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.     

Contact order revisited: influence of protein size on the folding rate

Guided by the recent success of empirical model predicting the folding rates of small two-state folding proteins from the relative contact order (CO) of their native structures, by a theoretical model of protein folding that predicts that logarithm of the folding rate decreases with the protein chain length L as L(2/3), and by the finding that the folding rates of multistate folding proteins strongly correlate with their sizes and have very bad correlation with CO, we reexamined the dependence of folding rate on CO and L in attempt to find a structural parameter that determines folding rates for the totality of proteins. We show that the Abs_CO = CO x L, is able to predict rather accurately folding rates for both two-state and multistate folding proteins, as well as short peptides, and that this Abs_CO scales with the protein chain length as L(0.70 +/- 0.07) for the totality of studied single-domain proteins and peptides.     

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.     

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.     

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.     

Comparison between long-range interactions and contact order in determining the folding rate of two-state proteins: application of long-range order to folding rate prediction.

The contact order is believed to be an important factor for understanding protein folding mechanisms. In our earlier work, we have shown that the long-range interactions play a vital role in protein folding. In this work, we analyzed the contribution of long-range contacts to determine the folding rate of two-state proteins. We found that the residues that are close in space and are separated by at least ten to 15 residues in sequence are important determinants of folding rates, suggesting the presence of a folding nucleus at an interval of approximately 25 residues. A novel parameter "long-range order" has been proposed to predict protein folding rates. This parameter shows as good a relationship with the folding rate of two-state proteins as contact order. Further, we examined the minimum limit of residue separation to determine the long-range contacts for different structural classes. We observed an excellent correlation between long-range order and folding rate for all classes of globular proteins. We suggest that in mixed-class proteins, a larger number of residues can serve as folding nuclei compared to all-alpha and all-beta proteins. A simple statistical method has been developed to predict the folding rates of two-state proteins using the long-range order that produces an agreement with experimental results that is better or comparable to other methods in the literature.     

The dynamical contact order: Protein folding rate parameters based on quantum conformational transitions

Protein folding is regarded as a quantum transition between the torsion states of a polypeptide chain. According to the quantum theory of conformational dynamics, we propose the dynamical contact order (DCO) defined as a characteristic of the contact described by the moment of inertia and the torsion potential energy of the polypeptide chain between contact residues. Consequently, the protein folding rate can be quantitatively studied from the point of view of dynamics. By comparing theoretical calculations and experimental data on the folding rate of 80 proteins, we successfully validate the view that protein folding is a quantum conformational transition. We conclude that (i) a correlation between the protein folding rate and the contact inertial moment exists; (ii) multi-state protein folding can be regarded as a quantum conformational transition similar to that of two-state proteins but with an intermediate delay. We have estimated the order of magnitude of the time delay; (iii) folding can be classified into two types, exergonic and endergonic. Most of the two-state proteins with higher folding rate are exergonic and most of the multi-state proteins with low folding rate are endergonic. The folding speed limit is determined by exergonic folding.     

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.     

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.     

Amino acid sequence predicts folding rate for middle-size two-state proteins

The significant correlation between protein folding rates and the sequence-predicted secondary structure suggests that folding rates are largely determined by the amino acid sequence. Here, we present a method for predicting the folding rates of proteins from sequences using the intrinsic properties of amino acids, which does not require any information on secondary structure prediction and structural topology. The contribution of residue to the folding rate is expressed by the residue's Omega value. For a given residue, its Omega depends on the amino acid properties (amino acid rigidity and dislike of amino acid for secondary structures). Our investigation achieves 82% correlation with folding rates determined experimentally for simple, two-state proteins studied until the present, suggesting that the amino acid sequence of a protein is an important determinant of the protein-folding rate and mechanism.     

Rapid cooperative two-state folding of a miniature alpha-beta protein and design of a thermostable variant

There is a great deal of interest in developing small stably folded miniature proteins. A limited number of these molecules have been described, however they typically have not been characterized in depth. In particular, almost no detailed studies of the thermodynamics and folding kinetics of these proteins have been reported. Here we describe detailed studies of the thermodynamics and kinetics of folding of a 39 residue mixed alpha-beta protein (NTL9(1-39)) derived from the N-terminal domain of the ribosomal protein L9. The protein folds cooperatively and rapidly in a two-state fashion to a native state typical of those found for normal globular proteins. At pH 5.4 in 20mM sodium acetate, 100mM NaCl the temperature of maximum stability is 6 degrees C, the t(m) is 65.3 degrees C, deltaH degrees (t(m)) is between 24.6 kcalmol(-1) and 26.3 kcalmol(-1), and deltaC(p) degrees is 0.38 kcalmol(-1)deg(-1). The thermodynamic parameters are in the range expected on the basis of per residue values determined from databases of globular proteins. H/2H exchange measurements reveal a set of amides that exchange via global unfolding, exactly as expected for a normal cooperatively folded globular protein. Kinetic measurements show that folding is two-state folding. The folding rate is 640 s(-1) and the value of deltaG degrees calculated from the folding and unfolding rates is in excellent agreement with the equilibrium value. A designed thermostable variant, generated by mutating K12 to M, was characterized and found to have a t(m) of 82 degrees C. Equilibrium and kinetic measurements demonstrate that its folding is cooperative and two-state.     

