Dynamics, Energetics and Structure in Protein Folding

Abstract

For many decades, protein folding experimentalists have worked with no information about the timescales of relevant protein folding motions and without methods for estimating the height of folding barriers. Experiments in protein folding have been interpreted using chemical models in which the folding process is characterized as a series of equilibria between two or more distinct states that interconvert with activated kinetics. Accordingly, the information to be extracted from experiment was circumscribed to apparent equilibrium constants and relative folding rates. Recent developments are changing this situation dramatically. The combination of fast-folding experiments with the development of analytical methods more closely connected to physical theory reveals that folding barriers in native conditions range from minimally high (~14 RT for the very slow folder AcP) to nonexisting. While slow-folding (i.e. 1 millisecond or longer) single domain proteins are expected to fold in a two-state fashion, microsecond-folding proteins should exhibit complex behavior arising from crossing marginal or negligible folding barriers. This realization opens a realm of exciting opportunities for experimentalists. The free energy surface of a protein with marginal (or no) barrier can be mapped using equilibrium experiments, which could resolve energetic from structural factors in folding. Kinetic experiments on these proteins provide the unique opportunity to measure folding dynamics directly. Furthermore, the complex distributions of time-dependent folding behaviors expected for these proteins might be accessible to single molecule measurements. Here, we discuss some of these recent developments in protein folding, emphasizing aspects that can serve as a guide for experimentalists interested in exploiting this new avenue of research.

In folding to their biologically active 3D structures, proteins must coordinate the vast number of degrees of freedom of their polypeptide chains by forming complex networks of non-covalent interactions. Therefore, understanding protein folding involves determining the relations between the energetics of weak interactions and protein conformation, and the collective chain dynamics that govern the search in conformational space. In modern rate theory, these issues are resolved by mapping the potential energy of the molecule as a function of the relevant coordinates. The dynamics are then represented as diffusion on such an energy surface(1, 2). For folding reactions, however, even the solvent-averaged free energy surface is hyper-dimensional due to the large number of relevant coordinates (i.e. thousands of atomic coordinates for a small protein)(3). Folding hypersurfaces should also be topographically complex due to frustration between the myriads of possible interactions(3, 4). Moreover, molecular simulations(5–8) and NMR dynamics experiments(9) indicate that protein conformational motions span a wide range of timescales (i.e. from picoseconds to milliseconds). The implication is that measuring these surfaces experimentally would require detecting single protein molecules with full atomic detail and sub-nanosecond resolution during long periods of time. Obviously, such measurements are well beyond our current experimental capabilities.

To complicate matters further, global protein folding reactions appear deceivingly simple in traditional thermodynamic and kinetic experiments(10). Accordingly, it has been customary to interpret folding experiments as elementary chemical reactions. Protein thermodynamics are then discussed in terms of chemical equilibria between a series of discrete macrostates(11), which interconvert with activated kinetics over arbitrarily high free energy barriers(12). The simplicity of the chemical approach has made it extremely popular in the folding field(10). However, interpreting protein folding as a simple chemical reaction carries a significant amount of baggage. The dimensionality and shape of the folding free energy surface are assumed implicitly and thus, cannot be investigated experimentally. Furthermore, the information obtainable from experiment is limited to equilibrium and rate constants in which energetic, dynamic, and structural factors are all intertwined. In other words, by describing folding in chemical terms we are giving up the opportunity to measure experimentally the very same fundamental properties that determine folding mechanisms and are most relevant in comparing with theory.

In this light, the divide existing between classical folding experimentalists and theorists should be hardly surprising(13). This divide has been bridged in part by the spectacular development of computer simulations(6–8, 14–16) together with a shift in experimental focus towards fast-folding proteins(17–22). The microsecond folding timescales of these proteins make possible to connect with modern simulations, which are finally reaching comparable timescales. Furthermore, work of fast-folding proteins has produced significant evidence indicating that folding barriers are small, sometimes event absent altogether. Such observations have confirmed critical theoretical predictions. Moreover, the existence of small folding barriers opens a new realm of opportunities for the experimental analysis of protein folding. With marginal folding barriers it becomes possible to extricate dynamic from energetic factors in kinetic experiments and resolve energetic contributions and structural ensembles in equilibrium folding experiments. Here we discuss recent developments in this area, which are paving the way for a shift from the traditional biochemical to a physical paradigm in experimental protein folding.

