Outlining folding nuclei in globular proteins

Our theoretical approach for prediction of folding/unfolding nuclei in three-dimensional protein structures is based on a search for free energy saddle points on networks of protein unfolding pathways. Under some approximations, this search is performed rapidly by dynamic programming and results in prediction of Phi values, which can be compared with those found experimentally. In this study, we optimize some details of the model (specifically, hydrogen atoms are taken into account in addition to heavy atoms), and compare the theoretically obtained and experimental Phi values (which characterize involvement of residues in folding nuclei) for all 17 proteins, where Phi values are now known for many residues. We show that the model provides good Phi value predictions for proteins whose structures have been determined by X-ray analysis (the average correlation coefficient is 0.65), with a more limited success for proteins whose structures have been determined by NMR techniques only (the average correlation coefficient is 0.34), and that the transition state free energies computed from the same model are in a good anticorrelation with logarithms of experimentally measured folding rates at mid-transition (the correlation coefficient is -0.73).     

Simple physical models connect theory and experiment in protein folding kinetics

Our understanding of the principles underlying the protein-folding problem can be tested by developing and characterizing simple models that make predictions which can be compared to experimental data. Here we extend our earlier model of folding free energy landscapes, in which each residue is considered to be either folded as in the native state or completely disordered, by investigating the role of additional factors representing hydrogen bonding and backbone torsion strain, and by using a hybrid between the master equation approach and the simple transition state theory to evaluate kinetics near the free energy barrier in greater detail. Model calculations of folding phi-values are compared to experimental data for 19 proteins, and for more than half of these, experimental data are reproduced with correlation coefficients between r=0.41 and 0.88; calculations of transition state free energy barriers correlate with rates measured for 37 single domain proteins (r=0.69). The model provides insight into the contribution of alternative-folding pathways, the validity of quasi-equilibrium treatments of the folding landscape, and the magnitude of the Arrhenius prefactor for protein folding. Finally, we discuss the limitations of simple native-state-based models, and as a more general test of such models, provide predictions of folding rates and mechanisms for a comprehensive set of over 400 small protein domains of known structure.     

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.     

Analyses of the folding properties of ferredoxin‐like fold proteins by means of a coarse‐grained Gō model: Relationship between the free energy profiles and folding cores

The folding mechanisms of proteins with multi‐state transitions, the role of the intermediate states, and the precise mechanism how each transition occurs are significant on‐going research issues. In this study, we investigate ferredoxin‐like fold proteins which have a simple topology and multi‐state transitions. We analyze the folding processes by means of a coarse‐grained Gō model. We are able to reproduce the differences in the folding mechanisms between U1A, which has a high‐free‐energy intermediate state, and ADA2h and S6, which fold into the native structure through two‐state transitions. The folding pathways of U1A, ADA2h, S6, and the S6 circular permutant, S6_p54‐55, are reproduced and compared with experimental observations. We show that the ferredoxin‐like fold contains two common regions consisting folding cores as predicted in other studies and that U1A produces an intermediate state due to the distinct cooperative folding of each core. However, because one of the cores of S6 loses its cooperativity and the two cores of ADA2h are tightly coupled, these proteins fold into the native structure through a two‐state mechanism. Proteins 2014; 82:954–965. © 2013 Wiley Periodicals, Inc.     

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

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

On the Prediction of Folding Nuclei in Globular Proteins

An approach to predicting folding nuclei in globular proteins with known three-dimensional structures is proposed. This approach is based on the pinpointing of the lowest saddle points on the barrier between the unfolded state and native structure on the free-energy landscape of a protein chain; the proposed technique uses the dynamic programming method. A comparison of calculation results with experimental data on the folding nuclei of 21 proteins shows that the model provides good Φ value predictions for protein structures determined by X-ray analysis and, less successfully, in structures determined by nuclear magnetic resonance. Consideration of the whole ensemble of transition states provides a better prediction of folding nuclei than consideration of only transition states with lowest free energies. In addition, we predict the location of folding nuclei in three-dimensional structures of some proteins whose folding kinetics is being studied, but there is no experimental evidence concerning their folding nuclei.     

