Combinatorial modeling of protein folding kinetics: free energy profiles and rates

A Kinetic Transition Network Model Reveals the Diversity of Protein Dimer Formation Mechanisms

Protein homodimers have been classified as three-state or two-state dimers depending on whether a folded monomer forms before association, but the details of the folding-binding mechanisms are poorly understood. Kinetic transition networks of conformational states have provided insight into the folding mechanisms of monomeric proteins, but extending such a network to two protein chains is challenging as all the relative positions and orientations of the chains need to be included, greatly increasing the number of degrees of freedom. Here, we present a simplification of the problem by grouping all states of the two chains into two layers: a dissociated and an associated layer. We combined our two-layer approach with the Wako-Saito-Muñoz-Eaton method and used Transition Path Theory to investigate the dimer formation kinetics of eight homodimers. The analysis reveals a remarkable diversity of dimer formation mechanisms. Induced folding, conformational selection, and rigid docking are often simultaneously at work, and their contribution depends on the protein concentration. Pre-folded structural elements are always present at the moment of association, and asymmetric binding mechanisms are common. Our two-layer network approach can be combined with various methods that generate discrete states, yielding new insights into the kinetics and pathways of flexible binding processes.

Accurate prediction of protein folding mechanisms by simple structure-based statistical mechanical models

Recent breakthroughs in highly accurate protein structure prediction using deep neural networks have made considerable progress in solving the structure prediction component of the 'protein folding problem'. However, predicting detailed mechanisms of how proteins fold into specific native structures remains challenging, especially for multidomain proteins constituting most of the proteomes. Here, we develop a simple structure-based statistical mechanical model that introduces nonlocal interactions driving the folding of multidomain proteins. Our model successfully predicts protein folding processes consistent with experiments, without the limitations of protein size and shape. Furthermore, slight modifications of the model allow prediction of disulfide-oxidative and disulfide-intact protein folding. These predictions depict details of the folding processes beyond reproducing experimental results and provide a rationale for the folding mechanisms. Thus, our physics-based models enable accurate prediction of protein folding mechanisms with low computational complexity, paving the way for solving the folding process component of the 'protein folding problem'.

Polyelectrolyte Influence on Beta-Hairpin Peptide Stability: A Simulation Study

Assemblies of proteins and charged macromolecules (polyelectrolytes) find important applications as pharmaceutical formulations, biocatalysts, and cell-contacting substrates. A key question is how the polymer component influences the structure and function of the protein. The present paper addresses the influence of charged polymers on the thermal stability of two model beta-hairpin-forming peptides through an all-atom, replica exchange molecular dynamics simulation. The (negatively charged) peptides consist of the terminal 16 amino acids of the B1 domain of Protein G (GB1) and a variant with three of the GB1 residues substituted with tryptophan (Tryptophan Zipper 4, or TZ4). A (cationic) lysine polymer is seen to thermally stabilize TZ4 and destabilize GB1, while a (also cationic) chitosan polymer slightly stabilizes GB1 but has essentially no effect on TZ4. Free energy profiles reveal folded and unfolded conformations to be separated by kinetic barriers generally acting in the direction of the thermodynamically favored state. Through application of an Ising-like statistical mechanical model, a mechanism is proposed based on competition between (indirect) entropic stabilization of folded versus unfolded states and (direct) competition for hydrogen-bonding and hydrophobic interactions. These findings have important implications to the design of polyelectrolyte-based materials for biomedical and biotechnological applications.

Thermodynamic architecture and conformational plasticity of GPCRs

G-protein-coupled receptors (GPCRs) are ubiquitous integral membrane proteins involved in diverse cellular signaling processes. Here, we carry out a large-scale ensemble thermodynamic study of 45 ligand-free GPCRs employing a structure-based statistical mechanical framework. We find that multiple partially structured states co-exist in the GPCR native ensemble, with the TM helices 1, 6 and 7 displaying varied folding status, and shaping the conformational landscape. Strongly coupled residues are anisotropically distributed, accounting for only 13% of the residues, illustrating that a large number of residues are inherently dynamic. Active-state GPCRs are characterized by reduced conformational heterogeneity with altered coupling-patterns distributed throughout the structural scaffold. In silico alanine-scanning mutagenesis reveals that extra- and intra-cellular faces of GPCRs are coupled thermodynamically, highlighting an exquisite structural specialization and the fluid nature of the intramolecular interaction network. The ensemble-based perturbation methodology presented here lays the foundation for understanding allosteric mechanisms and the effects of disease-causing mutations in GCPRs.

The Wako-Saitô-Muñoz-Eaton Model for Predicting Protein Folding and Dynamics

Despite the recent advances in the prediction of protein structures by deep neutral networks, the elucidation of protein-folding mechanisms remains challenging. A promising theory for describing protein folding is a coarse-grained statistical mechanical model called the Wako-Saitô-Muñoz-Eaton (WSME) model. The model can calculate the free-energy landscapes of proteins based on a three-dimensional structure with low computational complexity, thereby providing a comprehensive understanding of the folding pathways and the structure and stability of the intermediates and transition states involved in the folding reaction. In this review, we summarize previous and recent studies on protein folding and dynamics performed using the WSME model and discuss future challenges and prospects. The WSME model successfully predicted the folding mechanisms of small single-domain proteins and the effects of amino-acid substitutions on protein stability and folding in a manner that was consistent with experimental results. Furthermore, extended versions of the WSME model were applied to predict the folding mechanisms of multi-domain proteins and the conformational changes associated with protein function. Thus, the WSME model may contribute significantly to solving the protein-folding problem and is expected to be useful for predicting protein folding, stability, and dynamics in basic research and in industrial and medical applications.

Computational design of a thermolabile uracil-DNA glycosylase of Escherichia coli

Polymerase chain reaction (PCR) is a powerful tool to diagnose infectious diseases. Uracil DNA glycosylase (UDG) is broadly used to remove carryover contamination in PCR. However, UDG can contribute to false negative results when not inactivated completely, leading to DNA degradation during the amplification step. In this study, we designed novel thermolabile UDG derivatives by supercomputing molecular dynamic simulations and residual network analysis. Based on enzyme activity analysis, thermolability, thermal stability, and biochemical experiments of Escherichia coli-derived UDG and 22 derivatives, we uncovered that the UDG D43A mutant eliminated the false negative problem, demonstrated high efficiency, and offered great benefit for use in PCR diagnosis. We further obtained structural and thermodynamic insights into the role of the D43A mutation, including perturbed protein structure near D43; weakened pairwise interactions of D43 with K42, N46, and R80; and decreased melting temperature and native fraction of the UDG D43A mutant compared with wild-type UDG.

