Reproduced from Annu. Rev. Biophys. Biomol. Struct. 2000. 29:327-359.

This review describes how kinetic experiments using techniques with dramatically improved time resolution have contributed to understanding mechanisms in protein folding. Optical triggering with nanosecond laser pulses has made it possible to study the fastest-folding proteins as well as fundamental processes in folding for the first time. These include formation of α-helices, β-sheets, and contacts between residues distant in sequence, as well as overall collapse of the polypeptide chain. Improvements in the time resolution of mixing experiments and the use of dynamic nuclear magnetic resonance methods have also allowed kinetic studies of proteins that fold too fast (≳10³ s^-1) to be observed by conventional methods. Simple statistical mechanical models have been extremely useful in interpreting the experimental results. One of the surprises is that models originally developed for explaining the fast kinetics of secondary structure formation in isolated peptides are also successful in calculating folding rates of single domain proteins from their native three-dimensional structure.

Introduction and Overview

Protein folding is a subject that is attracting scientists from a wide range of disciplines (34). The interest arises from the explosion of information on protein sequences and the challenge of understanding one of the most fundamental biochemical processes. The protein-folding problem is generally divided into two parts. In the first the goal is to predict the three-dimensional structure from the amino acid sequence. The second part is the no less daunting task of understanding the relation between protein sequences and mechanisms of folding. In protein folding, the mechanism is the distribution of microscopic pathways that connect the myriad of structures of the denatured state with the unique structure of the native state.

Over the past decade major advances have occurred in both theoretical (7, 19, 29, 37, 50, 83, 85, 103, 115) and experimental approaches to investigating protein-folding mechanisms. A turning point in experimental studies was the recognition that it is essential to first characterize the kinetics and thermodynamics of folding in the simplest systems. The idea of studying small, single domain proteins without additional chemical complexity from disulfide bridges, metals, or other cofactors began with the equilibrium and kinetic experiments on chymotrypsin inhibitor 2 (CI2) (56). CI2 was found to fold and unfold as a simple two-state system with no kinetic intermediates. In subsequent work, the participation of individual residues in the transition state for folding was investigated by studying the relative effects of mutations on folding rates and equilibrium constants (35, 54). The studies on CI2 sparked considerable interest among experimentalists in carefully characterizing the kinetics, thermodynamics, and effects of mutations in other small, single domain proteins (34, 55).

A second major advance in experimental studies has resulted from the introduction of a new generation of experiments with dramatically improved time resolution. The contribution of these studies to our understanding of protein folding is the subject of this review. Until just a few years ago, folding kinetics were studied almost exclusively using stopped-flow techniques. In this experiment, protein folding is initiated by rapidly mixing a chemically denatured protein solution with a buffer to dilute the denaturant. The kinetics of folding are then monitored with one of several optical spectroscopic methods. Similarly, unfolding can be initiated by mixing a native protein solution with concentrated denaturant. Stopped flow experiments have yielded an enormous amount of valuable information that has provided the basis for much of what we know about the kinetics of folding and unfolding. It does, however, have a fundamental limitation—poor time resolution. The time required to mix solutions and move them into an observation cuvette is generally at least a few milliseconds. This is the so-called “dead time” of the instrument. A typical observation in many experiments has been that much, and in some cases all, of the spectroscopic changes associated with folding already occur within this dead time (64, 72, 95, 98). Furthermore, it was known for some time from studies on synthetic polymers that elementary processes such as α-helix formation are too fast to be observed in stopped-flow experiments (44).

A second motivation for developing fast kinetic methods has been to provide a much-needed reality check on computer simulations, which are flooding the protein-folding literature. Our conceptualization of the folding process is strongly influenced by the results of simulations, so it is important to perform real experiments on time scales that can be directly compared with the computer experiments. So far the most important insights have come from simulations of simplified representations of proteins in lattice and off-lattice models (19, 29, 83, 85, 103, 115). Such models provide simple, concrete examples that can be extremely helpful in clarifying both theoretical and experimental issues. However, the most detailed and potentially realistic kinetic simulations employ all-atom molecular dynamics calculations. With few exceptions (23, 24, 75), these calculations have been restricted by computer time to one or just a few trajectories of tens of nanoseconds or less. They are therefore not yet capable of direct simulation of protein folding (see, however, Ref. 30) but are rapidly approaching time scales long enough to simulate secondary structure formation.

Finally, development of fast methods has been motivated by theoretical studies that have introduced an energy landscape approach to protein folding (9). A particularly interesting result from calculations of free energy surfaces is that the barriers to protein folding should be quite small, suggesting that for the fastest folding proteins the barrier might disappear altogether. This barrierless or so-called downhill folding could produce unusual (i.e. nonexponential) kinetics (8). The energy landscape perspective also suggested that in many proteins the multiphasic kinetics observed in stopped-flow experiments arise from the escape of misfolded or partially folded molecules from traps in the landscape and not from a stepwise formation of native structure (8, 9). For such proteins the fast productive routes to the native state were yet to be resolved (61). A notable example is cytochrome c in which folding is multiexponential and requires hundreds of milliseconds (33, 110). In this case kinetic complexity results from transient covalent binding to the heme iron of a nonnative histidine almost 50 residues distant along the sequence from the native methionine ligand. Once binding of this histidine is blocked, folding is submillisecond, too fast to be observed by the stopped-flow method (33, 110).