Evidence for sequential barriers and obligatory intermediates in apparent two-state protein folding

Many small proteins fold fast and without detectable intermediates. This is frequently taken as evidence against the importance of partially folded states, which often transiently accumulate during folding of larger proteins. To get insight into the properties of free energy barriers in protein folding we analyzed experimental data from 23 proteins that were reported to show non-linear activation free-energy relationships. These non-linearities are generally interpreted in terms of broad transition barrier regions with a large number of energetically similar states. Our results argue against the presence of a single broad barrier region. They rather indicate that the non-linearities are caused by sequential folding pathways with consecutive distinct barriers and a few obligatory high-energy intermediates. In contrast to a broad barrier model the sequential model gives a consistent picture of the folding barriers for different variants of the same protein and when folding of a single protein is analyzed under different solvent conditions. The sequential model is also able to explain changes from linear to non-linear free energy relationships and from apparent two-state folding to folding through populated intermediates upon single point mutations or changes in the experimental conditions. These results suggest that the apparent discrepancy between two-state and multi-state folding originates in the relative stability of the intermediates, which argues for the importance of partially folded states in protein folding.     

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.     

Elastic properties of proteins: insight on the folding process and evolutionary selection of native structures

We carry out a theoretical study of the vibrational and relaxation properties of naturally occurring proteins with the purpose of characterizing both the folding and equilibrium thermodynamics. By means of a suitable model, we provide a full characterization of the spectrum and eigenmodes of vibration at various temperatures by merely exploiting the knowledge of the protein native structure. It is shown that the rate at which perturbations decay at the folding transition correlates well with experimental folding rates. This validation is carried out on a list of about 30 two-state folders. Furthermore, the qualitative analysis of residues mean square displacements (shown to reproduce crystallographic data accurately) provides a reliable and statistically accurate method to identify crucial folding sites/contacts. This novel strategy is validated against clinical data for human immunodeficiency virus type 1 (HIV-1) protease. Finally, we compare the spectra and eigenmodes of vibration of natural proteins against randomly generated compact structures and regular random graphs. The comparison reveals a distinctive enhanced flexibility of natural structures accompanied by slow relaxation times at the folding temperature. The fact that these properties are connected intimately to the presence and assembly of secondary motifs hints at the special criteria adopted by evolution in the selection of viable folds.     

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.     

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.     

The nature of the free energy barriers to two-state folding

We introduce a simple procedure to analyze the temperature dependence of the folding and unfolding rates of two-state proteins. We start from the simple transition-state-like rate expression: k = D(eff)exp(-DeltaG(TS)/RT), in which upper and lower bounds for the intra-chain effective diffusion coefficient (D(eff)) are obtained empirically using the timescales of elementary processes in protein folding. From the changes in DeltaG(TS) as a function of temperature, we calculate enthalpies and heat capacities of activation, together with the more elusive entropies of activation. We then estimate the conformational entropy of the transition state by extrapolation to the temperature at which the solvation entropy vanishes by cancellation between polar and apolar terms. This approach is based on the convergence temperatures for the entropy of solvating apolar (approximately 385 K) and polar groups (approximately 335 K), the assumption that the structural properties of the transition state are somewhere in between the unfolded and folded states, and the established relationship between observed heat capacity and solvent accessibility.1 To circumvent the lack of structural information about transition states, we use the empirically determined heat capacities of activation as constraints to identify the extreme values of the transition state conformational entropy that are consistent with experiment. The application of this simple approach to six two-state folding proteins for which there is temperature-dependent data available in the literature provides important clues about protein folding. For these six proteins, we obtain an average equilibrium cost in conformational entropy of -4.3 cal x mol(-1)K(-1)per residue, which is in close agreement to previous empirical and computational estimates of the same quantity. Furthermore, we find that all these proteins have a conformationally diverse transition state, with more than half of the conformational entropy of the unfolded state. In agreement with predictions from theory and computer simulations, the transition state signals the change from a regime dominated by loss in conformational entropy to one driven by the gain in stabilization free energy (i.e., including protein interactions and solvation effects). Moreover, the height of the barrier is determined by how much stabilization free energy is realized at that point, which is related to the relative contribution of local versus non-local interactions. A remarkable observation is that the fraction of conformational entropy per residue that is present in the transition state is very similar for the six proteins in this study. Based on this commonality, we propose that the observed change in thermodynamic regime is connected to a change in the pattern of structure formation: from one driven by formation of pairwise interactions to one dominated by coupling of the networks of interactions involved in forming the protein core. In this framework, the barrier to two-state folding is crossed when the folding protein reaches a "critical native density" that allows expulsion of remaining interstitial water and consolidation of the core. The principle of critical native density should be general for all two-state proteins, but can accommodate different folding mechanisms depending on the particularities of the structure and sequence.     

Critical nucleation size in the folding of small apparently two-state proteins

For apparently two-state proteins, we found that the size (number of folded residues) of a transition state is mostly encoded by the topology, defined by total contact distance (TCD) of the native state, and correlates with its folding rate. This is demonstrated by using a simple procedure to reduce the native structures of the 41 two-state proteins with native TCD as a constraint, and is further supported by analyzing the results of eight proteins from protein engineering studies. These results support the hypothesis that the major rate-limiting process in the folding of small apparently two-state proteins is the search for a critical number of residues with the topology close to that of the native state.