Local interactions and the optimization of protein folding

The role of local interactions in protein folding has recently been the subject of some controversy. Here we investigate an extension of Zwanzig's simple and general model of folding in which local and nonlocal interactions are represented by functions of single and multiple conformational degrees of freedom, respectively. The kinetics and thermodynamics of folding are studied for a series of energy functions in which the energy of the native structure is fixed, but the relative contributions of local and nonlocal interactions to this energy are varied over a broad range. For funnel shaped energy landscapes, we find that 1) the rate of folding increases, but the stability of the folded state decreases, as the contribution of local interactions to the energy of the native structure increases, and 2) the amount of native structure in the unfolded state and the transition state vary considerably with the local interaction strength. Simple exponential kinetics and a well-defined free energy barrier separating folded and unfolded states are observed when nonlocal interactions make an appreciable contribution to the energy of the native structure; in such cases a transition state theory type approximation yields reasonably accurate estimates of the folding rate. Bumps in the folding funnel near the native state, which could result from desolvation effects, side chain freezing, or the breaking of nonnative contacts, significantly alter the dependence of the folding rate on the local interaction strength: the rate of folding decreases when the local interaction strength is increased beyond a certain point. A survey of the distribution of strong contacts in the protein structure database suggests that evolutionary optimization has involved both kinetics and thermodynamics: strong contacts are enriched at both very short and very long sequence separations. Proteins 29:282–291, 1997. © 1997 Wiley-Liss, Inc.     

The osmolyte trimethylamine‐N‐oxide stabilizes the Fyn SH3 domain without altering the structure of its folding transition state

Trimethylamine‐N‐oxide (TMAO) is a naturally occurring osmolyte that stabilizes proteins against denaturation. Although the impact of TMAO on the folding thermodynamics of many proteins has been well characterized, far fewer studies have investigated its effects on protein folding kinetics. In particular, no previous studies have used Φ‐value analysis to determine whether TMAO may alter the structure of the folding transition state. Here we have measured the effects on folding kinetics of 16 different amino acid substitutions distributed across the structure of the Fyn SH3 domain both in the presence and absence of TMAO. The folding and unfolding rates in TMAO, on average, improved to equivalent degrees, with a twofold increase in the protein folding rate accompanied by a twofold decrease in the unfolding rate. Importantly, TMAO caused little alteration to the Φ‐values of the mutants tested, implying that this compound minimally perturbs the folding transition state structure. Furthermore, the solvent accessibility of the transition state was not altered as reflected in an absence of a TMAO‐induced change in the denaturant β factors. Through TMAO‐induced folding studies, a β factor of 0.5 was calculated for this compound, suggesting that the protein backbone, which is the target of action of TMAO, is 50% exposed in the transition state as compared to the native state. This finding is consistent with the equivalent effects of TMAO on the folding and unfolding rates. Through thermodynamic analysis of mutants, we also discovered that the stabilizing effect of TMAO is lessened with increasing temperature.     

Shape of the free energy barriers for protein folding probed by multiple perturbation analysis

The characterization of the free energy barriers has been a major goal in studies on the mechanism of protein folding. Testing the effect of mutations or denaturants on protein folding reactions revealed that transition state movement is rare, suggesting that folding barriers are robust and narrow maxima on the free energy landscape. Here we demonstrate that the application of multiple perturbations allows the observation of small transition state movements that escape detection in single perturbation experiments. We used tendamistat as a model protein to test the broadness of the free energy barriers. Tendamistat folds over two consecutive transition states and through a high-energy intermediate. Measuring the combined effect of temperature and denaturant on the position of the transition state in the wild-type protein and in several mutants revealed that the early transition state shows significant transition state movement. Its accessible surface area state becomes more native-like with destabilization of the native state by temperature. To the same extent, the entropy of the early transition state becomes more native-like with increasing denaturant concentration, in accordance with Hammond behavior. The position of the late transition state, in contrast, is much less sensitive to the applied perturbations. These results suggest that the barriers in protein folding become increasingly narrow as the folding polypeptide chain approaches the native state.     