With this background several groups of biophysical scientists working in the area of time-resolved spectroscopy were attracted to the protein-folding problem. Time-resolved spectroscopy with pulsed lasers had been quite successful in functional studies on proteins, so it seemed natural to take advantage of this powerful technology to investigate protein folding. It was, moreover, apparent that finding ways to initiate protein folding with laser pulses would provide dramatically improved time resolution. Recognizing the importance of experiments in which the concentration of chemical denaturant is rapidly changed to initiate folding or unfolding, a second line of investigation has been directed toward the development of methods to improve the time resolution in mixing experiments. These new, fast-folding techniques have permitted the investigation of fundamental processes in protein folding, including formation of α-helices, β-sheets, and contacts between residues distant in sequence as well as overall collapse of the polypeptide chain and folding of the very fastest proteins.

Although these new techniques have allowed the study of previously unobserved processes in protein folding, it is only quite recently that they have provided major new insights into folding mechanisms. These include a much deeper understanding of the mechanism of secondary structure formation, the introduction of the notion of an upper limit on the rate of protein folding (a “speed limit”), the discovery of unusual kinetics suggesting that very fast folding is continuously downhill in free energy, and direct observation of polypeptide collapse prior to formation of the native structure. One of the surprises from this work is that simple statistical mechanical models, originally developed for explaining the fast kinetics of secondary structure formation in isolated peptides, are remarkably successful in quantitatively explaining the kinetic behavior of small proteins. This finding has several important implications, including the possibility that the underlying physics of folding may be much simpler than previously thought.

Basic Ideas of Fast-Folding Methods

There have been several recent reviews on fast-folding techniques, so here we only present a very brief description of the basic physical ideas underlying these methods (14, 43, 52, 99). Methods for rapid initiation of protein folding can be roughly classified into three categories—photochemical triggering, temperature or pressure jump, and ultrarapid mixing methods. The first fast-folding study employed a photochemical trigger—the photodissociation of carbon monoxide from denatured cytochrome c (17, 59). This experiment takes advantage of the fact that CO binds much more strongly to the heme of the denatured protein than to the heme of the native protein. Photodissociation of CO initiates folding because the CO-free protein is much more stable (Figure 1). The experiment has unlimited time resolution for folding experiments because photodissociation occurs in less than one picosecond. A conceptually similar, but more general, method utilizes a photo-induced electron transfer reaction to initiate folding (89) (Figure 1). The first such experiment was also performed on cytochrome c (20, 73, 89, 113). The reduced form of cytochrome c is more stable than the oxidized form, primarily due to the increased stability of the bond between methionine 80 and the heme iron in the native protein. Optical excitation of a ruthenium bL-pyridine complex or NADH produces a long-lived excited state that is a powerful reductant. Injection of an electron reduces the heme iron of cytochrome c from ferric to ferrous in a few microseconds in a diffusion-limited bimolecular reaction and initiates folding. This technique has also been used to trigger folding of cytochrome b₅₆₂ (121, 122) and myoglobin (123).

Figure 1

Initiating the folding of reduced cytochrome c with photochemical triggers. Guanidinium chloride (GuHCl) unfolding curves for oxidized cytochrome c [Fe(III)], reduced cytochrome c [Fe(II)], and the carbon monoxide complex of reduced cytochrome c [Fe(II) (more...)

A different kind of photochemical trigger uses a photolabile disulfide (70, 118), which does not require a metal site, so in principle it can be used as a trigger for a much wider range of proteins. In this method, segments of a polypeptide chain are cross-linked by an aromatic disulfide. An appropriately engineered cross-link constrains the peptide or protein in a misfolded conformation, allowing folding to be triggered by sub-picosecond photo-cleavage of the disulfide bond.

A second class of optical triggers uses an intense laser pulse to produce a rapid temperature jump. This is a much more generally applicable method because it can be used to perturb the folding/unfolding equilibrium for any process that produces a significant enthalpy change. This was first done in a protein-folding experiment by heating a solution of a dye with a mode-locked picosecond laser operating at 532 nm (90). The time resolution of this experiment was ~70 ps—the time required for thermal diffusion from the hot dye molecule to the solvent. A much more commonly employed method has been to directly heat water 10–20°C by shifting the fundamental of a Q-switched Nd:YAG laser at 1064 nm to a longer wavelength where water strongly absorbs (3, 4, 116, 120). The solution is heated during the ~5 ns duration of the pulse by vibrational excitation of an O-H stretching overtone of water. The Eigen T-jump method of resistive heating with an electrical discharge has also been used to study folding kinetics with ~10 μs time resolution (81, 82). Another perturbation method, pressure jump, has recently been employed to study folding kinetics (57). In this experiment, a stack of piezo-electric crystals is used to change the pressure 100–200 bar in 50–100 μs. The pressure change alters the folding/unfolding equilibrium because of the associated volume change.

