5.4.1.1. Predicting Complex Protein Structures

Interactions between proteins are crucial to the functioning of all cells. While there is much experimental information being gathered regarding protein structures, many interactions are not fully understood and have to be modeled computationally. The topic of computational prediction of protein-protein structure remains to be solved and is one of the most active areas of research in bioinformatics and structural biology.

ZDOCK and RDOCK are two computer programs that address this problem, also known as protein docking.³⁷ ZDOCK is an initial stage protein docking program that performs a full search of the relative orientations of two molecules (referred to by convention as the ligand and receptor) to determine their best fit based on surface complementarity, electrostatics and desolvation. The efficiency of the algorithm is enhanced by discretizing the molecules onto a grid and performing a fast Fourier transform (FFT) to quickly explore the translational degrees of freedom.

RDOCK takes as input the ZDOCK predictions and improves them using two steps. The first step is to improve the energetics of the prediction and remove clashes by performing small movements of the predicted complex, using a program known as CHARMM. The second step is to rescore these minimized predictions with more detailed scoring functions for electrostatics and desolvation.

The combination of these two algorithms has been tested and verified with a benchmark set of proteins collected for use in testing docking algorithms. Now at version 2.0, this benchmark is publicly available and contains 87 test cases. These test cases cover a breadth of interactions, such as antibody-antigen, and cases involving significant conformational changes.

The ZDOCK-RDOCK programs have consistently performed well in the international docking competition CAPRI (Figure 5.1). Some notable predictions were for the Rotavirus VP6/Fab (50 of 52 contacting residues correctly predicted), and SAG-1/Fab complex (61 of 70 contacts correct), and the cellulosome cohesion-dockerin structure (50 of 55 contacts correct). In the first two cases, the number of contacts in the ZDOCK-RDOCK predictions were the highest among all participating groups.

FIGURE 5.1

The ZDOCK/RDOCK prediction for dockerin (in red) superposed on the crystal structure for CAPRI Target 13, cohesin/dockerin. SOURCE: Courtesy of Brian Pierce and Zhiping Weng, Boston University.

5.4.1.2. A Method to Discern a Functional Class of Proteins

The DNA-binding helix-turn-helix structural motif plays an essential role in a variety of cellular pathways that include transcription, DNA recombination and repair, and DNA replication. Current methods for identifying the motif rely on amino acid sequence, but since members of the motif belong to different sequence families that have no sequence homology to each other, these methods have been unable to identify all motif members.

A new method based on three-dimensional structure was created that involved the following steps:³⁸ (1) choosing a conserved component of the motif, (2) measuring structural features relative to that component, and (3) creating classification models by comparing measurements of structures known to contain the motif to measurements of structures known not to contain the motif. In this case, the conserved component chosen was the recognition helix (i.e., the alpha helix that makes sequence-specific contact with DNA), and two types of relevant measurements were the hydrophobic area of interaction between secondary structure elements (SSEs) and the relative solvent accessibility of SSEs.

With a classification model created, the entire Protein Data Bank of experimentally measured structures was searched and new examples of the motif were found that have no detected sequence homology with previously known examples. Two such examples are Esa1 histone acetyltransferase and isoflavone 4-O-methyltransferase. The result emphasizes an important utility of the approach: sequence-based methods used to discern a functional class of proteins may be supplemented through the use of a classification model based on three-dimensional structural information.

5.4.1.3. Molecular Docking

Using a simple, uniform representation of molecular surfaces that requires minimal parameterization, Jain³⁹ has constructed functions that are effective for scoring protein-ligand interactions, quantitatively comparing small molecules, and making comparisons of proteins in a manner that does not depend on protein backbone. These methods rely on computational approaches that are rooted in understanding the physics of molecular interactions, but whose functional forms do not resemble those used in physics-based approaches. That is, this problem can be treated as a pure computer science problem that can be solved using combinations of scoring and search or optimization techniques parameterized with the use of domain knowledge. The approach is as follows:

• Molecules are approximated as collections of spheres with fixed radii: H = 1.2; C = 1.6; N = 1.5; O = 1.4; S = 1.95; P = 1.9; F = 1.35; Cl = 1.8; Br = 1.95; I = 2.15.
• A labeling of the features of polar atoms is superimposed on the molecular representation: polarity, charge, and directional preference (Figure 5.2, subfigures A and B).
• A scoring function is derived that, given a protein and a ligand in some relative alignment, yields a prediction of the energy of interaction.
• The function is parameterized in terms of the pairwise distances between molecular surfaces.
• The dominant terms are a hydrophobic term that characterizes interactions between nonpolar atoms and a polar term that captures complementary polar contacts with proper directionality.
• The parameters of the function were derived from empirical binding data and 34 protein-ligand complexes that were experimentally determined.
• The scoring function is described in Figure 5.2, Subfigure C. The hydrophobic term peaks at approximately 0.1 unit with a slight surface interpenetration. The hydrophobic term for an ideal hydrogen bond peaks at 1.25 units, and a charged interaction (tertiary amine proton (+1.0) to a charged carboxylate (−0.5)) peaks at about 2.3 units. Note that this scoring function looks nothing like a force field derived from molecular mechanics.
• Figure 5.2, Subfigure D compares eight docking methods on screening efficiency using thymidine kinase as a docking target. For the test, 10 known ligands and 990 random ligands were used. Particularly at low false-positive rates (low database coverage), the scoring function approach shows substantial improvements over the other methods.

FIGURE 5.2

A Computational Approach to Molecular Docking. SOURCE: Courtesy of A.N. Jain, University of California, San Francisco.

5.4.1.4. Computational Analysis and Recognition of Functional and Structural Sites in Protein Structures⁴⁰

Structural genomics initiatives are producing a great increase in protein three-dimensional structures determined by X-ray and nuclear magnetic resonance technologies as well as those predicted by computational methods. A critical next step is to study the relationships between protein structures and functions. Studying structures individually entails the danger of identifying idiosyncratic rather than conserved features and the risk of missing important relationships that would be revealed by statistically pooling relevant data. The expected surfeit of protein structures provides an opportunity to develop computational methods for collectively examining multiple biological structures and extracting key biophysical and biochemical features, as well as methods for automatically recognizing these features in new protein structures.

Wei and Altman have developed an automated system known as FEATURE that statistically studies the important functional and structural sites in protein structures such as active sites, binding sites, disulfide bonding sites, and so forth. FEATURE collects all known examples of a type of site from the Protein Data Bank (PDB) as well as a number of control “nonsite” examples. For each of them, FEATURE computes the spatial distributions of a large set of defined biophysical and biochemical properties spanning multiple levels of details in order to capture conserved features beyond basic amino acid sequence similarity. It then uses a nonparametric statistical test, the Wilcoxin Rank Sum Test, to find the features that are characteristic of the sites, in the context of control nonsites. Figure 5.3 shows the statistical features of calcium binding sites.

FIGURE 5.3

Statistical features of calcium binding sites determined by FEATURE. The volumes in this case correspond to concentrate radial shells 1 Å in thickness around the calcium ion or a control nonsite location. The column shows properties that are statistically (more...)

By using a Bayesian scoring function that recognizes whether a local region within a three-dimensional structure is likely to be any of the sites and a scanning procedure that searches the whole structure for the sites, FEATURE can also provide an initial annotation of new protein structures. FEATURE has been shown to have good sensitivity and specificity in recognizing a diverse set of site types, including active sites, binding sites, and structural sites and is especially useful when the sites do not have conserved residues or residue geometry. Figure 5.4 shows the result of searching for ATP (adenosine triphosphate) binding sites in a protein structure.

FIGURE 5.4

Results of automatic scanning for ATP binding sites in the structure of casein kinase (PDB ID 1csn) using WebFEATURE, a freely available, Web-based server of FEATURE. The solid red dots show the prediction of FEATURE, they correspond correctly with the (more...)

5.4.2.1. Cellular Modeling and Simulation Efforts

Cellular simulation requires a theoretical framework for analyzing the interactions of molecular components, of modules made up of those components, and of systems in which such modules are linked to carry out a variety of functions. The theoretical goal is to quantitatively organize, analyze, and interpret complex data on cell biological processes, and experiments provide images, biochemical and electrophysiological data on the initial concentrations, kinetic rates, and transport properties of the molecules and cellular structures that are presumed to be the key components of a cellular event.⁴¹ A simulation embeds the relevant rate laws and rate constants for the biochemical transformations being modeled. Based on these laws and parameters, the model accepts as initial conditions the initial concentrations, diffusion coefficients, and locations of all molecules implicated in the transformation, and generates predictions for the concentration of all molecular species as a function of time and space. These predictions are compared against experiment, and the differences between prediction and experiment are used to further refine the model. If the system is perturbed by the addition of a ligand, electrical stimulus, or other experimental intervention, the model should be capable of predicting changes as well in the relevant spatiotemporal distributions of the molecules involved.

There are many different tools for simulating and analyzing models of cellular systems (Table 5.1). More general tools, such as Mathematica and MATLAB or other systems that can be used for solving systems of differential or stochastic-differential equations, can be used to develop simulations, and because these tools are commonly used by many researchers, their use facilitates the transfer of models among different researchers. Another approach is to link data gathering and biological information systems to software that can integrate and predict behavior of interacting components (currently, researchers are far from this goal, but see Box 5.5 and Box 5.6). Finally, several platform-independent model specification languages are under development that will facilitate greater sharing and interoperability. For example, SBML,⁴² Gepasi,⁴³ and CellML⁴⁴ are specialized systems for biological and biochemical modeling. Madonna⁴⁵ is a general-purpose system for solving a variety of equations (differential equations, integral equations, and so on).