Predicting and Simulating Mutational Effects on Protein Folding Kinetics

Mutational perturbations of protein structures, i.e., phi-value analysis, are commonly employed to probe the extent of involvement of a particular residue in the rate-determining step(s) of folding. This generally involves the measurement of folding thermodynamic parameters and kinetic rate constants for the wild-type and mutant proteins. While computational approaches have been reasonably successful in understanding and predicting the effect of mutations on folding thermodynamics, it has been challenging to explore the same on kinetics due to confounding structural, energetic, and dynamic factors. Accordingly, the frequent observation of fractional phi-values (mean of ~0.3) has resisted a precise and consistent interpretation. Here, we describe how to construct, parameterize, and employ a simple one-dimensional free energy surface model that is grounded in the basic tenets of the energy landscape theory to predict and simulate the effect of mutations on folding kinetics. As a proof of principle, we simulate one-dimensional free energy profiles of 806 mutations from 24 different proteins employing just the experimental destabilization as input, reproduce the relative unfolding activation free energies with a correlation of 0.91, and show that the mean phi-value of 0.3 essentially corresponds to the extent of stabilization energy gained at the barrier top while folding.

Folding Intermediates, Heterogeneous Native Ensembles and Protein Function

Single domain proteins fold via diverse mechanisms emphasizing the intricate relationship between energetics and structure, which is a direct consequence of functional constraints and demands imposed at the level of sequence. On the other hand, elucidating the interplay between folding mechanisms and function is challenging in large proteins, given the inherent shortcomings in identifying metastable states experimentally and the sampling limitations associated with computational methods. Here, we show that free energy profiles and surfaces of large systems (>150 residues), as predicted by a statistical mechanical model, display a wide array of folding mechanisms with ubiquitous folding intermediates and heterogeneous native ensembles. Importantly, residues around the ligand binding or enzyme active site display a larger tendency to partially unfold and this manifests as intermediates or excited states along the folding coordinate in ligand binding domains, transcription repressors, and representative enzymes from all the six classes, including the SARS-CoV-2 receptor binding domain (RBD) of the spike protein and the protease M^pro. It thus appears that it is relatively easier to distill the imprints of function on the folding landscape of larger proteins as opposed to smaller systems. We discuss how an understanding of energetic-entropic features in ordered proteins can pinpoint specific avenues through which folding mechanisms, populations of partially structured states and function can be engineered.

A hierarchy of coupling free energies underlie the thermodynamic and functional architecture of protein structures

Protein sequences and structures evolve by satisfying varied physical and biochemical constraints. This multi-level selection is enabled not just by the patterning of amino acids on the sequence, but also via coupling between residues in the native structure. Here, we employ an energetically detailed statistical mechanical model with millions of microstates to extract such long-range structural correlations, i.e. thermodynamic coupling free energies, from a diverse family of protein structures. We find that despite the intricate and anisotropic distribution of coupling patterns, the majority of residues (>70%) are only marginally coupled contributing to functional motions and catalysis. Physical origins of ‘sectors’, determinants of native ensemble heterogeneity in extant, ancient and designed proteins, and the basis for allostery emerge naturally from coupling free energies. The statistical framework highlights how evolutionary selection and optimization occur at the level of global interaction network for a given protein fold impacting folding, function, and allosteric outputs.

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Evolution of protein structures occurs at the level of global interaction network.

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More than 70% of the protein residues are weakly or marginally coupled.

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Functional ‘sector’ regions are a manifestation of marginal coupling.

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Coupling indices vary across the entire proteins in extant-ancient and natural-designed pairs.

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The proposed methodology can be used to understand allostery and epistasis.

First-passage times in protein folding: exploring the native-like states vs. overcoming the free energy barrier

Using a model β-hairpin protein as a representative example of simple two-state folders whose kinetics are uncomplicated by the presence of on- and off-pathway intermediates, it is studied how the search for the protein's native state among native-like states affects the folding kinetics. It is revealed that the first-passage time (FPT) distributions are essentially single-exponential not only for the times to overcome the free energy barrier between the unfolded and native-like states but also for the times to find the native state among the native-like ones. The FPT distributions of this type are observed through all studied two-state-like regimes of protein folding, varying from a regime close to two-state folding to a regime close to downhill folding. If the protein explores native-like states for a time much longer than the time to overcome the free energy barrier, which is characteristic of high temperatures, the resulting FPT distribution to reach the native state remains close to exponential but the mean FPT (MFPT) is determined not by the height of the free energy barrier but by the time to explore native-like states. In particular, the mean time to overcome the free energy barrier is in reasonable agreement with the Kramers rate formula and generally far shorter than the overall MFPT to reach the native state. The observed increase of the overall MFPT, as a result of longer exploration of native-like states, may lead to an overestimate of the height of the free energy barrier between the unfolded and folded states when it is calculated from the overall MFPT.

Modern Kinetics and Mechanism of Protein Folding: A Retrospective

Modern experimental kinetics of protein folding began in the early 1990s with the introduction of nanosecond laser pulses to trigger the folding reaction, providing an almost 10⁶-fold improvement in time resolution over the stopped-flow method being employed at the time. These experiments marked the beginning of the "fast-folding" subfield that enabled investigation of the kinetics of formation of secondary structural elements and disordered loops for the first time, as well as the fastest folding proteins. When I started to work on this subject, a fast folding protein was one that folded in milliseconds. There were, moreover, no analytical theoretical models and no atomistic or coarse-grained molecular dynamics simulations to describe the mechanism. Two of the most important discoveries from my lab since then are a protein that folds in hundreds of nanoseconds, as determined from nanosecond laser temperature experiments, and the discovery that the theoretically predicted barrier crossing time is about the same for proteins that differ in folding rates by 10⁴-fold, as determined from single molecule fluorescence measurements. We also developed what has been called the "Hückel model" of protein folding, which quantitatively explains a wide range of equilibrium and kinetic measurements. This retrospective traces the history of contributions to the "fast folding" subfield from my lab until about 3 years ago, when I left protein folding to spend the rest of my research career trying to discover an inexpensive drug for treating sickle cell disease.