Free Energy Projections and Reaction Coordinates

Hyperdimensional free-energy surfaces can be greatly simplified by projecting them onto one, or a few order parameters. The resulting low-dimensional free energy projections then provide a bridge to connect theory, simulation, and experiment. This approach, which was pioneered by Wolynes and collaborators(3, 4, 23, 24) and has become a very popular tool for the analysis of computer simulations(16, 25–27), explains the two-state-like folding exhibited by many proteins. It also predicts the existence of barrierless folding and other complex kinetic behaviors(3). In principle, the approach should be extensible to the direct interpretation of folding experiments. What is not so clear a priori is whether a surface of very low dimensionality, or even one-dimensional, can reproduce folding reactions effectively nor what order parameters might behave as reaction coordinates.

Here, it is interesting to note that the chemical two-state model is a simplified representation of a one-dimensional free energy surface with a high barrier separating two minima. Therefore, the apparent success of two-state analyses already suggests that even one-dimensional projections may suffice to describe coarse folding properties. The first direct evidence supporting the validity of one-dimensional folding surfaces has come from the analysis of simulations in the cubic lattice(28). Further support comes from the reasonable success in calculating relative folding rates from protein structures using one-dimensional free energy surfaces(29). More recently, the performance of one-dimensional free energy projections has been tested empirically for the case of α-helix formation(30). The analysis took advantage of a detailed kinetic theory for α-helix formation that explains quantitatively the complex observations of Gai and collaborators in laser-induced temperature-jump experiments of isotopically labeled helical peptides(31–33). This allows for direct comparison between experiment, detailed theory, and the free-energy projection approach. The kinetic theory is an extension of nucleation-elongation helix-coil equilibrium models in which helix segments nucleate at any point in the sequence and elongate from their ends(34). The nucleation-elongation mechanism produces complex kinetics with hundreds of thousands of coupled differential equations and myriads of possible pathways. Remarkably, the projection of the multidimensional free energy surface onto one order parameter (i.e. the number of helical peptide bonds) reproduced the observed kinetics with surprising accuracy. The agreement went to the level of matching the relative amplitudes for the characteristic slow and fast-phases of helix coil kinetics. Furthermore, the diffusion coefficient from the surface analysis was essentially identical to the elementary peptide bond rotation rate (~ 1/(2 ns)) in the kinetic theory, indicating that the number of helical peptide bonds is a good reaction coordinate for α-helix formation(30). These results validate a one-dimensional free energy projection on a folding-related problem for which the mechanism is reasonably well understood. The free energy surface analysis of α-helix formation also exemplifies some of the features expected for protein folding reactions with marginal barriers. Because the barrier is very small (~2 RT), the relaxation process is expected to exhibit two phases (Figure 1). The fast phase involves the relaxation in the broad helical minimum, which is diffusive. The somewhat slower phase consists in the re-equilibration between both sides of the barrier and thus, has diffusive and activated terms. This same kinetic behavior has recently been observed in fast-folding mutants of λ repressor(35). From the emergence of an additional, faster, kinetic phase in the mutants, Gruebele and collaborators concluded that these proteins folded by crossing a marginal folding barrier(35). The same mutants lost the slow kinetic phase when highly stabilized by cosolvents, suggestive of downhill folding under those conditions (36).

Figure 1.

One dimensional free energy surface for α-helix formation obtained by projecting the free energy of all of the species from the detailed kinetic model as a function of the number of helical peptide bonds. The arrows represent the main flux involved in each of the two observable kinetic phases.

Protein folding is obviously more complex than helix formation, involving large changes in overall shape and compaction. The larger complexity in folding is likely to result in diffusion coefficients that change with the value of the order parameter. These changes would reflect the different dynamic modes controlling overall protein motions and the increasing surface roughness as the protein becomes more folded (i.e. more compact)(3). Effective diffusion coefficients for folding on a one-dimensional energy surface are therefore expected to be complex functions of the order parameter, including both dynamic and energetic terms(28).

Motions in Protein Folding

Although most experimentalists have focused on the energetic and structural aspects of folding, the dynamic modes that govern polypeptide chain motions are critical components of folding mechanisms. The timescales for the relevant folding motions provide upper limits for the effective diffusion coefficient for folding, which are required to estimate barrier heights from folding rates using the simple rate expression:

$$k=D_{\mathit{eff}}exp\left(-\mathrm{\Delta }{G}^{‡}/RT\right)$$ equation (1)

This is far from being a trivial point. Without an independent estimate of either the diffusion coefficient or the barrier height, protein folding rates can only be analyzed in relative terms (e.g. differences between wild-type and mutant(37)). Furthermore, an Arrhenius analysis of the temperature dependence of the folding rate can be seriously misleading because: i) the barrier and the diffusion coefficient have temperature dependent and temperature independent contributions; ii) the hydrophobic effect results in large enthalpy-entropy compensations(38); iii) glassy dynamics may arise at low temperatures(3).