TABLE 5.1

Sample Simulation Programs.

Box 5.5

BioSPICE. BioSPICE, the Biological Simulation Program for Intra-Cellular Evaluation, is in essence a modeling framework that provides users with model components, tools, databases, and infrastructure to develop predictive dynamical models of cellular (more...)

Box 5.6

Cytoscape. A variety of computer-aided models has been developed to simulate biological networks, typically focusing on specific cellular processes or single pathways. Cytoscape is a modeling environment particularly suited to the analysis of global data (more...)

Rice and Stolovitzky describe the task of inferring signaling, metabolic, or gene regulatory pathways from experimental data as one of reverse engineering.⁴⁶ They note that automated, high-throughput methods that collect species- and tissue-specific datasets in large volume can help to deal with the risks in generalizing signaling pathways from one organism to another. At the same time, fully detailed kinetic models of intracellular processes are not generally feasible. Thus, one step is to consider models that describe network topology (i.e., that identify the interactions between nodes in the system—genes, proteins, metabolites, and so on). A model with more detail would describe network topology that is causally directional (i.e., that specifies which entities serve as input to others). Box 5.7 provides more detail.

Box 5.7

Pathway Reconstruction: A Systems Approach. On Topology. In this level, we are only concerned with identifying the interaction between nodes (genes, proteins, metabolites, etc.) in the system. The goal is the generation of a diagram of non-directional (more...)

An example of a cellular simulation environment is E-CELL, an open-source system for modeling biochemical and genetic processes. Organizationally, E-CELL is an international research project aimed at developing theoretical and functioning technologies to allow precise “whole cell” simulation; it is supported by the New Energy and Industrial Technology Development Organization (NEDO) of Japan.

E-CELL simulations allow a user to model hypothetical virtual cells by defining functions of proteins, protein-protein interactions, protein-DNA interactions, regulation of gene expression, and other features of cellular metabolism.⁴⁷ Based on reaction rules that are known through experiment and assumed concentrations of various molecules in various locations, E-CELL numerically integrates differential equations implicitly described in these reaction rules, resulting in changes over time in the concentrations of proteins, protein complexes, and other chemical compounds in the cell.

Developers hope E-CELL will ultimately allow investigators a cheap, fast way to screen drug candidates, study the effects of mutations or toxins, or simply probe the networks that govern cell behavior. One application of E-CELL has been to construct a model of a hypothetical cell capable of transcription, translation, energy production, and phospholipid synthesis with only 127 genes. Most of these genes were taken from Mycoplasma genitalium, the organism with the smallest known chromosome (the complete genome sequence is 580 kilobases).⁴⁸ E-CELL has also been used to construct a computer model of the human erythrocyte,⁴⁹ to estimate a gene regulatory network and signaling pathway involved in the circadian rhythm of Synechococcus sp. PCC 7942,⁵⁰ and to model mitochondrial energy metabolism and metabolic pathways in rice.⁵¹

Another cellular simulation environment is the Virtual Cell, developed at the University of Connecticut Health Center.⁵² The Virtual Cell is a tool for experimentalists and theoreticians for computationally testing hypotheses and models. To address a particular question, these mechanisms (chemical kinetics, membrane fluxes and reactions, ionic currents, and diffusion) are combined with a specific set of experimental conditions (geometry, spatial scale, time scale, stimuli) and applicable conservation laws to specify a concrete system of differential and algebraic equations. This experimental geometry may assume well-mixed compartments or a one-, two-, or three-dimensional spatial representation (e.g., experimental images from a microscope). Models are constructed from biochemical and electrophysiological data mapped to appropriate subcellular locations in images obtained from a microscope. A variety of modeling approximations are available including pseudo-steady state in time (infinite kinetic rates) or space (infinite diffusion or conductivity). In the case of spatial simulations, the results are mapped back to experimental images and can be analyzed by applying the arsenal of image-processing tools that is familiar to a cell biologist. Section 5.4.2.4 describes a study undertaken within the Virtual Cell framework.

Simulation models can be useful for many purposes. One important use is to facilitate an understanding of what design properties of an intracellular network are necessary for its function. For example, von Dassow et al.⁵³ used a simulation model of the gap and pair-rule gene network in Drosophila melanogaster to show that the structure of the network is sufficient to explain a great deal of the observed cellular patterning. In addition, they showed that the network behavior was robust to parameter variation upon the addition of hypothetical (but reasonable) elements to the known network. Thus, simulations can also be used to formally propose and justify new hypothetical mechanisms and predict new network elements.

Another use of simulation models is in exploring the nature of control in networks. An example of exploring network control with simulation is the work of Chen et al.⁵⁴ in elucidating the control of different phases of mitosis and explaining the impact of 50 different mutants on cellular decisions related to mitosis.

Simulations have also been used to model metabolic pathways. For example, Edwards and Palsson developed a constraint-based genome-scale simulation of Escherichia coli metabolism (Box 5.8). By applying successive constraints (stoichiometric, thermodynamic, and enzyme capacity constraints) to the metabolic network, it is possible to impose limits on cellular, biochemical, and systemic functions, thereby identifying all allowable solutions (i.e., those that do not violate the applicable constraints). Compared to the detailed theory-based models, such an approach has the major advantage that it does not require knowledge of the kinetics involved (since it is concerned only with steady-state function). (On the other hand, it is impossible to implement without genome-scale knowledge, because only genome-scale knowledge can bound the system in question.) Within the space of allowable solutions, a particular solution corresponds to the maximization of some selected function, such as cellular growth or a response to some environmental change. A more robust model accounting for a larger number of pathways is also described in Box 5.8.

Box 5.8

Escherichia coli Constraint-based Models. A. In Silico Model The Escherichia coli MG1655 genome has been completely sequenced. The annotated sequence, biochemical information, and other information were used to reconstruct the E. coli metabolic map. The (more...)

The Edwards and Palsson model has been used to predict the evolution of E. coli metabolism under a variety of environmental conditions. In the words of Ibarra et al., “When placed under growth selection pressure, the growth rate of E. coli on glycerol reproducibly evolved over 40 days, or about 700 generations, from a sub-optimal value to the optimal growth rate predicted from a whole-cell in silico model. These results open the possibility of using adaptive evolution of entire metabolic networks to realize metabolic states that have been determined a priori based on in silico analysis.”⁵⁵

Simulation models can also be used to test design ideas for engineering networks in cells. For example, very simple models have been used to provide insight into a genetic oscillator and a switch in E. coli.⁵⁶ Models have also been used to test designs for the control of cellular networks, as illustrated by Endy and Yin in using their T7 model to propose a pharmaceutical strategy for preventing both T7 propagation and the development of drug resistance through mutation.⁵⁷

Given observed cell behavior, simulation models can be used to suggest the necessity of a given regulatory motif or the sufficiency of known interactions to produce the phenomenon. For example, Qi et al. demonstrate the sufficiency of membrane energetics, protein diffusion, and receptor-binding kinetics to generate a particular dynamic pattern of protein location at the synapse between two immune cells.⁵⁸

The following sections describe several simulation studies in more detail.

5.4.2.2. Cell Cycle Regulation

Biological growth and reproduction depend ultimately on the cycle of DNA synthesis and physical separation of the replicate DNA molecules within individual cells. In eukaryotes, these processes are triggered by cyclin-dependent protein kinases (CDKs). In fission yeast, CDK activity (cdc2 = kinase subunit, cdc13 = cyclin subunit) is regulated by a network of protein interactions (Figure 5.5), including cyclin synthesis and degradation, phosphorylation of cdc2, and binding to an inhibitor.

FIGURE 5.5

The cell-cycle control system in fission yeast. This system can be divided into three modules, which regulate the transitions from G1 into S phase, from G2 into M phase, and exit from mitosis. SOURCE: J.J. Tyson, K. Chen, and B. Novak, “Network (more...)

A network of such complexity, with multiple feedback loops, cannot be understood thoroughly by casual intuition. Instead, the network is converted into a set of nonlinear differential equations, and the physiological implications of these equations are studied.⁵⁹ Numerical simulation of the equations (Figure 5.6) provides complete time courses of every component and can be interpreted in terms of observable events in the cell cycle. Simulations can be run, not only of wild-type cells but also of dozens of mutants constructed by deleting or overexpressing each component singly or in multiple combinations. From the observed phenotypes of these mutants it is possible to reverse-engineer the regulatory network and the set of kinetic constants associated with the component reactions.

FIGURE 5.6

Simulated time courses of cdc2 and related proteins during the cell cycle of fission yeast. Numerical integration of the full set of differential equations that describe the wiring diagram in Figure 5.5 yields these time courses. Time is expressed in (more...)

For understanding the dynamics of molecular regulatory systems, bifurcation theory is a powerful complement to numerical simulation. The bifurcation diagram in Figure 5.7 presents recurrent solutions (steady states and limit cycle oscillations) of the differential equations as functions of cell size. The control system has three characteristic steady states: low cdc2 activity (G1 = pre-replication), medium cdc2 activity (S/G2 = replication and post-replication), and high cdc2 activity (M = separation of replicated DNA molecules). G1 and S/G2 are stable steady states; M is unstable because of a negative feedback loop as shown in Figure 5.5 (cdc2-cdc13 activates Slp1, which degrades cdc13).

FIGURE 5.7

Bifurcation diagram for the full cell-cycle control network…. [T]he full diagram is not a simple sum of the bifurcation diagrams of its modules. In particular, oscillations around the M state are greatly modified in the composite control system. (more...)

