Using stochastic roadmap simulation to predict experimental quantities in protein folding kinetics: folding rates and phi-values.

This paper presents a new method for studying protein folding kinetics. It uses the recently introduced Stochastic Roadmap Simulation (SRS) method to estimate the transition state ensemble (TSE) and predict the rates and the Phi-values for protein folding. The new method was tested on 16 proteins, whose rates and Phi-values have been determined experimentally. Comparison with experimental data shows that our method estimates the TSE much more accurately than an existing method based on dynamic programming. This improvement leads to better folding-rate predictions. We also compute the mean first passage time of the unfolded states and show that the computed values correlate with experimentally determined folding rates. The results on Phi-value predictions are mixed, possibly due to the simple energy model used in the tests. This is the first time that results obtained from SRS have been compared against a substantial amount of experimental data. The results further validate the SRS method and indicate its potential as a general tool for studying protein folding kinetics.     

Nucleation‐based prediction of the protein folding rate and its correlation with the folding nucleus size

The problem of protein self‐organization is in the focus of current molecular biology studies. Although the general principles are understood, many details remain unclear. Specifically, protein folding rates are of interest because they dictate the rate of protein aggregation which underlies many human diseases. Here we offer predictions of protein folding rates and their correlation with folding nucleus sizes. We calculated free energies of the transition state and sizes of folding nuclei for 84 proteins and peptides whose other parameters were measured at the point of thermodynamic equilibrium between their unfolded and native states. We used the dynamic programming method where each residue was considered to be either as folded as in its native state or completely disordered. The calculated and measured folding rates showed a good correlation at the temperature mid‐transition point (the correlation coefficient was 0.75). Also, we pioneered in demonstrating a moderate (‐0.57) correlation coefficient between the calculated sizes of folding nuclei and the folding rates. Predictions made by different methods were compared. The established good correlation between the estimated free energy barrier and the experimentally found folding rate of each studied protein/peptide indicates that our model gives reliable results for the considered data set. Proteins 2012; © 2012 Wiley Periodicals, Inc.     

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).     

Calculation of mutational free energy changes in transition states for protein folding

Recent advances in experimental and computational methods have made it possible to determine with considerable accuracy the structures whose formation is rate limiting for the folding of some small proteins-the transition state ensemble, or TSE. We present a method to analyze and validate all-atom models of such structures. The method is based on the comparison of experimental data with the computation of the change in free energy of the TSE resulting from specific mutations. Each mutation is modeled individually in all members of an ensemble of transition state structures using a method originally developed to predict mutational changes in the stability of native proteins. We first apply this method to six proteins for which we have determined the TSEs with a technique that uses experimental mutational data (Phi-values) as restraints in the structure determination and find a highly significant correlation between the calculated free energy changes and those derived from experimental kinetic data. We then use the procedure to analyze transition state structures determined by molecular dynamics simulations of unfolding, again finding a high correlation. Finally, we use the method to estimate changes in folding rates of several hydrophobic core mutants of Fyn SH3. Taken together, these results show that the procedure developed here is a tool of general validity for analyzing, assessing, and improving the quality of the structures of transition states for protein folding.     

On prediction of folding nuclei in globular proteins

The approach described in this paper on the prediction of folding nuclei in globular proteins with known three dimensional structures is based on a search of the lowest saddle points through the barrier separating the unfolded state from the native structure on the free-energy landscape of protein chain. This search is performed by a dynamic programming method. Comparison of theoretical results with experimental data on the folding nuclei of two dozen of proteins shows that our model provides good phi value predictions for proteins whose structures have been determined by X-ray analysis, with a less limited success for proteins whose structures have been determined by NMR techniques only. Consideration of a full ensemble of transition states results in more successful prediction than consideration of only the transition states with the minimal free energy. In conclusion we have predicted the localization of folding nuclei for three dimensional protein structures for which kinetics of folding is studied now but the localization of folding nuclei is still unknown.     

