Protein folding rates estimated from contact predictions

Folding rates of small single-domain proteins that fold through simple two-state kinetics can be estimated from details of the three-dimensional protein structure. Previously, predictions of secondary structure had been exploited to predict folding rates from sequence. Here, we estimate two-state folding rates from predictions of internal residue-residue contacts in proteins of unknown structure. Our estimate is based on the correlation between the folding rate and the number of predicted long-range contacts normalized by the square of the protein length. It is well known that long-range order derived from known structures correlates with folding rates. The surprise was that estimates based on very noisy contact predictions were almost as accurate as the estimates based on known contacts. On average, our estimates were similar to those previously published from secondary structure predictions. The combination of these methods that exploit different sources of information improved performance. It appeared that the combined method reliably distinguished fast from slow two-state folders.     

Prediction of folding transition-state position (betaT) of small, two-state proteins from local secondary structure content

Folding kinetics of proteins is governed by the free energy and position of transition states. But attempts to predict the position of folding transition state on reaction pathway from protein structure have been met with only limited success, unlike the folding-rate prediction. Here, we find that the folding transition-state position is related to the secondary structure content of native two-state proteins. We present a simple method for predicting the transition-state position from their alpha-helix, turn and polyproline secondary structures. The method achieves 81% correlation with experiment over 24 small, two-state proteins, suggesting that the local secondary structure content, especially for content of alpha-helix, is a determinant of the solvent accessibility of the transition state ensemble and size of folding nucleus.     

Local secondary structure content predicts folding rates for simple,two-state proteins

Many single-domain proteins exhibit two-state folding kinetics, with folding rates that span more than six orders of magnitude. A quantity of much recent interest for such proteins is their contact order, the average separation in sequence between contacting residue pairs. Numerous studies have reached the surprising conclusion that contact order is well-correlated with the logarithm of the folding rate for these small, well-characterized molecules. Here, we investigate the physico-chemical basis for this finding by asking whether contact order is actually a composite number that measures the fraction of local secondary structure in the protein; viz. turns, helices, and hairpins. To pursue this question, we calculated the secondary structure content for 24 two-state proteins and obtained coefficients that predict their folding rates. The predicted rates correlate strongly with experimentally determined rates, comparable to the correlation with contact order. Further, these predicted folding rates are correlated strongly with contact order. Our results suggest that the folding rate of two-state proteins is a function of their local secondary structure content, consistent with the hierarchic model of protein folding. Accordingly, it should be possible to utilize secondary structure prediction methods to predict folding rates from sequence alone.     

Class-specific correlations between protein folding rate, structure-derived, and sequence-derived descriptors

Small single-domain proteins that fold by simple two-state kinetics have been shown to exhibit a wide variation in their folding rates. It has been proposed that folding mechanisms in these proteins are largely determined by the native-state topology, and a significant correlation between folding rate and measures of the average topological complexity, such as relative contact order (RCO), has been reported. We perform a statistical analysis of folding rate and RCO in all three major structural classes (alpha, beta, and alpha/beta) of small two-state proteins and of RCO in groups of analogous and homologous small single-domain proteins with the same topology. We also study correlation between folding rate and the average physicochemical properties of amino acid sequences in two-state proteins. Our results indicate that 1) helical proteins have statistically distinguishable, class-specific folding rates; 2) RCO accounts for essentially all the variation of folding rate in helical proteins, but for only a part of the variation in beta-sheet-containing proteins; and 3) only a small fraction of the protein topologies studied show a topology-specific RCO. We also report a highly significant correlation between the folding rate and average intrinsic structural propensities of protein sequences. These results suggest that intrinsic structural propensities may be an important determinant of the rate of folding in small two-state proteins.     

Folding with downhill behavior and low cooperativity of proteins

The downhill folding observed experimentally for a small protein BBL is studied using off-lattice Gō-like model. Our simulations show that the downhill folding has low cooperativity and is barrierless, which is consistent with the experimental findings. As an example of comparison in detail, the two-state folding behavior of proteins, for example, protein CI2, is also simulated. By observing the formation of contacts between the residues for these two proteins, it is found that the physical origin of the downhill folding is due to the deficiency of nonlocal contacts which determine the folding cooperatively. From a statistics on contacts of the native structures of 17 well-studied proteins and the calculation of their cooperativity factors kappa2 based on folding simulations, a strong correlation between the number of nonlocal contacts per residue NN and the factors kappa2 is obtained. Protein BBL with a value of NN = 0.73 has the lowest cooperativity factor kappa2 = 0.34 among all 17 proteins. A crossover around NNc approximately 0.9 could be defined to separate the two-state folders and the downhill folder roughly. A protein would behave downhill folding when its NN = NNc. For proteins with their NN values are about (or slightly larger than) NNc, the folding behaves with low cooperativity and the barriers are small, showing a weak two-state behavior or a downhill-like behavior. Furthermore, simulations on mutants of a two-state folder show that a mutant becomes a downhill folder when its NN is reduced to a value smaller than NNc. These could enable us to identify the downhill folding or the cooperative two-state folding behavior solely from the native structures of proteins.     

