Contact order dependent protein folding rates: kinetic consequences of a cooperative interplay between favorable nonlocal interactions and local conformational preferences

On the role of native contact cooperativity in protein folding

The consistency of energy landscape theory predictions with available experimental data, as well as direct evidence from molecular simulations, have shown that protein folding mechanisms are largely determined by the contacts present in the native structure. As expected, native contacts are generally energetically favorable. However, there are usually at least as many energetically favorable nonnative pairs owing to the greater number of possible nonnative interactions. This apparent frustration must therefore be reduced by the greater cooperativity of native interactions. In this work, we analyze the statistics of contacts in the unbiased all-atom folding trajectories obtained by Shaw and coworkers, focusing on the unfolded state. By computing mutual cooperativities between contacts formed in the unfolded state, we show that native contacts form the most cooperative pairs, while cooperativities among nonnative or between native and nonnative contacts are typically much less favorable or even anticooperative. Furthermore, we show that the largest network of cooperative interactions observed in the unfolded state consists mainly of native contacts, suggesting that this set of mutually reinforcing interactions has evolved to stabilize the native state.

Slowest-first protein translation scheme: Structural asymmetry and co-translational folding

Proteins are translated from the N to the C terminus, raising the basic question of how this innate directionality affects their evolution. To explore this question, we analyze 16,200 structures from the Protein Data Bank (PDB). We find remarkable enrichment of α helices at the C terminus and β strands at the N terminus. Furthermore, this α-β asymmetry correlates with sequence length and contact order, both determinants of folding rate, hinting at possible links to co-translational folding (CTF). Hence, we propose the "slowest-first" scheme, whereby protein sequences evolved structural asymmetry to accelerate CTF: the slowest of the cooperatively folding segments are positioned near the N terminus so they have more time to fold during translation. A phenomenological model predicts that CTF can be accelerated by asymmetry in folding rate, up to double the rate, when folding time is commensurate with translation time; analysis of the PDB predicts that structural asymmetry is indeed maximal in this regime. This correspondence is greater in prokaryotes, which generally require faster protein production. Altogether, this indicates that accelerating CTF is a substantial evolutionary force whose interplay with stability and functionality is encoded in secondary structure asymmetry.

Small-Angle X-ray Scattering Signatures of Conformational Heterogeneity and Homogeneity of Disordered Protein Ensembles

An accurate account of disordered protein conformations is of central importance to deciphering the physicochemical basis of biological functions of intrinsically disordered proteins and the folding-unfolding energetics of globular proteins. Physically, disordered ensembles of nonhomopolymeric polypeptides are expected to be heterogeneous, i.e., they should differ from those homogeneous ensembles of homopolymers that harbor an essentially unique relationship between average values of end-to-end distance R_EE and radius of gyration R_g. It was posited recently, however, that small-angle X-ray scattering (SAXS) data on conformational dimensions of disordered proteins can be rationalized almost exclusively by homopolymer ensembles. Assessing this perspective, chain-model simulations are used to evaluate the discriminatory power of SAXS-determined molecular form factors (MFFs) with regard to homogeneous versus heterogeneous ensembles. The general approach adopted here is not bound by any assumption about ensemble encodability, in that the postulated heterogeneous ensembles we evaluated are not restricted to those entailed by simple interaction schemes. Our analysis of MFFs for certain heterogeneous ensembles with more narrowly distributed R_EE and R_g indicates that while they deviate from MFFs of homogeneous ensembles, the differences can be rather small. Remarkably, some heterogeneous ensembles with asphericity and R_EE drastically different from those of homogeneous ensembles can nonetheless exhibit practically identical MFFs, demonstrating that SAXS MFFs do not afford unique characterizations of basic properties of conformational ensembles in general. In other words, the ensemble to MFF mapping is practically many-to-one and likely nonsmooth. Heteropolymeric variations of the R_EE-R_g relationship were further showcased using an analytical perturbation theory developed here for flexible heteropolymers. Ramifications of our findings for interpretation of experimental data are discussed.