When the time courses of size and cdc2 activity from Figure 5.6 are superimposed on the bifurcation diagram (curve labeled “size”), one sees how progress through the cell cycle is governed by the bifurcations that turn stable steady states into unstable steady states and/or stable oscillations. A mutation changes a specific rate constant, which changes the locations of the bifurcation points in Figure 5.7, which changes how cells progress through (or halt in) the cell cycle. By this route one can trace the dynamical consequences of genetic information all the way to observable cell behavior.

5.4.2.3. A Computational Model to Determine the Effects of SNPs in Human Pathophysiology of Red Blood Cells

The completion of the Human Genome Project has led to the construction of single nucleotide polymorphism (SNP) maps. Single nucleotide polymorphisms are common DNA sequence variations among individuals. A result of the construction of SNP maps is to determine the effects of SNPs on the development of disease(s) since sequence variations can lead to altered biological function or disease.

Currently, it is difficult to determine the causal relationship between the variations in sequence, SNPs, and the physiological function. One way to analyze this relationship is to create computational models or simulations of biological processes. Since erythrocyte (red blood cell) metabolism has been studied extensively over the years and many SNPs have been characterized, Jamshidi et al. used this information to build their computational models.⁶⁰

Two important metabolic enzymes, glucose-6-phosphate dehydrogenase (G6PD) and pyruvate kinase (PK), were studied for alterations in their kinetic properties in an in silico model to calculate the overall effect of SNPs on red blood cell function. Defects in these enzymes cause hemolytic anemia. Clinical data taken from the published literature were used for the measured values of the kinetic parameters. These values were then used in model simulations to determine whether a direct link could be established between the SNP and the disease (anemia).

The computational modeling revealed two results. For the G6PD and PK variants analyzed, there appeared to be no clear relationship between their kinetic properties as a function of sequence variation or SNP. However, upon assessment of overall biological function, a correlation was found between the sequence variation of G6PD and the severity of the clinical disease. Thus, in silico modeling of biological processes may aid in analysis and prediction of SNPs and pathophysiological conditions.

5.4.2.4. Spatial Inhomogeneities in Cellular Development

Simulation models can be used to provide insight into the significance of spatial inhomogeneities. For example, the interior of living cells does not resemble at all a uniform aqueous solution of dissolved chemicals, and yet this is the implicit assumption underlying many views of the cell. This assumption serves traditional biochemistry and molecular biology reasonably well, but research increasingly demonstrates that the physical locations of specific molecules are crucial. Multiprotein complexes act as machines for internal movements or as integrated circuits in signaling. Messenger RNA molecules are transported in a highly directed fashion to specific regions of the cell (in nerve axons, for example). Cells adopt highly complex shapes and undergo complex movements thanks to the matrix of protein filaments and associated proteins within their cytoplasm.

5.4.2.4.1. Unraveling the Physical Basis of Microtubule Structure and Stability

Microtubules are cylindrical polymers found in every eukaryotic cell. Microtubles play a role in cellular architecture and as molecular train tracks used to transport everything from chromosomes to drug molecules. An understanding of microtubule structure and function is key not just to unraveling fundamental mechanisms of the cell, but also to opening the way to the discovery of new antiparasitic and anticancer drugs.

Until now, researchers have known that the microtubules, constructed of units called protofilaments in a hollow, helical arrangement, are rigid but not static, and undergo periods of growth and sudden collapse. Yet the mechanism for this construction-destruction had eluded researchers.

Over the past several years, McCammon and his colleagues have pioneered the use of a combination of an atomically detailed model for a microtubule and large-scale computations using the adaptive Poisson-Boltzmann Solver to create a high-resolution, 1.25-million-atom map of the electrostatic interactions within the microtubule.⁶¹

More recently, David Sept and Nathan Baker of Washington University and McCammon used the same technique to successfully predict the helicity of the tubule with a striking correspondence to experimental observation.⁶² Based on the lateral interactions between protofilaments, they determined that the microtubule prefers to be in a configuration in which the protofilaments assemble with a seam at each turn, rather than spiraling smoothly upward with alpha and beta monomers wrapping the microtubule as if it were a barber's pole. At the end of each turn, a chain of alphas is trailed by a chain of betas, then after that turn, a chain of alphas, and so on. It is as if the red and white stripes on the barber's pole traded places with every twist (Figure 5.8).

FIGURE 5.8

The binding free energy between two protofilaments as a function of the subunit rise between adjacent dimmers. Sept et al. used electrostatic calculations to determine the binding energy between two protofilaments as a function of the subunit rise between (more...)

5.4.3.1. Cis-regulation of Transcription Activity as Process Control Computing

It has been known for some time that the genome contains both genes and cis-regulatory elements.⁶⁷ The presence or absence of particular combinations of these regulatory elements determines the extent to which specific genes are expressed (i.e., transcribed into specific proteins). In pioneering work undertaken by Davidson et al.,⁶⁸ it was shown that cis-regulation could—in the case of a specific gene—be viewed as a logical process analogous to a computer program that connected various inputs to a single output determining the precise level of transcription for that gene.

In particular, Davidson and his colleagues developed a high-level computer simulation of the cis-regulatory system governing the expression of the endo16 gene in the sea urchin (endo16 is a gut-specific gene of the sea urchin embryo). In this context, the term “high-level” means a highly abstracted representation, consisting at its core of 18 lines of code. This simulation enabled them to make predictions about the effect of specific manipulations of the various regulatory factors on endo16 transcription levels that could be tested against experiment.

Some of the inputs to the simulation were binary values. The value 1 indicated that a binding site was both present and productively occupied by the appropriate cis-regulatory factor. A 0 indicated that the site was mutationally destroyed or inactive because its factor was not present or was inactive. The other inputs to the simulation were continuous and varied with time, and represented outputs (protein concentrations) in other parts of the system. The output of this process in some cases was a continuous time-varying variable that regulated the extent to which the specific gene in question was transcribed.

Davidson et al. were able to confirm the predictions made by their computational model, concluding that all of the regulatory functions in question (and the resulting system properties) were encoded in the DNA sequence, and that the regulatory system described is capable of processing complex informational inputs and hence indicates the presence of a multifunctional organization of the endo16 cis-regulatory system.⁶⁹

The computational model undoubtedly provides a compact representation of the relationships between different inputs and different outputs.⁷⁰ Perhaps a more interesting question, however, is the extent to which it is meaningful to ascribe a computational function to the biochemical substrate underlying the regulatory system. Davidson et al. argue that the DNA sequence in this case specifies “what is essentially a hard-wired, analog computational device,” resulting in system properties that are “all explicitly specified in the genomic DNA sequence.”⁷¹

It is highly unlikely that the precise computational structure of endo16's regulatory system will generalize to the regulatory systems of other genes. From the perspective of the biologist, the reason is clear—organisms are not designed as general-purpose devices. Indeed, the evolutionary process virtually guarantees that individualized solutions and architectures will be abundant, because specific adaptations are the rule of the day. Nevertheless, insight into the computational behavior of the endo16 cis-regulatory system provides a new way of looking at biological behavior.

Can the regulatory systems of some other genes be cast in similar computational terms? If and when future work demonstrates that such casting is possible, it will become increasingly meaningful to view the genome as thousands of simple computational devices operating in tandem. Davidson's work suggests the possibility that a class of regulatory mechanisms, complex though they might be with respect to their behavior, may be governed by what are in essence hard-wired devices whose essential functionality can be understood in computational terms through a logic of operation that is in fact relatively simple at its core. Prior to Davidson's work and despite extensive research, the literature had not revealed any apparent regularity in the organization of regulatory elements or in the ways in which they interact to regulate gene expression.

Indeed, while many promoters appear either to have a simpler organization or to operate less logically than that of endo16, few promoters have been examined with the many precise quantitative assays that were carried out by Davidson et al., and nonquantitative assays would have completely missed most of the functions that the majority of the regulatory system's elements encode.⁷² So, it is at this point an open question whether this computational view has applicability beyond the specific case of endo16.

5.4.3.2. Genetic Regulatory Networks as Finite-state Automata

Trans-regulation (as contrasted to cis-regulation) is based on the notion that some genes can have regulatory effects on others.⁷³ In reality, the network of connections between genes that regulate and genes that are regulated is highly complex. In an attempt to gain insight into genetic regulatory networks from a gross oversimplification, Kaufmann proposed that actual genetic regulatory networks might be modeled as randomly connected Boolean networks.⁷⁴

Kaufmann's model made several simplifying assumptions:

• The total number of genes involved is N, a number of order 30,000.
• The number of genes that regulate a given target is a constant (call it K) for all regulated genes; K is a small integer.
• The regulatory signal associated with a connection or the expression of a gene is either on or off. (In fact, almost certainly it is not just the fact of a connection between genes that influences regulation, but rather the nature of that connection as a continuous time-varying value such as a molecular concentration over time.)
• Every gene is governed by the same transition rule (i.e., a Boolean function) that specifies its state (on or off) as a function of the activities of its K inputs at the immediately earlier time.
• The regulatory network operates synchronously (and, by implication, kinetics are unimportant).
• Secondary effects on genetic regulation arising from the nondigital characteristics of DNA (such as methylation) can be neglected.
• The genes that regulate and genes that are regulated (which may overlap) are connected at random.

Box 5.9 provides more details about this model. Because the model treats all genes as identical (i.e., all obey the same transition rule) and assigns connections between genes at random, it obviously lacks fidelity to any specific genome and cannot predict the biological phenomenology of any specific organism. Yet, it may provide insight into biological order that emerges from the structure of the genetic regulatory network itself.

Box 5.9

Finite-state Automata and a Comparison of Genetic Networks and Boolean Networks. In Kaufmann's Boolean representation of a genetic regulatory network, there are N genes, each with two states of activity (expressed or inhibited), and hence 2^N possible (more...)

