Topological determinants of protein unfolding rates

For proteins that fold by two-state kinetics, the folding and unfolding processes are believed to be closely related to their native structures. In particular, folding and unfolding rates are influenced by the native structures of proteins. Thus, we focus on finding important topological quantities from a protein structure that determine its unfolding rate. After constructing graphs from protein native structures, we investigate the relationships between unfolding rates and various topological quantities of the graphs. First, we find that the correlation between the unfolding rate and the contact order is not as prominent as in the case of the folding rate and the contact order. Next, we investigate the correlation between the unfolding rate and the clustering coefficient of the graph of a protein native structure, and observe no correlation between them. Finally, we find that a newly introduced quantity, the impact of edge removal per residue, has a good overall correlation with protein unfolding rates. The impact of edge removal is defined as the ratio of the change of the average path length to the edge removal probability. From these facts, we conclude that the protein unfolding process is closely related to the protein native structure.     

Unification of the folding mechanisms of non-two-state and two-state proteins

We have collected the kinetic folding data for non-two-state and two-state globular proteins reported in the literature, and investigated the relationships between the folding kinetics and the native three-dimensional structure of these proteins. The rate constants of formation of both the intermediate and the native state of non-two-state folders were found to be significantly correlated with protein chain length and native backbone topology, which is represented by the absolute contact order and sequence-distant native pairs. The folding rate of two-state folders, which is known to be correlated with the native backbone topology, apparently does not correlate significantly with protein chain length. On the basis of a comparison of the folding rates of the non-two-state and two-state folders, it was found that they are similarly dependent on the parameters that reflect the native backbone topology. This suggests that the mechanisms behind non-two-state and two-state folding are essentially identical. The present results lead us to propose a unified mechanism of protein folding, in which folding occurs in a hierarchical manner, reflecting the hierarchy of the native three-dimensional structure, as embodied in the case of non-two-state folding with an accumulation of the intermediate. Apparently, two-state folding is merely a simplified version of hierarchical folding caused either by an alteration in the rate-limiting step of folding or by destabilization of the intermediate.     

Probing possible downhill folding: native contact topology likely places a significant constraint on the folding cooperativity of proteins with approximately 40 residues

Experiments point to appreciable variations in folding cooperativity among natural proteins with approximately 40 residues, indicating that the behaviors of these proteins are valuable for delineating the contributing factors to cooperative folding. To explore the role of native topology in a protein's propensity to fold cooperatively and how native topology might constrain the degree of cooperativity achievable by a given set of physical interactions, we compared folding/unfolding kinetics simulated using three classes of native-centric C^α chain models with different interaction schemes. The approach was applied to two homologous 45-residue fragments from the peripheral subunit-binding domain family and a 39-residue fragment of the N-terminal domain of ribosomal protein L9. Free-energy profiles as functions of native contact number were computed to assess the heights of thermodynamic barriers to folding. In addition, chevron plots of folding/unfolding rates were constructed as functions of native stability to facilitate comparison with available experimental data. Although common Gō-like models with pairwise Lennard-Jones-type interactions generally fold less cooperatively than real proteins, the rank ordering of cooperativity predicted by these models is consistent with experiment for the proteins investigated, showing increasing folding cooperativity with increasing nonlocality of a protein's native contacts. Models that account for water-expulsion (desolvation) barriers and models with many-body (nonadditive) interactions generally entail higher degrees of folding cooperativity indicated by more linear model chevron plots, but the rank ordering of cooperativity remains unchanged. A robust, experimentally valid rank ordering of model folding cooperativity independent of the multiple native-centric interaction schemes tested here argues that native topology places significant constraints on how cooperatively a protein can fold.     