Use of chemical denaturants, such as urea and guanidinium chloride, has played a central role in protein-folding experiments. Almost all proteins can be unfolded and remain soluble at sufficiently high denaturant concentrations, so dilution of denaturants continues to be an extremely important way of initiating protein folding. For this reason, there has been considerable effort to improve the time-resolution in mixing experiments by using continuous flow methods. Two such methods have recently been applied to protein folding, one based on turbulent mixing (18, 88, 104–106, 112, 124) and another based on hydrodynamic focusing (63, 93). In the turbulent mixing method liquids are forced through a small gap at high velocities (96). The turbulence created by the high shear forces “breaks” the liquids into very small volume elements called turbulent eddies. Mixing is rapid (1–10 μs) because diffusion of the denaturant occurs over short distances. In this experiment the mixed liquids flow continuously at constant velocity. The kinetics are monitored at various positions along the jet emerging from the mixer, time being measured by the distance from the mixing region and the flow velocity. The first time point in the kinetic measurements is significantly longer than the mixing time because the mixed solution must typically flow through a volume that is inaccessible to the monitoring probe. Nevertheless, the dead time of these instruments is only 50–200 μs, an improvement of more than an order of magnitude over the stopped flow method.

The second rapid mixing method employs hydrodynamic focusing to create a micron or submicron-wide stream of protein solution flowing at constant velocity in contact with a surrounding flowing reservoir (63, 93). Mixing occurs by diffusion into and out of the narrow stream. Solutions have been focused to diameters as small as 50 nm, corresponding to ~1 μs mixing times (63). It should therefore be possible to obtain significant reduction in effective dead time compared to the turbulent mixing methods. An advantage of the hydrodynamic focusing method is that the flow is laminar. Consequently, the concentration of all species at all positions in the mixing device, including the mixing region, can in principle be calculated theoretically and also determined by experiments. This technique has allowed measurements of submillisecond, time-resolved small angle X-ray scattering for the determination of the radius of gyration of transient structures in protein folding (93).

Finally, protein-folding kinetics can be studied at equilibrium using dynamic nuclear magnetic resonance (NMR) methods (12, 13, 53). For a protein undergoing a simple two-state folding/unfolding transition, both the folding and unfolding rates can be derived from the measured lineshape if the resonant frequencies and transverse relaxation times for the two states are known. This method is useful for determining folding rates in the 10 μs-10 ms range under conditions where there is a significant population of both folded and unfolded states.

Elementary Processes in Folding

Contact Formation

Perhaps the simplest elementary process in protein folding is the formation of a contact between two residues of an unfolded polypeptide chain. In spite of its importance, this process was not investigated until the photodissociation experiments on cytochrome c (59). Time-resolved spectroscopy with nanosecond lasers was used to monitor heme absorption following photodissociation of the carbon monoxide complex of denatured reduced cytochrome c (Figure 1). The first change in absorption occurs with a relaxation time of ~3 μs and was identified as binding of methionine to the heme. Detailed kinetic modeling led to an estimate of the time constant for methionine binding and dissociation of ~40 μs and ~4 μs, respectively. In subsequent experiments, the binding of free methionine to the heme attached to a small fragment of the cytochrome c polypeptide (the 11–21 undecapeptide) was studied, and it was found that the bimolecular rate is ~2 × 10⁸ M^-1 s^-1, close to the diffusion limit (48, 49). Analysis of a simple two-step mechanism, in which binding occurs by first forming an encounter complex followed by formation of the heme-methionine covalent bond, showed that the observed unimolecular rate is almost purely diffusion limited. The heme iron is covalently bonded to a histidine at position 18, while the two methionines in the sequence are located at positions 65 and 80. The measured time of 40 μs is therefore the time required for diffusion-controlled contact between regions of the polypeptide separated by ~50 residues. Together with the length scaling from polymer theory (15, 45, 111), this number could be used to estimate the formation rate of a contact between residues separated by any number of residues. For example, ~3 μs was estimated for the formation of a short contact between residues separated by ~10 peptide bonds using the n^-3/2 scaling for a random walk chain (see below). A second important result from these experiments was the calculation of the effective diffusion coefficient ≈5 × 10^-7 cm² s^-1 for the relative motion of heme and methionines, assuming a Gaussian distance distribution (111). This value is comparable to the diffusion constant ≈(1.5–8) × 10^-7 cm² s^-1 for the relative motion of a donor and acceptor in unfolded ribonuclease A, determined from the effect of Förster excitation energy transfer on the donor fluorescence decay kinetics (11).

