal fractions,( 1394) reaction of chemicals with glutathione that is monitored using a fluorogenic competitor, and 2D NMR analysis (ALARM NMR) of binding to isotopically labeled human La antigen, whose two cysteins can react with chemicals or change the oxidation state in the presence of chemicals.( 1396) NMR can potentially play an important role in monitoring the binding modes of compounds in the active site of CYPs, thanks to the presence of a natural paramagnetic iron of the heme. The relaxation induced by unpaired electrons provides information about the distance to the iron, 1397,1398 in addition to the identification of the interacting parts of the molecule. The proton−iron distances can be used as constraints for the docking of ligands to CYPs.( 1399) The importance of CYP metabolism in the development of bioactive compounds has led to a continuous interest in prediction of all aspects of CYP interactions with chemicals,( 1400) including the induction of different CYP isoforms.( 1401) The induction process in rat hepatocytes was analyzed as a function of structure of 4- n-alkyl-methylenedioxybenzenes.( 1402) Metabolic rates and their inhibition were often analyzed as a function of structure or properties of substrates and inhibitors. In a limited series of substrates and inhibitors, CYP binding or metabolism were frequently found to correlate with electronic indices( 1403) or lipophilicity. 1404–1409 The correlations are series-specific, and different correlations for the same CYP are not uncommon.( 1410) Empirical correlations with large numbers of descriptors, processed by soft-modeling techniques, such as linear regression,( 1411) neural networks,( 1412) PLS regression, 1413,1414 and k-nearest-neighbor variable selection( 1415) extended the scope to larger sets of chemicals. Predictions for more-diverse chemicals required a 3D approach. The long absence of the mammalian CYP structures resulted in numerous ligand-based 3D-QSAR models of the CYP binding for both substrates 1416–1422 and inhibitors. 1423–1428 The 3D-QSAR models were combined with comparative structural models, which used bacterial, 1425,1429–1432 rabbit, 1433–1437 and human( 1438) CYP structures as templates. Peculiar properties of the CYP binding sites, including sizes that often exceed the size of common ligands, conformational changes, and the possibility of accommodation of two bound molecules, led to less-predictive docking than for those with other receptors. 1438,1439 The success rates were improved by re-parameterization of the ligand−iron interactions by knowledge-based potentials, and a new parameterization of lipophilicity contributions of the planar heme nitrogens.( 1440) The estimation of binding free energies 1378,1441 is complicated by the isolation of the binding site from the medium. The problematic exchange of the water molecules displaced by ligand binding with the medium was emulated by coupling of MD and Grand Canonical MC simulations.( 1442) A more precise description of the CYP metabolism and prediction of structures and formation rates of individual metabolites is a formidable task. Predictions of the sites of metabolism in the structures of metabolized molecules 1443,1444 were performed using quantum-mechanical (QM) reactivity calculations, 1403,1445QSAR approaches,( 1446) and pharmacophore-based procedures 1420,1430,1431,1433,1447 on substrate molecules, and docking( 1439) utilizing the structural information about the binding site. Practical software applications use either knowledge-based rules or a simplified simulation of the interaction between the pertinent CYP and the metabolized molecule. The former category includes the commercial expert systems MetabolExpert,( 1448) METEOR 1449–1452 and META.( 1453) The software tool MetaSite( 1454) is the only representative of the second category. The binding sites in the homology models of several CYP isoforms are characterized by the GRID-based descriptors and the substrate molecules are described by clustered atom distances.( 1455) A comparison of the resulting binding site and substrate fingerprints provides the accessibilities of individual hydrogens in the substrate molecule. The accessibilities are multiplied by estimated, semi-quantitative hydrogen abstraction energies to provide a probability of metabolism for each position.( 1456) 2.5.1.2. Other Enzymes Phase-II enzymes, hydrolases, and other metabolizing enzymes were studied less extensively than CYPs. For a limited series of compounds, the biotransformation rate parameter can be dependent linearly on a single property, such as the reactivity in the reaction imitating the pertinent process in the reaction center of the enzyme( 1457) (section ), relevant QM indices, 1458–1462 or the partition coefficient,( 1463) if the noncovalent binding to the enzyme is dependent on lipophilicity and all the other parameters are invariant. For diverse compounds, the ligand-based 1464,1465 or receptor-based 3D-QSAR methods( 1466) were applied. If the prediction of disposition of diverse chemicals in organisms is of interest, the in-vitro-measured metabolism parameters often may be the best choice, considering the complexity of the QSAR prediction of metabolism. The scale-up procedures 1467–1475 can be improved using the SBSP models. Kinetic issues are addressed in section . 2.5.2. Spontaneous Reactions This category of elimination processes is comprised of hydrolysis, as well as the reactions of chemicals with low-molecular-weight body constituents and macromolecules. Abundant are the reactions of chemicals with cellular nucleophiles such as free amino acids, glutathione, and proteins.( 1476) Spontaneous reactions do not require a preceding binding of the chemical to a macromolecule, and they are comparatively easy to imitate in vitro. The rate constants measured in the reaction mixtures mimicking biological conditions belong to the most-precise reactivity parameters. 1477,1478 The experimentally determined reaction rate parameters for a reference reactant ( k) can be used to assess the parameters for other reactants ( kx) in similar reactions, as, e.g., in nucleophilic additions to different acceptors:
The empirical coefficients A r and β r are dependent on the reactants and the reaction conditions. The reference reactants used include 4-( p-nitrobenzyl)pyridine, thiourea,( 1479) glycine,( 1480) cysteine,( 1481) and thioglycolic acid. 1482–1484 Empirical substituent constants and QM indices related to the electron density at the reaction centers in intact molecules and transition intermediates 1485–1489 can be used to substitute the rate parameters of the spontaneous biological reactions. The reactivity parameters were mainly applied in the QSAR studies of toxicities or other effects of alkylating agents. In these studies, the dependence of the effect on reactivity frequently exhibits an optimum. 1457,1490,1491 This behavior reflects the interplay of two phenomena, binding to the receptors and metabolism, which may both involve similar chemical reactions behaving according to the nonlinear dependence described in eq 5. While the receptor binding increases with reactivity, the disposition in the receptor surroundings is independent of low reactivity and decreases after reactivity exceeds a certain limit (see eq 13 in section ). The increase in reactivity of chemicals leads to more-pronounced effects up to the point when high reactivity causes a fast elimination of the compounds, which are no longer able to reach the receptors in meaningful concentrations. 3. Disposition of Chemicals The disposition of chemicals in biological systems has been experimentally analyzed at different levels of complexity, starting with membrane vesicles, subcellular organelles, cells, cell monolayers, biological sheets, up to organs and organisms. In this section, a brief overview of the most frequently studied biosystems and used techniques is provided. The coverage could not keep up with the abundance of published studies in all mentioned areas, but especially for the perfused organs (section ) and organisms (section ), and lists only the most important developments. The relationships between the transport or accumulation and physicochemical properties are analyzed, where available. In contrast to section , which focused on individual steps, in complex systems, several interactions between chemicals and biosystem components participate in the overall outcome. 3.1. Membrane Vesicles The vesicles form spontaneously under appropriate experimental conditions from complete membranes in tissue homogenates and are isolated from other cellular material by differential centrifugation. Historically, availability of these systems was associated with the progress in the biochemical separation techniques: the microsomal fractions were first prepared in the 1940s, while membrane vesicles appeared in the 1970s. Some of the membrane proteins in these preparations function as transporters or metabolizing enzymes, while others play more-passive roles. Integral and peripheral membrane proteins diminish the surface area available for passive transport, causing the process to become slower. The binding of chemicals to both integral and peripheral membrane proteins leads to a decrease in the free concentration of chemicals, thus reducing all types of transport. 3.1.1. Endoplasmic Reticulum Vesicles Microsomal vesicles are derived mainly from endoplasmic reticulum, but the cell membrane and the membranes from lysozomes, mitochondria, and Golgi apparatus are also present.( 1492) The majority of these minor microsomal components sediments out, along with intact cells, mitochondria, and nuclei, during centrifugation of a tissue homogenate at 9000 g. The resulting supernatant, which is called the S-9 fraction, contains mainly endoplasmic reticulum and soluble enzymes and is sometimes used for metabolic activation of chemicals to test biological effects of metabolites. 1493,1494 More-intense centrifugation of the supernatant at ~100 000 g pelletizes the microsomal membranes, which are usually of a reddish-brown color because of the high content of the CYP heme.( 1495) The vesicles were prepared from the liver, 1496–1501 brain, 1502,1503 lungs,( 1504) heart, 1505,1506 kidney, 1505,1507 intestine, 1508,1509 skeletal muscle, 1510,1511 placenta,( 1512) leukemic cells,( 1513) and other sources. The vesicular structure of the microsomes was demonstrated by electron microscopy.( 1514) The preparations were first used to characterize subcellular location of enzymes,( 1498) other macromolecules,( 1515) and metabolic processes. 1503,1504,1512 Microsomes have been mainly applied to the study of metabolism of chemicals. 1472,1475,1494,1516–1518 Although the microsomal data serve as the input for prediction of hepatic metabolism, 1469,1519,1520 the uptake and transport studies in the microsomes focused on physiological molecules, 1521–1524 and the data on chemicals are rather scarce. Because the substrates may access the CYP binding sites from the membranes,( 1387) the concentrations of chemicals in individual bilayer regions (sections and ) are important determinants of the overall kinetics. The assessment of stability of chemicals in microsomes or S-9 fractions with different levels of expressed CYP isozymes belongs to the standard components of metabolic( 1525) and drug−drug interaction( 1526) tests in preclinical drug-development process. The data have been processed by three categories of models: (i) predictions of in vivo metabolic clearance of individual chemicals, 1469,1519,1520 (ii) QSAR analyses of the rate of metabolism, and (iii) predictions of the site of metabolism. The QSAR models in the second category usually utilize empirical approaches based on numerous descriptors, which are calibrated by regression analyses 1411,1527 or by the variable selection based on the model-free k-nearest-neighbor intrapolation.( 1415) The predictions of metabolic sites in the third category are listed in section , because they are based on the structure of pertinent CYPs. 3.1.2. Cell Membrane Vesicles Different techniques were used to prepare vesicles from cell membranes of epithelial cells, either from apical, brush border membranes from the intestine, 1528–1530 renal tubules, 1531,1532 and placenta, 1533–1535 or from renal basolateral membranes. 1529,1536,1537 The brush border membrane vesicle form spontaneously by segmentation of the microvilli in the preparation process, so the correct polarity of the membrane is maintained.( 1538) They were mainly used for the monitoring of the active transport of physiologic molecules. Studies on the transport of chemicals are rare.( 1539) Red blood cells, which represent ~45% (hematocrit) of the blood volume, lack a nucleus and organelles. Their membranes are sometimes isolated and resealed to generate vesicles for the study of transport and accumulation of chemicals. Erythrocyte ghosts are prepared by hypotonic( 1540) or isotonic( 1541) lysis at 4° C, washing the membranes, and subsequent resealing at 37° C, whereby lipid asymmetry is aborted.( 1542) Membrane proteins of erythrocytes include several MRP efflux pumps (section ),( 1543) and the vesicles were used to measure the outward transport of amphiphilic molecules.( 1544) The ease of preparation makes the vesicles a popular system for the study of asymmetry of distribution of proteins( 1545) and lipids, 708,714,1546–1548 as well as for the measurement of the interactions of nonpolar( 77) and amphiphilic chemicals 82,103,335,441,1549–1553 with membranes. Accumulation in erythrocyte ghosts was analyzed via the PCA approach, using Volsurf descriptors.( 1136) 3.2. Subcellular Organelles The complexity of subcellular organelles is higher than that of membrane vesicles. Although both are covered by complete membranes, the organelles contain additional structures, which may bind the chemicals and complicate the interpretation of the disposition data. The most often used subcellular organelle is mitochondrion, which became the object of interest for the study of energy metabolism and the effects of compounds affecting this process, as well as for the analysis of the mechanism of apoptosis. Mitochondria are isolated by differential centrifugation from the cell homogenate, which is prepared by mechanic disruption or nitrogen pulsation,( 1554) most frequently from liver, 1555–1557 heart,( 1558) brown adipose tissue,( 1559) skeletal muscle,( 1560) neurons, 1554,1561 protozoa,( 1562) as well as from yeasts and other sources.( 1563) The preparations were used to measure the uptake 78,1564 and effects of uncouplers of oxidative phosphorylation and respiratory inhibitors. 78,1564–1576 The uptake into rat liver mitochondria was described by a conceptual pseudo-equilibrium model (section ) and used to infer membrane concentrations of the uncouplers and to generate conceptual QSARs for their effects. 78,1564For DNA intercalators, the interactions with isolated nuclei were studied and compared to the binding of the chemicals to DNA in intact cells. 