Hammond behavior versus ground state effects in protein folding: evidence for narrow free energy barriers and residual structure in unfolded states

Apparent transition state movement upon mutation or changes in solvent conditions is frequently observed in protein folding and is often interpreted in terms of Hammond behavior. This led to the conclusion that barrier regions in protein folding are broad maxima on the free energy landscape. Here, we use the concept of self-interaction and cross-interaction parameters to test experimental data of 21 well-characterized proteins for Hammond behavior. This allows us to characterize the origin of transition state movements along different reaction coordinates. Only one of the 21 proteins shows a small but coherent transition state movement in agreement with the Hammond postulate. In most proteins the structure of the transition state is insensitive to changes in protein stability. The apparent change in the position of the transition state upon mutation, which is frequently observed in phi-value analysis, is in most cases due to ground-state effects caused by structural changes in the unfolded state. This argues for significant residual structure in unfolded polypeptide chains of many proteins. Disruption of these residual interactions by mutation often leads to decreased folding rates, which implies that these interactions are still present in the transition state. The failure to detect Hammond behavior shows that the free energy barriers encountered by a folding polypeptide chain are generally rather narrow and robust maxima for all experimentally explorable reaction coordinates.     

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.     

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.     

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.     

Folding-unfolding energy change of a simple sphere model protein and an energy landscape of the folding process

We have calculated the free energy of a spherical model of a protein or part of a protein generated in the way of protein folding. Two spherical models are examined; one is a homogeneous model consisting of only one residue type—hydrophobic. The other is a heterogeneous model consisting of two residue types—strong hydrophobic and weak hydrophobic. Both models show a folding transition state, and the latter model reproduces the trend of the experimental folded-unfolded energy change. The heterogeneous model suggests that in the folding process of a protein of more than 70 residues, a specific region of the protein folds first to form a stable region, then the other residues follow the folding process. The energy landscape of folding of a small protein is approximately a funnel model, whereas a flatter energy landscape is suggested for larger proteins of more than 55–70 residues. Proteins 33:408–416, 1998. © 1998 Wiley-Liss, Inc.     

The cooperativity of burst phase reactions explored.

The denaturant-dependence of the major, observable relaxation rates for folding (kobs) of ribonuclease HI from Escherichia coli (RNase H) and phage T4 lysozyme (T4L) reveal that, for both proteins, folding begins with the rapid and transient accumulation of intermediate species in a "burst phase" which precedes the rate-limiting formation of the native state; this is evidenced by a "rollover" in the folding limb of the rate profiles (kobs versus denaturant, or chevron plot). These rate profiles are most simply described by a three-state mechanism (unfolded-to-intermediate-to-native), which implies that the burst phase represents a transition between two distinct thermodynamic states. It is shown here that the equilibrium properties of these burst phase reactions can be equally well modeled by a mechanism involving a continuum of states where the free energy of each state is linearly related to its m-value (the parameter describing the linear relationship between free energy and denaturant). A numerical model is also developed to describe the time evolution of such a system, which exhibits nearly perfect exponential behavior. Both models emphasize how a continuum of states operating under a linear free energy relationship may behave like a two state system. Such a scheme finds experimental justification from an interpretation of recent native state hydrogen exchange data. The analytical model described for a continuum can account for the observed kinetic profiles of several other model proteins. The results, however, appear context specific, suggesting that burst phase reactions are not entirely random and non-specific. The results reported in this study have important implications for the concept of cooperativity in protein folding reactions.     