The Anti-Aggregation Holdase Hsp33 Promotes the Formation of Folded Protein Structures

Holdase chaperones are known to be central to suppressing aggregation, but how they affect substrate conformations remains poorly understood. Here, we use optical tweezers to study how the holdase Hsp33 alters folding transitions within single maltose binding proteins and aggregation transitions between maltose binding protein substrates. Surprisingly, we find that Hsp33 not only suppresses aggregation but also guides the folding process. Two modes of action underlie these effects. First, Hsp33 binds unfolded chains, which suppresses aggregation between substrates and folding transitions within substrates. Second, Hsp33 binding promotes substrate states in which most of the chain is folded and modifies their structure, possibly by intercalating its intrinsically disordered regions. A statistical ensemble model shows how Hsp33 function results from the competition between these two contrasting effects. Our findings reveal an unexpectedly comprehensive functional repertoire for Hsp33 that may be more prevalent among holdases and dispels the notion of a strict chaperone hierarchy.

Thermodynamics and folding landscapes of large proteins from a statistical mechanical model

Statistical mechanical models that afford an intermediate resolution between macroscopic chemical models and all-atom simulations have been successful in capturing folding behaviors of many small single-domain proteins. However, the applicability of one such successful approach, the Wako-Saitô-Muñoz-Eaton (WSME) model, is limited by the size of the protein as the number of conformations grows exponentially with protein length. In this work, we surmount this size limitation by introducing a novel approximation that treats stretches of 3 or 4 residues as blocks, thus reducing the phase space by nearly three orders of magnitude. The performance of the 'bWSME' model is validated by comparing the predictions for a globular enzyme (RNase H) and a repeat protein (IκBα), against experimental observables and the model without block approximation. Finally, as a proof of concept, we predict the free-energy surface of the 370-residue, multi-domain maltose binding protein and identify an intermediate in good agreement with single-molecule force-spectroscopy measurements. The bWSME model can thus be employed as a quantitative predictive tool to explore the conformational landscapes of large proteins, extract the structural features of putative intermediates, identify parallel folding paths, and thus aid in the interpretation of both ensemble and single-molecule experiments.

Effect of Mutations on the Global and Site-Specific Stability and Folding of an Elementary Protein Structural Motif

Understanding the folding mechanism of proteins requires detailed knowledge of the roles of individual amino acid residues in stabilization of specific elements and local segments of the native structure. Recently, we have utilized the combination of circular dichroism (CD) and site-specific ¹³C isotopically edited infrared spectroscopy (IR) coupled with the Ising-like model for protein folding to map the thermal unfolding at the residue level of a de novo designed helix-turn-helix motif αtα. Here we use the same methodology to study how the sequence of local thermal unfolding is affected by selected mutations introduced into the most and least stable parts of the motif. Seven different mutants of αtα are screened to find substitutions with the most pronounced effects on the overall stability. Subsequently, thermal unfolding of two mutated αtα sequences is studied with site-specific resolution, using four distinct ¹³C isotopologues of each. The data are analyzed with the Ising-like model, which builds on a previous parametrization for the original αtα sequence and tests different ways of incorporating the amino acid substitution. We show that for both more and less stable mutants only the adjustment of all interaction parameters of the model can yield a satisfactory fit to the experimental data. The stabilizing and destabilizing mutations result, respectively, in a similar increase and decrease of the stability of all probed local segments, irrespective of their position with respect to the mutation site. Consequently, the relative order of their unfolding remains essentially unchanged. These results underline the importance of the interconnectivity of the stabilizing interaction network and cooperativity of the protein structure, which is evident even in a small motif with apparently noncooperative, heterogeneous unfolding. Overall, our findings are consistent with the native structure being the dominant factor in determining the folding mechanism, regardless of the details of its overall or local thermodynamic stabilization.

Peptide and Protein Structure Prediction with a Simplified Continuum Solvent Model

A continuum solvent model based on screened Coulomb potentials has been simplified and parametrized to sample native-like structures in replica-exchange simulations of each of six different peptides and miniproteins. Low-energy, native, and non-native structures were used to iteratively refine 11 parameter values. The centroid of the largest cluster of structures sampled in simulations initiated from an extended conformation represents the predicted structure. The main-chain rms deviation of this prediction from the experimental structure was 0.47 Å for the 12-residue Trp-zip2, 0.86 Å for the 14-residue MBH12, 2.53 Å for the 17-residue U(1-17)T9D, 2.03 Å for the 20-residue BS1, 1.08 Å for the 20-residue Trp-cage, and 3.64 Å for the 35-residue villin headpiece subdomain HP35. The centroid of the sixth largest cluster sampled for HP35 deviated by 0.91 Å. The CHARMM22/CMAP force field was used, with an additional ψ torsion term for residues other than glycine and proline. Six parameters govern the dielectric response of the continuum solvent, and four values of surface tension approximate nonpolar effects. An atom's self-energy and interaction energies are screened independently, each depending on whether the atom is part of a charged group, a neutral hydrogen-bonding main-chain group, or any other neutral group. The parameters inferred result in strong main-chain hydrogen bonds, consistent with the view that protein folding is dominated by the formation of these bonds. (1,2) Conformations of MBH12 and BS1 were excluded from the energy-function refinement, suggesting the parameters, referred to as SCP18, are transferable. An efficient estimate of solvent-accessible surface area is also described.

Transition-path properties for folding reactions in the limit of small barriers

Transition paths are of great interest because they encapsulate information about the mechanisms of barrier-crossing reactions. Analysis of experiments measuring biomolecular folding reactions has relied on expressions for properties of transition paths such as transition-path times and velocities that hold in the limit of large harmonic barriers, but real molecules often have relatively small barriers. Recent theoretical work presented more general expressions for transition-path properties. Here we extend this work, deriving expressions from the general case that can be applied to small harmonic barriers. We first compared the performance of small-barrier, large-barrier, and general solutions when applied to simulated transitions, focusing on improvements in estimates of the diffusion coefficient determined from transition times and velocities. We then applied these expressions to experimental data from force spectroscopy measurements of DNA hairpins. We found that the low-barrier approximation and exact solution reduced or resolved the small but systematic inconsistencies that had arisen from assuming large harmonic barriers, demonstrating the practical utility of the new equations for analyzing experimental data.