1577,1578 The results contribute to the analysis of subcellular distribution of studied chemicals. 3.3. Cells The accumulation of chemicals in cells has been studied since the end of the 19th century. The reports amply illustrate the thinking process, based on sparse and imprecise data. In the following, the most important milestones are listed in chronological order, with each paragraph covering about two decades. Plant cells and protoplasts, and the eggs of marine invertebrates were the first systems to be studied from the viewpoint of transport of chemicals, and the permeabilities were monitored as osmotic volume changes. 188,192 The Overton’s theory on the association between the permeability and lipophilicity( 1579) that led to the hypothesis on the lipid character of the membranes, was developed using the tadpole irritability data.( 192) In histology, uneven intracellular distribution of dyes, which were used to visualize various cell components for microscopic observation, was considered a consequence of different solubilities in cellular material (cited in ref ( 1580)). The partitioning of acids and bases in surrogate solvent/water systems exhibited increasing or decreasing sigmoidal dependences on the pH values of the aqueous phase.( 1580) With the objective of monitoring the transport of ionizable compounds,( 1581) the pH indicators were introduced into cells and used to measure the uptake of bases by sea urchin eggs.( 1582) To improve survival of the test material, cells containing natural pigments serving as the pH indicators were used. 1183,1583–1585 However, these studies did not use single cells; therefore, they are described in section . In large cells of algae ( Valonia, Laminara, Chara, Nitella and others), conductivity was monitored as a measure of the overall ion movement, and the effects of various chemicals on this parameter were studied. 1586,1587The size of the algae cells allowed a direct chemical analysis of the cellular content. 1588,1589 A similar analysis of cellular content was also applied to the protoplasts of oyster plant ( Rhoeo discolor). 1590,1591 Micromanipulation of amoeba cells in the microscope field allowed for the analysis of anion and cation effects, not only by the cell immersion but also by microinjection. 1592,1593 Transport of a series of acids was tested as hemolytic activity in the author’s own blood,( 679) and as the threshold concentrations for the sour taste.( 1594) The comparison with physicochemical properties of chemicals became more quantitative: the oil/water partition coefficients were used in the former study, and published data on charcoal adsorption in the second study. The first real correlations of permeabilities with oil/water partition coefficients were published by Collander( 521) for the transport of nonelectrolytes in algae Chara ceratophila and other cells. This work is remarkable also for other conclusions: the transport process was described by Fick’s law, and some small molecules permeated at a faster rate than expected, based on their partition coefficient. The latter observation formed the basis of the lipoid-sieve hypothesis, which assumes the presence of pores, allowing the passage of smaller molecules. The nonelectrolyte data in algae Nitella mucronata( 680) were fitted by a linear permeability−lipophilicity relationship and later were shown to adhere better to a parabolic equation.( 1595) The availability of radiolabeled chemicals opened the door for autoradiography. 1596–1604 For visualization of subcellular distribution, autoradiographic images were enlarged by light microscopy. 1605,1606 The use of electron microscopy in the transmission mode 1607,1608 or scanning mode( 1609) increased the resolution from ~0.2 μm, which is typical for light microscopy, by three orders of magnitude. The technique enabled important discoveries related to DNA, RNA, and proteins, but the location of diffusible, noncovalently bound chemicals could not be analyzed precisely, because of the artifacts introduced by sample preparation. To monitor the distribution of chemicals and drugs, the precision was greatly improved 1610,1611 by the introduction of precise cryosection( 1612) of freeze-dried samples,( 1613) which avoided fixation and embedding of the samples. Other techniques for monitoring membrane accumulation, including potentiometry, equilibrium dialysis, and 1H NMR spectrometry, have been recently reviewed.( 307) Confocal fluorescence microscopy allows monitoring of the kinetics of subcellular distribution of fluorescent compounds. 1614–1618 The high shutter speeds are of special importance for capturing the detailed kinetics of the transport process. 1619,1620 The rich spatial and temporal data represent an ideal test bed for the analyses by SBSP models. Partitioning into erythrocytes and leukocytes was used to provide a picture about the binding of chemicals in the blood. Notably, in classical pharmacokinetics the blood corpuscles represent the components of tissues as opposed to those of plasma. This anomaly is caused by the definition of the distribution volume, which is given as the ratio of the actual drug amount in the body and the total drug concentration in plasma. As a consequence, a drug that, after intravascular administration, does not leave the bloodstream but binds to erythrocytes or leukocytes will have a distribution volume that is larger than the plasma volume. Blood corpuscles are readily available, are easy to purify, have comparatively simple structures, and their shapes can be comfortably monitored. Their interactions with chemicals have different consequences, depending on the used concentrations of chemicals: simple partitioning, 1552,1621–1623 shape changes, 1624–1628 aggregation,( 1629) and even hemolysis. 1630,1631 Hemolytic activity of homologous N-alkylpiperidines and α-monoglycerides exhibit a bilinear dependence on lipophilicity, 8,64,706 indicating non-equilibrium transport, as decribed in detail in section . A linear lipophilicity−concentration profile was observed for the binding to erythrocytes for a series of barbiturates.( 1314) Understanding the uptake into epithelial cells or enterocytes lining the luminal surface of the intestine is important for orally dosed drugs. Enterocytes are highly polarized and feature completely different apical and basolateral membranes. Germinal cells in the crypts between the villi differ from nonproliferative columnar cells that cover the villi, through which the absorption proceeds. These cell populations( 1632) can be separated after the cells are isolated.( 1633) Upon isolation, enterocytes will round and depolarize. The cells respond extensively to environmental factors, are difficult to maintain in primary cultures, and have a short lifetime. Therefore, the uptake experiments are mostly performed using freshly isolated enterocytes.( 1634) Alternatively, everted rings of the entire intestine are used to measure the drug uptake, with the risk that the uptake into the cells other than enterocytes will affect the results. With proper oxygenation and nutrition, the rings can be maintained for several hours before the changes to the mucosa become observable.( 1635) The transport through the entire intestine in the form of everted sacs or sheets is described in section . Continuous monolayers of human intestinal adenocarcinoma Caco-2 cells, which are more comfortable to maintain and use, found widespread utility in assessing intestinal absorption in the drug-development process (section ).( 1636) The interest in the use of hepatocytes in disposition and metabolism studies stems from the importance of metabolism for drug pharmacokinetics. The uptake into hepatocytes 1474,1637 was studied frequently, because of its importance for the prediction of metabolism.( 1638) The importance of transporters in hepatocytes, as opposed to microsomes, for the prediction of transport in humans was emphasized. 1639,1640 Human hepatocytes can be engrafted into immunochemically compromised mice, where they repopulate up to 90% of the liver.( 1641) The animals can be used as a source of fresh human hepatocytes or directly for toxicity studies of metabolites. The uptake of chemicals into various cell types( 1642) was measured using the analysis of the medium after the cell separation by ultracentrifugation, ultrafiltration, gel filtration, and dialysis. The disadvantage of the uptake monitoring, the uncertainty about the localization of the chemical interacting with the cell either on the cell membrane or inside the cell, can be alleviated using the SBSP models (see sections − ). This approach was used to estimate the subcellular distribution and resolve the mechanism of uncoupling of oxidative phosphorylation for a series of carbonylcyanide phenylhydrazones in rat liver mitochondria,( 78) bacteria Paracoccus denitrificans, and leukemic P-388 cells.( 1643) Advanced kinetic approaches were applied to the modeling of the cellular uptake data, 1644,1645 although accumulation in the bilayers, which is an important process for the majority of chemicals, was not included in the analysis. 3.4. Cell Monolayers Some cell lines form confluent cell monolayers, imitating biological epithelia, when grown on porous supports. The monolayer cultures were initially used to study physiological processes occurring in renal 1646,1647 and intestinal 1648–1650 epithelia, and later adapted for the analysis of the distribution of chemicals. Transport across the layer separating two aqueous compartments is conveniently examined by measuring the compound concentration in the receptor compartment. Sometimes, accumulation in the cells is also assessed. The initial use of radiolabeled chemicals was largely replaced by UV-Vis absorption spectrometry and liquid chromatography analyses. The latter technique, especially when coupled with mass spectrometry, allows the detection of metabolites, and provides the opportunity to increase the throughput by simultaneous testing of several chemicals in a mixture.( 1651) The most frequently used monolayer system is represented by human colorectal adenocarcinoma cell line Caco-2 that undergoes epithelial differentiation and forms monolayers( 1652) of polarized cells( 1653) on porous supports. The process may take up to three weeks, and attempts were made to shorten this period to seven( 1654) or even fewer days.( 1655) Initial studies, focusing on the uptake of physiologic compounds, 1656,1657 were soon extended to the transport setup utilizing the porous supports.( 1658) The differentiated monolayer imitates intestinal epithelium and is widely used to assess the oral absorption potential of chemicals. 1659–1671 The unstirred aqueous layers affect permeation through the Caco-2 monolayer.( 1672) In several studies, the Caco-2 permeability was shown to be a predictor of intestinal absorption. 1673–1681 An approximate sigmoidal relationship between intestinal and Caco-2 permeabilities was observed for a limited set of compounds. 1682,1683 The relevance of the results for the prediction of intestinal absorption can be improved through the use of media that imitate the gut content in the apical compartment.( 1684) The throughput of experiments was increased by introducing multiwell systems( 1654) and cassette dosing of several chemicals.( 1685) The latter modification works well for passively transported chemicals, but saturable and competitive binding may invalidate the results if any of the protein-mediated mechanisms (section ) is involved. The Caco-2 monolayer system was used to study the transport of simple organic molecules (urea and mannitol as the markers of paracellular transport,( 1686) creatinine, erythritol, phosphonomethanoic acid,( 1687) amines, 1686,1688 amino acids, 1689,1690 carboxylates,( 1686) oxazolidin-2-ones( 1691)), adrenergic receptor antagonists, 1686,1692,1693 β-lactams,( 1694) peptides, 1664,1691,1695–1699,1699–1702 peptide prodrugs, ( 1703– 1706) and peptidomimetics, 1707,1708 as well as numerous other drugs 1683,1709–1714 and drug candidates. 1680,1711,1715,1716 The impact on absorption of chemicals of permeation enhancers, 1717–1720 excipients for drug dosage forms, 1710,1721–1724 and polymeric nanoparticles as drug delivery vehicles( 1725) was also analyzed. Although the Caco-2 cells in monolayer attain several morphological and functional characteristics of mature enterocytes, some differences do exist, primarily because of the colonic origin of the cell line. Those most important for the transport studies include very low expression of cytochrome P450,( 1726) especially that of the 3A4 isoenzyme that is important in drug metabolism,( 1651) a lower content of efflux pumps and transporters for active absorption of vitamins, bile acids, amino acids, nucleosides and peptides,( 1727) and tighter cell junctions reminiscent of those in the colon, as opposed to those in the small intestine. 1667,1728Variability of the Caco-2 monolayers, caused by cellular factors and culture conditions, complicates the interlaboratory comparisons. The parental cell line is heterogeneous, and different subpopulations may become prevalent, depending on the cultivation conditions.( 1729) Circumstances such as the number of passages,( 1730) the nature of support, 1731,1732 and the presence of nutrients( 1733) and induction factors 1651,1706 affect the characteristics of the prepared monolayers. Standardization of the experimental setup, including the cell source, cultivation conditions, and the use of media and solvents, would contribute greatly to the availability of reference-quality permeability data of chemicals, leading to more-general models and increased understanding of the structural dependence of the intestinal transport process. Depending on the structure of transported chemicals, the Caco-2 monolayer permeation is affected by any or all of the following processes: passive transcellular transport, 1692,1734 active transport 1733,1735 and efflux,( 1697) transport through tight junctions, 1688,1722,1736 endocytosis,( 1737) transcytosis,( 1663) and metabolism. 1666,1738 This variety, although providing the opportunity to study any of the involved mechanisms, complicates the correlation of the permeability data with physicochemical properties of transported chemicals. Several attempts have been made to discern the fluxes through individual routes. 1686–1688,1692,1728,1739,1740The polarity of the Caco-2 monolayers provides a simple means to detect active transport mechanisms, especially the efflux mediated by P-glycoprotein. For this purpose, the trans-monolayer fluxes in the apical-to-basolateral direction and backwards are compared. Identity of both fluxes between two identical aqueous phases indicates passive permeation. If the backward flux is significantly higher, energy-dependent, saturable, and the difference is abolished by a P-glycoprotein inhibitor such as verapamil, the efflux probably affects the permeation. 1700,1741 Mechanistic studies of active transport are performed preferably in the Caco-2 monolayers, because isolated intestinal enterocytes are fragile and have limited viability.