Fortunately, our understanding of protein folding motions has improved significantly in recent years. The detailed kinetic analysis of α-helix and β-hairpin formation indicates that local motions (i.e. peptide bond rotations) with timescales of ~ 1–10 nanoseconds control secondary structure formation(39, 40). These local motions presumably dominate the dynamics of expanded random coil polypeptides, in which there are not significant structural correlations. It is noteworthy that this peptide bond rotation rate already has an energetic term (i.e. activation energy) equivalent to ~20–25 kJ/mol(40), which arises from chain steric hindrance and changes in water viscosity. The rate of end to end contact formation in unstructured chains has been measured for several sequences and conditions, and with a variety of methods(41–45). The timescales cluster in the few tens of nanoseconds, providing a good estimate for the concerted motions leading to loop formation in protein folding(22). The overall motions of a denatured but collapsed polypeptide are even slower. A timescale of ~100 nanoseconds has been recently measured for the random collapse of a 40 residue protein(46). This collapse time scales to ~500 nanoseconds for a protein domain of 100 residues. The slower timescales of these motions reflect coupling of many local rotations into a single global mode, which slows down as the protein becomes more compact(46). The chain dynamics of collapsed denatured proteins are probably similar to the dynamics in early stages of folding, in which the protein is already compact but has few specific tertiary interactions.

As the protein becomes more folded and tertiary interactions more consolidated, chain motions should slow further down because changing conformation is likely to involve breaking preformed interactions. At this point the energetic terms in the diffusion coefficient (i.e. the roughness of the landscape) might be quite large(4). A dramatic demonstration of this effect is seen in the rate of collapse of unfolded cytochrome c, in which the strong non-specific interactions with the heme slow down the collapse rate to hundreds of microseconds(47–50). The experiments on the fast-folding mutants of λ repressor (see previous section) suggest that the effective diffusion coefficient for folding of this protein is ~1/(2 microseconds) at 340 K(35). A diffusion coefficient of ~1/(8 microseconds) has also been recently estimated for cyctochrome c at 298 K(51). It is thus clear that the average folding speed limit, and therefore the dynamic term to use in equation 1, is in the microsecond timescale for natural proteins near 298 K. These values are surprisingly close to the original estimate of Eaton and coworkers(52), which was based on the rate of contact formation in chemically unfolded cytochrome c, and about 7 orders of magnitude slower than the gas-phase frequency factor (k_BT/h).

It is important to keep in mind that the diffusion coefficient should depend on experimental conditions (e.g. temperature, solvent viscosity), protein structure, protein sequence and size, and the position of the top of the barrier along the reaction coordinate. For example, hyperfast-folding mutants or de novo designed proteins -in which secondary structure propensities are typically maximized- are likely to tip the top of the folding barrier towards the unfolded state, thereby changing the relevant dynamic regime to that of a less compact ensemble (i.e. faster dynamics). Some of these engineered proteins might fold in less than 1 microsecond(53). By the same token, the use of a simple rate equation (equation 1) yields reasonable results when the barrier is sufficiently high (i.e. several RT). In those conditions the diffusion coefficient reflects the dynamics at the top of the barrier. However, this treatment breaks down as the barrier heights decrease. For proteins with marginal barriers the rate becomes increasingly diffusive and separates into two observable phases (see Figure 1). Although it may seem counterintuitive, for downhill folding proteins the observed folding rate might actually slow down again. The reason being that under strongly native conditions the diffusive downhill relaxation will be dominated by slow dynamics near the native well. Equation 1 should only be used for slow folding proteins (millisecond or longer folding times) and while keeping in mind that the use of an average diffusion coefficient inevitably introduces error in the estimation of the folding barriers.