The cytochrome c experiments indicated that it would be interesting to develop a more generic method for investigating contact formation in unfolded polypeptides. One approach to this problem has been to label the ends of small peptides with probes that undergo triplet-triplet energy transfer upon contact (5). In this experiment the triplet donor was a three-ring organic compound, thioxanthone, attached to the N-terminus of peptide, while the triplet acceptor was naphthalene attached to the peptide as 1-naphthyl alanine (Figure 2). The triplet state of thioxanthone has a lifetime of ~30 μs and transfers its energy in a diffusion-limited process to the triplet state of naphthalene upon contact. A nice feature of this experiment is that energy transfer was clearly demonstrated by the disappearance of the thiox-anthone triplet-triplet absorption spectrum simultaneously with the appearance of the naphthalene triplet-triplet absorption spectrum. Peptides were investigated that contain 1–4 glycine/serine pairs, corresponding to a spacing of 3–9 peptide bonds between the donor and acceptor. Exponential kinetics was found in all cases, with time constants varying from 20 ns for the shortest peptide to 50 ns for the longest.

Figure 2

Using long-lived triplet states to measure rates of contact formation in peptides. In the triplet-triplet energy transfer experiment, an organic donor (thioxanthone, T) attached at one end of a peptide is optically excited to its long-lived triplet state. (more...)

One limitation of this triplet-triplet energy transfer method is that, with the current probes, positioning of the triplet states for efficient energy transfer requires the use of ethanol/water mixtures. Also, the use of extrinsic probes makes the study of proteins and the exploitation of site-directed mutagenesis more difficult. Another method has been developed for measuring contact formation rates that does not suffer from either of these limitations (67). In this method, contact formation was measured from the quenching of the triplet state of tryptophan at the C-terminus of a peptide by cysteine at the N-terminus (Figure 2). Earlier work had suggested that cysteine is a much more efficient triplet quencher than any other amino acid, exhibiting a near diffusion-limited bimolecular rate for the quenching of free tryptophan (42). The tryptophan triplet state is most probably depopulated by transfer of an electron to the sulfur of the quencher at distances close to van der Waals contact. The long lifetime (> 50 μs) of the tryptophan triplet state, moreover, permits quenching rates to be measured over a relatively wide dynamic range by measuring the decay of the triplet-triplet absorption spectrum. The peptides studied contained the repeating amino acid triplet—alanine/glycine/glutamine—to avoid any specific secondary structure formation. Varying the number of peptide bonds between tryptophan and cysteine from 4 to 19 yielded contact time constants of 30 ns to 150 ns, consistent with the values found in the triplet-triplet energy transfer study.

Although both kinds of triplet state experiments yield times ~50-fold smaller than the earlier estimate of ~3 μs for contact between residues separated by ~10 peptide bonds (49), they are not inconsistent with this prediction. At least two factors contribute to this difference. For an idealized chain with a Gaussian end-to-end equilibrium distribution, this rate is given by: 3Da/[(π/6)^1/2 〈r²〉^3/2]; where D is the effective relative diffusion constant of the two ends, a is the contact radius, and 〈r²〉 is the mean squared end-to-end distance (111). The contact radius for the tryptophan-cysteine interaction is larger than for the methionine-heme encounter complex of cytochrome c, and the heme is only accessible to the methionine on one side, so the effective value of a is larger in the quenching experiments. Also, the peptides are rich in glycine and therefore have smaller 〈r²〉 compared to cytochrome c, which has only an average of 1 glycine per 6–7 residues between the heme and methionines.

The above results suggest that tryptophan triplet lifetime measurements can provide a useful method for investigating the kinetics of forming a specific intramolecular contact in an unfolded or folding protein. They also show that this method can be used to address fundamental questions on the dynamical properties of polypeptides. Foremost among these are the length and composition dependence of contact formation rates. A maximum in the loop probability as a function of chain length has been predicted at about 10 residues for a generic polypeptide (15, 45, 114). In these calculations, formation of an end-to-end contact in longer chains is less probable because of the larger entropy decrease, while forming a contact in shorter chains is disfavored because of chain stiffness. In the simplest picture, the maximum in the loop probability should be reflected as a maximum in the contact formation rate [the rate of dissociation of the contacting polymer ends ( =3 D / a²) is independent of chain length]. No such maximum was observed in the glycine-containing peptides, presumably because of their larger flexibility compared to the generic peptide considered in the theoretical calculations. This question is now being studied using less flexible peptides (LJ Lapidus, unpublished results). Another important aspect of studies of contact formation, at least in small glycine-containing peptides, is that it occurs on a time scale that can be directly simulated by all-atom molecular dynamics calculations, simulations that have yet to be performed.