Some recommendations for the practitioner to improve the precision of experimentally determined protein folding rates and phi values

The mechanism by which proteins fold from an initially random conformation into a functional, native structure remains a major unsolved question in molecular biology. Of particular interest to the protein folding community is the structure that the protein adopts in the folding transition state (the highest free energy state on the pathway from unfolded to folded), as that state forms the barrier that defines the folding pathway. Unfortunately, however, unlike those of the initial, unfolded state and the final, folded state of the protein, the structure in the transition state cannot be directly assessed via experiment. Instead, experimentalists infer the structure of the transition state, often by estimating changes in its free energy by measuring the effects of amino acid substitutions on folding and unfolding rates (Phi-value analysis). In this article we show how to obtain more efficient estimates of these important quantities via improved experimental designs, and how to avoid common pitfalls in the analysis of kinetic data during the extraction of these parameters.     

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

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

Interpretation of protein folding psi values

The structural characterization of transition states is essential for understanding the mechanism of protein folding. Analyzing the effect of mutations on protein stability and folding kinetics in phi-value analysis is commonly used to gain information about the presence of side-chain interactions in transition states. Recently, specific binding of ligands to engineered binding sites was applied to monitor the formation of local structures in transition states (psi analysis). A surprising result from psi analysis was the presence of parallel folding pathways in all reported studies and a major discrepancy between phi and psi values measured in the same protein. Here, we show that psi values cannot be analyzed in the same way as other rate-equilibrium free energy relationships due to the involvement of bimolecular reactions that may have different dissociation constants for the native, unfolded and transition state. As a consequence, psi values reflect the relative binding energy (kappa) of the transition state only for the extreme values of kappa=0 or kappa=1. In all other cases, non-linear rate-equilibrium free-energy relationships (Leffler plots) are observed. This apparently indicates the presence of parallel folding pathways even if folding occurs over a single homogeneous transition state. Consequently, the results from Leffler plots do not yield information about the structural properties of the transition state. This explains the lack of agreement between results from psi analysis and other methods used to characterize protein folding transition states. We further show that the same considerations apply for the analysis of the effect of pH on protein folding.     

Analysis of the pH-dependent folding and stability of histidine point mutants allows characterization of the denatured state and transition state for protein folding

pH-Dependent studies of the folding kinetics and stability of a set of His to Gln point mutants were used to characterize the denatured state and transition state ensembles for the C-terminal domain of the ribosomal protein L9 (CTL9). CTL9 contains three histidine residues, two of which, H106 and H134, are buried in the native state, while the third, H144, is more exposed. Comparison of the pH-dependent stability calculated using the Tanford-Wyman linkage relationship to the measured values demonstrates that the apparent pK(a) values of the three histidine residues are not significantly perturbed in the denatured state ensemble. Kinetic measurements show that mutation of H134 has a larger effect on the folding process than does mutation of H106 and H144. The Phi-value for H134 is significantly larger than the Phi-values for the other histidine residues, which are near zero at both pH 5.45 and pH 8.0. The Phi-value for H134 is higher, 0.55, at pH 8.0 than at pH 5.45, 0.39. At pH 5.45, H134 is protonated in the unfolded state but deprotonated in the native state, while at pH 8.0 it is deprotonated in both. There is an excellent linear relationship between stability (logK) and folding rates (logk(f)) over the range of pH 5-9 for all mutants. From these plots, the ratio of DeltaQ( not equal)/DeltaQ can be calculated for each mutant. DeltaQ( not equal) is the difference in the number of protons bound to the transition state and to the unfolded state, while DeltaQ represents the difference between folded and denatured state. The linear plots indicate that the relative position of the transition state ensemble as judged by DeltaQ( not equal)/DeltaQ is independent of pH. The linkage analysis is consistent with the Phi-value analysis, showing that H134 is the most critical contributor to the development of pH-dependent interactions, including desolvation effects in the transition state ensemble.     