Sequence determinants of protein folding rates: positive correlation between contact energy and contact range indicates selection for fast folding

In comparison with intense investigation of the structural determinants of protein folding rates, the sequence features favoring fast folding have received little attention. Here, we investigate this subject using simple models of protein folding and a statistical analysis of the Protein Data Bank (PDB). The mean-field model by Plotkin and coworkers predicts that the folding rate is accelerated by stronger-than-average interactions at short distance along the sequence. We confirmed this prediction using the Finkelstein model of protein folding, which accounts for realistic features of polymer entropy. We then tested this prediction on the PDB. We found that native interactions are strongest at contact range l = 8. However, since short range contacts tend to be exposed and they are frequently formed in misfolded structures, selection for folding stability tends to make them less attractive, that is, stability and kinetics may have contrasting requirements. Using a recently proposed model, we predicted the relationship between contact range and contact energy based on buriedness and contact frequency. Deviations from this prediction induce a positive correlation between contact range and contact energy, that is, short range contacts are stronger than expected, for 2/3 of the proteins. This correlation increases with the absolute contact order (ACO), as expected if proteins that tend to fold slowly due to large ACO are subject to stronger selection for sequence features favoring fast folding. Our results suggest that the selective pressure for fast folding is detectable only for one third of the proteins in the PDB, in particular those with large contact order.     

Transition state contact orders correlate with protein folding rates

We have used molecular dynamics simulations restrained by experimental phi values derived from protein engineering experiments to determine the structures of the transition state ensembles of ten proteins that fold with two-state kinetics. For each of these proteins we then calculated the average contact order in the transition state ensemble and compared it with the corresponding experimental folding rate. The resulting correlation coefficient is similar to that computed for the contact orders of the native structures, supporting the use of native state contact orders for predicting folding rates. The native contacts in the transition state also correlate with those of the native state but are found to be about 30% lower. These results show that, despite the high levels of heterogeneity in the transition state ensemble, the large majority of contributing structures have native-like topologies and that the native state contact order captures this phenomenon.     

pH-jump-induced folding and unfolding studies of barstar: evidence for multiple folding and unfolding pathways.

Equilibrium and kinetic characterization of the high pH-induced unfolding transition of the small protein barstar have been carried out in the pH range 7-12. A mutant form of barstar, containing a single tryptophan, Trp 53, completely buried in the core of the native protein, has been used. It is shown that the protein undergoes reversible unfolding above pH 10. The pH 12 form (the D form) appears to be as unfolded as the form unfolded by 6 M guanidine hydrochloride (GdnHCl) at pH 7 (the U form): both forms have similar fluorescence and far-UV circular dichroism (CD) signals and have similar sizes, as determined by dynamic light scattering and size-exclusion chromatography. No residual structure is detected in the D form: addition of GdnHCl does not alter its fluorescence and far-UV CD properties. The fluorescence signal of Trp 53 has been used to monitor folding and unfolding kinetics. The kinetics of folding of the D form in the pH range 7-11 are complex and are described by four exponential processes, as are the kinetics of unfolding of the native state (N state) in the pH range 10.5-12. Each kinetic phase of folding decreases in rate with increase in pH from 7 to 10.85, and each kinetic phase of unfolding decreases in rate with decrease in pH from 12 to 10.85. At pH 10.85, the folding and unfolding rates for any particular kinetic phase are identical and minimal. The two slowest phases of folding and unfolding have identical kinetics whether measured by Trp 53 fluorescence or by mean residue ellipticity at 222 nm. Direct determination of the increase in the N state with time of folding at pH 7 and of the D form with time of unfolding at pH 12, by means of double-jump assays, show that between 85 and 95% of protein molecules fold or unfold via fast pathways between the two forms. The remaining 5-15% of protein molecules appear to fold or unfold via slower pathways, on which at least two intermediates accumulate. The mechanism of folding from the high pH-denatured D form is remarkably similar to the mechanism of folding from the urea or GdnHCl-denatured U form.     

Mutational analysis of acylphosphatase suggests the importance of topology and contact order in protein folding.

Muscle acylphosphatase (AcP) is a small protein that folds very slowly with two-state behavior. The conformational stability and the rates of folding and unfolding have been determined for a number of mutants of AcP in order to characterize the structure of the folding transition state. The results show that the transition state is an expanded version of the native protein, where most of the native interactions are partially established. The transition state of AcP turns out to be remarkably similar in structure to that of the activation domain of procarboxypeptidase A2 (ADA2h), a protein having the same overall topology but sharing only 13% sequence identity with AcP. This suggests that transition states are conserved between proteins with the same native fold. Comparison of the rates of folding of AcP and four other proteins with the same topology, including ADA2h, supports the concept that the average distance in sequence between interacting residues (that is, the contact order) is an important determinant of the rate of protein folding.     