Free Energy Barriers to Folding

The application of equation 1 parameterized with the empirical estimates of the diffusion coefficient (i.e. from 1/(20 nanoseconds) to 1/(5 microseconds)) to several proteins previously catalogued as two-state (CI2, protein L, FKBP12, Im9, CspB, and GCN4) indicated that their barrier heights in native conditions are typically < 10 RT(54). These barriers are sufficiently high to result in apparent two-state kinetics, in agreement with their classification as two-state-like. However, for some of them the barrier is low enough to suggest that strongly destabilizing mutations might result in departures from two-state behavior. The rate analysis for these 6 two-state proteins offered another valuable piece of information. Namely, it confirmed experimentally that folding barriers arise from the non-synchronous decrease in conformational entropy and gain in stabilization free energy (Figure 2). The origin of the folding barriers became apparent when the activation free energy as a function of temperature derived from equation 1 was analyzed thermodynamically with the approach developed by Freire and coworkers for equilibrium thermal unfolding(55). This analysis produced very similar results for the 6 proteins: at the top of the folding barrier these proteins had lost ~36% of their conformational entropy and only gained ~27% in stabilization free energy. This very small relative difference can still result in barriers of several RT because both compensating terms are large (i.e. for a protein of 65 residues with a cost in conformational entropy of 17.5 J.mol⁻¹.K⁻¹ per residue the 36/27 ratio produces a barrier of ~12 RT at a midpoint temperature of 300 K). Incidentally, the 27% stabilization energy derived from this analysis is in very close agreement with the average phi-value for these proteins(56). The striking similarity between six different proteins -which vary widely in size, structure, sequence and thermodynamic parameters- immediately suggested a fundamental principle at work. It would be of interest to further explore this idea by extending the analysis to other slow two-state folding proteins.

Figure 2.

Representation of a one-dimensional folding free energy surface: stabilization free energy (dashed line); conformational entropy contribution (dotted line); and global free energy (continuous line). The height of the free energy barrier is ~20 kJ/mol.

This simple analysis provides a quantitative picture of the nature and heights of folding barriers that is consistent with physical theory. It is important, however, to develop methods for extracting folding barrier heights that do not require an empirical estimate for the diffusion coefficient. Barrier heights obtained with independent methods can be compared to test for consistency in the results. Moreover, once the barrier height has been measured with a different approach, equation 1 could be used backwards to estimate effective diffusion coefficients from folding rates. Two independent methods that move in that direction have been recently developed.

The first procedure is based on scaling arguments from polymer physics. Scaling laws have been invoked to indicate that folding times should increase with N^v, where N is the number of residues in the protein and v an exponent smaller than 1(57–60). This predicted scaling behavior has become clearly apparent once a folding database with sufficient dynamic range in size was available(61). The correlation between rates and N gives a ruler to calculate the average increase in barrier height per residue, but it does not provide the absolute magnitude of the barriers. The magnitude was estimated by noticing that the 1/2 exponent proposed by Thirumalai(57) can be interpreted thermodynamically with the expression:

$$n_{\sigma }=\mathrm{\Delta }H(T)/\sqrt{R{T}^2\mathrm{\Delta }{C}_p}$$ equation (2)

where ΔH(T) is the equilibrium unfolding enthalpy at a given temperature and ΔC_p the change in heat capacity upon unfolding(61). n_σ represents the frequency at which the excess enthalpy fluctuations in the unfolded state reach enthalpy values characteristic of the native state. It is converted into free energy by calculating the height at n_σ standard deviations from the minimum in a harmonic potential. The free energies calculated with equation 2 are temperature dependent, but at 333 K their values scale with N with exactly the same slope as the folding times (the ruler). The matching of slopes suggests that at 333 K solvation terms in ΔH and ΔC_p compensate exactly in equation 2 so that the calculated free energies approach the real barrier heights. The barriers obtained with this method are consistent with the estimates from equation 1. In fact, a plot of the barrier heights calculated from equation 2 at 333 K versus the logarithm of experimental folding rates (in energy units) crosses the zero barrier at ~ 1/(1 microsecond), providing an independent estimate of the average diffusion coefficient for folding(61). Another interesting aspect of this analysis is that it is entirely based on thermodynamic properties. Thermodynamic parameters obtained in equilibrium unfolding experiments are everything that is needed to catalogue proteins according to their folding behavior. This is illustrated in Figure 3, which shows a plot of experimentally determined n_σ versus N for a series of proteins with midpoint temperatures near 333 K. The line at n_σ = 3 in the figure delimits the threshold separating two-state like proteins from those that fold crossing marginal (or no) barriers. The figure reveals that individual proteins follow the general trend quite closely. Moreover, it indicates that proteins with less than 55 residues are expected to fold by crossing marginal barriers. For many of the proteins below the threshold in Figure 3 there is kinetic data available showing folding in the microsecond timescale. It is also interesting that the de novo designed protein α3D (2A3D in Figure 3) is a clear outlier in the plot, featuring a much smaller n_σ than expected for its size. This observation suggests that the parameter n_σ could be used to gauge the compliance of designed proteins with their “natural” counterparts.

Figure 3.