Folding with a protein's native shortcut network

A complex network approach to protein folding is proposed, wherein a protein's contact map is reconceptualized as a network of shortcut edges, and folding is steered by a structural characteristic of this network. Shortcut networks are generated by a known message passing algorithm operating on protein residue networks. It is found that the shortcut networks of native structures (SCN0s) are relevant graph objects with which to study protein folding at a formal level. The logarithm form of their contact order (SCN0_lnCO) correlates significantly with folding rate of two-state and nontwo-state proteins. The clustering coefficient of SCN0s (C_SCN0 ) correlates significantly with folding rate, transition-state placement and stability of two-state folders. Reasonable folding pathways for several model proteins are produced when C_SCN0 is used to combine protein segments incrementally to form the native structure. The folding bias captured by C_SCN0 is detectable in non-native structures, as evidenced by Molecular Dynamics simulation generated configurations for the fast folding Villin-headpiece peptide. These results support the use of shortcut networks to investigate the role protein geometry plays in the folding of both small and large globular proteins, and have implications for the design of multibody interaction schemes in folding models. One facet of this geometry is the set of native shortcut triangles, whose attributes are found to be well-suited to identify dehydrated intraprotein areas in tight turns, or at the interface of different secondary structure elements.

Helical Propensity Affects the Conformational Properties of the Denatured State of Cytochrome c'

Changing the helical propensity of a polypeptide sequence might be expected to affect the conformational properties of the denatured state of a protein. To test this hypothesis, alanines at positions 83 and 87 near the center of helix 3 of cytochrome c' from Rhodopseudomonas palustris were mutated to serine to decrease the stability of this helix. A set of 13 single histidine variants in the A83S/A87S background were prepared to permit assessment of the conformational properties of the denatured state using histidine-loop formation in 3 M guanidine hydrochloride. The data are compared with previous histidine-heme loop formation data for wild-type cytochrome c'. As expected, destabilization of helix 3 decreases the global stabilities of the histidine variants in the A83S/A87S background relative to the wild-type background. Loop stability versus loop size data yields a scaling exponent of 2.1 ± 0.2, similar to the value of 2.3 ± 0.2 obtained for wild-type cytochrome c'. However, the stabilities of all histidine-heme loops, which contain the helix 3 sequence segment, are increased in the A83S/A87S background compared to the wild-type background. Rate constants for histidine-heme loop breakage are similar for the wild-type and A83S/A87S variants. However, for histidine-heme loops that contain the helix 3 sequence segment, the rate constants for loop formation increase in the A83S/A87S background compared to the wild-type background. Thus, residual helical structure appears to stiffen the polypeptide chain slowing loop formation in the denatured state. The implications of these results for protein folding mechanisms are discussed.

Topological and sequence information predict that foldons organize a partially overlapped and hierarchical structure

It has been suggested that proteins have substructures, called foldons, which can cooperatively fold into the native structure. However, several prior investigations define foldons in various ways, citing different foldon characteristics, thereby making the concept of a foldon ambiguous. In this study, we perform a Gō model simulation and analyze the characteristics of substructures that cooperatively fold into the native-like structure. Although some results do not agree well with the experimental evidence due to the simplicity of our coarse-grained model, our results strongly suggest that cooperatively folding units sometimes organize a partially overlapped and hierarchical structure. This view makes us easy to interpret some different proposal about the foldon as a difference of the hierarchical structure. On the basis of this finding, we present a new method to assign foldons and their hierarchy, using structural and sequence information. The results show that the foldons assigned by our method correspond to the intermediate structures identified by some experimental techniques. The new method makes it easy to predict whether a protein folds sequentially into the native structure or whether some foldons fold into the native structure in parallel.

How determinant is N-terminal to C-terminal coupling for protein folding?

The existence of native interactions between the protein termini is a major determinant of the free energy barrier in a two-state folding transition being therefore a critical modulator of protein folding cooperativity.

Capturing protein folding-relevant topology via absolute contact order variants

Absolute contact order is one of the simplest parameters used to predict protein folding rates. Many variants of contact order (CO) have been applied to highlight different aspects of contact neighborhoods and their relationship to folding. However, a systematic study of the influence of CO variants on correlation with folding rate has not been performed for a large combined set of multi- and two-state proteins. We explore different contact neighborhoods and resulting CO by varying the distance thresholds and weighting of sequence separation for heavy atom and residue-based counting methods for a set of 136 proteins diverse across folding and structural classes. We examine the changes in contact neighborhoods and compare correlations with our CO variants and the protein folding rates across our data set as well as by folding type and structural class. Different CO variants lead to the strongest correlations within each protein structural class. Our results demonstrate that backbone topology at a distance beyond where energetic interactions dominate is able to capture folding determinants, and suggest that more sensitive methods of characterizing contact relationships may improve ln kf prediction for diverse protein sets.