Protein topology affects the appearance of intermediates during the folding of proteins with a flavodoxin-like fold

The topology of a native protein influences the rate with which it is formed, but does topology affect the appearance of folding intermediates and their specific role in kinetic folding as well? This question is addressed by comparing the folding data recently obtained on apoflavodoxin from Azotobacter vinelandii with those available on all three other alpha-beta parallel proteins the kinetic folding mechanism of which has been studied, i.e. Anabaena apoflavodoxin, Fusarium solani pisi cutinase and CheY. Two kinetic folding intermediates, one on-pathway and the other off-pathway, seem to be present during the folding of proteins with an alpha-beta parallel, also called flavodoxin-like, topology. The on-pathway intermediate lies on a direct route from the unfolded to the native state of the protein involved. The off-pathway intermediate needs to unfold to allow the production of native protein. Available simulation data of the folding of CheY show the involvement of two intermediates with characteristics that resemble those of the two intermediates experimentally observed. Apparently, protein topology governs the appearance and kinetic roles of protein folding intermediates during the folding of proteins that have a flavodoxin-like fold.     

Prediction of folding rates and transition-state placement from native-state geometry

A variety of experimental and theoretical studies have established that the folding process of monomeric proteins is strongly influenced by the topology of the native state. In particular, folding times have been shown to correlate well with the contact order, a measure of contact locality. Our investigation focuses on identifying additional topologic properties that correlate with experimentally measurable quantities, such as folding rates and transition-state placement, for both two- and three-state folders. The validation against data from 40 experiments shows that a particular topological property that measures the interdependence of contacts, termed cliquishness or clustering coefficient, can account with statistically significant accuracy both for the transition state placement and especially for folding rates. The observed correlations can be further improved by optimally combining the distinct topological information captured by cliquishness and contact order.     

First principles prediction of protein folding rates

Experimental studies have demonstrated that many small, single-domain proteins fold via simple two-state kinetics. We present a first principles approach for predicting these experimentally determined folding rates. Our approach is based on a nucleation-condensation folding mechanism, where the rate-limiting step is a random, diffusive search for the native tertiary topology. To estimate the rates of folding for various proteins via this mechanism, we first determine the probability of randomly sampling a conformation with the native fold topology. Next, we convert these probabilities into folding rates by estimating the rate that a protein samples different topologies during diffusive folding. This topology-sampling rate is calculated using the Einstein diffusion equation in conjunction with an experimentally determined intra-protein diffusion constant. We have applied our prediction method to the 21 topologically distinct small proteins for which two-state rate data is available. For the 18 beta-sheet and mixed alpha-beta native proteins, we predict folding rates within an average factor of 4, even though the experimental rates vary by a factor of approximately 4 x 10(4). Interestingly, the experimental folding rates for the three four-helix bundle proteins are significantly underestimated by this approach, suggesting that proteins with significant helical content may fold by a faster, alternative mechanism. This method can be applied to any protein for which the structure is known and hence can be used to predict the folding rates of many proteins prior to experiment.     

Studies of Pseudomonas aeruginosa azurin mutants: cavities in beta-barrel do not affect refolding speed

Pseudomonas aeruginosa azurin is a blue-copper protein with a Greek-key fold. Removal of copper produces an apoprotein with the same structure as holoazurin. To address the effects on thermodynamic stability and folding dynamics caused by small cavities in a beta-barrel, we have studied the behavior of the apo-forms of wild-type and two mutant (His-46-Gly and His-117-Gly) azurins. The equilibrium- and kinetic-folding and unfolding reactions appear as two-state processes for all three proteins. The thermodynamic stability of the two mutants is significantly decreased as compared with the stability of wild-type azurin, in accord with cavities in or near the hydrophobic interior having an overall destabilizing effect. Large differences are also found in the unfolding rates: the mutants unfold much faster than wild-type azurin. In contrast, the folding-rate constants are almost identical for the three proteins and closely match the rate-constant predicted from the native-state topology of azurin. We conclude that the topology is more important than equilibrium stability in determining the folding speed of azurin.     

Extensive misfolding in the refolding reaction of alkaline ferrocytochrome C

This work describes an extensively misfolded kinetic intermediate in the folding of horse ferrocytochrome c. Under absolute native conditions, the alkali-unfolded protein liganded with carbon-monoxide exhibits misfolding. The misfolded product, apparently an off-pathway intermediate, requires large-scale unfolding in order to have a chance to fold correctly to the native state. The rate of unfolding of the misfolded intermediate limits the overall rate of protein folding. The high level of observed misfolding possibly results from a failure of the polypeptide chain to achieve by stochastic search the transition state relevant for successful folding. Such misfolding may be analogous to the failure of a sizable set of proteins in the intracellular milieu to fold to the functionally active native state.     