( 1742) Thanks to the importance of oral absorption in the drug development process, there has been considerable interest in the elucidation of the relationships between structure or properties of chemicals and their Caco-2 permeability. The correlations are obscured by the multitude of processes, which contribute to the overall outcome of the transport experiment. Obviously, any mechanistic knowledge is of great value for the development of reliable descriptions. Unfortunately, this knowledge was mostly used for explanation of outliers,( 1691) rather than for a systematic classification of the permeants for the correlations. In addition, the analyses are complicated by the fact that the determined permeability coefficient is not a complete characteristics of the transport process (section ), and the information about the lag time( 1743) and accumulation is not considered. The simplest models were applied to the data for compounds permeating without the use of protein carriers (i.e., by passive transport through enterocytes or through tight junctions). The former process received more attention because it can be applied to the majority of tested chemicals. The descriptors used in mostly linear correlations of permeability were the 1-octanol/water partition coefficients ( P) for neutral molecules, the distribution coefficient representing the P values for ionized molecules at the pH of experiment, 1691,1743,1744 the number of hydrogen bonds ( 1695) or more quantitative hydrogen-bond parameters,( 1745) polar surface area,( 1746) as calculated from all low-energy conformations,( 1747) from 1000 conformations obtained by an MD simulation,( 1748) from a single conformation,( 1749) or by a fast fragment-based method,( 1127) molecular size, 1702,1743–1745 charge, 1702,1712,1743,1744 and the partition coefficients for IAMs (section ). 536,1702,1707,1711 For several peptide series, the 1-octanol/water distribution coefficient Pi was not a good predictor of permeability, especially if the transport was significantly affected by efflux.( 1700) The correlations were mostly linear, with an exception of a sigmoidal relationship on log Pi, shifted by the contribution of paracellular transport parametrized by molecular weight. 1743,1744 Similar but more-scattered sigmoidal relationships were observed for the log Pi values measured in the n-hexadecane/water and PGDP/water systems for 51 diverse drugs.( 1683) Numerous descriptors (surface-related, 1750–1752 hydrogen bonding with log P( 1128) and charges( 1679)) were simultaneously correlated as a linear combination using PLS or multiple linear regression with the permeability of diverse compounds, demonstrating a large negative impact of hydrogen bonding, and positive impacts of lipophilicity and sometimes polarizability. A linear combination of a hydrogen-bond descriptor, log Pi, molecular weight, and an indicator variable for charge was used in a permeability−property correlation.( 1679) Volsurf descriptors were correlated with Caco-2 permeabilities using the PLS 655,1140–1142 and PCA( 1136) methods. The 3D-QSAR approaches, which are usually suitable for structure-specific protein-mediated interactions, were applied to the passive permeation of 38 inhibitors of rhinovirus replication.( 1753) Detailed simulations were performed for passive uptake and permeation of cells( 1754) and cell layers,( 1755) respectively, as affected by physicochemical properties. The models accommodated the ionization of chemicals and varying acidities of individual compartments, but they did not include accumulation of chemicals in the bilayers and used rather simplified relationships between the permeability coefficients and lipophilicities. The cooperation between P-glycoprotein (section ) and CYP3A4 (section ) at the apical membrane in creating a defense mechanism to prevent the entry of some chemicals into the body is an interesting concept that has important consequences for oral administration of drugs. 1756–1758 The idea is supported by similar broad substrate specificities of both systems. To improve the suitability of the Caco-2 monolayers for studying this mechanism and intestinal metabolism in general, the CYP3A4 expression can be elevated using vitamin D3 1651,1706 or other inducers. The metabolizing enzyme levels are generally higher in earlier passages of the cells in the process of preparing the monolayers.( 1730) However, if vitamin D3 is used for CYP3A4 induction, the cultures with higher passage numbers contain more enzymes.( 1759) The Caco-2 cells transfected with CYP3A4 represent an alternative option. 1760–1762 Other cell lines forming the polarized monolayer and having a high expression of both transporters and enzymes can be used to study their cooperation.( 1726) Among other than Caco-2 cellular monolayers, the Madin-Darby canine kidney (MDCK) cell line( 1763) has been deployed most frequently. The initial focus on tissue cultures that were suitable for the monitoring of viral propagation, 1764,1765 oncogenesis,( 1766) and physiological processes 1647,1767 was extended to the monolayers grown on a porous support, which were used to study the transport of drugs 1768,1769 and ions, imitating tubular reabsorption. 1770,1771 The MDCK cells, transfected with CYP3A4, are available.( 1762) Transport studies using MDCK cells have been performed less frequently than those with Caco-2 monolayers. The QSAR studies of drug( 1772) and peptide 1773,1774 permeation through the MDCK cell monolayer are of rather preliminary character. Other studied cellular monolayer systems include intestinal cell lines HT29-18-C1( 1775) and IEC-18,( 1776) as well as 2/4/A1 cells with reduced transporter activities( 1777) for the study of intestinal absorption, LLC PK1 renal epithelial cells( 1778) as an in vitro system for studying tubular reabsorption, brain microvessel endothelial cells 1779,1780 imitating the blood-brain barrier, and Hep G2 cells( 1781) for studying aspects of hepatobiliary clearance. The ability of the human intestinal goblet cell line HT29-H to produce mucus was deployed to assess the barrier properties of the mucus.( 1782) Diffusion of drugs in the mucus was analyzed and compared to that in water. 1783,1784 Cellular systems imitating skin( 1785) did not provide relevant absorption data, 1786,1787 but may find a more successful application in reconstructive surgery.( 1788) The use of these monolayers was much less frequent than that of Caco-2 cells and no QSAR studies were performed. 3.5. Biological Sheets Studies of transport through planar biological systems consisting of several different cell layers are briefly summarized in this section. The sheets include human, porcine, rhesus monkey, 1789,1790 and mouse skin, 1791,1792 dog and cat pericardia,( 1793) the walls of the intestine( 1794) and urinary bladder,( 1795) frog skin, goldfish gills,( 1796) and other planar objects that are suitable for the study of the transport of chemicals, because they can be mounted as barriers separating two media in diffusion chambers of Ussing 1797,1798 and Franz 1799,1800 designs. An air-lift system normally is used to provide a proper gas balance, and stirring is applied to reduce the diffusion barrier thickness and ensure homogenization of bulk media. Transport through the skin and the small intestine, although not fully representative of the real situation, are by far the most frequently studied processes in this category. The majority of the experiments with other sheets was made in the first half of the last century, focusing on general information about biological transport without placing much emphasis on the practical importance of the sheets that were used, such as frog skin, 1801–1803 toad urinary bladder,( 1795) or goldfish gills.( 1796) In quantitative studies, the transport was characterized by the permeability coefficients PC. Several relationships for the dependence of the PC on the structure of transported chemicals are noted below. They usually make simplifying assumption about the structure of the studied sheet. The expression that takes the multilamellar sheet structure into account is given as eq 51 in section . 3.5.1. Skin Transdermal absorption can be studied in various skin preparations, which provide a more realistic picture of events than the phospholipid systems (section ); although some steps are still missing and some may be superfluous (section ). The process was studied in subcutaneous rat tissue( 1804) and hairless mouse skin, 1791,1792 as well as in human skin obtained from amputations and cadavers. Macroscopic dimensions allow mechanical separation of individual skin layers, 1805,1806 tape stripping,( 1807) sectioning of the skin layers at different depths, 1808,1809 and visualization of permeant positions in the cross sections. 1809–1812 The main barrier for percutaneous absorption of most chemicals is the stratum corneum 1813–1815—the surface lamina consisting of several layers of corneocytes, which are embedded in a lipid matrix represented by multiple bilayers separating the cells. 161,1816–1819 The corneocytes are filled with water and microfibrillar keratin, and they are surrounded by a keratinized envelope, 1820,1821 which is covered by a monolayer of covalently bound lipids( 1822) that play a role in cell adhesion.( 1823) Chemicals could permeate the stratum corneum via diffusion through the corneocytes,( 1824) the voids in continuous lipid bilayers (as described in section ), aqueous pores resulting from either imperfections in lipid bilayers 163,397 or reorganization of the bilayer forming the transitional opening lined with the headgroups, 1825,1826 and shunts represented by hair follicles and sweat ducts, as well as by lateral diffusion along the plane of the bilayer.( 163) Relative contributions of these parallel mechanisms to transdermal transport of a specific chemical are dependent on its physicochemical properties, although the latter mechanism is considered dominant for drugs.( 1820) Impermeable keratinized corneocytes in the topmost layers of the stratum corneum impede the transport by creating a tortuous diffusion path for the lipid-based mechanisms. This fact makes percutaneous absorption a unique process, with the paracellular mechanism prevailing for most chemicals. The rate of absorption can be increased using chemical enhancers( 1827) or iontophoresis. 1828,1829The importance of the transdermal route of administration for some drugs( 1830) led to the development of modeling approaches for single compounds 1805,1831,1832 and QSARs for sets of chemicals. Simple rules regarding molecular weight, aqueous solubility, melting point, and the number of hydrogen-bonding atoms were formulated to classify the permeants as fast and slow.( 1833) Additive contributions of structural fragments to transdermal permeability were determined.( 1834) For chemicals traversing the skin via the lipid-based paracellular route, the permeability coefficient was expressed as the product of the partition coefficient and diffusion coefficient,( 1813) as known for the solubility-diffusion mechanism for transport through a single bilayer (section ). The simplest property-based QSARs reflect this fact and describe the permeability coefficient as a power function of the partition coefficient (see the Collander equation 1 in section ) in surrogate systems,( 1835) including the stratum corneum/water system,( 1836) for compounds of similar sizes. For more-diverse compounds, the fits improved via the addition of molecular weight, 1837,1838 molecular volume,( 1839) hydrogen-bond parameters, 1836,1840,1841 and atomic charges,( 1842) which should characterize the structural dependencies of the diffusion coefficients. Importance of the difference between logarithms of 1-octanol/water and alkane/water partition coefficients, formally equal to the 1-octanol/alkane partition coefficient and characterizing hydrogen bonding (section ), was also examined.( 1843) Acceptable correlations for heterogeneous chemicals were obtained using Abraham’s solvatochromic equation. 1844–1846 The interesting observation that asymmetric positions of hydroxyl groups in polyphenols led to a slower transdermal transport than a symmetric distribution( 1847) might be related to the role of amphiphilicity of the asymmetric molecules in the trans-bilayer transport (section ). Complexity of the process was indicated by the need to break down the sets of compounds to obtain good linear relationships.( 1848) Additional equations were summarized in detailed reviews of simple QSAR approaches to transdermal permeability. 1838,1849More-detailed models that take into account the structure of the stratum corneum and tortuosity of the lipid route were developed, 163,1850 describing not only the steady-state transdermal flux but also the lag time. Simplified two-dimensional (2D) models 164,1851,1852 were extended to more-realistic 3D representations. The presence of several pathways, each traversed to the extent given by the properties of chemicals, was considered. 1853–1855 3.5.2. Small Intestine The second-most frequently studied sheet was the small intestine, thanks to the importance of intestinal absorption for chemotherapy. Experiments in this area started at the beginning of the last century. 1856,1857 The intestine was used in the form of everted sacs( 1858) or mounted as a barrier separating two solutions in side-by-side diffusion chambers. 1859–1863 The latter setup utilized either the entire intestine 1864–1866 or mucosal sheets. 1867–1873 The viability of the tissue must be carefully monitored.( 1874) The in vitro systems were used to study the absorption of small sets of chemicals. 1687,1868,1875–1877 Other intestine-based test systems include brush-border vesicles (section ), isolated enterocytes, and the rings of the entire intestine (section ), which are used to measure the uptake of chemicals, as opposed to the transport across the intestine. Intestinal absorption of a chemical requires that the molecules cross only a small part of the intestinal structures, namely, from enterocytes to blood capillaries and lymphatic capillaries (lacteals) embedded in the villi of the mucosa, not the entire intestinal wall. Therefore, the relevance, for intestinal absorption, of the in vitro studies of transport through the complete intestinal wall is questionable. The use of mucosal sheets isolated from the small intestine only reduces the presence of the intestinal structures, which do not need to be crossed by absorbed molecules, but does not solve the problem. On the other hand, the intestine-based systems allow the study of the effects of unstirred layers( 1878) and intestinal metabolism, as well as a comparison of absorption in different segments of the intestine.( 1676) The QSAR studies of intestinal absorption usually utilize the convenient Caco-2 monolayer permeabilities (section ) or the more-relevant in situ and in vivo data (section ), and the data obtained in the in vitro intestinal preparations are used less often. The permeabilities of a small set of diverse drugs in rat intestinal sacks were analyzed using the linear regression analysis and neural networks with the retention factors on the IAM columns (section ), molecular volume, and other physicochemical properties as the descriptors.