Plot of the n_σ parameter versus the number of aminoacids (N) for several proteins with T_m near 333 K. Proteins are named with their pdb code. The continuous line shows the average n_σ at 333 K obtained using a ΔH = 2.92 kJ/mol per residue and ΔC_p = 58 J/(mol.K) per residue(83). The two dotted lines show the average n_σ at 318 K and 348 K to provide the swath of T_m values for the proteins used in the plot. The dashed line at n_σ = 3 sets an approximate threshold between two-state-like and marginal barriers.

In the second method, thermodynamic barrier heights for folding are obtained from the analysis of DSC data(62). The fundamental idea here is to extract the probability density as a function of an order parameter by taking advantage of the known connection between the DSC thermogram of a protein and its folding partition function. Measuring the probability density is equivalent to mapping the complete one-dimensional free energy surface for folding (not just the difference between the minimum and the saddle point). In a thermogram without additional sources of fluctuations (i.e. experimental noise, protein vibrational modes, and solvent-protein fluctuations) the probability density is obtained by calculating the inverse Laplace transform. When handling a real protein DSC thermogram, however, one must fit the experimental data to an idealized mathematical description of the free energy surface. In this particular case the free energy surface was represented by a Landau polynomial using enthalpy as the order parameter(62). The procedure is quite simple and involves fewer fitting parameters than even a chemical two-state fit. Of course, its great advantage is that it provides the folding free energy surface and with it the folding barrier height. Because it is an equilibrium method, it is ideally suited to study proteins with small barriers for which there is significant probability density at the saddle point. Indeed, the application of this procedure to many of the proteins with n_σ < 3 in Figure 3 has confirmed that these proteins have marginal folding barriers or fold globally downhill(63). Perhaps more surprisingly, the method seems to effectively detect barriers of up to ~ 8 RT (20 kJ/mol), indicating that it can be used for many of the available two-state-like proteins. Furthermore, the thermodynamic barrier heights measured with this analysis strongly correlate with the experimental folding rates with a slope close to 1(63). This important observation supports the validity of the free energy surface analysis of protein folding and suggests that folding energy landscapes of natural proteins are very smooth. The interesting question is then whether protein landscapes are fundamentally smooth (due to intrinsic properties of the interactions driving folding) or have been designed smooth by natural selection. This question can now be addressed experimentally by comparing the folding energy surfaces of natural versus de novo designed proteins(64).

Manifestations of Folding with Marginal (or No) Barriers

From the developments described in the previous sections, it is apparent that a significant number of proteins that have been, and will be investigated, are likely to have marginal folding barriers. Proteins of small size (i.e. smaller than 55 residues), particularly α-helical, are likely to have marginal folding barriers. All proteins that fold in the microsecond timescale are expected to fall directly into this category or even in the global downhill extreme case (see next section). An important practical issue is how to distinguish between proteins that cross marginal barriers from those that can be reasonably well treated as two-state systems. This is not only important from the standpoint of reclassifying some of the proteins previously described as two-state, but also to help in rationalizing observed deviations from two-state-like behavior. In the chemical paradigm, some of the deviations from 2-state behavior are dealt with by introducing an additional chemical state (i.e. an intermediate). A typical example is the interpretation of two observed kinetic phases as an indication of a kinetically populated intermediate. Other deviations are just ignored, or blamed on experimental uncertainties(65). In principle, the marginal barrier and two-state folding regimes could be simply distinguished by measuring the free energy barrier in a DSC experiment. However, it is also convenient to define an additional set of empirical criteria based on standard protein folding experiments. This set of criteria could be used as a checklist for experimentalists. Recent efforts along these lines have crystallized into a series of empirical criteria.