Effects of desolvation barriers and sidechains on local–nonlocal coupling and chevron behaviors in coarse-grained models of protein folding

Coarse-grained protein chain models with desolvation barriers or sidechains lead to stronger local–nonlocal coupling and more linear chevron plots.

Spatial ranges of driving forces are a key determinant of protein folding cooperativity and rate diversity

The physical basis of two-state-like folding transitions and the tremendous diversity in folding rates is elucidated by directly simulating the folding kinetics of 52 representative proteins. Relative to the results from a common modeling approach, the diversity of the simulated folding rates can be increased from ~10(2.1) to the experimental ~10(6.0) by a modest decrease in the spatial range of the attractive potential. The required theoretical range is consistent with desolvation physics and is notably much more permissive than that needed for two-state-like homopolymer collapse.

How difficult is it to fold a knotted protein? In silico insights from surface-tethered folding experiments

We explore the effect of surface tethering on the folding process of a lattice protein that contains a trefoil knot in its native structure via Monte Carlo simulations. We show that the outcome of the tethering experiment depends critically on which terminus is used to link the protein to a chemically inert plane. In particular, if surface tethering occurs at the bead that is closer to the knotted core the folding rate becomes exceedingly slow and the protein is not able to find the native structure in all the attempted folding trajectories. Such low folding efficiency is also apparent from the analysis of the probability of knot formation, p(knot), as a function of nativeness. Indeed, p(knot) increases abruptly from ∼0 to ∼1 only when the protein has more than 80% of its native contacts formed, showing that a highly compact conformation must undergo substantial structural re-arrangement in order to get effectively knotted. When the protein is surface tethered by the bead that is placed more far away from the knotted core p(knot) is higher than in the other folding setups (including folding in the bulk), especially if conformations are highly native-like. These results show that the mobility of the terminus closest to the knotted core is critical for successful folding of trefoil proteins, which, in turn, highlights the importance of a knotting mechanism that is based on a threading movement of this terminus through a knotting loop. The results reported here predict that if this movement is blocked, knotting occurs via an alternative mechanism, the so-called spindle mechanism, which is prone to misfolding. Our simulations show that in the three considered folding setups the formation of the knot is typically a late event in the folding process. We discuss the implications of our findings for co-translational folding of knotted trefoils.

Why do protein folding rates correlate with metrics of native topology?

For almost 15 years, the experimental correlation between protein folding rates and the contact order parameter has been under scrutiny. Here, we use a simple simulation model combined with a native-centric interaction potential to investigate the physical roots of this empirical observation. We simulate a large set of circular permutants, thus eliminating dependencies of the folding rate on other protein properties (e.g. stability). We show that the rate-contact order correlation is a consequence of the fact that, in high contact order structures, the contact order of the transition state ensemble closely mirrors the contact order of the native state. This happens because, in these structures, the native topology is represented in the transition state through the formation of a network of tertiary interactions that are distinctively long-ranged.

Cooperativity, local-nonlocal coupling, and nonnative interactions: principles of protein folding from coarse-grained models

Coarse-grained, self-contained polymer models are powerful tools in the study of protein folding. They are also essential to assess predictions from less rigorous theoretical approaches that lack an explicit-chain representation. Here we review advances in coarse-grained modeling of cooperative protein folding, noting in particular that the Levinthal paradox was raised in response to the experimental discovery of two-state-like folding in the late 1960s, rather than to the problem of conformational search per se. Comparisons between theory and experiment indicate a prominent role of desolvation barriers in cooperative folding, which likely emerges generally from a coupling between local conformational preferences and nonlocal packing interactions. Many of these principles have been elucidated by native-centric models, wherein nonnative interactions may be treated perturbatively. We discuss these developments as well as recent applications of coarse-grained chain modeling to knotted proteins and to intrinsically disordered proteins.