( 1879) Human jejunum permeabilities were correlated with the polar surface area of a small set of permeants.( 1127) 3.6. Tissues In SBSP, tissues are viewed as spatially arranged collections of similar cells that contain some extracellular fluid. Historically, the first studies were performed with the tissues containing natural pigments changing colors in response to the internal pH. Examples include the pieces of mantle edge of the hermaphroditic nudibranch Chromodoris zebra,( 1584) and the testis filaments of a sea cucumber.( 1583) The testis and intestines, after being expulsed in a defense act or harvested, regenerate upon the return of the animal to sea water. The importance of the complete dose-response curve in the tissue transport studies, as opposed to the measurements at a single concentration, for the characterization of relative transport rates of individual acids was noted.( 1584) Permeabilities of acids and bases were measured using the flower petals containing pH-sensitive pigments.( 1585) With the availability of radiolabeled chemicals, tracking the distribution at the levels of tissues, organs, or organisms was made possible by autoradiography. 1596–1604 The first attempts at monitoring the distribution in entire animals were made using natural isotopes.( 1880) Further developments of autoradiography at cellular level are described in section . Tissues have been used in two forms: slices for the uptake and metabolism studies 1881,1882 and tissue homogenates for equilibrium binding studies.( 1883) The most frequently used tissue slices are the fresh or vitrified precision-cut slices from the rat liver, which were used to study metabolism of chemicals. 1881,1882 Metabolism in these systems differs from that in hepatocytes, because of the diffusion limitations in the former system.( 1884) This phenomenon could provide a basis for the analysis of the distribution process. The data from both slices and hepatocytes were used for prediction of in vivo metabolic clearance.( 1519) Brain slices were used to analyze metabolism( 1885) and the mechanisms of learning and addiction. 1886–1889Tissue homogenates were first used by anesthesiologists who wanted to understand the partitioning of anesthetic gases between blood and tissues, especially the brain.( 1883) The technique was also used in occupational toxicology to assess the exposure risks of industrial chemicals( 1890) and in environmental toxicology to understand disposition of chemicals in fish.( 1891) Depending on the structures and properties of chemicals, the transport rates into tissues vary over a broad range. For quickly transported chemicals, the distribution is characterized by the tissue/plasma or tissue/blood partition coefficients, which are measured using tissue and blood samples. For slower transport, which is observed most frequently in perfused organs, more-sophisticated, nonequilibrium descriptions are required. In the following, we will focus on equilibrium binding to tissue constituents. For compounds quickly transported through membranes, tissue/plasma partition coefficients ( PTP) are important characteristics, which are defined as the ratio of the average concentration of a chemical in the respective tissue and the concentration in plasma. The PTP values represent pseudo-equilibrium inputs in physiologically-based pharmacokinetics (PBPK) models of quickly transported chemicals. The relation between PTP and elementary processes is given by eq 57 or eq 58 in section . A sum of the PTP values for all tissues in the body, each weighted by the tissue volume, is equal to the volume of distribution.( 1892) Therefore, the product of the PTP value and the volume of the tissue can be viewed as the distribution volume of the tissue. Historically, the PTP values were first measured( 1883) and utilized in QSAR studies for simple chemicals such as rare and other gases (H 2, N 2, CO, CO 2, NO),( 1893) gaseous narcotics, alkanes,( 1894) and alcohols. These data correlated well with properties of the chemicals, 1895–1897 especially using a model-based nonlinear function of lipophilicity expressed as a surrogate partition coefficient (section ).( 1898) Several approaches were developed for modeling the data for more-diverse compounds. The solvatochromic equation, which describes solvation-related phenomena as a linear combination of solute excess molar refractivity, dipolarity/polarizability, hydrogen-bond donor and acceptor abilities, and McGowan characteristic volume, was frequently applied to PTP or similar parameters. 1119,1121,1124,1125,1846,1899–1904 Generally, the correlations were worse than those for one-phase( 1905) or two-phase systems, 1906,1907 for which the solvatochromic equation was originally devised. In addition to more-pronounced data variability in biological samples, there are two other potential sources of error. First, protein binding for diverse sets cannot be described as a structure-nonspecific process, using the same approach as for solvation and partitioning. Second, the equation that is used applies to one- or two- phase systems and cannot be used for tissues that contain multiple phases. The correlation equation should contain similar terms as eq 57 or eq 58 in section , and the solvatochromic equation should be applied to each phase separately. These changes would make the correlation equation more complex and nonlinear in optimized coefficients, but the number of independent variables would not increase. The situation is even more complicated for the tissue/blood partition coefficients, which were also frequently modeled using the solvatochromic equation. 1901,1902 As compared with plasma, the blood contains additional phases in the corpuscles. A series of publications advocated the use of the vegetable oil/water partition coefficients in modeling the tissue distribution. 665,1908,1909 Vegetable oil, although composed mainly of triglycerides, contains other lipid species, varying by the source and preparation conditions, which may significantly affect the partitioning process. A better-defined triglyceride phase, such as triolein 192,562,563 or trioctanoylglycerol (see section ) would help standardize the results and provide a good surrogate for adipose tissue. The partitioning to phospholipids was treated as if they were composed of 30% vegetable oil (triglycerides) and 70% water. This combination does not seem to create a good surrogate system, because the fatty acid chains differ in structure from triglycerides and water is not a good surrogate for the headgroup region, despite its high water content (see section ). The estimates of the accumulation in neutral phospholipids, assuming the 30:70 lipid/water composition, were also used in otherwise-sound models for tissue partitioning of basic drugs.( 1910) Later, in a study of the volume of distribution,( 559) vegetable oil was replaced by 1-octanol for nonadipose tissues. Interactions with proteins were treated by using the measured fraction unbound in plasma,( 1911) and assuming that a similar protein binding is taking place in tissues. Albumin, although present in tissues, does not represent their major component (sections and ). Consequently, this assumption could affect the descriptive and predictive abilities of the models. The used equations are mostly conceptual and similar in form to eqs 57 and 58. A major drawback is the use of the implausible assumption regarding the proportionality or even equal magnitudes of the partition coefficients in surrogate solvents and biological systems. An application of the Collander equation to relate the partitioning in surrogate and biological phases via adjustable coefficients usually provides better results. Conceptual equations reminiscent of eqs 57 and 58 were also used for the correlation of the tissue/blood partition coefficients of both neutral( 1912) and ionizable compounds. 1913–1915 While the tissues were decomposed into lipids, proteins, and water, the blood was considered as a homogeneous phase, thus neglecting the partitioning into the membranes of blood corpuscles and lipoproteins, as well as plasma protein binding. This omission could affect the results, especially for compounds with higher affinities to blood components. Extrathermodynamic relationships were used to describe the partition coefficients and proteins association constants. In the latter case, this description could be problematic for more-diverse compounds. The approach was further refined by inclusion of neutral lipids, as well as neutral and acidic phospholipids, and was applied to tissue/plasma partition coefficients.( 543) The experimental values of fraction unbound in plasma were used to describe protein binding in plasma and also in interstitial fluid, which has a similar protein composition. The binding to intracellular proteins was correlated with experimental partition coefficients for phospholipids,( 1916) which limits the application to the studied chemicals. All published approaches to the correlation of tissue partitioning of chemicals with their structures and properties treat the interactions with proteins as structure-nonspecific processes, which are related to the properties of entire molecules. This assumption limits the applicability of the techniques to sets of homologues. For larger and more-diverse sets of chemicals, protein binding must be treated as a structure-specific process, using 3D-QSAR techniques, as we recently demonstrated for the binding to the proteins of extracellular matrix (section ). 1307,13093.7. Organs When transport rates are comparable with those of metabolism or excretion, the distribution models become more complex and require kinetic data for characterization. These experiments are usually performed in perfused organs, because they have longer lifetimes than isolated tissues. Perfused organs such as frog legs;( 1917) frog, cat, and dog hearts; 1918–1921 dog livers, 1922–1927 and rabbit and pig ears, 1928,1929 have been used to study the effects of physiological molecules since the beginning of the last century. Currently, the organ preparations are used in situations when simpler models provide a very limited picture of the disposition events. Typical examples are the skin and intestine, which can be mounted as sheet barriers in the diffusion chambers (sections and ), but the results are not completely representative of the situation in vivo. In both cases, more rigorous results are obtained with perfused preparations, which better imitate the actual transport routes. On the other hand, the cost and labor of these preparations limits the number of available data. Organ preparations can be used in either ex vivo or in situ conditions, the difference being the disconnection from the natural bloodstream of the test animal in the former case. The ex vivo setup allows a better control of the vascular perfusion but often results in a shorter lifetime of the preparation. The in situ preparation preserves a natural environment of the studied organ and maintains many in vivo features, while providing controlled inputs and outputs, which allow application of precise analytical techniques and the use of the mass balance in the description. Intestinal absorption rate is one of the key parameters of orally dosed drugs. The absorption route from the intestinal content through the epithelial cells and connective tissue into the capillaries inside the villi cannot be precisely duplicated in in vitro intestine-based systems, such as the brush-border membrane vesicles (part ), isolated enterocytes, everted rings of the entire intestine (section ), and the entire intestinal wall or mucosal sheets (section ). Instead, it requires the use of comparatively complex ex vivo and in situ preparations, or in vivo conditions. The ex vivo, isolated intestine with adjacent vasculature, perfused both luminally and vascularly though the superior mesenteric artery and the portal vein, provides a better control of the experimental conditions, although the viability is shorter. 1930–1934 A preparation of the ex vivo perfused rat intestine and liver was described.( 1935) Different models were developed and evaluated for the description of absorption from the perfused intestine.( 1936) A structure−permeability relationship was described for steroid absorption in perfused rabbit ileum.( 1937) A similar controversy between simple in vitro systems and reality applies to transdermal absorption, because the bilayer stacks of skin phospholipids (section ), cell monolayers (section ), and sheet-like skin preparations (section ) do not capture all details of the in vivo situation. To reproduce the absorption route from the surface of stratum corneum into the capillaries, several perfused skin preparations were described, 1938–1942 of which the perfused skin flap technique for pigs, 1943–1946 weaning pigs,( 1947) or horses( 1948) received the most attention. Small regions of skin, sometimes with tumors,( 1947) which are primarily perfused by a single superficial artery, are incised and formed into a tubelike shape around the artery. Later, the artery is cannulated, the flap is removed from the body and placed into a tissue chamber, where the venous perfusate is collected and analyzed. Specialized models for the description of the recorded data have been developed for single chemicals( 1949) and mixtures.( 1950) In situ preparations, where the artificially perfused organ remains attached to the systemic circulation, include the rat intestine, 1864,1951–1959 kidney,( 1960) and liver. 1961–1969 The small intestine can be perfused in situ in humans, using stoppers to define a perfused segment.( 1970) The externalized rat kidney was used for a direct monitoring of renal clearance of fluorescent compounds.( 1971) The methods utilizing perfused organs have been used extensively in the drug-development process. 1636,1972–1977Intestinal permeability in rats was studied using in situ techniques for small sets of compounds,( 1978) and a sigmoidal relationship on lipophilicity, containing a molecular weight term, was formulated.( 1979) Quantitative studies of distribution in terms of structure and properties of compounds are limited to small sets of homologous chemicals, 1967,1980–1986 because of the cost and labor associated with the use of perfusion techniques. Because of the importance of the process in the metabolism of chemicals, distribution in the liver has been modeled using advanced techniques, which take into account spatial differences of the compound’s and enzymes’ concentrations in individual organ parts. 1638,1961,1962,1964,1987–1996 Interestingly, the heterogeneity at the cellular level, which is caused by different accumulation in the membranes and aqueous phases, was not considered, although the importance of the binding to endoplasmic reticulum for the overall metabolism rate has been recognized.