The dependence of the observed folding behavior on the specific experimental probe is possibly the best-known indicator of folding with marginal barriers(66). If folding barriers are very small or absent, perturbations in the free energy surface lead to large displacements in the position and depth of its minima. The properties of the ensemble should change in complex patterns leading to potentially different observations with techniques that probe different structural features. The differences can be observed in both equilibrium and kinetic experiments(66). In its equilibrium version, this concept was exploited for the characterization of global downhill folding(67)(see next section). The kinetic manifestations have been thoroughly investigated in recent experiments and simulations(68–70). The apparent lack of probe-dependence is also frequently used as indication of two-state folding, but a negative criterion is always a much weaker criterion(65). A thorough application of nuclear magnetic resonance can eliminate this limitation altogether (see next section), but it is also possible to exploit the spectroscopic properties of some of the commonly used low-resolution techniques. For α-helical proteins, for example, circular dichroism (CD) is particularly convenient. Exciton effects from the coupling of amide dipoles in the α-helix structure make its CD spectrum change linearly as the helix length increases(71). Therefore, the spectroscopic analysis of protein CD spectra can provide independent estimates of the α-helix content and average helix length. Figure 4 shows the results of such analysis for myoglobin (panel A) and the downhill folding protein BBL (panel B). The CD spectra of myoglobin at different temperatures are consistent with a helix signal of constant length (~11 residues) and a sigmoidal decrease in helix content, as expected for an all or none transition. For BBL however, the helix content decays sigmoidally while the helix length decreases almost linearly across the whole temperature range. This observation is the basis for the CD wavelength dependence previously reported for BBL(65, 67), and indicates that the 3 helices of BBL are unfolding in the complex manner expected for barrierless unfolding. A similar analysis in several other α-helical proteins suggests that is a robust criterion for distinguishing between two-state and marginal (or no) barrier folding (unpublished data).

Figure 4.

Analysis of far-uv CD spectra as a function of protein unfolding for the two state protein myoglobin (A) and the global downhill protein BBL (B): average helical content (black squares, left scale); average helix length (open circles, right scale). Experiments were performed at 20 mM phosphate buffer pH 7.0. Each CD spectrum was fitted to the expression: $\langle \theta (\lambda )\rangle =x_H·\left({\theta }_{\lambda }^{\infty }·(1-l_H/k_{\lambda })\right)+(1-x_H)·{\theta }_{\lambda }^{\mathit{coil}}$ where x_H is the fraction helix, l_H is the average helix length, θ^∞is the mean residue ellipticity of an infinite length helix as a function of wavelength (obtained from Chen et al.(71)), k is the wavelength dependent helix dependence of the CD spectrum (from ref (71) and then further fitted to the lowest temperature CD spectrum with the known x_H and l_H from the structure. The latter was required to fine tune structural details on each particular protein. θ^coil is the mean residue ellipticity as a function of wavelength of a random coil (taken as the CD spectrum of the protein in fully denaturing conditions).

Another manifestation of folding with marginal barriers is the observation of significant unfolding occurring in the pre- and post-transition regions of the still sigmoidal unfolding curve. This behavior can be detected in a standard chemical two-state analysis by inspecting the physical properties of the native and unfolded phenomenological baselines. In a truly two-state system, the baselines should represent the properties of each of the two states as a function of the unfolding variable. If the folding reaction involves crossing a marginal (or no) barrier, these baselines embed some degree of unfolding and thus, have values indicating structural changes as a function of the unfolding variable. The expected changes are highly sloped baselines that might even cross near the unfolding transition. This effect has been documented in CD baselines(72), but it is in the DSC experiment that becomes very informative(62). The reason is that the DSC baselines directly report the energy fluctuations of the system and the difference between the “native” and “unfolded” baselines renders the ΔC_p of unfolding. Furthermore, baseline effects on DSC experiments can be analyzed theoretically and have, indeed, been the subject of intense theoretical investigations by Chan and collaborators(73). Figure 5 shows DSC thermograms for free energy surfaces with a high barrier (~ 18 kJ/mol at the midpoint, see inset), a marginal barrier (~3 kJ/mol at the midpoint, see inset), and fully downhill. The three DSC thermograms can be well fit to a chemical two-state model (continuous lines), as has been discussed before for experimental DSC data(62, 65). However, comparison with the true theoretical DSC baseline reveals that the fitted “native” baseline becomes increasingly sloped and shifts to higher values (i.e. more disorder) as the barrier decreases. Furthermore, the “native” and “unfolded” baselines cross at increasingly lower temperatures. In fact, the baselines cross in the middle of the unfolding process for proteins with marginal barriers. This is unphysical because it implies that the ΔC_p changes sign in the middle of the transition. Thermograms with these features should be rigorously interpreted with the new methods for DSC analysis (see previous section). However, even a simple two-state analysis of DSC data (measured in absolute heat capacity units so that baselines can be compared to pre-tabulated values) provides a first useful check.

Figure 5.

Simulated DSC thermograms for proteins with different folding barriers: A) two-state-like (midpoint barrier ~18 kJ/mol); B) Marginal barrier (midpoint barrier ~2 kJ/mol); C) Global downhill (no barrier at midpoint). The continuous curves are two-state fits to the simulated data with continuous and dashed lines representing the phenomenological baselines for the “native” and “unfolded” states, respectively. The dotted line shows the theoretical native heat capacity. Insets shows the free energy barrier at the midpoint.