Simulations of the operation of this model yielded interesting behavior, which depends on the values of N and K. For K = 1 or K > 5, the behavior of the network exhibits little interesting order, where order is defined in terms of fixed cycles known as attractors. If K = 1, the networks are static, with the number of attractors exponential in the size of the network and the cycle length approaching unity. If K > 5, there are few attractors, and it is the cycle length that is exponential in the size of the network. However, for K = 2, the network does exhibit order that has potential biological significance—both the number of attractors and the cycle length are proportional to N^1/2.⁷⁵

What might be the biological significance of these results?

• The trajectory of an attractor through its successive states would reflect the fact that, over time, different genes are expressed in a biological organism.
• The fact that there are multiple attractors within the same genome suggests that multiple biological structures might exist, even within the same organism, corresponding to the genome being in one of these attractor states. An obvious candidate for such structures would be multiple cell types. That is, this analysis suggests that a cell type corresponds to a given state cycle attractor, and the different attractors to the different cell types of the organism. Another possibility is that different but similar attractors correspond to cells in different states (e.g., disease state, resting state, perturbed state).
• The fact that an attractor is cyclic suggests that it may be related to cyclic behavior in a biological organism. If cell types can be identified with attractors, the cyclic trajectory in phase space of an attractor may correspond to the cell cycle in which a cell divides.
• States that can be moved from one trajectory (for one attractor) to another trajectory (and another attractor) by changing a single state variable are not robust and may represent the phenomenon that small, apparently minor perturbations to a cell's environment may kick it into a different state.
• The square root of the number of genes in the human genome (around 30,000) is 173. Under the assumption of K = 2 scaling, this would correspond to the number of cyclic attractors and thus to the number of cell types in the human body. This is not far from the number of cell types actually observed (about 200). Such a result may be numerological coincidence or rooted in the fact that nearly all cells in a given organism (even across eukaryotes) share the same basic housekeeping mechanisms (metabolism, cell-cycle control, cytoskeletal construction and deconstruction, and so on), or it may reflect phenotypic structure driven by the large-scale connectivity in the overall genetic regulatory network. More work will be needed to investigate these possibilities.⁷⁶Box 5.10 provides one view on experimental work that might be relevant.

Box 5.10

Testing the Potential Relevance of the Boolean Network Model. Because of the extreme simplifications embedded in the Boolean network model, detailed predictions (e.g., genes A and B turn on gene C) are unlikely to be possible. Instead, the utility of (more...)

To illustrate the potential value of Boolean networks as a model for genetic regulatory networks, consider their application to understanding the etiology of cancer.⁷⁷ Specifically, cancer is widely believed to be a pathology of the hereditary apparatus. However, it has been clear for some time that single-cause, single-effect etiologies cannot account for all or nearly all occurrences of cancer.⁷⁸

If the correspondence between attractor and cell is assumed, malignancy can be viewed as an attractor similar in most ways to that associated with a normal cell,⁷⁹ and the transition from normal to malignant is represented by a “phase transition” from one attractor to another. Such a transition might be induced by an external event (radiation, chemical exposure, lack of nutrients, and so on).

As one illustration, Szallasi and Liang argue that changes in large-scale gene expression patterns associated with conversion to malignancy depend on the nature of attractor transition in the underlying genetic network in three ways:

1. A specific oncogene can induce changes in the state of downstream genes (i.e., genes for which the oncogene is part of their regulatory network) and transition rules for those genes without driving the system from one attractor to another one. If this is true, inhibition of the oncogene will result a reversion of those downstream changes and a consequent normal phenotype. In some cases, just such phenomenology has been suggested,⁸⁰ although whether or not this mechanism is the basis of some forms of human cancer is unknown as yet.
2. A specific oncogene could force the system to leave one attractor and flow into another one. The new attractor might have a much shorter cycle time (implying rapid cell division and reproduction) and/or be more resistant to outside perturbations (implying difficulty in killing those cells). In this case, inhibition of the oncogene would not result in reversion to a normal cellular state.
3. A set of “partial” oncogenes may force the system into a new attractor. In this case, no individual partial oncogene would induce a phenotypical change by itself—however, the phenomenology associated with a new attractor would be similar.

These different scenarios have implications for both research and therapy. From a research perspective, the operation of the second and third mechanisms implies that the network's trajectory through state space is entirely different, a fact that would impede the effectiveness of traditional methodologies that focus on one or a few regulatory pathways or oncogenes. From a therapeutic standpoint, the operation of the latter two mechanisms implies that a focus on “knocking out the causal oncogene” is not likely to be very effective.

5.4.3.3. Genetic Regulation as Circuits

Genetic networks can also be modeled as electrical circuits.⁸¹ In some ways, the electrical circuit analogy is almost irresistible, as can be seen from a glance at any of the known regulatory pathways: the tangle of links and nodes could easily pass for a circuit diagram of Intel's latest Pentium chip. For example, McAdams and Shapiro described the regulatory network that governs the course of a λ-phage infection in E. coli as a circuit, and included factors such as time delays, which are critical in biological networks (gene transcription and translation are not instantaneous, for example) and indeed, in electrical networks, as well.

More generally, nature's designs for the cellular circuitry seems to draw on any number of techniques that are very familiar from engineering: “The biochemical logic in genetic regulatory circuits provides real-time regulatory control [via positive and negative feedback loops], implements a branching decision logic, and executes stored programs [in the DNA] that guide cellular differentiation extending over many cell generations.”⁸²Table 5.2 describes some of the similarities.

TABLE 5.2

Points of Similarity Between Genetic Logic and Electronic Digital Logic in Computer Chips.

Of course, taking an engineering view of biological circuits does not make understanding them trivial. For example, consider that cellular regulatory circuits implement a complex adaptive control system. Understanding this system is greatly complicated by the fact that at the biochemical implementation level, the distinction between the controlling mechanisms and the controlled processes is not as clear as it is when such control is engineered into a human-designed artifact. In a biochemical environment, control reactions and controlled functions are composed of intermingled molecules interacting in ways that make identification of roles much more complex.

Nor does the analogy to electrical circuits always carry over perfectly. Because critical molecules are often present in the cell in extremely small quantities, to take the most notable example, certain critical reactions are subject to large statistical fluctuations, meaning that they proceed in fits and starts, much more erratically than their electrical counterparts.

5.4.3.4. Combinatorial Synthesis of Genetic Networks⁸³

Guet et al. have demonstrated the feasibility of creating synthetic networks, composed of well-characterized genetic elements, that provide a framework for understanding how diverse phenotypical functionality can (but does not always) arise from changes in network topology rather than changes in the elements themselves. This functionality includes networks that exhibit the behavior associated with negative and positive feedback loops, oscillators, and toggle switches. By showing that functionality can change dramatically due to changes in topology, Guet et al. argue that once a simple set of genes and regulatory elements is in place, it is possible to jump discontinuously from one functional phenotype to another using the same “toolkit” of genes simply by modifying the regulatory connections. Such discontinuous changes are different from the more gradual effects driven by successive point mutations.

Such discontinuities reflect the nonlinear nature of genetic networks. Furthermore, the topology of connectivity of a network does not necessarily determine its behavior uniquely, and the behavior of even simple networks built out of a few well-characterized components cannot always be inferred from connectivity diagrams alone. Because genetic networks are nonlinear (and stochastic as well), the unknown details of interactions between components might be of crucial importance to understanding their functions. Combinatorially developed libraries of simple networks may thus be useful in uncovering the existence of additional regulatory mechanisms and exploring the limits of quantitative modeling of cellular systems.

The system of Guet et al. uses a small number of elements restricted to a single type of interaction (transcriptional regulation), but the range of biochemical interactions can be extended by including other modular genetic elements. For example, the approach can be extended to include linking input and output through cell-cell signaling molecules, such as those involved in quorum sensing. Also, this combinatorial strategy can be used to search for other dynamic behaviors such as switches, sensors, oscillators, and amplifiers, as well as for high-level structural properties such as robustness or noise resistance.

5.4.3.5. Identifying Systems Responses by Combining Experimental Data with Biological Network Information

Mawuenyega et al. have developed a method to identify specific subnetworks in large biological networks.⁸⁴ A biological network is constructed by identifying components (genes, proteins, transcription factors, chemicals) and interactions between components (protein-protein, protein-DNA, signal transduction, gene expression, catalysis) from genome context information as well as from external sources (databases, literature, and direct interaction with experimentalists). By superimposing experimental data such as expression values or identified proteins, it is possible to identify a best-scored subnetwork in the large biological network. This subnetwork is known as the response network, identifying a system's response with respect to the experimental scenario and data used.

Proteomic mass spectroscopy (MS) analysis was used to identify and characterize 1,044 Mycobacterium tuberculosis (TB) proteins and their corresponding cellular locations. From these 1,044 identified, 70 proteins were selected that are known to function in lipid biosynthesis (20) and fatty acid degradation (50). It is striking that the identified proteins involved in fatty acid degradation were distributed between the different cellular compartments in an almost exclusive fashion (e.g., in the subnetwork centered on fadB2 and fadB3) (Figure 5.9).

FIGURE 5.9

Fatty acid degradation network. SOURCE: Courtesy of Christian Forst, Los Alamos National Laboratories, December 8, 2004.

In addition, Forst and colleagues performed a response network analysis of mycobacterium tuberculosis to isoniazid (INH) drug treatment.⁸⁵ The entirety of the FAS-II fatty acid synthase group (except acpM, which was not included in the interaction data used to construct the original, whole network) showed up in the INH response network, all with significant up-regulation (Figure 5.10). Furthermore, the specific removal of these genes (kasA, kasB, fabD, accD6) from the initial set of genes did not affect their presence in the INH response subnetwork: the newly calculated network continued to contain each of them. Forst concluded that INH does directly interfere with the FAS-II fatty acid production pathway, in confirmation of earlier results.