( 1472) 3.8. Organisms Historically, the first technique that allowed the distribution of chemicals at the level of organisms to be tracked was autoradiography, 1596–1604 which was initially performed using natural isotopes.( 1880) In addition to autoradiography, 1603,1997–1999 the distribution of chemicals in entire organisms can be directly and noninvasively monitored by magnetic resonance imaging based on 7Li,( 2000) 13C,( 2001) 19F,( 2002) and proton nuclei, 2003–2006 positron emission tomography,( 2007) and specific imaging techniques for fluorescent compounds in nude mice. 2008–2012Indirect techniques, based on the extraction of chemicals from the body fluid and tissue samples, have been used much more extensively, especially in the areas of drug development and environmental toxicology. The resulting data have been processed by pharmacokinetic modeling methods. A brief overview of the development of these approaches( 2013) is provided below. Pseudo-equilibrium accumulation of a chemical in an aquatic organism 2014,2015 is usually characterized by the bioconcentration factor (BCF),( 2016) given as the ratio of the chemical concentrations in an organism and the surrounding aqueous phase. BCFs were frequently and successfully correlated with the reference partition coefficients, 1891,2017–2027 aqueous solubilities, 2017,2021,2023 and acidities.( 2028) For very lipophilic compounds, nonlinearities were observed in the relationships with lipophilicities 2029–2033 and reactivities.( 2034) Nonlinear dependences on the reference partition coefficients were also observed for the uptake and release rate constants. 60,2035,2036 Clearances were inversely related to the reference partition coefficients. 2024,2035 For reactive chemicals, the toxicities were conceptually correlated with the reference partition coefficients and reactivies.( 1478) Because the chemicals studied in environmental toxicology usually have much simpler structures than drugs, research in this area contributed several interesting ideas to SBSP. The disposition of chemicals in animals and humans has been studied mainly in relation to chemotherapy. The history of classical pharmacokinetics is summarized in Figure , and more details are given in the following text. The notion of a relationship between an anesthetic effect and the concentration of ether in the brain dates back to the middle of the 19th century.( 2037) The kinetics of disposition of chemicals has been studied quantitatively in animals and humans and documented since the second decade of the 20th century. 2038–2048 During the same time period, the Michaelis−Menten description of saturable kinetics( 2049) and the one-compartment model( 2050) were developed. The 1930s brought the two-compartment models( 2051) and the key concepts of the mean residence time,( 2052) clearance, 2053,2054 and the volume of distribution.( 2055) The one-compartment model and two-compartment models were later applied to the disposition kinetics of drugs 2056–2058 and radioactive substances 2059–2064 in the organisms. Since the 1950s, kinetic models of multiple dosage regimens 2056,2065–2068 have had a broad impact on how therapeutic interventions are performed. The research area was called pharmacokinetics( 2058) and developed( 2013) into a key pharmacy discipline( 75) that is taught in all pharmacy programs around the world. More-detailed descriptions of the fates of chemicals in the body, the PBPK models, were spearheaded by physiologists, who routinely monitored the blood flows and the concentrations of oxygen, carbon dioxide, and anesthetic gases in individual organs.( 2069) The compartments in PBPK models are individual organs or their specified sets, in contrast to the models of classical pharmacokinetics, which utilize loosely defined compartments, the number and function of which is dictated by the fit of the model to the data. Principles of PBPK modeling were formulated in the 1960s. 2070–2073The pseudo-equilibrium models of SBSP in section provide useful descriptions for clearance, the elimination rate, and the volume of distribution. The QSAR models for individual pharmacokinetic parameters are summarized and compared with the pseudo-equilibrium models in sections − . 4. Disposition Function in QSAR The fates and effects of chemicals in biosystems are governed by a myriad of interactions. As outlined in section , some interactions can be described using global, conformation-averaged properties of the molecules, and the descriptions of other interactions require the knowledge of precise conformations. SBSP attempts to sort the interactions, lump together kinetically similar processes, identify the rate- and extent-determining steps, and systematically formulate model-based QSARs and their time-dependent forms (QSTARs, where T represents time), which we call the SBSP models. The SBSP models serve two main purposes: prediction of distribution and/or bioactivity of untested chemicals in biosystems, and elucidation of the mechanism of action. In the drug development process, the predictions are used in lead finding and mainly in the optimization of lead structures by derivatization. Regardless of the size and complexity of the biosystem, formulation of a model-based QSTAR for an effect of chemicals requires appropriate descriptions for three steps of action of a chemical as shown in Figure : (i) disposition in the receptor surroundings, (ii) the interaction with receptor, and (iii) transformation of the receptor modification into biological response. In most cases, the loss of the chemical caused by its interaction with the receptors is negligibly small. Consequently, step (i) can be described separately. Linearity of the processes is an important concept that greatly simplifies the description of the fates of chemicals in biosystems. Linear (or, in other words, first-order) processes have the rate and/or extent proportional to the concentration of the chemical (denoted as [C]). Linearity is a plausible approximation, because the concentrations of chemicals in living biosystems are generally low. If the processes that control the distribution (such as transport, membrane accumulation, binding to body constituents, and elimination) are linear, the time course of [C] in the receptor surroundings can be expressed symbolically as
where c0 is the initial concentration, proportional to the dose, of the chemical in the entry compartment, and A(pp, t) is the disposition function, with the time of exposure ( t) and physicochemical properties (pp) of both the chemicals and the biosystem as variables.( 143) The term “disposition function” has been used in classical pharmacokinetics,( 2074) in a sense similar to that illustrated by eq 6, but without the property variables. The actual form of the disposition function is dependent on the complexity of the studied biosystem and on the properties of analyzed chemicals. Using eq 6, the kinetics of the interaction between the receptor and a chemical can be described for various mechanisms of this step. 54,143,2075 For the simplest mechanism of the chemical−receptor interaction that (i) consists of a fast and reversible 1:1 interaction, (ii) is characterized by the association constant K, and (iii) leads to a primary effect that (iv) is an immediate consequence of the receptor modification and (v) is proportional to the fraction of the occupied receptors,( 143) the effects are related to K and the disposition function as follows:
The predefined relative intensity of the chemical’s effect ( X) is a number between 0 and 1 (frequently, X = 0.5). The experimentally determined isoeffective concentration cX, eliciting the fraction X of the maximum effect, serves as a measure of the potency of individual chemicals. Equation 7 can be easily converted to a model-based QSTAR equation, if the chemical−receptor association constant and the disposition function are expressed via the structures and physicochemical properties of the chemicals. The bioactivity directly mirrors the disposition of a set of chemicals in the receptor surroundings, if K in eq 7 is invariant for all studied chemicals. In the rest of this communication, when comparing the derived form of the disposition function with biological effect data, as opposed to the disposition data, we will assume that these conditions are met, and that bioactivity and disposition are proportional at each moment. The relationships correlating potency with the chemical−receptor association constant K and the disposition function A(pp i, t) for more-complicated mechanisms of the chemical−receptor interaction can be found elsewhere. 54,143,2075The form of the disposition function is heavily affected by the size and complexity of the biosystem, as well as by the kinetics and types of the interactions of chemicals with the components of the biosystem. The construction of the disposition functions for various scenarios is described in the following parts, starting with simple cases, and gradually progressing to more-complex situations. 5. Numerical Simulations for Multi-Bilayer Systems Some of the simplest model systems that imitate small biosystems are represented by a series of the alternating aqueous phases and the bilayer cores, in which the chemicals, which do not interact with the headgroups, are transported passively. Movement of molecules in a multimembrane system can be envisaged as a series of partitioning steps through alternating aqueous and lipid compartments,( 2076) as shown in Figure . Protein binding and elimination that influence the disposition of chemicals are also depicted. The majority of experimental systems used for the monitoring of the interactions with chemicals are actually multimembrane systems: some subcellular organelles; suspensions of cells from organisms and some microorganisms, such as gram-negative bacteria, yeasts, and fungi (section ); cell monolayers (section ); and various physiological sheets that form the walls of the blood capillaries, urinary bladder, gallbladder, stomach, and the intestine (section ). The approaches to the description of the disposition of chemicals via the disposition function, defined by eq 6, can be classified as numerical simulations and explicit descriptions, and are summarized in sections and , respectively. Note that all the results presented below are valid for chemical disposition following a single dose, except those in section , which is devoted to continuous dosing. The situations after repeated doses would be completely different and could be described by applying the principle of superposition, using the results for the single dose, as known in classical pharmacokinetics. 2056,2065–2068To understand the contributions of the individual processes to chemical disposition quantitatively, it is advantageous to examine them separately. 5.1. Pure Transport From the viewpoint of chemical distribution, biosystems are generally open, because of irreversible elimination processes. Under certain conditions, however, the concept of a closed biosystem is useful. Providing that the reversible processes (transport and protein binding) are much faster than irreversible elimination, a closed model properly represents the in vivo situation for the time interval until elimination becomes observable. Among the processes summarized in Figure , transport has long been studied intensively as the primary cause of nonlinearities in the concentration−lipophilicity profiles of chemicals.( 2077) Hansch and Fujita( 56) assumed, with remarkable intuition, that the probability of the occurrence of molecules inside a biosystem after a predetermined time interval could follow a Gaussian distribution, with respect to lipophilicity, expressed as the logarithm of the 1-octanol/water partition coefficient, P. The resulting parabolic dependence of the intracellular concentration or bioactivity on lipophilicity,
proved to be rather robust and usable. Here, c is the actual concentration in an intracellular compartment, cX is the isoeffective concentration that characterizes the bioactivity, and A, B, and C are adjustable coefficients that are optimized by linear regression analysis. Coefficient A is usually negative, so the function has a concave shape, with the maximum marking the optimum lipophilicity. Hansch and Clayton( 6) compiled almost 200 parabolic dependencies of bioactivity on lipophilicity. A searchable QSAR database that includes 553 concave dependencies is available.( 1) The first mathematical treatment of the distribution versus properties problem was performed by Penniston et al.( 58) Because the experimentally verified dependencies of the transfer rate parameters li and lo (Figure in section ) on the partition coefficient (eqs 3 in section ) were not available at that time, the authors used the assumption lilo = 1. Their system consisted of the aqueous and lipid compartments with identical volumes and interfacial areas. The intuitively assumed fast equilibration of chemicals in the compartment volumes proved to be one of the most important simplifying steps in the SBSP modeling. This assumption is justified by the small distances ( x), which the molecules of chemicals cross in the subcellular compartments. The typical distance, considering the average cell size, is smaller than δ = 10 −5 m and reaches the nanometer dimensions in the bilayer headgroup and core regions (see section ). The diffusion coefficients of the low-molecular-weight compounds are of the order of 10 −9 m 2/s in most liquids.( 2078) Fluorescence recovery after photobleaching and other experiments showed that small molecules and macromolecules up to 100 kDa exhibited diffusion coefficients in the cytoplasm that are only 4−5 times smaller than those in saline.( 2079) The time t that the molecules require to cross the small subcellular aqueous compartments then can be estimated( 2080) as t = δ 2/(2 D), i.e., t = 0.2 s, under these conditions. The equilibration times inside the bilayer regions are shorter by several orders of magnitude. The equilibration process in the miniature compartments is practically instantaneous on the timescale of most experiments. This fact warrants the representation of a biosystem by a set of alternating aqueous phases and membranes, with the molecules of chemicals diffusing quickly in bulk water and in the bilayer headgroup and core regions. The three phases have much lower density than subregions 2 and 3 in the bilayers (section , Figure ). The stochastic kinetic effects of the crowded environment( 2081) seem to be far more important for larger macromolecules than for chemicals. The differential equations for the fast intracompartmental equilibration, similar to eqs 35 in section , were integrated numerically.( 58) The concave concentration−lipophilicity dependencies, consisting of two linear parts connected by a curve, observed after a constant distribution period, were interpreted as a support for the already known parabola.( 56) McFarland( 59) used a probabilistic approach to describe the movement of chemicals in the Penniston model.( 58) For constant transport periods, symmetrical concave concentration−lipophilicity relationships with linear ascending and descending parts were obtained. The relationships were not suitable for fitting to real data, because the maximum concentration was always attained by the chemical with equal affinities for both core and aqueous phases, i.e., having the core/water partition coefficient P = 1. Kubinyi( 64) re-examined the McFarland’s approach and considered the substantial differences in the volumes of aqueous and lipid phases in biosystems. The resulting concave concentration−lipophilicity profiles for aqueous phases were also symmetrical, but they were not restricted in the peak position.( 59) Assuming that the receptor binding is also lipophilicity-dependent and making all the coefficients A, B, C, and D freely adjustable, Kubinyi obtained the asymmetrical bilinear equation:
Equation 9 has been shown to fit the concentration−lipophilicity profiles generated by numerical simulations of chemical transport with the experimentally measured dependencies of the transfer rate parameters li and lo on lipophilicity, as described by eqs 3 in section . If a reference partition coefficient is used instead of the model partition coefficient, the Collander exponent β must be used with each P term in eq 9, according to eq 1 in section . The magnitude of β determines the sharpness of the curved stretch that joins the two linear parts in the concentration−lipophilicity plots( 8) described by eq 9. The ability to describe the simulation results gives eq 9 the stature of a model-based description of the transport, although its original derivation started from rather intuitive concepts. As will be mentioned in sections and , eq 9 also describes the equilibrium distribution. This versatility makes the bilinear equation a valuable tool for describing the relationships between bioactivity and lipophilicity, which was used by many researchers.( 30) A linearized biexponential expression that also describes the bilinear dependencies was recently introduced.( 2082) Dearden( 65) further developed the Penniston model( 58) and drew attention to the importance of the exposure time in the transport simulations and, generally, QSAR. The most-significant results include the behavior of the optimum lipophilicity (log P of the compound with the maximum concentration in the concentration−lipophilicity profiles) in dependence on the exposure time and on the number of the partitioning steps, as well as the dependence of the time to maximal response on lipophilicity. 28,46Van de Waterbeemd( 2083) described the transport in a three-compartment model with equal compartment volumes analytically. He extended the results for the initial distribution period to multiple bilayer systems and showed that the concentrations are proportional to the product of the transfer rate parameters of involved steps. 2084,2085Some attempts 67,2086 at description of the concentration−lipophilicity profiles were based on the use of a reduced time τ (where τ = lot, with lo being the transfer rate parameter for the membrane−water direction and t being time; see Figure in section ). The reduced time τ is proportional to the real time t only when lo is a constant, which holds for P ![[double less-than sign]](/corehtml/pmc/pmcents/x226A.gif) 1/B t, as follows from eqs 3 in section . Thus, the presented results are valid just for hydrophilic compounds. We examined the complete time course of the chemical transport in closed systems with identical lipid phases. 2087–2089 For this purpose, the set of linear differential equations corresponding to the reduced equations described by eqs 47 (section ) was integrated numerically. The transfer rate parameters used are dependent on the partition coefficient as given by eqs 3 with the experimentally determined values( 631) of the coefficients A t and B t. The results are illustrated in Figure , using the kinetics of the passive chemical transport in a 10-compartment system, composed of five aqueous phases, interspersed with five bilayer cores. Before the start of the simulation, the compounds are present only in the first aqueous compartment. The concentrations are plotted as a function of the reference partition coefficient ( P) and the exposure time ( t). The kinetics of distribution is dependent on the position and nature of the sampling compartment. In the third compartment, representing the aqueous phase immediately following the first bilayer core, the concentrations of the chemicals initially increase and later decrease to their equilibrium magnitudes, resulting, for a certain time period, in a remarkable accumulation of the chemicals above their equilibrium levels. This phenomenon was observed in the first half of the compartments (i.e., in compartments 2−5 in this case), regardless of their aqueous or lipid nature. The decline of the concentration caused by pure transport, taking the molecules to deeper compartments (no elimination was considered in this case), can be important in biosystems that contain many compartments, such as tissues, organs, and organisms (sections − , respectively). The situation in the second half of the compartments, which are more distant from the site of administration, is completely different, as illustrated by the transport kinetics in the sixth compartment, which is shown in Figure . The concentrations asymptotically approach the equilibrium magnitudes, and no accumulation above the equilibrium levels is observed. The concentration−lipophilicity−time profiles, similar to those shown in Figures and , can be described by semiempirical equations. We call these equations semiempirical, because they currently do not have an obvious mechanistic underpinning. In contrast to empirical equations, our equations are nonlinear in optimized coefficients and provide an almost-perfect fit to simulated data. Polynomials, which are typically used in empirical equations, do not perform well in SBSP models, because they are not able to properly fit planar areas in the surface plots as seen in Figures and , and linear stretches in two-dimensional plots (see Figures 17−25, shown in this section through section ). The semiempirical equations represent an important step enabling a straightforward use of the simulations in quantitative characterization of biological data. The excellent fits to simulation data help select a proper form of the empirical equation for fitting the experimental data. The simulated data subsets with constant P values in Figures and were first fitted by eq 11 (see below), and the optimized coefficients A, B, C, and D were subsequently fitted as a function of P and the exposure time t. This approach resulted in the following empirical equation:
The overall fit to the data simulated for the third compartment of a 10-compartment system, shown in Figure , was characterized by the following coefficients. The exponents β 1 and β 3 were linearly dependent on log P, with slopes of (5.604 ± 4.3101) × 10 −2 and (8.607 ± 2.982) × 10 −2, respectively, and the same intercept (1.681 ± 0.0129). The values of the other adjustable coefficients, as optimized by nonlinear regression analysis,( 443) are summarized in Table 5. All coefficients are statistically significant. The overall fit is very satisfactory, as shown in Figure and by the values of statistical indices: number of experimental points, n = 1212; the squared correlation coefficient (equal to the percentage of explained variance), r2 = 0.992; the standard error of the fit, s = 0.063; and the Fisher test value, F = 18 210. These statistical indices will be used throughout the paper, to characterize the quality of a fit. For comparison, the best typical empirical equation (a polynomial with cross terms) with the same number of optimized coefficients as eq 10, achieved values of r2 = 0.743, s = 0.375, and F = 389.2. | Table 5Optimized Values of Adjustable Coefficients in eq 10 Describing the Transport Kinetics in Compartments 3 and 6, as shown in Figures and , Respectively |
In the sixth compartment, no accumulation above the equilibrium concentrations was observed, as seen in Figure . The data were fitted with the second term in eq 10 (see Table 5). The fit was also very satisfactory, as illustrated by Figure and the values of the statistical indices ( n = 780, r2 = 0.996, s = 0.029, and F = 49 147). The best polynomial with the same number of optimized coefficients achieved values of r2 = 0.815, s = 0.196, and F = 1731. The compounds with different lipophilicities reach the equilibrium at different times. Figures and show that the total period of distribution can be subdivided into three parts: the nonequilibrium period (log t < 2.5 in this case; no compounds have reached the equilibrium), the mixed period (2.5 < log t < 5), and the equilibrium period (log t > 5; all compounds have attained equilibrium). The dependencies of the concentrations or effects on lipophilicity profiles are most frequently measured after a constant exposure time for all chemicals. The profiles are different for individual distribution periods and will be described in detail below. To facilitate the discussion, an empirical equation describing all simulated profiles( 2087) is introduced at this point:
Equation 11 describes the multilinear dependencies with up to four linear parts and can be considered an extended form of the bilinear equation described by eq 9. The model partition coefficient was substituted by a reference partition coefficient using the Collander equation 1 in section . The first term on the right side is written in a way that emphasizes the substitution of the partition coefficients. The connection between the adjustable coefficients A, B, and C and the shape of the corresponding curves is outlined in Figure . The coefficients A and Bi determine the slopes of linear parts, as shown in the upper panel of Figure . The coefficients Ci are associated with the position of the curvatures. The Collander exponent β modulates the sharpness of the curvatures. The upper curve, which is characteristic for the mixed period of distribution, is described using the integer values of the coefficients A and Bi, which are selected from available combinations, summarized in Table 6. The lower curve is typical for open systems. The leftmost and rightmost linear parts still maintain the integer values that are specified in Table 6. However, the slopes of two linear parts that characterize the convex dip are no longer confined to the integer values and, consequently, the coefficients B1 and B2 are freely adjustable. The adjustable coefficients of eq 11, for the dependencies shown in Figure and other cases discussed in sections − , were fitted to the experimental data using nonlinear regression analysis, deploying the Fletcher and Powell algorithm.( 2090) | Table 6Integer Slopes of Linear Parts of the Concentration−Lipophilicity Profiles (Figures −) and the Values of the Coefficients A, B1, B2, and B3 in eq 11 for the ith Compartment during Individual Distribution Periods |
5.1.1. Nonequilibrium Period The majority of previously published studies by other authors on the description of movement of chemicals through a series of alternating aqueous phases and membranes focus exclusively on the nonequilibrium period of transport when none of the studied chemicals have achieved the partitioning equilibrium. We also simulated this period and compared the results with those obtained in the most advanced studies by Kubinyi. 8,64 The results of the simulations for the nonequilibrium period were practically identical, despite different integration methods. However, the interpretation of the results and their application to experimental data were somewhat different. Kubinyi made all the coefficients in eq 9 freely adjustable, whereas, in our approach, a tighter adherence to the simulation results had been chosen. The resulting concentration−lipophilicity profiles for individual compartments are given in Figure . For the first two compartments, the curves exhibit the zero slope values in some linear parts. Other curves consist of linear ascending and descending parts, connected with a rounded apical part. The curves are symmetrical for the aqueous phases (odd compartment numbers) and asymmetrical for the membranes (even compartment numbers). The slopes of the linear parts (Figures − and A) are integers, which are characteristic of the corresponding compartments (Table 6). 8,2083,2087 This fact significantly promotes elucidation of the action mechanisms of bioactive chemicals, as the shape of the relationship between bioactivity and lipophilicity under the nonequilibrium conditions could indicate the sequential number and the nature of the compartment, where the receptors for the studied bioactivity are localized. This phenomenon( 2091) is illustrated in Figure , showing the dependence of mutagenic activity of N-substituted alkyl amides of 3-(5-nitro-2-furyl)-acrylic acid ( 59 in Chart 5) after 1-h incubation with two strains of Salmonella typhimurium TA100 (rfa + and rfa −) on lipophilicity. The strains differ in regard to the structure of the cell walls. In contrast to the rfa − strain, which has an exposed cell membrane, the cells of the rfa + strain are covered by a lipopolysaccharidic complex that contains the second lipid bilayer, in addition to the cell membrane. The mutagenic effects of nitrofurans are likely caused by unstable metabolites formed in the course of enzymatic reduction of the nitro group.( 2092) Because the compounds are n-alkyl derivatives, it is plausible that they do not differ in regard to the electronic properties of the nitro group and, consequently, in the rate of reduction. The data were fitted by eq 11 with i = 1; the optimized coefficient values are given in the first two rows of Table 7 and the corresponding lines are shown in Figure . The integer values of coefficients A and B1 in eq 11 indicate (cf. Table 6, nonequilibrium period) that the receptors are located in the fourth compartment of the rfa − strain and in the sixth compartment of the rfa + strain (cf. Figure ). The receptor compartment would correspond to the lipophilic DNA-core in both cases, because the cells of the rfa + strain are covered by two membranes, and the rfa − strain only has the cell membrane. | Table 7Optimized Values of the Adjustable Coefficients in eq 11 with i = 1 Describing the Dependencies between Bioactivity and Lipophilicity during the Nonequilibrium Period of Distribution |