What have we learned from the studies of two-state folders, and what are the unanswered questions about two-state protein folding?

Small proteins with globular structures often fold by simple all-or-none mechanisms, both in an equilibrium and a kinetic sense, despite the very large number of partly folded conformations available. This type of 'two-state' folding will be discussed in terms of experimental tests, underlying molecular mechanisms, and limits to two-state behavior. Factors that appear to be important for two-state folding include topology (sequence distance of contacts in the native structure), molecular cooperativity and local energy distribution. Because their local stability distributions and cooperativities can be dissected and analyzed separately from topological features, recent studies of the folding of symmetric proteins will be discussed as a means to better understand the origins of two-state folding.

Information entropy as a measure of nonexponentiality of waiting-time distributions

It is shown that the information entropy based on waiting-time distributions (WTDs) offers a natural and robust measure of nonexponentiality of the distributions in the form of the Schrödinger-Brillouin negentropy, or equivalently the Kullback-Leibler divergence, and has a straightforward interpretation in terms of transition state theory. Other measures of nonexponentiality of WTDs, based on comparison of the standard deviation and the median with the mean waiting time, are also discussed. The theoretical analysis is illustrated with results from protein folding studies.

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(alpha) 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.

Theoretical and experimental demonstration of the importance of specific nonnative interactions in protein folding

Many experimental and theoretical studies have suggested a significant role for nonnative interactions in protein folding pathways, but the energetic contributions of these interactions are not well understood. We have addressed the energetics and the position specificity of nonnative hydrophobic interactions by developing a continuum coarse-grained chain model with a native-centric potential augmented by sequence-dependent hydrophobic interactions. By modeling the effect of different hydrophobicity values at various positions in the Fyn SH3 domain, we predicted energetically significant nonnative interactions that led to acceleration or deceleration of the folding rate depending on whether they were more populated in the transition state or unfolded state. These nonnative contacts were centered on position 53 in the Fyn SH3 domain, which lies in an exposed position in a 3(10)-helix. The energetic importance of the predicted nonnative interactions was confirmed experimentally by folding kinetics studies combined with double mutant thermodynamic cycles. By attaining agreement of theoretical and experimental investigations, this study provides a compelling demonstration that specific nonnative interactions can significantly influence folding energetics. Moreover, we show that a coarse-grained model with a simple consideration of hydrophobicity is sufficient for the accurate prediction of kinetically important nonnative interactions.

Folding domain B of protein A on a dynamically partitioned free energy landscape

The B domain of staphylococcal protein A (BdpA) is a small helical protein that has been studied intensively in kinetics experiments and detailed computer simulations that include explicit water. The simulations indicate that BdpA needs to reorganize in crossing the transition barrier to facilitate folding its C-terminal helix (H3) onto the nucleus formed from helices H1 and H2. This process suggests frustration between two partially ordered forms of the protein, but recent varphi value measurements indicate that the transition structure is relatively constant over a broad range of temperatures. Here we develop a simplistic model to investigate the folding transition in which properties of the free energy landscape can be quantitatively compared with experimental data. The model is a continuation of the Muñoz-Eaton model to include the intermittency of contacts between structured parts of the protein, and the results compare variations in the landscape with denaturant and temperature to varphi value measurements and chevron plots of the kinetic rates. The topography of the model landscape (in particular, the feature of frustration) is consistent with detailed simulations even though variations in the varphi values are close to measured values. The transition barrier is smaller than indicated by the chevron data, but it agrees in order of magnitude with a similar alpha-carbon type of model. Discrepancies with the chevron plots are investigated from the point of view of solvent effects, and an approach is suggested to account for solvent participation in the model.

Loop-closure principles in protein folding

Simple theoretical concepts and models have been helpful to understand the folding rates and routes of single-domain proteins. As reviewed in this article, a physical principle that appears to underly these models is loop closure.

Rate determining factors in protein model structures

Previous research has shown a strong correlation of protein folding rates to the native state geometry, yet a complete explanation for this dependence is still lacking. Here we study the rate-geometry relationship with a simple statistical physics model, and focus on two classes of model geometries, representing ideal parallel and antiparallel structures. We find that the logarithm of the rate shows an almost perfect linear correlation with the "absolute contact order", but the slope depends on the particular class considered. We discuss these findings in the light of experimental results.