Switching Protein Conformational Substates by Protonation and Mutation

Protein modules that regulate the availability and conformational status of transcription factors determine the rapidity, duration, and magnitude of cellular response to changing conditions. One such system is the single-gene product Cnu, a four-helix bundle transcription co-repressor, which acts as a molecular thermosensor regulating the expression of virulence genes in enterobacteriaceae through modulation of its native conformational ensemble. Cnu and related genes have also been implicated in pH-dependent expression of virulence genes. We hypothesize that protonation of a conserved buried histidine (H45) in Cnu promotes large electrostatic frustration, thus disturbing the H-NS, a transcription factor, binding face. Spectroscopic and calorimetric methods reveal that H45 exhibits a suppressed p K_a of ∼5.1, the protonation of which switches the conformation to an alternate native ensemble in which the fourth helix is disordered. The population redistribution can also be achieved through a mutation H45V, which does not display any switching behavior at pH values greater than 4. The Wako-Saitô-Muñoz-Eaton (WSME) statistical mechanical model predicts specific differences in the conformations and fluctuations of the fourth and first helices of Cnu determining the observed pH response. We validate these predictions through fluorescence lifetime measurements of a sole tryptophan, highlighting the presence of both native and non-native interactions in the regions adjoining the binding face of Cnu. Our combined experimental-computational study thus shows that Cnu acts both as a thermo- and pH-sensor orchestrated via a subtle but quantifiable balance between the weak packing of a structural element and protonation of a buried histidine that promotes electrostatic frustration.

Structural Determinants of Misfolding in Multidomain Proteins

Recent single molecule experiments, using either atomic force microscopy (AFM) or Förster resonance energy transfer (FRET) have shown that multidomain proteins containing tandem repeats may form stable misfolded structures. Topology-based simulation models have been used successfully to generate models for these structures with domain-swapped features, fully consistent with the available data. However, it is also known that some multidomain protein folds exhibit no evidence for misfolding, even when adjacent domains have identical sequences. Here we pose the question: what factors influence the propensity of a given fold to undergo domain-swapped misfolding? Using a coarse-grained simulation model, we can reproduce the known propensities of multidomain proteins to form domain-swapped misfolds, where data is available. Contrary to what might be naively expected based on the previously described misfolding mechanism, we find that the extent of misfolding is not determined by the relative folding rates or barrier heights for forming the domains present in the initial intermediates leading to folded or misfolded structures. Instead, it appears that the propensity is more closely related to the relative stability of the domains present in folded and misfolded intermediates. We show that these findings can be rationalized if the folded and misfolded domains are part of the same folding funnel, with commitment to one structure or the other occurring only at a relatively late stage of folding. Nonetheless, the results are still fully consistent with the kinetic models previously proposed to explain misfolding, with a specific interpretation of the observed rate coefficients. Finally, we investigate the relation between interdomain linker length and misfolding, and propose a simple alchemical model to predict the propensity for domain-swapped misfolding of multidomain proteins.

Building a minimum frustration framework for brain functions over long time scales

The minimum frustration principle (MFP) is a computational approach stating that, over the long time scales of evolution, proteins' free energy decreases more than expected by thermodynamical constraints as their amino acids assume conformations progressively closer to the lowest energetic state. This Review shows that this general principle, borrowed from protein folding dynamics, can also be fruitfully applied to nervous function. Highlighting the foremost role of energetic requirements, macromolecular dynamics, and above all intertwined time scales in brain activity, the MFP elucidates a wide range of mental processes from sensations to memory retrieval. Brain functions are compared with trajectories that, over long nervous time scales, are attracted toward the low-energy bottom of funnel-like structures characterized by both robustness and plasticity. We discuss how the principle, derived explicitly from evolution and selection of a funneling structure from microdynamics of contacts, is unlike other brain models equipped with energy landscapes, such as the Bayesian and free energy principles and the Hopfield networks. In summary, we make available a novel approach to brain function cast in a biologically informed fashion, with the potential to be operationalized and assessed empirically. © 2016 Wiley Periodicals, Inc.

Imprints of function on the folding landscape: functional role for an intermediate in a conserved eukaryotic binding protein

In the computational characterization of single domain protein folding, the effective free energies of numerous microstates are projected onto few collective degrees of freedom that in turn serve as well-defined reaction coordinates. In this regard, one-dimensional (1D) free energy profiles are widely used mainly for their simplicity. Since folding and functional landscapes are interlinked, how well can these reduced representations capture the structural and dynamic features of functional states while being simultaneously consistent with experimental observables? We investigate this issue by characterizing the folding of the four-helix bundle bovine acyl-CoA binding protein (bACBP), which exhibits complex equilibrium and kinetic behaviours, employing an Ising-like statistical mechanical model and molecular simulations. We show that the features of the 1D free energy profile are sufficient to quantitatively reproduce multiple experimental observations including millisecond chevron-like kinetics and temperature dependence, a microsecond fast phase, barrier heights, unfolded state movements, the intermediate structure and average ϕ-values. Importantly, we find that the structural features of the native-like intermediate (partial disorder in helix 1) are intricately linked to a unique interplay between packing and electrostatics in this domain. By comparison with available experimental data, we propose that this intermediate determines the promiscuous functional behaviour of bACBP that exhibits broad substrate specificity. Our results present evidence to the possibility of employing the statistical mechanical model and the resulting 1D free energy profile to not just understand folding mechanisms but to even extract features of functionally relevant states and their energetic origins.

Probing the Folding-Unfolding Transition of a Thermophilic Protein, MTH1880

The folding mechanism of typical proteins has been studied widely, while our understanding of the origin of the high stability of thermophilic proteins is still elusive. Of particular interest is how an atypical thermophilic protein with a novel fold maintains its structure and stability under extreme conditions. Folding-unfolding transitions of MTH1880, a thermophilic protein from Methanobacterium thermoautotrophicum, induced by heat, urea, and GdnHCl, were investigated using spectroscopic techniques including circular dichorism, fluorescence, NMR combined with molecular dynamics (MD) simulations. Our results suggest that MTH1880 undergoes a two-state N to D transition and it is extremely stable against temperature and denaturants. The reversibility of refolding was confirmed by spectroscopic methods and size exclusion chromatography. We found that the hyper-stability of the thermophilic MTH1880 protein originates from an extensive network of both electrostatic and hydrophobic interactions coordinated by the central β-sheet. Spectroscopic measurements, in combination with computational simulations, have helped to clarify the thermodynamic and structural basis for hyper-stability of the novel thermophilic protein MTH1880.