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.     

Folding Nuclei in Proteins

When a protein folds or unfolds, it passes through many half-folded microstates. Only a few of them can accumulate and be seen experimentally, and this happens only when the folding (or unfolding) occurs far from the point of thermodynamic equilibrium between the native and denatured states. The universal features of folding, though, are observed in the vicinity of the equilibrium point. Here the two-state transition proceeds without any accumulation of metastable intermediates, and only the transition state (folding nucleus) is outlined by its key influence on the folding/unfolding kinetics. This review covers recent experimental and theoretical studies of folding nuclei.     

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.     

Non-native interactions play an effective role in protein folding dynamics

Systematic Monte Carlo simulations of simple lattice models show that the final stage of protein folding is an ordered process where native contacts get locked (i.e., the residues come into contact and remain in contact for the duration of the folding process) in a well‐defined order. The detailed study of the folding dynamics of protein‐like sequences designed as to exhibit different contact energy distributions, as well as different degrees of sequence optimization (i.e., participation of non‐native interactions in the folding process), reveals significant differences in the corresponding locking scenarios—the collection of native contacts and their average locking times, which are largely ascribable to the dynamics of non‐native contacts. Furthermore, strong evidence for a positive role played by non‐native contacts at an early folding stage was also found. Interestingly, for topologically simple target structures, a positive interplay between native and non‐native contacts is observed also toward the end of the folding process, suggesting that non‐native contacts may indeed affect the overall folding process. For target models exhibiting clear two‐state kinetics, the relation between the nucleation mechanism of folding and the locking scenario is investigated. Our results suggest that the stabilization of the folding transition state can be achieved through the establishment of a very small network of native contacts that are the first to lock during the folding process.     

Identifying folding nucleus based on residue contact networks of proteins

In the native structure of a protein, all the residues are tightly parked together in a specific order following its folding and every residue contacts with some spatially neighbor residues. A residue contact network can be constructed by defining the residues as nodes and the native contacts as edges. During the folding of small single-domain proteins, there is a set of contacts (or bonds), defined as the folding nucleus (FN), which is formed around the transition state, i.e., a rate-limiting barrier located at about the middle between the unfolded states and the native state on the free energy landscape. Such a FN plays an essential role in the folding dynamics and the residues, which form the related contacts called as folding nucleus residues (FNRs). In this work, the FNRs in proteins are identified by using quantities which characterize the topology of residue contact networks of proteins. By comparing the specificities of residues with the network quantities K(R), L(R), and D(R), up to 90% FNRs of six typical proteins found experimentally are identified. It is found that the FNRs behave the full-closeness centrals rather than degree or closeness centers in the residue contact network, implying that they are important to the folding cooperativity of proteins. Our study shows that the FNRs can be identified solely from the native structures of proteins based on the analysis of residue contact network without any knowledge of the transition state ensemble.     

Side-chain dynamics and protein folding

The processes by which protein side chains reach equilibrium during a folding reaction are investigated using both lattice and all-atom simulations. We find that rates of side-chain relaxation exhibit a distribution over the protein structure, with the fastest relaxing side chains located in positions kinetically important for folding. Traversal of the major folding transition state corresponds to the freezing of a small number of side chains, belonging to the folding nucleus, whereas the rest of the protein proceeds toward equilibrium via backbone fluctuations around the native fold. The postnucleation processes by which side chains relax are characterized by very slow dynamics and many barrier crossings, and thus resemble the behavior of a glass.     

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.