Statistical analyses of protein folding rates from the view of quantum transition

Understanding protein folding rate is the primary key to unlock the fundamental physics underlying protein structure and its folding mechanism.Especially,the temperature dependence of the folding rate remains unsolved in the literature.Starting from the assumption that protein folding is an event of quantum transition between molecular conformations,we calculated the folding rate for all two-state proteins in a database and studied their temperature dependencies.The non-Arrhenius temperature relation for 16 proteins,whose experimental data had previously been available,was successfully interpreted by comparing the Arrhenius plot with the first-principle calculation.A statistical formula for the prediction of two-state protein folding rate was proposed based on quantum folding theory.The statistical comparisons of the folding rates for 65 two-state proteins were carried out,and the theoretical vs.experimental correlation coefficient was 0.73.Moreover,the maximum and the minimum folding rates given by the theory were consistent with the experimental results.     

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.     

Secondary structure length as a determinant of folding rate of proteins with two- and three-state kinetics

We present a simple method for determining the folding rates of two- and three-state proteins from the number of residues in their secondary structures (secondary structure length). The method is based on the hypothesis that two- and three-state foldings share a common pattern. Three-state proteins first condense into metastable intermediates, subsequent forming of alpha-helices, turns, and beta-sheets at slow rate-limiting step. The folding rate of such proteins anticorrelate with the length of these beta-secondary structures. It is also assumed that in two-state folding, rapidly folded alpha-helices and turns may facilitate formation of fleeting unobservable intermediates and thus show two-state behavior. There is an inverse relationship between the folding rate and the length of beta-sheets and loops. Our study achieves 94.0 and 88.1% correlations with folding rates determined experimentally for 21 three- and 38 two-state proteins, respectively, suggesting that protein-folding rates are determined by the secondary structure length. The kinetic kinds are selected on the basis of a competitive formation of hydrophobic collapse and alpha-structure in early intermediates.     

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.     

Thermodynamics and kinetics of protein folding: an evolutionary perspective

This article appeals to an evolutionary model which postulates that primordial proteins were described by small polypeptide chains which (i) lack disulfide bridges, and (ii) display slow folding rates with multi-state kinetics, to determine relations between structural properties of proteins and their folding kinetics. We parameterize the energy landscape of proteins in terms of thermodynamic activation variables. The model studies evolutionary changes in these thermodynamic parameters, and we invoke relations between these activation variables and structural properties of the protein to predict the following correspondence between protein structure and folding kinetics. 1. Proteins with inter- and intra-chain disulfide bridges: large variability in both folding rates and stability of intermediates, multi-state kinetics. 2. Proteins which lack inter and intra-chain disulfide bridges. 2.1 Single-domain chains: fast folding rates; unstable intermediates; two-state kinetics. 2.2 Multi-domain monomers: intermediate rates; metastable intermediates; multi-state kinetics. 2.3 Multi-domain oligomers: slow rates; metastable intermediates; multi-state kinetics. The evolutionary model thus provides a kinetic characterization of one important subfamily of proteins which we describe by the following properties: Folding dynamics of single-domain proteins which lack disulfide bridges are described by two-state kinetics. Folding rate of this class of proteins is positively correlated with the thermodynamic stability of the folded state.     

Exploring the relationship between funneled energy landscapes and two-state protein folding

It should take an astronomical time span for unfolded protein chains to find their native state based on an unguided conformational random search. The experimental observation that folding is fast can be rationalized by assuming that protein energy landscapes are sloped towards the native state minimum, such that rapid folding can proceed from virtually any point in conformational space. Folding transitions often exhibit two-state behavior, involving extensively disordered and highly structured conformers as the only two observable kinetic species. This study employs a simple Brownian dynamics model of "protein particles" moving in a spherically symmetrical potential. As expected, the presence of an overall slope towards the native state minimum is an effective means to speed up folding. However, the two-state nature of the transition is eradicated if a significant energetic bias extends too far into the non-native conformational space. The breakdown of two-state cooperativity under these conditions is caused by a continuous conformational drift of the unfolded proteins. Ideal two-state behavior can only be maintained on surfaces exhibiting large regions that are energetically flat, a result that is supported by other recent data in the literature (Kaya and Chan, Proteins: Struct Funct Genet 2003;52:510-523). Rapid two-state folding requires energy landscapes exhibiting the following features: (i) A large region in conformational space that is energetically flat, thus allowing for a significant degree of random sampling, such that unfolded proteins can retain a random coil structure; (ii) a trapping area that is strongly sloped towards the native state minimum.     