FIGURE 5.10

The isoniazid (INH) response network. Red nodes indicate up-regulated genes. Blue nodes indicate down-regulated genes. SOURCE: Courtesy of Christian Forst, Los Alamos National Laboratories, December 8, 2004.

5.4.4.1. Multiscale Physiological Modeling⁸⁸

Physiological modeling is the modeling of biological units at a level of aggregation larger than that of an individual cell. Biological units can be successively decomposed into subunits (e.g., an organism may consist of subsystems for circulatory, pulmonary, digestive, and cognitive function; a digestive system may consist of esophagus, stomach, and intestines; and so on down to the level of organelles within cells and molecular functions within organelles), and every unit depends on the coordinated interaction of its subunits.

Given the complexity of physiological modeling, it makes sense to replicate this natural organization. Thus, models of tissue, organs, and even entire organisms are relevant subjects of physiological modeling. Functional behavior in each of these entities depends on activity at all spatial and temporal scales associated with structure from protein to cell to tissue to organ to whole organism (Box 5.11) and requires the integration of interacting physiological processes such as regulation, growth, signaling, metabolism, excitation, contraction, and transport processes. One term sometimes used for work that involves such integration is “physiome” (or by analogy to genomics,“physiomics”).⁸⁹

Box 5.11

Levels of Biological Organization. One helpful approach is to consider a set of different, but interrelated, levels of biological organization: Organ system, in which the entire organ can be represented by a lumped-parameter systems model that can be (more...)

Integration of such models presents many intellectual challenges. Following McCulloch and Huber,⁹⁰ it is helpful to consider two different types of integration. Structural integration implies integration across physical scales of biological organization from protein to whole organism, while functional integration refers to the integrated representation of interacting physiological processes. Structurally integrative models (e.g., models of molecular dynamics and other strategies that predict protein function from structure) are driven by first principles and hence tend to be computation-intensive. Because they are based on first principles, they impose constraints on the space of possible organismic models. Functionally integrative models are strongly data-driven and therefore data-intensive, and are needed to bridge the multiple time and space scales of substructures within an organism without leaving the problem computationally intractable. Box 5.12 provides a number of examples of intersection between structurally and functionally integrated models.

Box 5.12

Examples of Intersection Between Structurally and Functionally Integrated Models. There are a number of examples of intersection between structurally and functionally integrated models, including the following: Linkage of biochemical networks and spatially (more...)

Predictive simulations of subcomponents at various levels of the hierarchy of complexity are generally based on physicochemical first principles. Integrating such simulations, of which micromechanical tissue models and molecular dynamics models are examples, with each other across scales of biological organization is highly computationally intensive (and requires a computational infrastructure that enables distributed and heterogeneous computational resources to participate in the integration and facilitates the modular addition of new models and levels of organization).

5.4.4.2. Hematology (Leukemia)

Childhood acute lymphoblastic leukemia (ALL) is a lethal but highly treatable disease. However, successful treatment depends on the ability to deliver the correct intensity of therapy. Improper intensity can result in an excess of deaths caused by toxicity, decreased mental function over the long term, and undertreatment for high-risk cases.

The appropriate intensity is determined today through an extensive—and expensive—range of procedures including morphology, immunophenotyping, cytogenetics, and molecular diagnostics. However, Limsoon Wong has developed a relatively inexpensive single-platform microarray test that uses gene expression profiling to identify each of the known clinically important subgroups of childhood ALL (Figure 5.11) and hence the appropriate intensity of treatment.⁹¹ This is confirmed using computer-assisted supervised learning algorithms, in which an overall diagnostic accuracy of 96 percent was achieved in a blinded test sample. To determine whether expression profiling at diagnosis might further help identify those patients who are likely to relapse up to 4 years later, the expression profiles of four groups of leukemic samples with different outcomes were compared. Distinct gene expression profiles for each of these groups were identified.

FIGURE 5.11

Microarray expression groupings indicating known clinically important subgroups of childhood acute lymphoblastic leukemia (ALL). Note in particular the second column from the right, labeled “novel.” In this instance, the hierarchical clustering (more...)

5.4.4.3. Immunology

The immune system provides protection for human beings from pathogens. (For purposes of this discussion, the immune system of interest here refers to the adaptive immune system. The human body also has an innate immune system that provides a first response to pathogens that is essentially independent of the specific pathogen—in essence, its role is to give the adaptive immune system time to build a more specific response.) To do so, the immune system must first identify an entity within the body as a harmful pathogen that it should attack or eliminate and then mount a response that does so.

In principle, the identification of harmful pathogens might be based on a list of known pathogens. If an entity is found within the human body that is sufficiently similar to a known pathogen, it could be marked for later attack and destruction. However, a list-based approach to pathogen identification suffers from two major weaknesses. First, any such list would have to be large enough to include most of the possible pathogens that an organism might encounter in its lifetime; some estimates of the number of different foreign molecules that the human immune system is capable of recognizing are as high as 10¹⁶.⁹² Second, because pathogens evolve (and, thus, new pathogens are created), an a priori list could never be complete.

Accordingly, nature has developed an alternative mechanism for pathogen identification based on the notion of “self” versus “nonself.” In this paradigm, entities or substances that are recognized as self are deemed harmless, while those that are nonself are regarded as potentially dangerous. Thus, the immune system has developed a variety of mechanisms to differentiate between these two categories. Note that this distinction is highly simplistic, as not all nonself entities are bad for the human body (e.g., transplanted organs that replace original organs damaged beyond repair). Nevertheless, the self-nonself distinction is not a bad point of departure for understanding the human immune system.

The immune system relies on a process that generates detectors for a subset of possible pathogens and constantly turns over those detectors for new detectors capable of identifying a different set of pathogens. When the immune system identifies a pathogen, it selects one of several immunological mechanisms (e.g., those associated with the different immunoglobulin [Ig] groups) to eliminate it. Furthermore, the immune system retains memory of the pathogen, in the form of detectors that are specifically configured for high affinity to that pathogen. Such memory enables the immune system to confer long-lasting resistance (immunity) to pathogens that may be encountered in the future and to mount a stronger response to such future encounters.

Many of the broad outlines of the immune system are believed to be understood, and computational modeling of the immune system has shed important light on its detailed workings, as described in Box 5.13. A medical application of simulation models in immunology has been to evaluate the effects of revaccinating someone yearly for influenza. Because of the phenomenon of immune memory, a vaccine that is too similar to a prior year's vaccine will be eliminated rapidly by the immune response (a negative interference effect). A simulation model by Smith et al. has examined this effect and suggests some circumstances under which individuals who are vaccinated annually will have greater or less protection than those with a first-time vaccination.⁹³ The Smith et al. results also suggested that in the production of flu vaccine, a choice among otherwise equivalent strains (i.e., strains thought to be equally good guesses of the upcoming epidemic strain and equally appropriate for manufacture) should be resolved in favor of the strain that is most different from the one used in the previous year, because this choice would reduce the effects of negative interference and thus potentially increase vaccine efficacy in recipients of repeat vaccines.

Box 5.13

Modeling in Immunology. In basic immunology, issues related to mutation also have been the focus of mathematical modeling and intense experimentation…. [For example,] during the course of an immune response, B lymphocytes within germinal centers (more...)

5.4.4.4. The Heart

The heart is an organ of primary importance in vertebrates, and heart disease is one of the primary causes of death in the Western world. At the same time, the heart is an organ of high complexity. Although it is in essence an impulsive pump, it is a pump that must operate continuously and repair itself if necessary while in operation. Its output must be regulated according to various physiological conditions in the body, and its performance is affected by the characteristics of the arterial and vein networks to which it is connected.

The heart brings together many subsystems that interact mutually through fundamental physiological processes. As a general rule, physiological processes have both functional and structural dimensions. For example, cells are functionally specialized—blood cells and myocytes (heart cells) do different things. Furthermore, blood cells and heart cells are themselves part of a collective of other blood cells and heart cells; thus, the structure within which an individual cell is embedded is relevant.

An integrated computational model of the heart would bring together all of the relevant physiological processes (Box 5.14).⁹⁴ Were such a model available, it would be possible to investigate common heart diseases and to probe cardiac structure and function in different places in the heart—a point of some significance in light of the fact that heart failure is usually regional and nonhomogeneous. The graphic in Box 5.14 emphasizes functional integration in the heart, and the majority of functional interactions take place at the scale of the single cell. However, an organism's behavior depends on interactions that span many orders of magnitude of space and time (from molecular structures and events to whole-organ anatomy and physiology). Thus, high-fidelity modeling of an organism or organ system within an organism demands the integration of information across similar scales.

Box 5.14

Computational Modeling of the Heart. … [Integrative cardiac modelling has sought] to integrate data and theories on the anatomy and structure, hemodynamics and metabolism, mechanics and electrophysiology, regulation and control of the normal and (more...)

An example of a functional model of a single cell is the work of Winslow et al. in modeling the cardiac ventricular cell, and specifically the relationship between various current flows in the cell and its contractile behavior.⁹⁵ In particular, Winslow has used this model to show that the reduced contractility (i.e., reduction in the strength with which a ventricular muscle contracts, which is associated with heart failure) is caused largely by changes in the calcium ion currents in those cells, rather than changes in potassium ion currents as was widely speculated before this work (Example 1 in Box 5.15). Such an insight suggests that the development of drugs to cope with heart failure would thus be better focused on those that can regulate calcium flow. Examples 2 and 3 in Box 5.15 illustrate some of the scientific insights that can be gained with a computational model integrated across functional and structural lines.