Monomer topology defines folding speed of heptamer

Small monomeric proteins often fold in apparent two-state processes with folding speeds dictated by their native-state topology. Here we test, for the first time, the influence of monomer topology on the folding speed of an oligomeric protein: the heptameric cochaperonin protein 10 (cpn10), which in the native state has seven beta-barrel subunits noncovalently assembled through beta-strand pairing. Cpn10 is a particularly useful model because equilibrium-unfolding experiments have revealed that the denatured state in urea is that of a nonnative heptamer. Surprisingly, refolding of the nonnative cpn10 heptamer is a simple two-state kinetic process with a folding-rate constant in water (2.1 sec(-1); pH 7.0, 20 degrees C) that is in excellent agreement with the prediction based on the native-state topology of the cpn10 monomer. Thus, the monomers appear to fold as independent units, with a speed that correlates with topology, although the C and N termini are trapped in beta-strand pairing with neighboring subunits. In contrast, refolding of unfolded cpn10 monomers is dominated by a slow association step.     

On the role of some conserved and nonconserved amino acid residues in the transitional state and intermediate of apomyoglobin folding

The contributions of some amino acid residues in the A, B, G, and H helices to the formation of the folding nucleus and folding intermediate of apomyoglobin were estimated. The effects of point substitutions of Ala for hydrophobic amino acid residues on the structural stability of the native (N) protein and its folding intermediate (I), as well as on the folding/unfolding rates for four mutant apomyoglobin forms, were studied. The equilibrium and kinetic studies of the folding/unfolding rates of these mutant proteins in a wide range of urea concentrations demonstrated that their native state was considerably destabilized as compared with the wild-type protein, whereas the stability of the intermediate state changed moderately. It was shown that the amino acid residues in the A, G, and H helices contributed insignificantly to the stabilization of the apomyoglobin folding nucleus in the rate-limiting I ? N transition, taking place after the formation of the intermediate, whereas the residue of the B helix was of great importance in the formation of the folding nucleus in this transition.     

Stability and folding rates of domains spanning the large A-band super-repeat of titin

Titin is a very large (>3 MDa) protein found in striated muscle where it is believed to participate in myogenesis and passive tension. A prominent feature in the A-band portion of titin is the presence of an 11-domain super-repeat of immunoglobulin superfamily and fibronectin-type-III-like domains. Seven overlapping constructs from human cardiac titin, each consisting of two or three domains and together spanning the entire 11-domain super-repeat, have been expressed in Escherichia coli. Fluorescence unfolding experiments and circular dichroism spectroscopy have been used to measure folding stabilities for each of the constructs and to assign unfolding rates for each super-repeat domain. Immunoglobulin superfamily domains were found to fold correctly only in the presence of their C-terminal fibronectin type II domain, suggesting close and possibly rigid association between these units. The domain stabilities, which range from 8.6 to 42 kJ mol(-1) under physiological conditions, correlate with previously reported mechanical forces required to unfold titin domains. Individual domains vary greatly in their rates of unfolding, with a range of unfolding rate constants between 2.6 x 10(-6) and 1.2 s(-1). This variation in folding behavior is likely to be an important determinant in ensuring independent folding of domains in multi-domain proteins such as titin.     

The Limited Role of Nonnative Contacts in the Folding Pathways of a Lattice Protein