Computing the transition state populations in simple protein models

We describe the master equation method for computing the kinetics of protein folding. We illustrate the method using a simple Go model. Presently most models of two-state fast-folding protein folding kinetics invoke the classical idea of a transition state to explain why there is a single exponential decay in time. However, if proteins fold via funnel-shaped energy landscapes, as predicted by many theoretical studies, then it raises the question of what is the transition state. Is it a specific structure, or a small ensemble of structures, as is expected from classical transition state theory? Or is it more like the denatured states of proteins, a very broad ensemble? The answer that is usually obtained depends on the assumptions made about the transition state. The present method is a rigorous way to find transition states, without assumptions or approximations, even for very nonclassical shapes of energy landscapes. We illustrate the method here, showing how the transition states in two-state protein folding can be very broad ensembles. © 2002 Wiley Periodicals, Inc. Biopolymers 68: 35–46, 2003     

Improved Gō-like models demonstrate the robustness of protein folding mechanisms towards non-native interactions

The use of simple theoretical models has provided a considerable contribution to our present understanding of the means by which proteins adopt their native fold from the plethora of available unfolded states. A common assumption in building computationally tractable models has been the neglect of stabilizing non-native interactions in the class of models described as "Gō-like." The focus of this study is the characterization of the folding of a number of proteins via a Gō-like model, which aims to map a maximal amount of information reflecting the protein sequence onto a "minimalist" skeleton. This model is shown to contain sufficient information to reproduce the folding transition states of a number of proteins, including topologically analogous proteins that fold via different transition states. Remarkably, these models also demonstrate consistency with the general features of folding transition states thought to be stabilized by non-native interactions. This suggests that native interactions are the primary determinant of most protein folding transition states, and that non-native interactions lead only to local structural perturbations. A prediction is also included for an asymmetrical folding transition state of bacteriophage lambda protein W, which has yet to be subjected to experimental characterization.     

Dimensions,energetics, and denaturant effects of the protein unstructured state

Determining the energetics of the unfolded state of a protein is essential for understanding the folding mechanics of ordered proteins and the structure–function relation of intrinsically disordered proteins. Here, we adopt a coil‐globule transition theory to develop a general scheme to extract interaction and free energy information from single‐molecule fluorescence resonance energy transfer spectroscopy. By combining protein stability data, we have determined the free energy difference between the native state and the maximally collapsed denatured state in a number of systems, providing insight on the specific/nonspecific interactions in protein folding. Both the transfer and binding models of the denaturant effects are demonstrated to account for the revealed linear dependence of inter‐residue interactions on the denaturant concentration, and are thus compatible under the coil‐globule transition theory to further determine the dimension and free energy of the conformational ensemble of the unfolded state. The scaling behaviors and the effective θ‐state are also discussed.     

Modeling disordered regions in proteins using Rosetta

Protein structure prediction methods such as Rosetta search for the lowest energy conformation of the polypeptide chain. However, the experimentally observed native state is at a minimum of the free energy, rather than the energy. The neglect of the missing configurational entropy contribution to the free energy can be partially justified by the assumption that the entropies of alternative folded states, while very much less than unfolded states, are not too different from one another, and hence can be to a first approximation neglected when searching for the lowest free energy state. The shortcomings of current structure prediction methods may be due in part to the breakdown of this assumption. Particularly problematic are proteins with significant disordered regions which do not populate single low energy conformations even in the native state. We describe two approaches within the Rosetta structure modeling methodology for treating such regions. The first does not require advance knowledge of the regions likely to be disordered; instead these are identified by minimizing a simple free energy function used previously to model protein folding landscapes and transition states. In this model, residues can be either completely ordered or completely disordered; they are considered disordered if the gain in entropy outweighs the loss of favorable energetic interactions with the rest of the protein chain. The second approach requires identification in advance of the disordered regions either from sequence alone using for example the DISOPRED server or from experimental data such as NMR chemical shifts. During Rosetta structure prediction calculations the disordered regions make only unfavorable repulsive contributions to the total energy. We find that the second approach has greater practical utility and illustrate this with examples from de novo structure prediction, NMR structure calculation, and comparative modeling.     