Reduction of All-Atom Protein Folding Dynamics to One-Dimensional Diffusion

Theoretical models have often modeled protein folding dynamics as diffusion on a low-dimensional free energy surface, a remarkable simplification. However, the accuracy of such an approximation and the number of dimensions required were not clear. For all-atom folding simulations of ten small proteins in explicit solvent we show that the folding dynamics can indeed be accurately described as diffusion on just a single coordinate, the fraction of native contacts (Q). The diffusion models reproduce both folding rates, and finer details such as transition-path durations and diffusive propagators. The Q-averaged diffusion coefficients decrease with chain length, as anticipated from energy landscape theory. Although the Q-diffusion model does not capture transition-path durations for the protein NuG2, we show that this can be accomplished by designing an improved coordinate Qopt. Overall, one-dimensional diffusion on a suitable coordinate turns out to be a remarkably faithful model for the dynamics of the proteins considered.

Sequence, structure, and cooperativity in folding of elementary protein structural motifs

Residue-level unfolding of two helix-turn-helix proteins--one naturally occurring and one de novo designed--is reconstructed from multiple sets of site-specific (13)C isotopically edited infrared (IR) and circular dichroism (CD) data using Ising-like statistical-mechanical models. Several model variants are parameterized to test the importance of sequence-specific interactions (approximated by Miyazawa-Jernigan statistical potentials), local structural flexibility (derived from the ensemble of NMR structures), interhelical hydrogen bonds, and native contacts separated by intervening disordered regions (through the Wako-Saitô-Muñoz-Eaton scheme, which disallows such configurations). The models are optimized by directly simulating experimental observables: CD ellipticity at 222 nm for model proteins and their fragments and (13)C-amide I' bands for multiple isotopologues of each protein. We find that data can be quantitatively reproduced by the model that allows two interacting segments flanking a disordered loop (double sequence approximation) and incorporates flexibility in the native contact maps, but neither sequence-specific interactions nor hydrogen bonds are required. The near-identical free energy profiles as a function of the global order parameter are consistent with expected similar folding kinetics for nearly identical structures. However, the predicted folding mechanism for the two motifs is different, reflecting the order of local stability. We introduce free energy profiles for "experimental" reaction coordinates--namely, the degree of local folding as sensed by site-specific (13)C-edited IR, which highlight folding heterogeneity and contrast its overall, average description with the detailed, local picture.

Folding RaCe: a robust method for predicting changes in protein folding rates upon point mutations

MOTIVATION

Protein engineering methods are commonly employed to decipher the folding mechanism of proteins and enzymes. However, such experiments are exceedingly time and resource intensive. It would therefore be advantageous to develop a simple computational tool to predict changes in folding rates upon mutations. Such a method should be able to rapidly provide the sequence position and chemical nature to modulate through mutation, to effect a particular change in rate. This can be of importance in protein folding, function or mechanistic studies.

RESULTS

We have developed a robust knowledge-based methodology to predict the changes in folding rates upon mutations formulated from amino and acid properties using multiple linear regression approach. We benchmarked this method against an experimental database of 790 point mutations from 26 two-state proteins. Mutants were first classified according to secondary structure, accessible surface area and position along the primary sequence. Three prime amino acid features eliciting the best relationship with folding rates change were then shortlisted for each class along with an optimized window length. We obtained a self-consistent mean absolute error of 0.36 s(-1) and a mean Pearson correlation coefficient (PCC) of 0.81. Jack-knife test resulted in a MAE of 0.42 s(-1) and a PCC of 0.73. Moreover, our method highlights the importance of outlier(s) detection and studying their implications in the folding mechanism.

AVAILABILITY AND IMPLEMENTATION

A web server 'Folding RaCe' has been developed and is available at http://www.iitm.ac.in/bioinfo/proteinfolding/foldingrace.html.

CONTACT

gromiha@iitm.ac.in

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

General mechanism of two-state protein folding kinetics

We describe here a general model of the kinetic mechanism of protein folding. In the Foldon Funnel Model, proteins fold in units of secondary structures, which form sequentially along the folding pathway, stabilized by tertiary interactions. The model predicts that the free energy landscape has a volcano shape, rather than a simple funnel, that folding is two-state (single-exponential) when secondary structures are intrinsically unstable, and that each structure along the folding path is a transition state for the previous structure. It shows how sequential pathways are consistent with multiple stochastic routes on funnel landscapes, and it gives good agreement with the 9 order of magnitude dependence of folding rates on protein size for a set of 93 proteins, at the same time it is consistent with the near independence of folding equilibrium constant on size. This model gives estimates of folding rates of proteomes, leading to a median folding time in Escherichia coli of about 5 s.

Comparing a simple theoretical model for protein folding with all-atom molecular dynamics simulations

Advances in computing have enabled microsecond all-atom molecular dynamics trajectories of protein folding that can be used to compare with and test critical assumptions of theoretical models. We show that recent simulations by the Shaw group (10, 11, 14, 15) are consistent with a key assumption of an Ising-like theoretical model that native structure grows in only a few regions of the amino acid sequence as folding progresses. The distribution of mechanisms predicted by simulating the master equation of this native-centric model for the benchmark villin subdomain, with only two adjustable thermodynamic parameters and one temperature-dependent kinetic parameter, is remarkably similar to the distribution in the molecular dynamics trajectories.

Native contacts determine protein folding mechanisms in atomistic simulations

The recent availability of long equilibrium simulations of protein folding in atomistic detail for more than 10 proteins allows us to identify the key interactions driving folding. We find that the collective fraction of native amino acid contacts, Q, captures remarkably well the transition states for all the proteins with a folding free energy barrier. Going beyond this global picture, we devise two different measures to quantify the importance of individual interresidue contacts in the folding mechanism: (i) the log-ratio of lifetimes of contacts during folding transition paths and in the unfolded state and (ii) a Bayesian measure of how predictive the formation of each contact is for being on a transition path. Both of these measures indicate that native, or near-native, contacts are important for determining mechanism, as might be expected. More remarkably, however, we found that for almost all the proteins, with the designed protein α3D being a notable exception, nonnative contacts play no significant part in determining folding mechanisms.