Models of protein energetics that neglect interactions between amino acids that are not adjacent in the native state, such as the Gō model, encode or underlie many influential ideas on protein folding. Implicit in this simplification is a crucial assumption that has never been critically evaluated in a broad context: Detailed mechanisms of protein folding are not biased by nonnative contacts, typically argued to be a consequence of sequence design and/or topology. Here we present, using computer simulations of a well-studied lattice heteropolymer model, the first systematic test of this oft-assumed correspondence over the statistically significant range of hundreds of thousands of amino acid sequences that fold to the same native structure. Contrary to previous conjectures, we find a multiplicity of folding mechanisms, suggesting that Gō-like models cannot be justified by considerations of topology alone. Instead, we find that the crucial factor in discriminating among topological pathways is the heterogeneity of native contact energies: The order in which native contacts accumulate is profoundly insensitive to omission of nonnative interactions, provided that native contact heterogeneity is retained. This robustness holds over a surprisingly wide range of folding rates for our designed sequences. Mirroring predictions based on the principle of minimum frustration, fast-folding sequences match their Gō-like counterparts in both topological mechanism and transit times. Less optimized sequences dwell much longer in the unfolded state and/or off-pathway intermediates than do Gō-like models. For dynamics that bridge unfolded and unfolded states, however, even slow folders exhibit topological mechanisms and transit times nearly identical with those of their Gō-like counterparts. Our results do not imply a direct correspondence between folding trajectories of Gō-like models and those of real proteins, but they do help to clarify key topological and energetic assumptions that are commonly used to justify such caricatures.     

A simple measure of native-state topology and chain connectivity predicts the folding rates of two-state proteins with and without crosslinks

The folding rates of two-state proteins have been found to correlate with simple measures of native-state topology. The most prominent among these measures is the relative contact order (CO), which is the average CO, or localness, of all contacts in the native protein structure, divided by the chain length. Here, we test whether such measures can be generalized to capture the effect of chain crosslinks on the folding rate. Crosslinks change the chain connectivity and therefore also the localness of some of the native contacts. These changes in localness can be taken into account by the graph-theoretical concept of effective contact order (ECO). The relative ECO, however, the natural extension of the relative CO for proteins with crosslinks, overestimates the changes in the folding rates caused by crosslinks. We suggest here a novel measure of native-state topology, the relative logCO, and its natural extension, the relative logECO. The relative logCO is the average value for the logarithm of the CO of all contacts, divided by the logarithm of the chain length. The relative log(E)CO reproduces the folding rates of a set of 26 two-state proteins without crosslinks with essentially the same high correlation coefficient as the relative CO. In addition, it also captures the folding rates of eight two-state proteins with crosslinks.     

Folding of DsbB in mixed micelles: a kinetic analysis of the stability of a bacterial membrane protein

Measuring the stability of integrated membrane proteins under equilibrium conditions is hampered by the nature of the proteins' amphiphilic environment. While intrinsic fluorescence is a useful probe for structural changes in water-soluble proteins, the fluorescence of membrane proteins is sensitive to changes in lipid and detergent composition. As an attempt to overcome this problem, I present a kinetic analysis of the folding of a membrane protein, disulfide bond reducing protein B (DsbB), in a mixed micelle system consisting of varying molar ratios of sodium dodecyl sulfate (SDS) and dodecyl maltoside (DM). This analysis incorporates both folding and unfolding rates, making it possible to determine both the stability of the native state and the process by which the protein folds. Refolding and unfolding occur on the second to millisecond timescale and involve only one relaxation phase, when monitored by conventional stopped-flow. The kinetic data indicate that denaturation occurs around 0.3 mole fraction of SDS, in agreement with CD analysis and acrylamide quenching data. The rate constants have been fit to a three-state folding scheme involving the SDS-denatured state, the native state and an unfolding intermediate that accumulates only under unfolding conditions at high mole fractions of SDS. The stability of DsbB is around 4.4 kcal/mol in DM, and this is halved upon reduction of the two periplasmic disulfide bonds, and is sensitive to mutagenesis. With the caveat that kinetic data are always open to alternative interpretations, time-resolved studies in mixed micelles provide a useful approach to measure membrane protein stability over a wide range of concentrations of SDS and DM, as well as a framework for the future characterization of the DsbB folding mechanism.     

Folding rates and low-entropy-loss routes of two-state proteins

We develop a simple model for computing the rates and routes of folding of two-state proteins from the contact maps of their native structures. The model is based on the graph-theoretical concept of effective contact order (ECO). The model predicts that proteins fold by "zipping up" in a sequence of small-loop-closure events, depending on the native chain fold. Using a simple equation, with a few physical rate parameters, we obtain a good correlation with the folding rates of 24 two-state folding proteins. The model rationalizes data from Phi-value analysis that have been interpreted in terms of delocalized or polarized transition states. This model indicates how much of protein folding may take place in parallel, not along a single reaction coordinate or with a single transition state.