Influence of the chain stiffness on the thermodynamics of a Gō-type model for protein folding

The relative importance of local and long range interactions in the characteristics of the protein folding process has long been a matter of controversy. Computer simulations based on Gō-type models have been widely used to study this topic, but without much agreement on which type of interactions is more relevant for the foldability of a protein. In this work, the authors also employ a topology-based potential and simulation model to analyze the influence of local and long range interactions on the thermodynamics of the folding transition. The former are mainly used to control the degree of flexibility (or stiffness) of the chain, mostly appreciable in the unfolded (noncompact) state. Our results show the different effects that local and nonlocal interactions have on the entropy and the energy of the system. This implies that a balance between both types of interactions is required, so that a free energy barrier exists between the native and the denatured states. The variations in the contribution of both types of interactions have also a direct effect on the stability of the chain conformations, including the possible appearance of thermodynamic folding intermediates.

The role of protein homochirality in shaping the energy landscape of folding

The homochirality, or isotacticity, of the natural amino acids facilitates the formation of regular secondary structures such as alpha-helices and beta-sheets. However, many examples exist in nature where novel polypeptide topologies use both l- and d-amino acids. In this study, we explore how stereochemistry of the polypeptide backbone influences basic properties such as compactness and the size of fold space by simulating both lattice and all-atom polypeptide chains. We formulate a rectangular lattice chain model in both two and three dimensions, where monomers are chiral, having the effect of restricting local conformation. Syndiotactic chains with alternating chirality of adjacent monomers have a very large ensemble of accessible conformations characterized predominantly by extended structures. Isotactic chains on the other hand, have far fewer possible conformations and a significant fraction of these are compact. Syndiotactic chains are often unable to access maximally compact states available to their isotactic counterparts of the same length. Similar features are observed in all-atom models of isotactic versus syndiotactic polyalanine. Our results suggest that protein isotacticity has evolved to increase the enthalpy of chain collapse by facilitating compact helical states and to reduce the entropic cost of folding by restricting the size of the unfolded ensemble of competing states.

Cooperativity and the origins of rapid, single-exponential kinetics in protein folding

The folding of naturally occurring, single-domain proteins is usually well described as a simple, single-exponential process lacking significant trapped states. Here we further explore the hypothesis that the smooth energy landscape this implies, and the rapid kinetics it engenders, arises due to the extraordinary thermodynamic cooperativity of protein folding. Studying Miyazawa-Jernigan lattice polymers, we find that, even under conditions where the folding energy landscape is relatively optimized (designed sequences folding at their temperature of maximum folding rate), the folding of protein-like heteropolymers is accelerated when their thermodynamic cooperativity is enhanced by enhancing the nonadditivity of their energy potentials. At lower temperatures, where kinetic traps presumably play a more significant role in defining folding rates, we observe still greater cooperativity-induced acceleration. Consistent with these observations, we find that the folding kinetics of our computational models more closely approximates single-exponential behavior as their cooperativity approaches optimal levels. These observations suggest that the rapid folding of naturally occurring proteins is, in part, a consequence of their remarkably cooperative folding.

Direct correlation between proteins' folding rates and their amino acid compositions: An ab initio folding rate prediction

Discovering the mechanism of protein folding, in molecular biology, is a great challenge. A key step to this end is to find factors that correlate with protein folding rates. Over the past few years, many empirical parameters, such as contact order, long-range order, total contact distance, secondary structure contents, have been developed to reflect the correlation between folding rates and protein tertiary or secondary structures. However, the correlation between proteins' folding rates and their amino acid compositions has not been explored. In the present work, we examined systematically the correlation between proteins' folding rates and their amino acid compositions for two-state and multistate folders and found that different amino acids contributed differently to the folding progress. The relation between the amino acids' molecular weight and degeneracy and the folding rates was examined, and the role of hydrophobicity in the protein folding process was also inspected. As a consequence, a new indicator called composition index was derived, which takes no structure factors into account and is merely determined by the amino acid composition of a protein. Such an indicator is found to be highly correlated with the protein's folding rate (r > 0.7). From the results of this work, three points of concluding remarks are evident. (1) Two-state folders and multistate folders have different rate-determining amino acids. (2) The main determining information of a protein's folding rate is largely reflected in its amino acid composition. (3) Composition index may be the best predictor for an ab initio protein folding rate prediction directly from protein sequence from the standpoint of practical application.