Chain length is the main determinant of the folding rate for proteins with three-state folding kinetics

We demonstrate that chain length is the main determinant of the folding rate for proteins with the three-state folding kinetics. The logarithm of their folding rate in water (k(f)) strongly anticorrelates with their chain length L (the correlation coefficient being -0.80). At the same time, the chain length has no correlation with the folding rate for two-state folding proteins (the correlation coefficient is -0.07). Another significant difference of these two groups of proteins is a strong anticorrelation between the folding rate and Baker's "relative contact order" for the two-state folders and the complete absence of such correlation for the three-state folders.     

Folding mechanism of all-beta globular proteins

Elucidating the mechanism for the fast folding of proteins is a challenging task. In our earlier work, we introduced the concept of "long-range order" and related it successfully to protein folding rates. In this article, we propose a new hypothesis for the folding of two-state all-beta proteins. The mechanism is based on the formation of a hydrophobic core, propagation of beta-strands, and the establishment of hydrogen bonds. Our hypothesis has been strengthened by the observation of a folding nucleus in beta-strands and the hydrogen-bonding network between residues in beta-strands. Our insights on protein folding show an excellent agreement with experimental observations.     

Detection and structure determination of an equilibrium unfolding intermediate of Rd-apocytochrome b562: native fold with non-native hydrophobic interactions

The absence of detectable kinetic and equilibrium folding intermediates by optical probes is commonly taken to indicate that protein folding is a two-state process. However, for some small proteins with apparent two-state behavior, unfolding intermediates have been identified in native-state hydrogen exchange or kinetic unfolding experiments monitored by nuclear magnetic resonance. Rd-apocytochrome b(562), a four-helix bundle, is one such protein. Here, we found another unfolding intermediate for Rd-apocytochrome b(562). It is based on a cooperative transition of (15)N chemical shifts of amide protons as a function of urea concentrations before the global unfolding. We have solved the high-resolution structure of the protein at 2.8 M urea, which is after this cooperative transition but before the global unfolding. All four helices remained intact, but a number of hydrophobic core residues repacked. This intermediate provides a possible structural interpretation for the kinetic unfolding intermediates observed using nuclear magnetic resonance methods for several proteins and has important implications for theoretical studies of protein folding.     

A comparison of the folding kinetics of a small,artificially selected DNA aptamer with those of equivalently simple naturally occurring proteins

The folding of larger proteins generally differs from the folding of similarly large nucleic acids in the number and stability of the intermediates involved. To date, however, no similar comparison has been made between the folding of smaller proteins, which typically fold without well-populated intermediates, and the folding of small, simple nucleic acids. In response, in this study, we compare the folding of a 38-base DNA aptamer with the folding of a set of equivalently simple proteins. We find that, as is true for the large majority of simple, single domain proteins, the aptamer folds through a concerted, millisecond-scale process lacking well-populated intermediates. Perhaps surprisingly, the observed folding rate falls within error of a previously described relationship between the folding kinetics of single-domain proteins and their native state topology. Likewise, similarly to single-domain proteins, the aptamer exhibits a relatively low urea-derived Tanford β, suggesting that its folding transition state is modestly ordered. In contrast to this, however, and in contrast to the behavior of proteins, ϕ-value analysis suggests that the aptamer''s folding transition state is highly ordered, a discrepancy that presumably reflects the markedly more important role that secondary structure formation plays in the folding of nucleic acids. This difference notwithstanding, the similarities that we observe between the two-state folding of single-domain proteins and the two-state folding of this similarly simple DNA presumably reflect properties that are universal in the folding of all sufficiently cooperative heteropolymers irrespective of their chemical details.     

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.     

Criteria for downhill protein folding: calorimetry, chevron plot, kinetic relaxation, and single-molecule radius of gyration in chain models with subdued degrees of cooperativity

Recent investigations of possible downhill folding of small proteins such as BBL have focused on the thermodynamics of non-two-state, "barrierless" folding/denaturation transitions. Downhill folding is noncooperative and thermodynamically "one-state," a phenomenon underpinned by a unimodal conformational distribution over chain properties such as enthalpy, hydrophobic exposure, and conformational dimension. In contrast, corresponding distributions for cooperative two-state folding are bimodal with well-separated population peaks. Using simplified atomic modeling of a three-helix bundle-in a scheme that accounts for hydrophobic interactions and hydrogen bonding-and coarse-grained C(alpha) models of four real proteins with various degrees of cooperativity, we evaluate the effectiveness of several observables at defining the underlying distribution. Bimodal distributions generally lead to sharper transitions, with a higher heat capacity peak at the transition midpoint, compared with unimodal distributions. However, the observation of a sigmoidal transition is not a reliable criterion for two-state behavior, and the heat capacity baselines, used to determine the van't Hoff and calorimetric enthalpies of the transition, can introduce ambiguity. Interestingly we find that, if the distribution of the single-molecule radius of gyration were available, it would permit discrimination between unimodal and bimodal underlying distributions. We investigate kinetic implications of thermodynamic noncooperativity using Langevin dynamics. Despite substantial chevron rollovers, the relaxation of the models considered is essentially single-exponential over an extended range of native stabilities. Consistent with experiments, significant deviations from single-exponential behavior occur only under strongly folding conditions.