Helix-Coil Transition

Although both the thermodynamics and kinetics of the helix-coil transition have been studied experimentally and theoretically for over 40 years, it was not until the late 1980s that a significant number of peptides of the size and composition found in proteins were shown to be helical in pure aqueous solvents. Since that time extensive equilibrium studies of α-helix formation in isolated peptides have been carried out and theoretically analyzed (16, 71, 78). Peptides rich in alanine are particularly suitable because of their high helix-forming propensity. This together with their ready availability have made them prime candidates for kinetic investigations using fast kinetic techniques.

In the first such study a 21-residue peptide having the sequence X-(A)₅ (AAARA)₃-A-NH₂ (X = succinyl) was investigated (120). A nanosecond laser temperature jump was used to perturb the helix-coil equilibrium, and infrared absorption measurements in the amide I region were used to monitor the decrease in helix content. The relaxation following the temperature jump was found to be biphasic with an unresolved (< 10 ns) amplitude and a larger amplitude 160 ns relaxation. Subsequent laser T-jump measurements with a fluorescent probe at the N-terminus (methyl amino benzoic acid instead of the succinyl group) showed only a single fast relaxation at ~20 ns (116). In this study a simple nucleation/propagation model was developed to explain both the infrared and fluorescence experiments. This model suggested that the fast relaxation results from helix propagation and partial melting of stretches of helix, while the slow relaxation corresponds to crossing a helix nucleation barrier (Figure 3). This barrier crossing occurs in both directions, corresponding to nucleation and growth of a new stretch of helix in one direction, and complete melting of a preexisting stretch of helix in the reverse direction. However, the model predicted that the N-terminal fluorescent probe would also show a comparable amplitude for the slow barrier crossing process, which was not observed.

Figure 3

Processes contributing to temperature-dependent changes in the helix content of alanine peptides. An increase in temperature produces both an increased number of random coil peptides, shown on the right, and a decrease in the average helix length of helical (more...)

This problem prompted further investigation of helix-coil transition kinetics in a very similar peptide, but one having an intrinsic fluorescent probe (117). In this peptide, with sequence Ac-WAAAH(AAARA)₃-A-NH₂, formation of the first turn of helix is signaled by quenching of tryptophan (W) fluorescence by the protonated histidine (H) (Figure 3). Both the temperature (117) and viscosity dependence (GS Jas, unpublished results) of the relaxation kinetics were studied. Again, only a single relaxation was observed, but with a rate (1/(220 ns)) very close to the slower phase observed in the infrared experiments on a very similar peptide described above. To understand this result, the model developed in the study of a β-hairpin was used (77, 80) (Figure 4), which takes into account both the side chain interactions and the secondary structure propensities of individual residues (117). According to the model, in phases corresponding to both, helix propagation and partial melting are slowed in this peptide. The much lower helical propensities of histidine and tryptophan, together with the histidine-arginine (i, i + 4) repulsion, slow incorporation of the 5 N-terminal residues into a helix growing from the C-terminal side, while melting of the N-terminus is slowed by the stabilizing tryptophan-histidine (i, i + 4) interaction. The net result is that the fast relaxation is slower and with a much smaller amplitude, so that the observed kinetics are dominated by crossing the nucleation barrier (117).

Figure 4

Model for formation of β-hairpin from protein GB1. (a) Structure. (b) Free energy profile. The free energy calculated from a simple statistical mechanical model is plotted as a function of the number of native peptide bonds in the modified single (more...)

A similar peptide without the tryptophan-histidine probe has also recently been studied using time-resolved resonance Raman spectra measured with far UV (204 nm) excitation to probe helix content (68). The Raman spectrum of the helical peptide exhibits much weaker intensities for the amide II, amide III, and Cα-H bending bands than does the random coil, so the Raman spectrum provides several independent measures of helical content. A single exponential relaxation was observed over the temperature range studied (310 K–337 K), with τ = 250 ns at 300 K and an apparent activation energy of 7 kcal/mol. These results are very similar to those obtained from the fluorescence and infrared absorption experiments. No fast relaxation could be detected in these experiments even though the time resolution was ~3 ns.

With a statistical mechanical model, quantitative predictions of helix-coil kinetics in other peptides are straightforward. Using the same parameters from fitting the equilibrium and kinetic data on the tryptophan/histidine peptide, the kinetics of the infrared experiments were simulated (117). Excellent agreement with experiment was obtained, with calculated relaxation times of 4 ns and 220 ns and an amplitude ratio for the fast and slow relaxations of approximately 1:4. More recently the model has been modified to predict the kinetics of peptides having the length and composition found in helices of proteins (V Muñoz, unpublished results) incorporating the more detailed equilibrium description of the helix-coil transition contained in the program AGADIR (78, 79). These calculations show that relaxation times for 15–20 residue peptides span the range from a few nanoseconds for peptides having less than 1% helix at equilibrium to ~1 μs for peptides containing > 75% helix. The relaxation time is much shorter for very unstable peptides because melting, which is dominating the relaxation in these peptides, is downhill in free energy. On the other hand, the relaxation time for stable, protein-like sequences is longer than that of alanine-based peptides of comparable stability. This is a consequence of two factors. The entropy loss associated with forming a turn is greater for nonalanine peptides (alanine has the highest helical propensity), producing a larger nucleation barrier. To compensate for the lower helical propensity, stable helices require stronger side chain interactions, and these interactions must be broken for complete melting to occur, increasing the barrier in the reverse direction.