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.

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.

Loop-closure events during protein folding: rationalizing the shape of Phi-value distributions

In the past years, the folding kinetics of many small single-domain proteins has been characterized by mutational Phi-value analysis. In this article, a simple, essentially parameter-free model is introduced which derives folding routes from native structures by minimizing the entropic loop-closure cost during folding. The model predicts characteristic folding sequences of structural elements such as helices and beta-strand pairings. Based on few simple rules, the kinetic impact of these structural elements is estimated from the routes and compared to average experimental Phi-values for the helices and strands of 15 small, well-characterized proteins. The comparison leads on average to a correlation coefficient of 0.62 for all proteins with polarized Phi-value distributions, and 0.74 if distributions with negative average Phi-values are excluded. The diffuse Phi-value distributions of the remaining proteins are reproduced correctly. The model shows that Phi-value distributions, averaged over secondary structural elements, can often be traced back to entropic loop-closure events, but also indicates energetic preferences in the case of a few proteins governed by parallel folding processes.

Topology-dependent protein folding rates analyzed by a stereochemical model

It is an experimental fact that gross topological parameters of the native structure of small proteins presenting two-state kinetics, as relative contact order chi, correlate with the logarithm of their respective folding rate constant kappa(f). However, reported results show specific cases for which the (chi,log kappa(f)) dependence does not follow the overall trend of the entire collection of experimental data. Therefore, an interesting point to be clarified is to what extent the native topology alone can explain these exceptional data. In this work, the structural determinants of the folding kinetics are investigated by means of a 27-mer lattice model, in that each native is represented by a compact self-avoiding (CSA) configuration. The hydrophobic effect and steric constraints are taken as basic ingredients of the folding mechanism, and each CSA configuration is characterized according to its composition of specific patterns (resembling basic structural elements such as loops, sheets, and helices). Our results suggest that (i) folding rate constants are largely influenced by topological details of the native structure, as configurational pattern types and their combinations, and (ii) global parameters, as the relative contact order, may not be effective to detect them. Distinct pattern types and their combinations are determinants of what we call here the "content of secondary-type" structure (sigma) of the native: high sigma implies a large kappa(f). The largest part of all CSA configurations presents a mix of distinct structural patterns, which determine the chixlog kappa(f) linear dependence: Those structures not presenting a proper chi-dependent balance of patterns have their folding kinetics affected with respect to the pretense linear correlation between chi and log kappa(f). The basic physical mechanism relating sigma and kappa(f) involves the concept of cooperativity: If the native is composed of patterns producing a spatial order rich in effective short-range contacts, a properly designed sequence undertakes a fast folding process. On the other hand, the presence of some structural patterns, such as long loops, may reduce substantially the folding performance. This fact is illustrated through natives having a very similar topology but presenting a distinct folding rate kappa(f), and by analyzing structures having the same chi but different sigma.

A critical assessment of the topomer search model of protein folding using a continuum explicit-chain model with extensive conformational sampling

Recently, a series of closely related theoretical constructs termed the "topomer search model" (TSM) has been proposed for the folding mechanism of small, single-domain proteins. A basic assumption of the proposed scenarios is that the rate-limiting step in folding is an essentially unbiased, diffusive search for a conformational state called the native topomer defined by an overall native-like topological pattern. Successes in correlating TSM-predicted folding rates with that of real proteins have been interpreted as experimental support for the model. To better delineate the physics entailed, key TSM concepts are examined here using extensive Langevin dynamics simulations of continuum C(alpha) chain models. The theoretical native topomers of four experimentally well-studied two-state proteins are characterized. Consistent with the TSM perspective, we found that the sizes of the native topomers increase with experimental folding rate. However, a careful determination of the corresponding probabilities that the native topomers are populated during a random search fails to reproduce the previously predicted folding rates. Instead, our results indicate that an unbiased TSM search for the native topomer amounts to a Levinthal-like process that would take an impossibly long average time to complete. Furthermore, intraprotein contacts in all four native topomers considered exhibit no apparent correlation with the experimental phi-values determined from the folding kinetics of these proteins. Thus, the present findings suggest that certain basic, generic yet essential energetic features in protein folding are not accounted for by TSM scenarios to date.