Box 5.15

Illustrations of Functional Models of Cellular Behavior. Example 1: Results from Single-cell Modeling Winslow et al. have developed and applied a model of the normal and failing canine ventricular myocyte to analysis of the functional significance of (more...)

Integrating these various perspectives on the heart (and other organs as well) is the mission of the Physiome Project, which seeks to construct models that incorporate the detailed anatomy and tissue structure of an organ in a way that allows the inclusion of cell-based models and spatial structure and distribution of proteins. The Physiome project has developed a computational framework for integrating the electrical, mechanical, and biochemical functions of the heart:⁹⁶

• The underlying anatomical descriptions are based on finite element techniques, and orthotropic constitutive laws based on the measured fiber-sheet structure of myocardial tissue drive the dynamics of the large deformation soft-tissue mechanics involved.
• Patterns of electrical current flow in the heart are computed using reaction-diffusion equations on a grid of deforming material points which access systems of ordinary differential equations representing the cellular processes underlying the cardiac action potential; these result in representations of the activation wavefronts that spread around the heart and initiate contraction.
• Coronary blood flow is computed based on the Navier-Stokes equations in a system of branching blood vessels embedded in the deforming myocardium and the delivery of oxygen and metabolites is coupled to the energy-dependent cellular processes.

These models of different cardiac phenomena have been been implemented with “horizontal” integration of mechanics, electrical activation and metabolism, together with “vertical” integration from cell to tissue to organ. Thus, these models can be said to deconstruct an organ into a set of (submodels for) constituent functions, with explicit feedback and connection between them represented in the overall model of the whole organ.

5.4.5.1. The Broad Landscape of Computational Neuroscience

Neuroscience seeks to probe the details of the brain and the mechanisms by which the nervous systems develops, is organized, processes information, and establishes mental abilities. Research in neuroscience spans many levels of organization, from atomic and molecular events on the order of one-tenth to one nanometer, up to the entire nervous system on the order of a meter or more. In addition, there are on the order of 10¹¹ neurons and thousands to tens of thousands of synapses per neuron.

Information processing in the brain occurs through the interactions and spread of chemical and electrical signals both within and among neurons. Acting within the extensive but intricate architecture of the neurons and their interconnections, the mechanisms are nonlinear and span a wide range of spatial and temporal scales.⁹⁷ Understanding how the nervous system and brain work thus requires an interdisciplinary approach to the challenging multiscale integration of experimental data, computational data, and theory.

It is helpful to describe the nervous system's functional processes and their mechanisms at several different levels of detail, depending on the goal of a given effort. Table 5.3 and Figure 5.12 describe the numerous spatial and temporal scales relevant to neuroscience research, and provide some indication of the complexity of such research.

TABLE 5.3

Scales of Neuroscience Research.

FIGURE 5.12

Temporal and spatial scales of neuroscience research. SOURCE: Courtesy of Christof Koch, Caltech.

To illustrate, a low level of analysis might involve consideration of individual neurons. In this analysis, functional properties of neurons such as electronic structure, nerve cell connections (synapses), and voltage-gated ion channels are important. At a higher level, it is recognized that individual neurons connect in networks—an analysis at this level examines how individual neurons interact to form functioning circuits. The mathematics of dynamic systems and visual neuroscience are notably relevant at this level. At a still higher level, individual networks—each with its own specific architecture and information-processing capabilities—interact to form neural nets and carry out cognition, speech perception, and imaging. At this level, computational analysis of nervous system networks and (connectionist) modeling of psychological processes is the primary focus.

Computational neuroscience provides the basis for testing models of the nervous system's functional processes and their mechanisms, and computational modeling at several levels of detail is important, depending on the purposes of a given effort. Box 5.16 describes simulators that operate at different levels of detail for different purposes.

Box 5.16

Simulators for Computational Neuroscience. The nervous system is extraordinarily complex. A single cubic centimeter in the brain's cerebral cortex contains on the order of 5 billion synapses, and these differ in size and shape. The transmission of chemical (more...)

5.4.5.2. Large-scale Neural Modeling⁹⁸

To better understand a system as complex as the human brain, it is necessary to develop techniques and tools for supporting large-scale, similarly complex simulations. Recent advances in understanding how single neurons represent the world,⁹⁹ how large populations of neurons cooperate to build more complex representations,¹⁰⁰ and how neurobiological systems compute functions over their representations make large-scale neural modeling a highly anticipated next step.

Recent theoretical work has suggested that it is possible to generally characterize the dynamics, representation, and computational properties of any neural population (Figure 5.13).¹⁰¹ Applications of these methods have been used successfully to generate models of working memory, rodent navigational tasks (path integration; see Figure 5.14), eye position control, representation of self-motion, lamprey and fish motor control, and deductive reasoning (Figure 5.15).

FIGURE 5.13

A generic neural subsystem. The diagram depicts the mathematical analysis of a neural subsystem and its mapping onto the biological system—a population of neurons. Labels outside the gray boxes indicate the relevant biological structures and processes. (more...)

FIGURE 5.14

Rodent navigation. These figures depict the behavior of a neurally realistic simulation of the path integrator in a rat. The simulation was generated by using a single (recurrent) generic neural subsystem. (A) When the simulation is given random noise, (more...)

FIGURE 5.15

System for learning and performing deductive reasoning. The graphic describes the proposed system used during solution of the Wason card selection task; see P.C. Wason and P.N. Johnson-Laird, Psychology of Reasoning: Structure and Content, Harvard University (more...)

Box 5.17 illustrates the use of computational modeling to understand how dopamine functions in the prefrontal cortex. The box also illustrates the often-present tension between those who believe that simple models (in this case, advocates of a connectionist model) can provide useful insight and those who believe that simple models cannot capture the implications of the complex dynamics of individual neurons and their synapses and that the addition of considerable biophysical and physiological detail is needed for real understanding. Many of these models require large numbers of individual, spiking neurons to be simulated concurrently, which results in significant computational demands. In addition, calculating the necessary connection weights requires the inversion of extremely large matrices. Thus, high-performance computing resources are essential for expanding these simulations to include more neural tissue, and hence more complex neural function.

Box 5.17

Computational Perspectives on Dopamine Function in the Prefrontal Cortex. Connectionist Models of Dopamine Neuromodulation A long-held hypothesis suggests that catecholamine neurotransmitters, including dopamine (DA), modulate target neuron responses, (more...)

5.4.5.3. Muscular Control

Muscles are controlled by action potentials—brief, rapid depolarizations of membranes in nerves and muscles. The timing of action potentials transmitted from motor neurons coordinates the contraction of the muscles they innervate. Rhythmic activity of the nervous system often takes the form of complex bursting oscillations in which intervals of action potential firing and quiescent intervals of membrane activity alternate. The relative timing of action potentials generated by different neurons is a key ingredient in the function of the nervous system.

Changes in the electrical potential of membranes are mediated by ion channels that selectively permit the flow of ions such as sodium, calcium, and potassium across the membrane. Individual channels are protein complexes containing membrane-spanning pores that open and close randomly at rates that depend on many factors. Cellular and network models of membrane potential represent these systems as electrical circuits in which voltage gated channels function as “nonlinear” resistors whose conductance depends on membrane potential. Information is transmitted from one neuron to another through synapses where action potentials trigger the release of neurotransmitters that bind to channels of adjacent cells, stimulating changes in the ionic currents of these cells. (The action potential is the basic neuronal signaling “packet” of ionic flow through a cell membrane.)

The most basic model of this mechanism is the Hodgkin-Huxley model, which refers to a set of differential equations that describe the action potential.¹⁰² Specifically, the Hodgkin-Huxley equations express quantitatively the feedback entailed in the relationship between changing ionic flows and changing membrane potential. Originally based on data collected from experiments on the giant axon of the squid, the physical model used is that of a membrane separating two infinite regions, each of which is homogeneous on its side of the membrane.

In the nervous system, different kinds of ions pass through the membrane, and the flow of ions through these channels is voltage dependent. In the model, a circuit is used to represent the ion flows and potential differences that drive ion flow. The semipermeable cell membrane separating the interior of the cell from the extracellular liquid is modeled as a capacitor, and each ion channel is modeled as a separately variable resistor. In series with each variable resistor is a battery representing the Nernst potential arising from the difference in ion concentration on each side of the membrane. All of these components are connected in parallel and are driven by a time-varying current source to ground. If a time-varying input current is injected into the cell, it may add further charge on the capacitor, or the added charge may leak through the channels in the cell membrane. Because of active ion transport through the cell membrane, the ion concentration inside the cell is different from that in the extracellular liquid. The potential generated by the difference in ion concentration is represented by a battery.

Elementary circuit theory allows the construction of a set of differential equations relating the different ion currents to the potential difference across the membrane. Using this set of differential equations, certain essential features of neural behavior can be modeled. For example, assuming appropriate parameter values, a constant input current larger than a certain critical value and turned on at a given instant of time results in the potential difference across the membrane taking the form of a regular spike train—which is reminiscent of how a real neuron fires. More realistic current inputs (e.g., stochastic ones) result in a much more realistic-looking output.

Despite lack of information about much of the cellular and molecular basis of neuronal excitation at the time, Hodgkin and Huxley were able to provide a relatively accurate quantitative description of how an action potential was generated by voltage-dependent ionic conductivities. The Hodgkin-Huxley model provided the basis for research for more than five decades, spinning off a new field of neurophysiology: in large part, this field rests on the foundation created by their model. Recent research on membrane ion channels can be related directly to the seminal ideas and (more importantly) precise mechanism that their model described.