Predictions from an Ising-like Statistical Mechanical Model on the Dynamic and Thermodynamic Effects of Protein Surface Electrostatics

Charged residues on the surface of a protein are known hot-spots for post-translational modification, protein/ligand-binding, and tuning conformational stabilities. Recent experimental evidence points to the fact that surface electrostatics can also modulate thermodynamic barriers and hence folding mechanisms. To probe for this behavior across different proteins, we develop a novel version of the Wako-Saitô-Muñoz-Eaton (WSME) model in which we include an electrostatic potential term in the energy function while simplifying the treatment of solvation free energy. Both of the energy terms are obtained by quantitatively fitting the model to differential scanning calorimetry (DSC) experiments that carry critical information on the protein partition function. We characterize four sets of structural/functional homologues (HEWL/BLA, CspB, engrailed, α-spectrin) either by fitting the experimental data of a single domain in the homologous set and predicting the conformational behavior of the rest with the same set of parameters or by performing semiblind predictions. The model with the added electrostatic term is able to successfully reproduce the order of thermodynamic stabilities and relaxation rates of most of the homologues. In parallel, we predict diverse conformational features including a wide range of thermodynamic barriers (∼9-40 kJ/mol), broad native ensembles in helical proteins, structured unfolded states and intermediates, rugged folding landscapes, and further provide an independent protein-specific estimate of the folding speed limit at 298 K (1/(7-300 μs)). Our results are evidence that protein surface electrostatics can be tailored to not only engineer stabilities but also folding mechanisms and the ruggedness of the underlying landscape.

Statistical mechanics of protein allostery: roles of backbone and side-chain structural fluctuations

A statistical mechanical model of allosteric transition of proteins is developed by extending the structure-based model of protein folding to cases that a protein has two different native conformations. Partition function is calculated exactly within the model and free-energy surfaces associated with allostery are derived. In this paper, the model of allosteric transition proposed in a previous paper [Proc. Natl. Acad. Sci. U.S.A 134, 7775 (2010)] is reformulated to describe both fluctuation in side-chain configurations and that in backbone structures in a balanced way. The model is applied to example proteins, Ras, calmodulin, and CheY: Ras undergoes the allosteric transition between guanosine diphosphate (GDP)-bound and guanosine triphosphate (GTP)-bound forms, and the model results show that the GDP-bound form is stabilized enough to prevent unnecessary signal transmission, but the conformation in the GTP-bound state bears large fluctuation in side-chain configurations, which may help to bind multiple target proteins for multiple pathways of signaling. The calculated results of calmodulin show the scenario of sequential ordering in Ca(2+) binding and the associated allosteric conformational change, which are realized though the sequential appearing of pre-existing structural fluctuations, i.e., fluctuations to show structures suitable to bind Ca(2+) before its binding. Here, the pre-existing fluctuations to accept the second and third Ca(2+) ions are dominated by the side-chain fluctuation. In CheY, the calculated side-chain fluctuation of Tyr106 is coordinated with the backbone structural change in the β4-α4 loop, which explains the pre-existing Y-T coupling process in this protein. Ability of the model to explain allosteric transitions of example proteins supports the view that the large entropic effects lower the free-energy barrier of allosteric transition.

Estimation of protein folding free energy barriers from calorimetric data by multi-model Bayesian analysis

The realization that folding free energy barriers can be small enough to result in significant population of the species at the barrier top has sprouted in several methods to estimate folding barriers from equilibrium experiments. Some of these approaches are based on fitting the experimental thermogram measured by differential scanning calorimetry (DSC) to a one-dimensional representation of the folding free-energy surface (FES). Different physical models have been used to represent the FES: (1) a Landau quartic polynomial as a function of the total enthalpy, which acts as an order parameter; (2) the projection onto a structural order parameter (i.e. number of native residues or native contacts) of the free energy of all the conformations generated by Ising-like statistical mechanical models; and (3) mean-field models that define conformational entropy and stabilization energy as functions of a continuous local order parameter. The fundamental question that emerges is how can we obtain robust, model-independent estimates of the thermodynamic folding barrier from the analysis of DSC experiments. Here we address this issue by comparing the performance of various FES models in interpreting the thermogram of a protein with a marginal folding barrier. We chose the small α-helical protein PDD, which folds-unfolds in microseconds crossing a free energy barrier previously estimated as ~1 RT. The fits of the PDD thermogram to the various models and assumptions produce FES with a consistently small free energy barrier separating the folded and unfolded ensembles. However, the fits vary in quality as well as in the estimated barrier. Applying Bayesian probabilistic analysis we rank the fit performance using a statistically rigorous criterion that leads to a global estimate of the folding barrier and its precision, which for PDD is 1.3 ± 0.4 kJ mol(-1). This result confirms that PDD folds over a minor barrier consistent with the downhill folding regime. We have further validated the multi-model Bayesian approach through the analysis of two additional protein systems: gpW, a midsize single-domain with α + β topology that also folds in microseconds and has been previously catalogued as a downhill folder, and α-spectrin SH3, a domain of similar size but with a β-barrel fold, slow-folding kinetics and two-state-like thermodynamics. From a general viewpoint, the Bayesian analysis developed here results in a statistically robust, virtually model-independent, method to estimate the thermodynamic free-energy barriers to protein folding from DSC thermograms. Our method appears to be sufficiently accurate to consistently detect small differences in the barrier height, and thus opens up the possibility of characterizing experimentally the changes in thermodynamic folding barriers induced by single-point mutations on proteins within the downhill regime.

Making connections between ultrafast protein folding kinetics and molecular dynamics simulations

Determining the rate of forming the truly folded conformation of ultrafast folding proteins is an important issue for both experiments and simulations. The double-norleucine mutant of the 35-residue villin subdomain is the focus of recent computer simulations with atomistic molecular dynamics because it is currently the fastest folding protein. The folding kinetics of this protein have been measured in laser temperature-jump experiments using tryptophan fluorescence as a probe of overall folding. The conclusion from the simulations, however, is that the rate determined by fluorescence is significantly larger than the rate of overall folding. We have therefore employed an independent experimental method to determine the folding rate. The decay of the tryptophan triplet-state in photoselection experiments was used to monitor the change in the unfolded population for a sequence of the villin subdomain with one amino acid difference from that of the laser temperature-jump experiments, but with almost identical equilibrium properties. Folding times obtained in a two-state analysis of the results from the two methods at denaturant concentrations varying from 1.5-6.0 M guanidinium chloride are in excellent agreement, with an average difference of only 20%. Polynomial extrapolation of all the data to zero denaturant yields a folding time of 220 (+100,-70) ns at 283 K, suggesting that under these conditions the barrier between folded and unfolded states has effectively disappeared--the so-called "downhill scenario."