A surprising result has recently been reported from stopped-flow kinetic studies of helix formation monitored by circular dichroism (CD) in a 16-residue alanine-based peptide (21). In this study helix formation was initiated by dilution from 6 M guanidinium chloride. All of the equilibrium circular dichroism change was found to occur in a ~100 ms relaxation. To reconcile these results with the T-jump experiments, it was argued that at this high concentration of denaturant the peptide is completely unfolded and that the 100 ms relaxation corresponds to helix nucleation, while the much faster relaxation observed in the T-jump experiments arises from unzipping of existing helical sequences. If this explanation were correct, then the helix nucleation rate would be several orders of magnitude slower than previously thought. In an attempt to reconcile these discrepant observations, we reexamined the trypophan-histidine peptide using both stopped-flow and T-jump experiments. The stopped-flow experiments showed that helix formation as monitored by the tryptophan fluorescence is complete within the 3 millisecond dead time, while precise measurements of the kinetic amplitudes in T-jump experiments confirm that the fluorescence reaches its equilibrium value in less than a microsecond (117; GS Jas, unpublished results). Similar results were also obtained in the infrared (120) and Raman (68) studies. We should also point out that the statistical mechanical model predicts that the kinetics of helix formation from the completely unfolded state exhibit almost exactly the same relaxation time as that observed in the T-jump perturbation experiment. The origin of the 100 ms circular dichroism change remains unclear.

Helix-coil kinetics have also been observed in laser T-jump experiments on proteins. An interesting example is the study on apomyoglobin using infrared spectroscopy (31, 40, 41). By monitoring the infrared absorption at different wavelengths, it was possible not only to determine the change in total helix content, but also to distinguish between changes in solvent exposed helices and helices that are packed against other helices. Following a temperature jump, partially unfolded apomyoglobin exhibits a multiphasic relaxation. A rapid helix melting/formation relaxation at 50 ns has maximal amplitude at wavelengths characteristic of solvent-exposed helix (possibly helices C, D, or E in the denatured state). A slower relaxation at 120 μs has maximal amplitude at wavelengths for packed helices and may represent the cooperative unfolding and refolding of the core of the protein containing helices A, G, and H (31, 40, 41).

What is the relation between real and computer experiments on helix-coil kinetics? As mentioned in the introduction, one of the motivations for studying the kinetics of the helix-coil transition is that it occurs on time scales that are becoming accessible to all-atom molecular dynamics simulations. Ideally one would like to observe the helix form and completely melt many times in a single trajectory. To have adequate statistics to obtain equilibrium populations and rate constants, a large number of such trajectories would be required. Such simulations have recently been carried out on a β heptapeptide in methanol, which forms a left-handed 3₁-helix (in a β peptide a methylene group is inserted in the backbone between Cα and NH). The helix melts and reforms several times during the course of the 50 ns simulation (23, 24). The analysis of the residence time in helix and coil states yields an equilibrium constant close to the experimental value. Unfortunately, the kinetics of helix formation in this heptapeptide have not yet been studied experimentally.

No kinetic study by molecular dynamics is available yet for alanine peptides, but there is a hint from a single trajectory on a 13-residue alanine peptide in explicit water that accurate rates can be calculated (22). This alanine helix unfolds in the simulation at 373 K in 200 ps, compared to 700 ps obtained by extrapolating the experimental data to 373 K using the statistical mechanical model for this peptide (117). Another fundamental kinetic quantity that could be readily obtained by molecular dynamics is the time to add an alanine residue to a growing helix. This time comes directly from the modeling of the kinetic data and is found to be ~600 ps at 300 K, varying from ~50 ps at 373 K to ~2 ns at 273 K (117).

A number of important issues concerning the kinetics of the helix-coil transition remain to be resolved. These include the direct observation of the rate of helix growth and the exploration of the effects on the kinetics of amino acid substitution and chain length. While estimates of the growth rate can be obtained from models, it should be possible to directly observe this process by using time-resolved infrared or Raman measurements and a T-jump method with subnanosecond time resolution. Such studies might be facilitated by using peptides constrained by chemical modification to have prenucleated helices.