The “plain vanilla” Hodgkin-Huxley model is still interesting today. For example, a recent study demonstrated previously unobserved dynamics in the Hodgkin-Huxley model, namely, the existence of chaotic solutions in the model with its original parameters.¹⁰³ The significance of chaos in this context is that the excitability of a neural membrane with respect to firing is likely to be more complex than can be explained by a simple sub- or super-threshold potential.

Simulation and mathematical analysis of models have become essential tools in investigations of the complicated processes underlying rhythm generation in the nervous system. There are many types of channels and synapses. The number of channels and synapses and their locations distinguish different types of neurons from one another. Simulation of networks consisting of model neurons with specified conductances and synapses enables researchers to test their intuitions regarding how these networks function. Simulations also lead to predictions of the effects of neuromodulators and disorders that affect the electrical excitability of the systems. Nonetheless, simulation alone is not sufficient to determine the information we would like to extract from these models. The models have large numbers of parameters, many of which are difficult or impossible to measure, and the goal is to determine how the system behavior depends on the values of all of these parameters.

Dynamical systems theory provides a conceptual framework for characterizing rhythms. This theory explains why there are only a small number of dynamical mechanisms that initiate or terminate bursts of action potentials, and it provides the foundations for algorithms that compute parameter space maps delineating regions with different dynamical behaviors. The presence of multiple time scales is an important ingredient of this analysis because the rates at which different families of channels respond to changes in membrane potential or ligand concentration vary over several orders of magnitude.

Figure 5.16 illustrates this type of analysis using a model for bursting in the pre-Bötzinger complex, a neural network in the brain stem that controls respiration. The first panel shows voltage recordings from intracellular recordings of a medullar slice from neonatal rats. Butera and colleagues measured conductances in this preparation and constructed a model for this system.¹⁰⁴ Simulations of the burst ing rhythms displayed by this model are shown in the second panel. The third panel shows a map of the simulated trajectory that illustrates the relationship of the bursting to slow and fast variables in the system.

FIGURE 5.16

Bursting in the pre-Botzinger complex. Panel 1: Example of voltage-dependent properties of pre-Bötzinger complex (pre-BötC) inspiratory bursting neurons. Traces show whole-cell patch-clamp recordings from a single candidate pacemaker neuron (more...)

5.4.5.4. Synaptic Transmission

The intercellular signaling process of synaptic transmission is a much-studied problem. Much has been learned about synaptic structure and function through the classical techniques of neuropharmacology, electron microscopy (EM) neuroanatomy, and electrophysiology, and correlation of the observations made through these various techniques has led to the development of computational models of synaptic microphysiology. However, the scope of previous modeling attempts has been limited by available computing power, modeling framework, and lack of high-resolution three-dimensional ultrastructural data in an appropriate machine representation.

What has been missing is an appropriate set of tools for acquiring, building, simulating, and analyzing biophysically realistic models of subcellular microdomains. Coggan et al. have developed and used a suite of such computational tools to build a realistic computational model of nicotinic synaptic transmission based on serial electron tomograms of a chick ciliary ganglion somatic spine mat.¹⁰⁵

The chick ciliary ganglion somatic spine mat is a complex system with more than one type of neurotransmitter receptor, possible alternative locations for transmitter release, and a tortuous synaptic geometry that includes a spine mat and calyx-type nerve terminal. Highly accurate models of the synaptic ultrastructure are obtained through large-scale, high-resolution electron tomography; computer-aided methods for extracting accurate surfaces and defining in-silico representations of their molecular properties; and physiological underpinnings from a variety of studies conducted by the involved laboratories and from the literature.

These data are then used as the framework for advanced simulations using MCell running on high-performance supercomputers as well as distributed or grid-based computational resources. This project pushes development of tools for acquisition of improved large-scale tomographic reconstructions of cellular interfaces down to supramolecular scales. It also drives improvements in the software tools both for the distribution of molecular components within the surface models extracted from the tomographic reconstructions and for the deposition and retrieval of relevant information for the MCell simulator (Box 5.18) in the tomography and Cell-Centered Database (CCDB) environment.

Box 5.18

The MCell Simulator. MCell is a general Monte Carlo simulator of cellular microphysiology. MCell simulations provide insights into the behavior and variability of real systems comprising finite numbers of molecules interacting in spatially complex environments. (more...)

Realistic modeling of synaptic microphysiology (as illustrated in Figure 5.17) requires the following:

FIGURE 5.17

Constructing the geometry of a chick ciliary ganglion (CG) somatic spine mat model. A serial EM tomogram of a CG spine mat was obtained at ~4 nm per voxel resolution. The serial tomogram encompassed a volume of ~27 mm³ (~3 µm × 3 µm (more...)

1. Acquisition of high-resolution, three-dimensional synaptic ultrastructure—this is accomplished with serial EM tomography.
2. Segmentation of pre- and postsynaptic membrane from the tomographic volume—this is accomplished using the tracing tool in Xvoxtrace.
3. Three-dimensional reconstruction of the membrane surface topology to form a triangle mesh—this is accomplished using the marching cubes isosurface extraction tool in Xvoxtrace.
4. Subdivision of the membrane surface meshes into physiologically relevant regions (e.g., spine versus nonspine membrane and PSD [phosphorylation site domain] versus non-PSD regions)—this is accomplished using the mesh tagging tool in DReAMM.
5. Placement of effector molecules (e.g., receptors, enzymes, reuptake transporters) onto membrane surfaces with the desired distribution and density—this is accomplished using the MCell model description language (MDL). Effector distribution and density may be determined by labeling and imaging studies.
6. Specification of the diffusion constant, quantity, and location of neurotransmitter release—this is accomplished using MCell MDL.
7. Specification of the reaction mechanisms and kinetic rate constants governing the mass action kinetics interaction of neurotransmitter and effector molecules—this is accomplished using MCell MDL.
8. Specification of what quantitative measures should be made during the simulation—this is accomplished using MCell MDL.
9. Simulation of the defined system—this is accomplished using the MCell compute kernel.
10. Analysis of the results at various points in the parameter space defined by the system—this is accomplished using analysis tools of the investigator's discretion.

Analysis of miniature excitatory postsynaptic currents (mEPSCs) recorded in electrophysiological experiments shows that mEPSCs in the CG somatic spine mat occur in a broad spectrum of amplitudes, rise times, and fall times. The differential kinetics and complementary distributions of α3 and α7 nAChRs are expected to lead to mEPSCs whose characteristics are highly dependent on the location of neurotransmitter release within the spine mat. Realistic simulation makes it possible to explore and quantify the degree to which this hypothesis is true and to make quantitative comparisons of the simulation and electrophysiological results. Figure 5.18 summarizes the results of simulations designed to explore the limits of mEPSC behavior by virtue of the choice of neurotransmitter release locations. The results not only confirm the qualitative expectations at each site but also predict their quantitative behavior, allowing fine discriminations to be made.

FIGURE 5.18

Summary of synaptic transmission simulations. (A) Location of selected transmitter release sites and their associated simulated mEPSC traces, each decomposed into their α3 and α7 nAChR components. Each trace is the average of 100 simulations (more...)

The process briefly outlined above represents a significant advance in the ability to create realistic computational models of subcellular microdomains from actual cellular ultrastructure. The preliminary results presented are just the beginning of exciting computational experiments that can now be performed on the CG model in an effort to illuminate and inform further bench experiments. Among all of the things learned, perhaps the most important is which of the physical characteristics of the CG are the least constrained, are least understood, and have the greatest impact on synaptic function. Specifically, the results clearly demonstrate that synaptic geometry, receptor distribution, and vesicle release location each have a profound quantitative impact on the efficacy of the postsynaptic response. This means that attention to accuracy in the model-building process must be a prime concern.

5.4.5.5. Neuropsychiatry¹⁰⁶

The field of computational neuropsychiatry has been exploding with applications of large-deformation brain mapping technology that provide mechanisms for discovering neuropsychiatric disorders of many types. The hippocampus is a region of the brain (depicted in green in Figure 5.19) that has been implicated in schizophrenia and other neurodegenerative diseases such as Alzheimer's. Using large-deformation brain mapping tools in computational anatomy, researchers can define, visualize, and measure the volume and shape of the hippocampus. These methods allow for precise assessment of changes in hippocampal formation.

FIGURE 5.19

The hippocampus in situ. SOURCE: Courtesy of Michael Miller, Johns Hopkins University.

Researchers at the Center for Imaging Science (CIS) used mapping tools to compare the left and right hippocampi (Figure 5.20) in 15 pairs of schizophrenic and control subjects. In the schizophrenic subjects, deformations were localized to hippocampal subregions that send projections to the prefrontal cortex. The deformations strongly distinguish schizophrenic subjects from control subjects. The pictures indicate inward deformations by cooler colors, outward deformations by warmer colors, and little deformation by a neutral green color. These results support the current hypothesis that schizophrenia involves a disturbance of hippocampal-prefrontal connections.

FIGURE 5.20

Left and right hippocampuses. SOURCE: Courtesy of Michael Miller, Johns Hopkins University.

In a separate study, CIS researchers also compared asymmetry between the left and right hippocampi. The left and the right side of normal brains develop at different rates. Structures on both sides of the brain are similar, but not identical. This is normal brain asymmetry. If a different asymmetry pattern exists in schizophrenic subjects, it may indicate a disturbance of the left-right balance during early stages of brain development. Researchers found that the left hippocampus was narrower along the outside edge than the right hippocampus. This asymmetry was similar in schizophrenic and normal subjects (Figure 5.21, left image). However, further comparison revealed a significant difference in asymmetry patterns of the hippocampal area called the subiculum (Figure 5.21, right image). People with schizophrenia tend to have a more pronounced depression and a downward bend in the surface of that structure.