Transferable Coarse-Grained Potential for De Novo Protein Folding and Design

Protein folding and design are major biophysical problems, the solution of which would lead to important applications especially in medicine. Here we provide evidence of how a novel parametrization of the Caterpillar model may be used for both quantitative protein design and folding. With computer simulations it is shown that, for a large set of real protein structures, the model produces designed sequences with similar physical properties to the corresponding natural occurring sequences. The designed sequences require further experimental testing. For an independent set of proteins, previously used as benchmark, the correct folded structure of both the designed and the natural sequences is also demonstrated. The equilibrium folding properties are characterized by free energy calculations. The resulting free energy profiles not only are consistent among natural and designed proteins, but also show a remarkable precision when the folded structures are compared to the experimentally determined ones. Ultimately, the updated Caterpillar model is unique in the combination of its fundamental three features: its simplicity, its ability to produce natural foldable designed sequences, and its structure prediction precision. It is also remarkable that low frustration sequences can be obtained with such a simple and universal design procedure, and that the folding of natural proteins shows funnelled free energy landscapes without the need of any potentials based on the native structure.     

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.     

Towards a consistent modeling of protein thermodynamic and kinetic cooperativity: how applicable is the transition state picture to folding and unfolding?

To what extent do general features of folding/unfolding kinetics of small globular proteins follow from their thermodynamic properties? To address this question, we investigate a new simplified protein chain model that embodies a cooperative interplay between local conformational preferences and hydrophobic burial. The present four-helix-bundle 55mer model exhibits protein-like calorimetric two-state cooperativity. It rationalizes native-state hydrogen exchange observations. Our analysis indicates that a coherent, self-consistent physical account of both the thermodynamic and kinetic properties of the model leads naturally to the concept of a native state ensemble that encompasses considerable conformational fluctuations. Such a multiple-conformation native state is seen to involve conformational states similar to those revealed by native-state hydrogen exchange. Many of these conformational states are predicted to lie below native baselines commonly used in interpreting calorimetric data. Folding and unfolding kinetics are studied under a range of intrachain interaction strengths as in experimental chevron plots. Kinetically determined transition midpoints match well with their thermodynamic counterparts. Kinetic relaxations are found to be essentially single-exponential over an extended range of model interaction strengths. This includes the entire unfolding regime and a significant part of a folding regime with a chevron rollover, as has been observed for real proteins that fold with non-two-state kinetics. The transition state picture of protein folding and unfolding is evaluated by comparing thermodynamic free energy profiles with actual kinetic rates. These analyses suggest that some chevron rollovers may arise from an internal frictional effect that increasingly impedes chain motions with more native conditions, rather than being caused by discrete deadtime folding intermediates or shifts of the transition state peak as previously posited.     

Engineering strand-specific DNA nicking enzymes from the type IIS restriction endonucleases BsaI, BsmBI, and BsmAI

To determine the extent to which protein folding rates and free energy landscapes have been shaped by natural selection, we have examined the folding kinetics of five proteins generated using computational design methods and, hence, never exposed to natural selection. Four of these proteins are complete computer-generated redesigns of naturally occurring structures and the fifth protein, called Top7, has a computer-generated fold not yet observed in nature. We find that three of the four redesigned proteins fold much faster than their naturally occurring counterparts. While natural selection thus does not appear to operate on protein folding rates, the majority of the designed proteins unfold considerably faster than their naturally occurring counterparts, suggesting possible selection for a high free energy barrier to unfolding. In contrast to almost all naturally occurring proteins of less than 100 residues but consistent with simple computational models, the folding energy landscape for Top7 appears to be quite complex, suggesting the smooth energy landscapes and highly cooperative folding transitions observed for small naturally occurring proteins may also reflect the workings of natural selection.     

Protein folding transition states probed by loop extension

We propose a new way to characterize protein folding transition states by (1) insertion of one or more residues into an unstructured protein loop, (2) measurement of the effect on protein folding kinetics and thermodynamics, and (3) analysis of the results in terms of a rate-equilibrium free energy relationship, alpha(Loop). alpha(Loop) reports on the fraction of molecules that form the perturbed loop in the transition state. Interpretation of the changes in equilibrium free energy using standard polymer theory can help detect residual structure in the unfolded state. We illustrate our approach with data for the model proteins CI2 and the alpha spectrin SH3 domain.