Fast Protein Folding

Protein-Folding Rates from a Simple Model

What factors determine whether a protein will be a fast or slow folder? Is there any relation between folding rate and the final native structure? It is now possible to address these questions because of the availability of a sufficiently large set of data on the folding of small single domain proteins. We pointed out in the introduction to this review that the study of CI2 marked a turning point in experimental studies on protein folding (56). The 64-residue version of this protein, which has neither disulfide bridges nor cis-prolines, behaves like a perfect two-state system. For a two-state protein, only two populations of molecules are detected in both equilibrium and kinetic experiments (126). One state corresponds to molecules having the native structure and the second to the ensemble of conformations that make up the denatured state. Over 20 proteins are now known to exhibit two-state behavior, and the results of these studies have been summarized in a recent review (55). Their folding rates span more than four orders of magnitude, from (200 μs)^-1 for monomeric λ repressor to (4 s)^-1 for muscle acyl phosphatase (Figure 5).

Figure 5

Schematic structures of monomeric λ repressor (left) and muscle acyl phosphatase (right).

In searching for empirical correlations in this set of kinetic data, a key observation was made: The rates of folding show a significant correlation with a simple measure of protein topology, the so-called “contact order” (1, 92). The relative contact order is the mean separation in sequence between residues that are in contact in the three-dimensional structure, normalized by the total number of amino acids in the protein. For proteins with many long-range interactions, such as muscle acyl phosphatase, the contact order is large and folding is relatively slow. For α-helical proteins, such as monomeric λ repressor, local (i, i + 4) interactions are much more abundant The contact order is therefore small and folding is observed to be relatively much faster.

The observation of a correlation between folding rates and such a simple measure of structure suggested that a simple physical model may be capable of calculating the rates of protein folding. A simple statistical mechanical model had already been developed for explaining the kinetics of β-hairpin formation, which as pointed out earlier, contains most of the basic features of folding a small protein. It was therefore logical to apply this model to two-state proteins (76). The analysis of the β-hairpin revealed that for this molecule the single sequence approximation, which allows only one stretch of native structure simultaneously in each molecule, produces results very close to the complete model, which imposes no restrictions on the number of stretches (125). To account for the larger size of the molecule in applying the model to proteins, this approximation was relaxed to allow two and three simultaneous stretches. For 18 proteins that exhibit two-state behavior experimentally the free energy as a function of the number of native peptide bonds, the natural reaction coordinate in this model, shows only two deep minima separated by a free energy barrier. Furthermore, free energy profiles for four other proteins, known to populate more than two states in kinetic experiments, show additional deep minima. The striking finding was that the calculated free energy barriers to folding are small for fast-folding proteins and large for slow-folding proteins (Figure 6a,b). The rates calculated for the 18 two-state proteins from diffusion on these free energy profiles, moreover, show a remarkably good correlation with the experimentally determined rates (Figure 6c). It is perhaps surprising that even approximate relative rates can be calculated using a single reaction coordinate for such a complex system. However, the absolute folding rate of a lattice protein with ~10¹⁶ possible structures was accurately calculated from diffusion on a one-dimensional free energy profile (108).

Figure 6

A simple statistical model for protein folding (from 76). As with the hairpin, structures are classified by the backbone conformation in which each peptide bond can adopt either a native or nonnative conformation as defined by the pair of flanking ψ, (more...)

In addition to ignoring much polymer physics, there are several obvious weaknesses to these simple calculations. One is a restrictive assumption of the model that does not allow native interactions between residues unless the intervening chain has its native conformation. A second is the naive treatment of the entropy of the denatured state. The denatured state appears as a sharp minimum in the free energy profiles (Figure 6a,b), instead of the broad minimum that would appear with no restriction on the distribution of native peptide bonds (125). Other weaknesses are the use of a simplified free energy function in which native propensities and contact energies are sequence independent and solvation effects are ignored. Nevertheless, the success of the simple model in calculating rates is an important result, for it suggests that the underlying physics determining folding rates is simpler than previously thought. It appears that the folding rate is determined largely by the distribution and strength of contacts in the native three-dimensional structure. As pointed out earlier in our discussion of β-hairpin formation (Figure 4b), compensation of the conformational entropy loss by strong stabilizing interactions earlier along the reaction coordinate lowers the free energy barrier and allows faster folding.

What about mechanism? Calculations with this simple model contain an enormous amount of detail on microscopic structural pathways between the native and denatured state. For these pathways it is possible to identify the structures that, once formed, rapidly proceed to the native state. This is the so-called transition state ensemble. Detailed structural information on this transition state ensemble can in principle be obtained from experimental measurements of the relative effects of mutations on the rates and equilibrium constants for folding (so-called φ values, defined as Δlnk_folding/ ΔlnK_equilibrium)(34, 35, 54, 84). This model, like two other simple models with similar approximations (2, 36), shows qualitative agreement with mutation experiments. However, thus far, these simple models do not give good quantitative agreement between the observed and calculated φ values, so it is somewhat premature to use them to describe structural pathways in any detail. Accurate calculation of φ values remains an important goal, and more realistic theoretical models (107) as well as knowledge of the detailed structure of the mutated protein may be necessary. The task would also be made easier by experimental studies in which there are several different mutations at each of the studied positions. Predicting φ values theoretically is a crucial issue because the mutation experiments probably hold the key to elucidating mechanism in structural detail.