FIGURE 5.21

Asymmetry in schizophrenia. SOURCE: Michael Miller, Johns Hopkins University.

As part of Washington University's Healthy Aging and Senile Dementia (HASD) program, CIS researchers have also applied brain mapping tools to assess the structure of the hippocampus in older human subjects (depicted in Figure 5.22). They compared measurements of hippocampal volume and shape in 18 subjects with early dementia of the Alzheimer type (DAT) with 18 healthy elderly and 15 younger control subjects. Hippocampal volume loss and shape deformities observed in subjects with DAT distinguished them from both elderly and younger control subjects. The pattern of hippocampal deformities in subjects with DAT was largely symmetric and suggested damage to the CA1 hippocampal subfield.

FIGURE 5.22

Hippocampal structure in normal aging (left) versus in Alzheimer's disease patients (right). SOURCE: Courtesy of Michael Miller, Johns Hopkins University.

Hippocampal shape changes were also observed in healthy elderly subjects, which distinguished them from healthy younger subjects. These shape changes occurred in a pattern distinct from the pattern seen in DAT and were not associated with substantial volume loss. These assessments indicate that hippocampal volume and shape derived from computational anatomy large deformation brain mapping tools may be useful in distinguishing early DAT from healthy aging.

5.4.8.1. Commonalities Between Evolution and Ecology

No two fields in biology encompass such a broad range of levels of biological organization as ecology and evolutionary biology. Although the two fields ask different questions, they both contend with factors of space and time, and share common theories about relationships between individuals, populations, and communities. The two intertwined fields view these relationships in different ways. Evolutionary biologists want to understand and quantify the effect of environment (e.g., natural selection) on individuals and populations; ecologists want to understanding the role of individuals and populations in shaping their environment (ecological inheritance, niche construction).

The two fields encompass a diverse assemblage of topics with applications in resource management, epidemiology, and global change. In these fields, data have been relatively difficult to collect in ways that relate directly to mathematical or computational models, although this has been changing over the past 10 years. Thus, both fields have relied heavily on theory to advance their insights. In fact, ecology and evolution have been the substrate for the development of important mathematical concepts. The quantitative study of biological inheritance and evolution provided the context for statistics, probability theory, stochasticity, and dynamical systems theory.

Among the fundamental questions in the study of evolution are those that seek to know the relative strengths of natural selection, genetic drift, dispersal processes, and genetic recombination in shaping the genome of a population—essentially the forces that provide genetic variability in a species. Both ecologists and evolutionary biologists want to know how these forces lead to morphological changes, speciation, and ultimately, survival over time. The fields seek theory, models, and data that can account for genetic changes over time in large heterogeneous populations in which genetic information is exchanged routinely in an environment that also exerts its influence and changes over time.

In addition to interest in genetic variability and fitness within a single species, the two fields are interested in relationships between multiple species. In ecology, this manifests itself in questions of how the individual forces of variability within and between species affect their relative ability to compete for resources and space that leads to their survival or extinction—in other words, forces that determines the biodiversity of an ecosystem (i.e., a set of biological organisms interacting among themselves and their environment). Ecologists want to understand what determines the minimum viable population size for a given population, the role of keystone species in determining the diversity of the ecosystem, and the role of diversity in preservation of the ecosystem.

For evolutionary biologists, questions regarding relationships between species focus on trying to understand the flow of genetic information over long periods of time as a measure of the relatedness of different species and the effects of selection on the genetic contribution to phenotypes. Among the great mysteries for evolutionary biologists is whether and how evolution relates to organismal development, an interaction for which no descriptive language currently exists.

How will ecologists and evolutionary biologists answer these questions? These fields have had few tools to monitor interactions in real time. But new opportunities have emerged in areas from genomics to satellite imaging and in new capabilities for the computer simulation of complex models.

5.4.8.2. Examples from Evolution

A plethora of genomic data is beginning to help untangle the relationship between traits, genes, developmental processes, and environments. The data will serve as the substrate from which new statistical conclusions can be drawn, for example, new methods for identifying inherited gene sequences such as those related to disease. To answer question about the process of genome rearrangement, the possibility of comparing gene sequences from multiple organisms provides the basis for testing tools that discern repeatable patterns and elucidate linkages.

As more detailed DNA and protein sequence information is compiled for more genes in more organisms, computational algorithms for estimating parameters of evolution have become extremely complex. New techniques will be needed to handle the likelihood functions and produce satisfactory statistics in a reasonable amount of time. Studies of the role of environmental and genetic plasticity in trait development will involve large-scale simulations of networks of linked genes and their interacting products. Such simulations may well suggest new approaches to such old problems as the nature-nurture dichotomy for human behaviors.

New techniques and the availability of more powerful computers have also led to the development of highly detailed models in which a wide variety of components and mechanisms can be incorporated. Among these are individual unit models that attempt to follow every individual in a population over time, thereby providing insight into dynamical behavior (Box 5.22).

Box 5.22

The Dynamics of Evolution. Avida is a simulation software system developed at the Digital Life Laboratory at the California Institute of Technology. In it, digital organisms have genomes comprised of a sequence of instructions that operate on a virtual (more...)

Levin argues that such models are “imitation[s] of reality that represent at best individual realization of complex processes in which stochasticity, contingency, and nonlinearity underlie a diversity of possible outcomes.”¹⁰⁹ From the collective behaviors of individual units arise the observable dynamics of the system. “The challenge, then, is to develop mechanistic models that begin from what is understood about the interactions of the individual units, and to use computation and analysis to explain emergent behavior in terms of the statistical mechanics of ensembles of such units.” Such models must extrapolate from the effects of change on individual plants and animals to changes in the distribution of individuals over longer time scales and broader space scales and hence in community-level patterns and the fluxes of nutrients.

5.4.8.2.1. Reconstruction of the Saccharomyces Phylogenetic Tree

Although the basic structure and mechanisms underlying evolution and genetics are known in principle, there are many complexities that force researchers into computational approaches in order to gain insight. Box 5.23 addresses complexities such as multiple loci, spatial factors, and the role of frequency dependence in evolution, and discusses a computational perspective on the evolution of altruism, a behavioral characteristic that is counterintuitive in the context of individual organisms doing all that they can to gain advantage in the face of selection pressures.

Box 5.23

Genetic Complexities in Evolutionary Processes. The dynamics of alleles at single loci are well understood, but the dynamics of alleles at two loci are still not completely understood, even in the deterministic case. As a rule, two-locus models require (more...)

Along these lines, a particularly interesting work on the reconstruction of phylogenies was reported in 2003 by Rokas et al.¹¹⁰ One of the primary goals of evolutionary research has been understanding the historical relationships between living organisms—reconstruction of the phylogenetic tree of life. A primary difficulty in phylogenetic reconstruction is that different single-gene datasets often result in different and incongruent phylogenies. Such incongruences occur in analyses at all taxonomic levels, from phylogenies of closely related species to relationships between major classes or phyla and higher taxonomic groups.

Many factors, both analytical and biological, may cause incongruence. To overcome the effect of some of these factors, analysis of concatenated datasets has been used. However, phylogenetic analyses of different sets of concatenated genes do not always converge on the same tree, and some studies have yielded results at odds with widely accepted phylogenies.

Rokas et al. exploited genome sequence data for seven Saccharomyces species and for the outgroup fungus Candida albicans to construct a phylogenetic tree. Their results suggested that datasets consisting of a single gene or a small number of concatenated genes had a significant probability of supporting conflicting topologies, but that use of the entire dataset of concatenated genes resulted in a single, fully resolved phylogeny with the maximum likelihood. In addition, all alternative topologies resulting from single-gene analyses were rejected with high probability. In other words, even though the individual genes examined supported alternative trees, the concatenated data exclusively supported a single tree. They concluded that “the maximum support for a single topology regardless of method of analysis is strongly suggestive of the power of large data sets in overcoming the incongruence present in single-gene analyses.”

5.4.8.3. Examples from Ecology¹²²

Simulation-based study of an ecosystem considers the dynamic behavior of systems of individual organisms as they respond to each other and to environmental stimuli and pressures (e.g., climate) and examines the behavior of the ecosystem in aggregate terms. However, no individual run of such a simulation can be expected to predict the detailed behavior of each individual organism within an ecosystem. Rather, the appropriate test of a simulation's fidelity is the extent to which it can, through a process of judicious averaging of many runs, predict features that are associated with aggregation at many levels of spatial and/or temporal detail. These more qualitative features provide the basis for descriptions of ecosystem dynamics that are robust across a variety of dynamical scenarios that are different at a detailed level and also provide high-level descriptions that can be more readily interpreted by researchers.

Because of the general applicability of the approach described above, simulations of dynamical behavior can be developed for aggregations of any organisms as long as they can be informed by adequate understandings of individual-level behavior and the implications of such behavior for interactions with other individuals and with the environment.

Note also the key role played by ecosystem heterogeneity. Spatial heterogeneity is one obvious way in which nonuniform distributions play a role. But in biodiversity, functional heterogeneity is also important. In particular, essential ecosystem functions such as the maintenance of fluxes of certain nutrients and pollutants, the mediation of climate and weather, and the stabilization of coastlines may depend not on the behavior of all species within the ecosystem but rather on a limited subset of these species. If biodiversity is to be maintained, the most fragile and functionally critical subsets species must be identified and understood.

The mathematical and computational challenges range from techniques for representing and accessing datasets, to algorithms for simulation of large-scale spatially stochastic, multivariate systems, to the development and analysis of simplified description. Novel data acquisition tools (e.g., a satellite-based geographic information system that records changes for insertion in the simulations) would be welcome in a field that is relatively data poor.