Table 7 also contains some other examples of eq 11 with i = 1 and the integer values of the coefficients A and B1 fitted to experimental dependencies of bioactivity 2093,2094 on lipophilicity. The concave dependencies were also fitted quite well by the parabolic equation 8 and the bilinear equation 9 with freely adjustable coefficients A and B (the results not shown). Nevertheless, the adherence of our approach to the results of model simulations brings an additional advantage, especially in the possibility to locate the compartment, where the interactions of the chemicals with the receptor occur. 5.1.2. Equilibrium Period In the closed system without elimination (as illustrated in Figure , if all elimination processes are omitted), the transport of chemicals will continue up to the achievement of the partitioning equilibrium (e.g., for log t > 5 in Figures and ). The bilayer cores of individual membranes may have different solvation properties, and the differences are assumed to be minimal, as discussed in section . Nevertheless, this factor may lead to the multiple partition coefficients Pi (up to a total number M). For ionizable chemicals, differing compositions of the aqueous phases also generate multiple apparent partition coefficients, and this situation will be treated in section . The corresponding expressions for the concentration−lipophilicity profiles can be derived easily from the definition of the partition coefficient ( Pi = cMi/ cW):
where n is the total amount, VM the total volume of the bilayers, and VW the total volume of water. The corresponding expression for the concentrations in the ith membrane is obtained as shown in eq 12, multiplied by Pi. Equation 12 truly describes the partitioning equilibrium and can be used to assess the accuracy of the integration method used for numerical simulations. The concentration−lipophilicity profiles resulting from eq 12 were approximated by the concave, bilinear curves, with the slopes of the linear parts having non-integer values.( 8) If all bilayer cores have identical solvation properties (M = 1), eq 12 corresponds to a special form of the bilinear equation 9 and the multilinear equation described by eq 11. The equilibrium concentration−lipophilicity dependencies are bilinear on the bilogarithmic scale. The slopes in the linear parts are equal to 1 and 0 (left to right − hydrophilic compounds to lipophilic compounds) for the membranes, and 0 and −1 for the aqueous phases. The dependence for the membranes has a similar shape as the red curve in Figure A. This curve, flipped horizontally, defines the shape of the dependence for the aqueous phases. 5.1.3. Mixed Period In this period, the fastest chemicals with the optimum partition coefficients have already reached the equilibrium, while other chemicals, which are either more lipophilic or more hydrophilic, are still diffusing. The concentration−lipophilicity profile in this case is a combination of the equilibrium and nonequilibrium profiles. As can be seen in Figure B, the mixed profiles consist of four linear parts, connected by the curved portions. The characteristic integer slopes of the linear parts are summarized in Table 6. The joining portions are mostly U-shaped. The S-shaped and Z-shaped parts are observed only for the first half of the compartments, where the chemicals temporarily accumulate in the concentrations exceeding their equilibrium values, as seen in Figure . Equation 11 with i ≤ 3 is suitable only for the description of the curves that contain the U-shaped connecting parts. A description of the curves with the S-shaped and Z-shaped parts would require the addition of more nonlinear terms to eq 11. The experimental examples of the complete concentration−lipophilicity profiles typical for the mixed period of distribution are rather rare, because they are only observed if the tested chemicals cover a broad range of the lipophilicity scale. Nevertheless, the fragments consisting of more than two linear parts connected by curved portions were discernible in some published concave profiles. The data usually did not cover the sufficient range of lipophilicity, and only three linear parts were seen. Such profiles were observed among those for the growth inhibition of 17 fungal strains and 14 bacterial strains by n-alkylamines with 4−18 carbons.( 2093) The resulting descriptions by the bilinear equation described by eq 9 and the trilinear equation described by eq 11 with i = 2 are summarized in Tables 8 and 9, respectively. The statistical indices in Table 8 deviate slightly from the published values,( 27) because of the more-precise log P values( 544) used in the present study. The differences in statistical indices between eqs 9 and 11 enable an assessment of the degree of trilinearity present in the experimental data. Both equations provide almost perfect fits. The data exhibit small but systematic deviations from the bilinear equation described by eq 9, as illustrated in Figure for the growth inhibition of Ctenomyces mentagrophytes.( 2093) The fact that this behavior occurred in 12 of the 17 fungal strains, as compared to only 1 of the 14 bacterial strains, speaks in favor of the hypothesis about the mixed period of distribution. In contrast to bacteria, fungi contain intracellular membranes, in addition to the cell membrane. Therefore, to achieve the partitioning equilibrium, the chemicals need more time in fungi than in bacteria. The mixed period of distribution has been observed in other experimental datasets. 2087,2088 | Table 8Optimized Values of the Adjustable Coefficients in eq 9 Describing the Relationships between Antimicrobial Activity(2093) and Lipophilicity of n-Alkyl Amines (C4H9NH2−C18H37NH2) for the Nonequilibrium Period of Distribution(16)a |
| Table 9Optimized Values of the Adjustable Coefficients in eq 11 with i = 2, A = 2, B1= −1, B2= −3, Describing the Dependencies of Antimicrobial Activity of n-Alkyl Amines (C4H9NH2−C18H37NH2) on Lipophilicity during Mixed Period of Distribution( (more ...) |
For the analyzed data of a homologous series of chemicals, the differences between bilinear and trilinear fits are small and play a minor role in the quality of the fit. They are more important from the mechanistic viewpoint, because they confirm the existence of the mixed period of distribution. As illustrated in the next part, dramatic changes have been observed in the simulated concentration−lipophilicity profiles during the mixed distribution period, if elimination has a role. 5.2. Transport with Elimination In principle, biosystems must be treated as open systems, because of irreversible elimination. In multicellular organisms, elimination of chemicals includes metabolism and excretion, via specialized organs, into urine, sweat, expired air, or bile and feces. In the cell suspensions and tissues, elimination is represented by catalyzed or spontaneous metabolic reactions. Efflux, which could be viewed as a cell-level excretion process, is treated in SBSP as a part of carrier-mediated transport. Consequently, in this section, which involves cellular systems, elimination is synonymous with metabolism. Elimination is one of the key processes in the disposition of chemicals, and it has a pronounced effect on the concentration−lipophilicity profiles. The phenomenon was first described by Dearden and Townend,( 65) who studied the Penniston transport model( 58) after prolonged periods of distribution. For a constant exposure time, they observed a distortion of the initially concave concentration-lipophilicity profiles in the region of optimum lipophilicity. The maximum of the initially bilinear curve decreased at longer times, forming a double-peaked dependence, as shown in the blue curve in Figure C. Similar dependencies were also observed in other simulations.( 706) The minimum between the two maxima arises from the following situation. The compounds with optimum lipophilicity are transported at the highest rates and are present in the highest concentrations in the internal compartments where elimination occurs. Because the rate of elimination is proportional to the concentration, these chemicals are eliminated most rapidly, and their concentrations decrease faster than those of slowly transported compounds. To compare behaviors of the open and corresponding closed models,( 2089) the concentration−lipophilicity profiles are shown for a four-compartment closed system (see Figure A), as well as for the situations when the system is made open via the introduction of hydrolysis proceeding in both extracellular and intracellular aqueous phases (see Figure B) and metabolism, which is confined to the intracellular space (see Figure C). The dependencies in the closed and open systems have similar shapes in the initial period of distribution, corresponding to the nonequilibrium period in the closed system. The curves are bilinear with the integer values of the slopes, which are equal to those given in Table 6 for the nonequilibrium period. The influence of elimination becomes noticeable at the time when the mixed period of distribution starts in the corresponding closed system (the blue curve in Figure A). The lipophilicity of the most rapidly eliminated chemicals, determining the position of the deformation in the concentration−lipophilicity profile, is dependent on the compartment where elimination occurs. Elimination proceeding in both aqueous compartments, e.g., hydrolysis, affects preferentially hydrophilic compounds, and the concentration−lipophilicity profiles drop in the region of low lipophilicity (Figure B, blue and red curves). They remain more or less bilinear, with the slopes in the distorted part having non-integer values, which are much higher than the original integer values of the nonequilibrium profiles (Table 6). Equation 9, with the optimized coefficient A and the coefficient B, which is dependent on A to maintain the integer slope of the rightmost part of the profile (cf. Figure ), describes each with sufficient precision. The situation is completely different when only intracellular elimination is considered. The maxima of the curves are distorted, because the chemicals with the optimum lipophilicity are eliminated at the fastest pace. The concentration−lipophilicity profiles have two maxima separated by a minimum (Figure C) and can be described by eq 11 with i = 3. Coefficients B1 and B2 are optimized, and coefficient B3 is dependent on B1 and B2 to maintain the integer slope value of the rightmost linear part (the association between the slope values and the magnitudes of coefficients A and B is depicted in Figure ). During the initial period of distribution, the overall shapes of the curves are similar, regardless of elimination. However, the curves are shifted along the log c-axis, depending on the rate and location of the elimination process. The simulation results were accurately described by the following modification of eq 11:
where k is the elimination rate parameter. The results for the fourth compartment of a four-compartment system are presented in Figure . Evidently, elimination becomes important when the value of the rate parameter k exceeds a certain limit that decreases with prolongation of the time of distribution. The dependence of the intracellular concentration on the elimination rate parameter k is bilinear, with slopes of 0 and −1. For a real situation, the rate parameter k can be substituted by the reactivity of the chemical in a reference reaction mimicking the metabolic elimination process, or by other experimental or computational reactivity parameters, according to eq 5 in section . 6. Explicit Descriptions for Multi-Bilayer Systems Some biologically relevant systems are sufficiently simple to be described by explicit solutions to the set of differential equations 47 (section ), which characterize the disposition of chemicals in multimembrane systems. Complexity can be reduced by considering a few compartments or by neglecting the reverse transport processes, if applicable. Both simplifications result in several useful descriptions. The advantage of the explicit descriptions is that the transfer rate parameters l do not need to be strictly limited to the passive diffusion and can be replaced by characteristics of active transport mechanisms in a linear regime. The explicit descriptions represent realistic yet comparatively simple SBSP models. 6.1. Transport through a Single Bilayer A description of transport through the single membrane is relevant for simple systems such as gram-positive bacteria and unilamellar vesicles, as well as for various absorption simulators, if the studied sheet only consists of one lipophilic phase in a transport simulator( 2095) or a single phospholipid bilayer, such as BLM (section ), if the chemicals do not significantly interact with the headgroups. The frequently published applications of single-membrane-based mathematical treatments to the description of transport through various physiological sheets of multimembrane nature (e.g., the intestine or skin; see section ) will hardly promote our understanding of the processes. The rate of transport through a single membrane is usually characterized by the permeability coefficient PC (Figure , row 1, section ), relating the trans-membrane flux of the chemical to the surface area and the difference in the concentrations in the donor and acceptor aqueous phases. The magnitude of PC is equal to the reciprocal value of the total resistance, which can be calculated in a laminary system as the sum of the reciprocal values of the diffusional resistances in individual laminae. The classical solubility−diffusion model (section ) uses just a single lamina. The partial resistance in the ith lamina can be written as 1000,2096Ri = hi/(D iPi), where h is the effective thickness of the lamina, D the diffusion coefficient, and P the partition coefficient for the ith lamina, with respect to the external phase (if both phases are identical, P = 1). This generalization has been frequently used for the construction of membrane transport models differing in the number and quality of the laminae, 2097–2103 as well as for the calculation of the permeability coefficients from the MD simulation data. 292,588,1005 The membrane transport models mostly result in the dependence of the permeability coefficient on the partition coefficient, which initially linearly increases and later becomes stable, such as, e.g., the black curve in Figure . Some reports document the effect of the solute size on permeability. 2104,2105 A comprehensive treatment showed that the solubility−diffusion mechanism (section ) overpredicts the permeabilities through the liquid-ordered bilayers, and the correction factors were proportional to the surface density of phosholipids( 506) and to the minimal cross-sectional area of the seven studied permeants.( 73) It can be shown that, for the transport of non-amphiphilic chemicals through a symmetrical bilayer (Figure , rows 2 and 3), PC = li/2, where li is the water-to-bilayer transfer rate parameter.