Biological Relevance of Folding Kinetics

From the preceding discussion we have seen that kinetics is important for a physical understanding of how proteins fold. However, is there any direct biological implication of folding kinetics? We should say at the outset that there are no answers to this question that can be rigorously supported, but there are a number of interesting speculations. One subject under study is whether proteins with similar structure but no sequence homology have the same folding mechanism. This is, in principle, independent of whether these proteins are products of divergent or convergent evolution. To explore this idea, sequence and structure alignments have been used to search for clusters of conserved residues that are not necessary for either function or stability. The assumption is that these residues, if they appear, correspond to “nuclei” that determine the folding rate (74).

A more general question is whether there is any pressure on folding rates other than that required for the appropriate stability. There must be some constraint, for newly synthesized polypeptide chains must be able to fold within a certain time to be able to perform their biological function and to avoid proteolysis and aggregation. What is the minimal folding time required for biological function? In Escherichia coli, proteins are synthesized at a rate of 10–15 amino-acids per second. Assuming an average molecular mass of 40,000 daltons (~350 amino-acids) for E. coli proteins, the ribosomes require on average 30 sec to synthesize a protein molecule (69). One could argue that folding speed will not be a limiting factor as long as it is comparable to the rate of protein synthesis. If there is indeed biological pressure, we should expect proteins to fold no slower than 1/minute. All known folding rates for two-state proteins are faster than this minimum speed (55) (Figure 6). Other processes coupled to protein folding, such as proline isomerization and disulfide formation, could slow folding considerably, but specialized enzymes catalyze these reactions in vivo (101). The role of molecular chaperones, such as GroEL and GroES, can also be rationalized in this light. Theoretical estimates indicate that the known chaperones only assist folding of ~5% of the existing proteins (69). One of their roles might be to ensure folding of a small subset of biologically critical proteins that would otherwise fold with rates slower than the one-minute biological minimum.

A still open issue, however, is whether it is difficult for evolution to find sequences that can form stable 3-D structures faster than this biological minimum speed. This question has been approached using combinatorial chemistry. Heavily mutated variants of the IgG binding domain of protein L were found to fold at similar, or even faster, rates than the wild-type protein (62). This result has been used to suggest that for single domain proteins, it is not difficult to find sequences that fold “biologically fast.” It would be important to further pursue this idea with experiments on other proteins, and, ideally, to explore large regions of sequence space that are not limited to the neighborhood of naturally occurring sequences. A potential drawback of combinatorial methods is that they generally involve biological selection. It is therefore possible that the method is finding fast folding proteins because they are the only ones that are biologically viable.

Does protein function ever require folding much faster than one minute? For λ repressor, which folds in 200 μs, it has been argued that very fast folding is required for stability to compensate fast unfolding, fast unfolding being required for efficient regulation by proteolysis (13). We do not know the rate of protease binding to unfolded states of proteins, but it could be argued that what really determines the rate of proteolysis of a target protein is the residence time in the unfolded state. Fast folding could then be a strategy to avoid proteolysis while maintaining the low stability necessary for its function (e.g. binding to DNA). Related to this is the interesting suggestion that there is evolutionary pressure for proteins to be marginally stable in order to have the conformational flexibility required for function (58).

A corollary to the idea of folding speed as an evolutionary pressure is the suggestion that to avoid aggregation in the cell, compact states may be required to form much more quickly than the final native structure (49). This idea is consistent with the finding that polypeptide collapse is submillisecond (see above). In a typical E. coli cell in logarithmic phase with a volume of ~10^-15 liter, there are ~35,000 ribosomes (69), which are producing ~1000 protein molecules per second. We can make a rough estimate of the maximum folding time that is compatible with minimal aggregation. If we assume that aggregation is nonspecific, irreversible, and diffusion limited (~10⁸ M^-1 s^-1) (119), folding in less than ~10 milliseconds would result in aggregation of less than ~1% of newly synthesized proteins. Although this calculation is very approximate for many reasons, it does support the idea that competition between aggregation and folding may provide additional selection pressure for fast-folding sequences or for sequences that collapse quickly to form compact denatured states that do not aggregate. Understanding this competition in more detail may be important for investigations of the pathophysiology of several human diseases (28, 60, 66).

Finally, is there any connection between evolutionary pressure on folding speed and the catalogue of naturally occurring protein structures? The model for protein folding discussed above would suggest that topologically simple structures are expected to fold faster than complicated structures. A large number of geometrically feasible and stable structures may not be observed in nature because they would fold too slowly to be consistent with the constraints of synthesis and aggregation discussed above. This could be a factor responsible for the limited number and distribution of distinct protein folds that have been observed so far (74, 127, 128).

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*

Present address: Department of Chemistry and Biochemistry, and Center for Biological Structure and Molecular Organization, University of Maryland, College Park, MD 20742.

†

Present address: Department of Physics, University of Florida, Gainesville, FL 32611-8440.