Quantitative prediction of protein folding behaviors from a simple statistical model

The statistical nature of the protein folding process requires the use of equally detailed yet simple models that lend themselves to characterize experiments. One such model is the Wako-Saitô-Muñoz-Eaton model, that we extend here to include solvation effects (WSME-S), introduced via empirical terms. We employ the novel version to analyze the folding of two proteins, gpW and SH3, that have similar size and thermodynamic stability but with the former folding 3 orders of magnitude faster than SH3. A quantitative analysis reveals that gpW presents at most marginal barriers, in contrast to SH3 that folds following a simple two-state approximation. We reproduce the observed experimental differences in melting temperature in gpW as seen by different experimental spectroscopic probes and the shape of the rate-temperature plot. In parallel, we predict the folding complexity expected in gpW from the analysis of both the residue-level thermodynamics and kinetics. SH3 serves as a stringent control with neither folding complexity nor dispersion in melting temperatures being observed. The extended model presented here serves as an ideal tool not only to characterize folding data but also to make experimentally testable predictions.

Evidence for a partially structured state of the amylin monomer

Islet amyloid polypeptide (amylin) is the main component in amyloid deposits formed in type II diabetes. We used triplet quenching to probe the dynamics of contact formation between the N-terminal disulfide loop and a C-terminal tryptophan in monomeric amylins from human and rat. Quenching rates measured in the absence of denaturant are four times larger than those in 6 M guanidinium chloride, indicating a decrease in the average end-to-end distance (collapse) at low denaturant concentrations. We were surprised to find an even greater (sevenfold) increase in quenching rates on removal of denaturant for a hydrophilic control peptide containing the disulfide loop compared to the same peptide without the loop (twofold change). These results suggest that collapse is driven by backbone-backbone and backbone-side chain interactions involving the disulfide loop portion of the chain rather than by the formation of side-chain hydrophobic contacts. Molecular dynamics simulations of the control peptide show that the collapse results from hydrogen-bonding interactions between the central residues of the chain and the disulfide loop. The quenching experiments also indicate that the monomer of the human, amyloidogenic form of amylin is more compact than the rat form, which does not form amyloid. We discuss these newly observed differences between human and rat amylin in solution and their possible relation to aggregation and to the physiological function of amylin binding to the calcitonin receptor.

Experimental evidence for a frustrated energy landscape in a three-helix-bundle protein family

Energy landscape theory is a powerful tool for understanding the structure and dynamics of complex molecular systems, in particular biological macromolecules. The primary sequence of a protein defines its free-energy landscape and thus determines the folding pathway and the rate constants of folding and unfolding, as well as the protein's native structure. Theory has shown that roughness in the energy landscape will lead to slower folding, but derivation of detailed experimental descriptions of this landscape is challenging. Simple folding models show that folding is significantly influenced by chain entropy; proteins in which the contacts are local fold quickly, owing to the low entropy cost of forming stabilizing, native contacts during folding. For some protein families, stability is also a determinant of folding rate constants. Where these simple metrics fail to predict folding behaviour, it is probable that there are features in the energy landscape that are unusual. Such general observations cannot explain the folding behaviour of the R15, R16 and R17 domains of alpha-spectrin. R15 folds approximately 3,000 times faster than its homologues, although they have similar structures, stabilities and, as far as can be determined, transition-state stabilities. Here we show that landscape roughness (internal friction) is responsible for the slower folding and unfolding of R16 and R17. We use chimaeric domains to demonstrate that this internal friction is a property of the core, and suggest that frustration in the landscape of the slow-folding spectrin domains may be due to misdocking of the long helices during folding. Theoretical studies have suggested that rugged landscapes will result in slower folding; here we show experimentally that such a phenomenon directly influences the folding kinetics of a 'normal' protein, that is, one with a significant energy barrier that folds on a relatively slow, millisecond-second, timescale.

Nearly symmetrical proteins: folding pathways and transition states

The folding pathways of the B domain of protein A have been the subject of many experimental and computational studies. Based on a statistical mechanical model, it has been suggested that the native state symmetry leads to multiple pathways, highly dependent on temperature and denaturant concentration. Experiments, however, have not confirmed this scenario. By considering four nearly symmetrical proteins, one of them being the above molecule, here we show that, if contact energies are properly taken into account, a different picture emerges from kinetic simulations of the above-mentioned model. This is characterized by a dominant folding pathway, which is consistent with the most recent experimental results. Given the simplicity of the model, we also report on a direct sampling of the transition state.

Reactive flux and folding pathways in network models of coarse-grained protein dynamics

The reactive flux between folded and unfolded states of a two-state protein, whose coarse-grained dynamics is described by a master equation, is expressed in terms of the commitment or splitting probabilities of the microstates in the bottleneck region. This allows one to determine how much each transition through a dividing surface contributes to the reactive flux. By repeating the analysis for a series of dividing surfaces or, alternatively, by partitioning the reactive flux into contributions of unidirectional pathways that connect reactants and products, insight can be gained into the mechanism of protein folding. Our results for the flux in a network with complex connectivity, obtained using the discrete counterpart of Kramers' theory of activated rate processes, show that the number of reactive transitions is typically much smaller than the total number of transitions that cross a dividing surface at equilibrium.

Chemical, physical, and theoretical kinetics of an ultrafast folding protein

An extensive set of equilibrium and kinetic data is presented and analyzed for an ultrafast folding protein--the villin subdomain. The equilibrium data consist of the excess heat capacity, tryptophan fluorescence quantum yield, and natural circular-dichroism spectrum as a function of temperature, and the kinetic data consist of time courses of the quantum yield from nanosecond-laser temperature-jump experiments. The data are well fit with three kinds of models--a three-state chemical-kinetics model, a physical-kinetics model, and an Ising-like theoretical model that considers 10(5) possible conformations (microstates). In both the physical-kinetics and theoretical models, folding is described as diffusion on a one-dimensional free-energy surface. In the physical-kinetics model the reaction coordinate is unspecified, whereas in the theoretical model, order parameters, either the fraction of native contacts or the number of native residues, are used as reaction coordinates. The validity of these two reaction coordinates is demonstrated from calculation of the splitting probability from the rate matrix of the master equation for all 10(5) microstates. The analysis of the data on site-directed mutants using the chemical-kinetics model provides information on the structure of the transition-state ensemble; the physical-kinetics model allows an estimate of the height of the free-energy barrier separating the folded and unfolded states; and the theoretical model provides a detailed picture of the free-energy surface and a residue-by-residue description of the evolution of the folded structure, yet contains many fewer adjustable parameters than either the chemical- or physical-kinetics models.