The analysis of other classical equilibrium and kinetic experiments with the chemical two-state model provide additional criteria for marginal (or no) barrier folding. One such example is obtained from double perturbation equilibrium unfolding experiments. The coupling between two denaturing agents (e.g. temperature and chemical denaturant) is defined by straightforward Maxwell relationships for a two-state system(74). For a protein with a marginal folding barrier, the coupling is more complex because the two agents change the properties of the ensemble in different ways(72). This phenomenon can be observed by simply plotting the change in unfolding enthalpy at a reference temperature (e.g. 298 K) as a function of the other denaturing agent (e.g. urea). The plot should be linear for a two-state system, while a protein with marginal or no barrier produces a curved profile(65). Another interesting criterion of two-state versus marginal (or no) barrier is provided by the sensitivity to chemical denaturants. A two-state protein should display the same sensitivity to chemical denaturants (i.e. m-value) in equilibrium and kinetic experiments. By the same token, conservative single-point mutations of a two-state protein are expected to have m-values similar to the wild-type(37). It has been recently found that proteins with marginal (or no) free energy barriers are characterized by an underestimation of the equilibrium m-value when measured kinetically. The underestimation is directly connected to the height of the barrier so that it becomes more severe the smaller the barrier. The same effect is observed by comparing between mutants of the same protein. Namely, the faster the relaxation rate of the mutant at its chemical midpoint, the smaller the m-value observed kinetically (A.N., U.D., and V.M., unpublished results). This is a simple criterion that can be applied directly to data analyzed in the traditional chemical two-state fashion.

Finally, it is important to investigate some of the classical kinetic tests for two-state behavior. This can be done by simulating the kinetics of two-state, marginal barrier, and downhill folding as diffusion on the free energy surfaces shown in the insets of Figure 5.. Figure 6 shows some of the results of such simulations at 324 K. The simulations show clearly that for a protein with a large barrier the relaxation rate is exponential and identical whether it is measured in the folding or in the unfolding direction. The global downhill protein folds much faster and with a relaxation that approximates a slightly stretched exponential. The marginal barrier protein displays a biexponential relaxation with two phases that differ by approximately one order of magnitude. For both marginal and global downhill, the apparent relaxation is faster when measured in the folding direction (panel A). The deviations from exponential kinetics are apparent even in noisy experimental kinetics traces (see panel B). For the marginal barrier case, deviations concentrate on the early times (missing the fast phase), while they are spread throughout the whole time course for global downhill. These results show that biexponential relaxations are a trademark of folding with marginal barriers. The two-phases merge again into a more complex relaxation decay when the barrier disappears. The observation of biexponential kinetics has been documented in several fast-folding proteins(35, 75–80). Gruebele and coworkers have interpreted this behavior as arising from a marginal barrier in the engineered λ repressor(35). However, a more common trend has been to discuss this observation in terms of a folding intermediate (i.e. a three-state system) (75, 76, 78–80). Given what we presently know about folding diffusion coefficients, the marginal barrier interpretation is much more likely to be correct for fast-folding proteins. Of course, this interpretation would be further strengthened if the protein complies with some of the other criteria for marginal barriers outlined above.

Figure 6.

Folding kinetics at different barrier heights: A) normalized relaxation kinetics in the folding direction (dotted lines) and unfolding direction (continuous lines) to the same final condition (near the midpoint); B) normalized relaxation kinetics with added noise and best fits to single exponential decays.

Global Downhill (One-State) Folding and Molecular Rheostats

The progress made in recent years in our understanding of folding barriers puts the experimental identification of global downhill, or one-state, folding(67) in the right perspective. Global downhill folding results from a free energy surface with only one well at all conditions. In highly stabilizing conditions, the minimum of the well is located on the far right side of the reaction coordinate (i.e. native structure). As the native bias decreases, the minimum progressively shifts towards lower values of the reaction coordinate (i.e. more disorder). Under this one-state scenario, the unfolding transition is structurally continuous and the conformational ensemble populated at each condition corresponds to a different stage of the folding reaction(67). The experiments and analyses described above demonstrate that global downhill and two-state-like folding constitute the two extremes of a continuous folding scale in which barriers heights range from significantly high (~15 RT) to non-existing. Single domain proteins that fold in the millisecond or longer timescales are likely to behave in a two-state fashion. On the other hand, fast-folding small domains might fold crossing marginal barriers and thus, display an apparently more complex folding behavior. The barrier might be negligible for some of the small fast-folding proteins, leading to the global downhill folding exhibited by BBL.