( 2106) The permeability coefficient PC (Figure , row 1) characterizes the steady-state rate of passage through the membrane. For most chemicals, the direct use of PC to calculate the time course of the concentration in the acceptor aqueous phase leads to substantial errors. The reason is that the concept resulting in the definition of the permeability coefficient considers the membrane an inert diaphragm. As illustrated in Figure , this assumption is not plausible for the majority of chemicals, which accumulate to some level in the membranes. The membranes represent a small volume fraction of the biosystem. Nevertheless, even considering the volume of the surrounding aqueous phases being a thousand times larger than that of the membranes, a chemical with the value of the membrane/water partition coefficient of ~1000 will partition in equal amounts in the aqueous phases and in the membranes. The equilibrium concentrations calculated without the consideration of membrane accumulation would be ~2 times higher than the experimentally observed quantities. Membrane accumulation can be taken into account via the use of at least two transfer rate parameters: one for the transfer of the chemical into the membrane and one for its exit ( li and lo, respectively). The transfer rate parameters are characterized in section ; their most important attributes are the dependencies on the partition coefficients, as described by eqs 3. The representation of the membrane by a single hydrocarbon core is a valid approximation for the chemicals, which do not interact with the headgroups. The explicit solutions to differential equations 47 were published for the simple water−core−water system (section ),( 2095) and for the system containing an additional lipophilic phase.( 2107) The concentration−lipophilicity profiles for individual distribution periods t, as generated by the explicit solutions, are similar to those given in Figure . 6.2. Unidirectional Transport The disposition of chemicals involving only unidirectional transport often represents real situations, at least for a certain time period.( 2108) Transport in any multimembrane system proceeds initially in one direction, away from the site of administration. The duration of the period of unidirectionality is primarily dependent on the length of the sequence of compartments, as well as on the rates and equilibria of the processes, which reduce the free concentrations in individual compartments either reversibly (noncovalent binding to macromolecules, section ) or irreversibly (metabolism and excretion, section ). The approximation may occasionally be sufficiently precise even for the complete period of distribution, especially in two borderline cases: (i) for large systems, where the total number of compartments approaches several hundred, as described in section for the muscles, connective tissues, and fat; and (ii) for rapidly biotransformed chemicals, even in the biosystems with a few bilayers, as in the cell suspensions. 6.2.1. Model Description The model system depicted in Figure in section , if the transport is considered unidirectional, is described by the set of linear differential equations 47 (section ) with a bi-diagonal matrix B. The elements above the diagonal represent the backward process and vanish for the unidirectional transport. If all compartments of the model are different (also known as the normal case), the free concentration in the mth compartment can be expressed 2064,2109 as a multi-exponential function of the exposure time t:
Here, n is the dose. The eigenvalue λi (eq 49, section ) comprises the traits of the ith compartment: the volume Vi, the interfacial area Si connecting it to the ( i + 1)th compartment (in the flux terms Fi,i+1, c.f., eq 46), the pH value of the phase (in the dissociation terms dij, c.f., eqs 36 and 46), and the concentration of protein binding sites sij (in the binding terms bijk, c.f., eq 38), as well as the properties of the tested chemicals, such as the rate parameters for the transfer from water to the core and backwards ( li and lo, respectively, in the terms Fi,i+1, c.f., eqs 35 and 46), the ionization constant to the jth degree (in dij), ion pairing with the kth counterion (in the pairing term pijk, c.f., eqs 36 and 46), the elimination rate parameters kijk (in Ei, defined in eq 46), and affinities for proteins Kij, (in bijk, c.f., eq 38). The descriptions equivalent to eq 14 were published for simpler scenarios when protein binding or elimination was not significant. 2110–2112 Equation 14 supports the previously published conclusion about the proportionality between the concentration at a given time and the product of the rate parameters of involved transport processes.( 2083) The cellular structure of tissues leads to the situation in which all membranes and all aqueous phases have the same characteristics (also known as the degenerate case). This phenomenon may also be observed in isolated mammalian cells and eukaryotic microbial cells, because they are subdivided into the sets of similarly sized compartments by endoplasmic reticulum and other intracellular membranes. The time courses of the drug concentrations in all compartments, preceding the cellular structure with repeated pairs of identical compartments, are described by eq 14. Inside the cellular structure, recurrent formulae should be applied, as has been shown for a reduced scenario.( 2112) Equation 14 and the recurrent formulae can be used in two ways: (i) for a direct construction of the SBSP models in smaller systems, such as suspensions of bacteria or mammalian cells, which comprise a few compartments; and (ii) as a basis for a derivation of more-manageable limit expressions, when the compartment number is large. 6.2.2. Concentration−Lipophilicity Profiles Equation 14, with the transfer rate parameters expressed by eqs 3, which is valid for chemicals that are not accumulating in the headgroups and at the interface, describes transport( 2113) as being combined with elimination( 2111) and protein binding.( 2112) For pure transport and transport with elimination, the generated concentration−lipophilicity profiles have been shown to be identical to those obtained by numerical simulations. The influence of protein binding on disposition of chemicals has only been investigated in SBSP models using the explicit relationship described by eq 14.( 2110) As the first approximation, protein binding was assumed to be dependent on lipophilicity, according to eq 4 in section . More-precise models will require replacement of the association constants Kij, which are contained in the terms bijk in eq 14, by the 3D-QSAR models for protein binding, as reviewed in sections − . All adjustable coefficients, such as those in eq 14, as well as those in the 3D-QSAR model, may need to be optimized simultaneously. The corresponding concentration−lipophilicity profiles for shorter periods, corresponding to the nonequilibrium distribution period in closed systems, are depicted in Figure . The bilinear profiles without the influence of protein binding have characteristic integer slopes (see Table 6 in section ). Lipophilicity-dependent protein binding changes the slope of the decreasing linear part, as can be seen in Figure . This fact could explain why the experimentally observed smooth concentration−lipophilicity profiles cannot always be described using eq 11 with the integer values of the adjustable coefficients, as given in Table 6. The dependencies for a plummeting right-hand part, which are similar to those in Figure , are accurately described using a simple empirical relation:( 2114)
Here, A, B, and C are optimized coefficients. Equation 15 provided excellent fits to several published datasets.( 2114) The results for the longer distribution periods are plotted in Figure . Analogous curves for pure transport (Figure A) have been described in sections and . The initially bilinear concentration−lipophilicity profiles are distorted after longer distribution periods in the region of optimum lipophilicity and form the double-peaked curves. Lipophilicity-dependent protein binding (Figure B) decreases the actual drug concentrations at shorter times. For the longer distribution periods, the depot function of proteins for chemicals with log P > 0 starts to play a role. The slowdown of the transport of the molecules with the optimum lipophilicities is the reason why the minima in Figure B become shallower than those without the protein influence (Figure A). The depot effect is more pronounced in the presence of elimination (Figure C). While hydrophilic compounds have been eliminated at longer times and the left-hand peaks have completely disappeared, lipophilic compounds are still present in the system, because of the depot function of protein binding. 6.3. Steady-State Distribution Continuous dosing at a steady rate is frequently used for therapeutic purposes, in the form of intravenous infusions for critical drugs or via the steady-release dosage forms, producing long-lasting effects. Environmental scenarios with practically invariant pollution levels also provide a steady input. If the continuous input of chemicals is maintained at the same rate for a sufficiently long period, the system will ultimately attain the steady state, manifested by the time-invariant concentrations in all compartments, which are depicted in Figure in section . Consequently, all the left-hand-side derivatives in SBSP models, which are represented by eqs 46 in section , vanish. To describe the situation, eqs 47 are not needed in this case, because the factors that are used to convert eqs 46 to eqs 47 would cancel each other in the solutions of the algebraic equations resulting from eqs 46 for the steady-state conditions. Because of a sustained concentration pressure from the site of dosing, reverse transport is negligible, and the back-flux terms Fi,i−1 and Fi+1,i in eqs 46 can be omitted. This simplification affords a compact solution for the steady-state concentration of the free non-ionized molecules in the ith compartment (see the numbering in Figure in section ), obtained by back-substitution:( 2115)
Here, c1 is the invariant concentration of the free non-ionized chemical in entry (aqueous) compartment 1, and the flux and elimination terms F and E are defined in eqs 46. The last compartment is usually an aqueous phase, so the sequence number of this compartment ( N in eq 16) is an odd number, according to the numbering scheme in Figure (section ). If the studied biosystem, such as a tissue, consists of the cells of the same type with similar intracellular compartments, the flux terms F are practically identical for the transport from all aqueous phases ( FA, odd initial subscript i) and from all bilayers ( FM, even initial subscript i). For this common situation, eq 16 can be simplified to eq 17 for the aqueous phases and eq 18 for the bilayers,( 2115) with the compartment numbering given in Figure in section :
The dependencies on the lipophilicity of the free, non-ionized steady-state concentrations, which were attained during continuous dosing, are illustrated in Figures and . The transfer rate parameters li and lo contained in the flux terms were expressed using eqs 3 in section , so the results are valid for the chemicals, which do not accumulate in the headgroups and at the interfaces. Interestingly, no peaks are apparent in the bilogarithmic dependencies; rather, the concentrations of chemicals are, after an initial linear increase for hydrophilic chemicals, independent of lipophilicity for aqueous phases, and rise at a lower rate for membranes. The slopes of the leftmost linear parts are dependent on the compartment number (see Figure ). The break-point log P value is affected less by the compartment number (see Figure ) than by the magnitude of the elimination rate (Figure ). Apparently, the peaks in the concentration−lipophilicity profiles after single doses (e.g., Figures −) are caused by a slow transport of lipophilic compounds. As can be seen in Figures and , this phenomenon does not exist under the continuous dosing regimens, after the distribution reaches the steady state. However, there is a string attached to the last phrase. The duration of the transient period, before the steady state is established, varies with lipophilicity, and can be very long for extremely hydrophilic and extremely lipophilic compounds. Achievement of the steady state for chemicals with a broad range of lipophilicities requires long exposure periods, which are typical for environmental scenarios but may be much longer than the common infusion periods of drugs. Consequently, in real situations, a type of concave concentration−lipophilicity profile can also be expected for the continuous dosing of drugs, if the lipophilicity range is sufficiently broad. The concentrations of very hydrophilic and very lipophilic drugs would be lower than expected on the basis of the steady-state profiles, which are shown in Figures and . The concentrations in individual phases, as expressed by eqs 17 and 18, are of interest, when the studied chemicals act by a specific mechanism on well-defined receptors, which are localized in spatially limited compartments. However, environmental pollutants frequently elicit their effects in all phases by mechanisms, which cannot be tracked back to the modification of a single receptor, such as polar and nonpolar narcosis, and alkylation of body constituents. 1487,2116,2117 These effects are caused by interactions proceeding in all aqueous phases or in all membranes; consequently, the average concentrations in all aqueous phases and all membranes are important for their descriptions. If individual steady-state concentrations in membranes are enumerated according to eq 18 and used to calculate the average concentrations, the result contains a practically infinite ( N may reach several hundred; see section ) geometric sequence with the argument x = FAFM/( FA + EA)( FM + EM) < 1 and the powers 0, 1,..., ( N − 3)/2. The sum of such a sequence is equal to 1/(1 − x) and can be used to calculate the average concentrations for all ( N − 1)/2 membranes (eq 20).( 2115) For the aqueous phases, the first term with the power of 0 (i.e., term = 1) is missing and the sum is then [1/(1 − x)] − 1. The average concentration for all ( N − 1)/2 aqueous phases, except the first phase, which maintains the constant concentration c1, is given by eq 19:( 2115)
Here, N is the number of compartments. The dependencies of the average concentrations in the aqueous phases and membranes on lipophilicity are shown in Figure . In aqueous phases, saturation-type curves are observed, whereas biphasic increasing dependencies result for the membranes. The elimination rate affects the average aqueous concentrations much more significantly than the average membrane concentrations. This fact is consistent with experimental observations regarding pollutants acting by narcosis, which is presumably elicited by a membrane perturbation. The toxicity of these chemicals is linearly dependent on lipophilicity, with a slope that approaches unity. The dependencies are perfectly linear for persistent pollutants, and reactive chemicals exhibit only modest deviations from the line. 2118–2121 |
Bilayer Systems
. A special effort has been made to compare and integrate, where feasible, the approaches to the analysis of the same phenomenon in different areas.
![[big up triangle, open]](/corehtml/pmc/pmcents/x25B3.gif)
) alcohols, (![[triangle]](/corehtml/pmc/pmcents/dtrif.gif)
) anilines, and (![[large star]](/corehtml/pmc/pmcents/x2605.gif)
) polar compounds with no hydrogen-bond