Fast method for estimating the energy distribution of globular states of proteins

By an enumeration study, we show that the energy distributions of a lattice protein sequence on all possible compact lattice configurations can be approximated by the energy distribution of shuffled sequences on a given lattice structure. We also show that the random energy model (REM) gives a good analytical approximation for the energy distribution of shuffled sequences on lattice structures. For real proteins, when a gapped threading method is used, REM calculations systematically underestimate the mean value of the energy distributions. We found that this discrepancy can be roughly compensated by a linear correction obtained from empirical fits. This result can be used to greatly reduce the computational effort in protein threading calculations.

The Gō model revisited: Native structure and the geometric coupling between local and long-range contacts

Monte Carlo simulations show that long-range interactions play a major role in determining the folding rates of 48-mer three-dimensional lattice polymers modeled by the Gō potential. For three target structures with different native geometries we found a sharp increase in the folding time when the relative contribution of the long-range interactions to the native state's energy is decreased from approximately 50% towards zero. However, the dispersion of the simulated folding times is strongly dependent on native geometry and Gō polymers folding to one of the target structures exhibits folding times spanning three orders of magnitude. We have also found that, depending on the target geometry, a strong geometric coupling may exist between local and long-range contacts, which means that, when this coupling exists, the formation of long-range contacts is forced by the previous formation of local contacts. The absence of a strong geometric coupling results in a kinetics that is more sensitive to the interaction energy parameters; in this case, the formation of local contacts is not capable of promoting the establishment of long-range ones when the latter are strongly penalized energetically and this results in longer folding times.

Chevron behavior and isostable enthalpic barriers in protein folding: successes and limitations of simple Gō-like modeling

It has been demonstrated that a "near-Levinthal" cooperative mechanism, whereby the common Gō interaction scheme is augmented by an extra favorability for the native state as a whole, can lead to apparent two-state folding/unfolding kinetics over a broad range of native stabilities in lattice models of proteins. Here such a mechanism is shown to be generalizable to a simplified continuum (off-lattice) Langevin dynamics model with a Calpha protein chain representation, with the resulting chevron plots exhibiting an extended quasilinear regime reminiscent of that of apparent two-state real proteins. Similarly high degrees of cooperativity are possible in Gō-like continuum models with rudimentary pairwise desolvation barriers as well. In these models, cooperativity increases with increasing desolvation barrier height, suggesting strongly that two-state-like folding/unfolding kinetics would be achievable when the pairwise desolvation barrier becomes sufficiently high. Besides cooperativity, another generic folding property of interest that has emerged from published experiments on several apparent two-state proteins is that their folding relaxation under constant native stability (isostability) conditions is essentially Arrhenius, entailing high intrinsic enthalpic folding barriers of approximately 17-30 kcal/mol. Based on a new analysis of published data on barnase, here we propose that a similar property should also apply to a certain class of non-two-state proteins that fold with chevron rollovers. However, several continuum Gō-like constructs considered here fail to predict any significant intrinsic enthalpic folding barrier under isostability conditions; thus the physical origin of such barriers in real proteins remains to be elucidated.