The discovery of one-state folding provides a unique opportunity to resolve folding mechanisms experimentally. The absence of a free energy barrier eliminates many of the technical limitations inherent to two-state folding. The three resolution requirements (structure, time, statistics) can be split apart for global downhill folding because the properties of the ensemble are easily tuned. High-resolution NMR experiments can be used in equilibrium to characterize the average structural properties of the protein ensemble with atomic resolution. At different stability conditions the ensemble populates specific subgroups of conformations corresponding to the various stages of the reaction. The apparent unfolding curves of individual protein atoms can then show distinct behaviors depending on how their chemical shifts are perturbed by the varying structural features of the ensemble(65, 66). If the degree of thermodynamic coupling between protein atoms is simply determined by their spatial proximity in the native structure, the distribution of atomic unfolding behaviors would be approximately Gaussian and span the global unfolding transition. Figure 7 depicts this scenario, in which the darker gray area defines the swath corresponding to behaviors within one standard deviation of the average unfolding curve (the circled line) and the lighter gray delimits the maximal spread expected for individual atoms. Measurement of 158 atomic unfolding curves for the protein BBL using NMR shows a distribution such as that of Figure 7(81). In addition to providing a structural confirmation of global downhill folding, these observations can be exploited to measure folding cooperativity and map out the networks of interactions that stabilize protein structure(81). The NMR results in BBL demonstrate that is possible to resolve folding mechanisms in equilibrium experiments. Similarly, kinetic experiments of downhill folding could be used to directly measure folding diffusion coefficients as a function of the reaction coordinate. Global downhill folding is also well suited for analysis with single molecule spectroscopic methods because it eliminates the need of resolving the sharp folding-unfolding transitions inherent to two-state folding.

Figure 7.

Equilibrium global downhill unfolding at atomic resolution. The circled line represents the average atomic unfolding behavior, or global unfolding behavior (i.e. the behavior observed with a low-resolution technique). The dark gray swath corresponds to atomic unfolding behaviors within one standard deviation of the average. The light gray swaths represent the maximal spread of individual atomic unfolding behaviors, which should roughly correspond to the broadness of the global unfolding transition.

The possible biological implications of one-state folding are also of interest. The classical view of protein function, in which proteins are switched from fully on to fully off by a discrete conformational change, is tightly connected to the two-state folding mechanism. In fact, it has been commonly assumed that proteins work as binary switches in a natural extension of their two-state (cooperative) folding behavior. However, from the analysis of folding free energy barriers we now know that two-state folding is not an intrinsic property of proteins. Rather, it seems to be the product of careful selection by natural evolution, exactly as it was originally proposed by the energy landscape approach to protein folding(3). If evolutionary pressure is an important factor in achieving two-state folding, it is only natural to infer that the observation of one-state protein folding responds to the same evolutionary forces. The question is then: what could the biological role of one-state folding proteins be? Their complex conformational behavior immediately suggests a role as molecular rheostats. We could think of one-state proteins working as instantaneous sensors, potentiometers, and oscillators. All of these functionalities could be engineered by coupling a signal to the continuous conformational changes characteristic of one-state folding. For example, an oscillator that synchronizes the action of three enzymes in a multistep reaction could be simply achieved by binding of the first and third enzymes to different conformational sub-ensembles of a one-state (single well) domain in the intermediate enzyme. The same oscillatory behavior can be exploited mechanically as a recoiling mechanism. Indeed, the molecular rheostat idea was put forward as an explanation for the complex functional roles that the small global downhill folder BBL plays in the oxo-glutarate dehydrogenase multienzyme complex(67). Other functionalities could be easily imagined: building a single molecule instantaneous sensor simply requires coupling the binding of the molecule to be detected to the folding of a one-state domain so that increased ligand concentrations induce a continuous change in conformation (and thus a gradual response). To explore these possibilities in more depth it is important to identify other one-state folding domains with potential to work biologically as rheostats. From a functional standpoint, small domains that perform multiple functions within large supramolecular assemblies are promising targets. From a folding standpoint, these small proteins will be more likely to fold in a one-state fashion if their structure is mostly α-helical and have very few aromatic residues in their hydrophobic core. Wang and collaborators have recently proposed a simple structural criterion(82), which could be used in combination with the other ideas outlined here in the pursuit of new examples of global downhill folding.

ABBREVIATIONS

3D

Three-dimensional

NMR

Nuclear Magnetic Resonance

CD

Circular Dichroism

DSC

Differential Scanning Calorimetry

AcP

Muscle Acyl Phosphatase