Desolvation is a likely origin of robust enthalpic barriers to protein folding

Experimental data from global analyses of temperature (T) and denaturant dependence of the folding rates of small proteins led to an intrinsic enthalpic folding barrier hypothesis: to a good approximation, the T-dependence of folding rate under constant native stability conditions is Arrhenius. Furthermore, for a given protein, the slope of isostability folding rate versus 1/T is essentially independent of native stability. This hypothesis implies a simple relationship between chevron and Eyring plots of folding that is easily discernible when both sets of rates are expressed as functions of native stability. Using experimental data in the literature, we verify the predicted chevron-Eyring relationship for 14 proteins and determine their intrinsic enthalpic folding barriers, which vary approximately from 15 kcal/mol to 40 kcal/mol for different proteins. These enthalpic barriers do not appear to correlate with folding rates, but they exhibit correlation with equilibrium unfolding enthalpy at room temperature. Intrinsic enthalpic barriers with similarly high magnitudes apply as well to at least two cases of peptide-peptide and peptide-protein association, suggesting that these barriers are a hallmark of certain general and fundamental kinetic processes during folding and binding. Using a class of explicit-chain C(alpha) protein models with constant elementary enthalpic desolvation barriers between C(alpha) positions, we show that small microscopic pairwise desolvation barriers, which are a direct consequence of the particulate nature of water, can act cooperatively to give rise to a significant overall enthalpic barrier to folding. This theoretical finding provides a physical rationalization for the high intrinsic enthalpic barriers in protein folding energetics. Ramifications of entropy-enthalpy compensation in hydrophobic association for the height of enthalpic desolvation barrier are discussed.

The complex kinetics of protein folding in wide temperature ranges

The complex protein folding kinetics in wide temperature ranges is studied through diffusive dynamics on the underlying energy landscape. The well-known kinetic chevron rollover behavior is recovered from the mean first passage time, with the U-shape dependence on temperature. The fastest folding temperature T0 is found to be smaller than the folding transition temperature Tf. We found that the fluctuations of the kinetics through the distribution of first passage time show rather universal behavior, from high-temperature exponential Poissonian kinetics to the relatively low-temperature highly non-exponential kinetics. The transition temperature is at Tk and T0 < Tk < Tf. In certain low-temperature regimes, a power law behavior at long time emerges. At very low temperatures (lower than trapping transition temperature T < T0/(4 approximately 6)), the kinetics is an exponential Poissonian process again.

A Statistical Model for Predicting Protein Folding Rates from Amino Acid Sequence with Structural Class Information

Prediction of protein folding rates from amino acid sequences is one of the most important challenges in molecular biology. In this work, I have related the protein folding rates with physical-chemical, energetic and conformational properties of amino acid residues. I found that the classification of proteins into different structural classes shows an excellent correlation between amino acid properties and folding rates of two- and three-state proteins, indicating the importance of native state topology in determining the protein folding rates. I have formulated a simple linear regression model for predicting the protein folding rates from amino acid sequences along with structural class information and obtained an excellent agreement between predicted and experimentally observed folding rates of proteins; the correlation coefficients are 0.99, 0.96 and 0.95, respectively, for all-alpha, all-beta and mixed class proteins. This is the first available method, which is capable of predicting the protein folding rates just from the amino acid sequence with the aid of generic amino acid properties and structural class information.

Native geometry and the dynamics of protein folding

In this paper, we investigate the role of native geometry on the kinetics of protein folding based on simple lattice models and Monte Carlo simulations. Results obtained within the scope of the Miyazawa-Jernigan indicate the existence of two dynamical folding regimes depending on the protein chain length. For chains larger than 80 amino acids, the folding performance is sensitive to the native state's conformation. Smaller chains, with less than 80 amino acids, fold via two-state kinetics and exhibit a significant correlation between the contact order parameter and the logarithmic folding times. In particular, chains with N=48 amino acids were found to belong to two broad classes of folding, characterized by different cooperativity, depending on the contact order parameter. Preliminary results based on the Go model show that the effect of long-range contact interaction strength in the folding kinetics is largely dependent on the native state's geometry.

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.

Three-body interactions improve the prediction of rate and mechanism in protein folding models

Here we study the effects of many-body interactions on rate and mechanism in protein folding by using the results of molecular dynamics simulations on numerous coarse-grained Calpha-model single-domain proteins. After adding three-body interactions explicitly as a perturbation to a Gō-like Hamiltonian with native pairwise interactions only, we have found (i) a significantly increased correlation with experimental phi values and folding rates, (ii) a stronger correlation of folding rate with contact order, matching the experimental range in rates when the fraction of three-body energy in the native state is approximately 20%, and (iii) a considerably larger amount of three-body energy present in chymotripsin inhibitor than in the other proteins studied.

Using protein folding rates to test protein folding theories

The fastest simple, kinetically two-state protein folds a million times more rapidly than the slowest. Here we review many recent theories of protein folding kinetics in terms of their ability to qualitatively rationalize, if not quantitatively predict, this fundamental experimental observation.