Measuring internal friction of an ultrafast-folding protein

Nanosecond laser T-jump was used to measure the viscosity dependence of the folding kinetics of the villin subdomain under conditions where the viscogen has no effect on its equilibrium properties. The dependence of the unfolding/refolding relaxation time on solvent viscosity indicates a major contribution to the dynamics from internal friction. The internal friction increases with increasing temperature, suggesting a shift in the transition state along the reaction coordinate toward the native state with more compact structures, and therefore, a smaller diffusion coefficient due to increased landscape roughness. Fitting the data with an Ising-like model yields a relatively small position dependence for the diffusion coefficient. This finding is consistent with the excellent correlation found between experimental and calculated folding rates based on free energy barrier heights using the same diffusion coefficient for every protein.

Cooperativity, connectivity, and folding pathways of multidomain proteins

Multidomain proteins are ubiquitous in both prokaryotic and eukaryotic proteomes. Study on protein folding, however, has concentrated more on the isolated single domains of proteins, and there have been relatively few systematic studies on the effects of domain-domain interactions on folding. We here discuss this issue by examining human gammaD-crystallin, spore coat protein S, and a tandem array of the R16 and R17 domains of spectrin as example proteins by using a structure-based model of folding. The calculated results consistently explain the experimental data on folding pathways and effects of mutational perturbations, supporting the view that the connectivity of two domains and the distribution of domain-domain interactions in the native conformation are factors to determine kinetic and equilibrium properties of cooperative folding.

Hydrogen bond networks determine emergent mechanical and thermodynamic properties across a protein family

Background

Gram-negative bacteria use periplasmic-binding proteins (bPBP) to transport nutrients through the periplasm. Despite immense diversity within the recognized substrates, all members of the family share a common fold that includes two domains that are separated by a conserved hinge. The hinge allows the protein to cycle between open (apo) and closed (ligated) conformations. Conformational changes within the proteins depend on a complex interplay of mechanical and thermodynamic response, which is manifested as an increase in thermal stability and decrease of flexibility upon ligand binding.

Results

We use a distance constraint model (DCM) to quantify the give and take between thermodynamic stability and mechanical flexibility across the bPBP family. Quantitative stability/flexibility relationships (QSFR) are readily evaluated because the DCM links mechanical and thermodynamic properties. We have previously demonstrated that QSFR is moderately conserved across a mesophilic/thermophilic RNase H pair, whereas the observed variance indicated that different enthalpy-entropy mechanisms allow similar mechanical response at their respective melting temperatures. Our predictions of heat capacity and free energy show marked diversity across the bPBP family. While backbone flexibility metrics are mostly conserved, cooperativity correlation (long-range couplings) also demonstrate considerable amount of variation. Upon ligand removal, heat capacity, melting point, and mechanical rigidity are, as expected, lowered. Nevertheless, significant differences are found in molecular cooperativity correlations that can be explained by the detailed nature of the hydrogen bond network.

Conclusion

Non-trivial mechanical and thermodynamic variation across the family is explained by differences within the underlying H-bond networks. The mechanism is simple; variation within the H-bond networks result in altered mechanical linkage properties that directly affect intrinsic flexibility. Moreover, varying numbers of H-bonds and their strengths control the likelihood for energetic fluctuations as H-bonds break and reform, thus directly affecting thermodynamic properties. Consequently, these results demonstrate how unexpected large differences, especially within cooperativity correlation, emerge from subtle differences within the underlying H-bond network. This inference is consistent with well-known results that show allosteric response within a family generally varies significantly. Identifying the hydrogen bond network as a critical determining factor for these large variances may lead to new methods that can predict such effects.

Temperature-dependent downhill unfolding of ubiquitin. II. Modeling the free energy surface

To provide evidence for the interpretation of temperature-dependent unfolding kinetics and the downhill unfolding scenario presented in the accompanying experimental article (Part I), the free energy surface of ubiquitin unfolding is calculated using statistical mechanical models of the Muñoz-Eaton (ME) form. The models allow only two states for each amino acid residue, folded or unfolded, and permutations of these states generate an ensemble of microstates. One-dimensional free energy curves are calculated using the number of folded residues as a reaction coordinate. The proposed sequential unfolding of ubiquitin's beta-sheet is tested by mapping the free energy onto two reaction coordinates inspired by the experiment as follows: the number of folded residues in ubiquitin's stable beta-strands I and II and those of the less stable strands III-V. Although the original ME model successfully captures folding features of zipper-like one-dimensional folders, it misses important tertiary interactions between residues that are far from each other in primary sequence. To take tertiary contacts into account, partially folded microstates based on a spherical growth model are included in the calculation and compared with the original model. By calculating the folding probability of each residue for a given point on the free energy surface, the unfolding pathway of ubiquitin is visualized. At low temperature, thermal unfolding occurs along a sequential unfolding pathway as follows: disruption of the beta-strands III-V followed by unfolding of the strands I and II. At high temperature, multiple unfolding routes are formed. The heterogeneity of the transition state explains the global nonexponential unfolding observed in the T-jump experiment at high temperature. The calculation also reports a high stability for the alpha-helix of ubiquitin.

Estimating free-energy barrier heights for an ultrafast folding protein from calorimetric and kinetic data

Differential scanning calorimetry was used to measure the temperature dependence of the absolute heat capacity of the 35-residue subdomain of the villin headpiece, a protein that folds in 5 mus and is therefore assumed to have a small free-energy barrier separating folded and unfolded states. To obtain an estimate of the barrier height from the calorimetric data, two models, a variable-barrier model and an Ising-like model, were used to fit the heat capacity in excess of the folded state over the temperature range 15-125 degrees C. The variable-barrier model is based on an empirical mathematical form for the density of states, with four adjustable parameters and the enthalpy (H) as a reaction coordinate. The Ising-like model is based on the inter-residue contact map of the X-ray structure with exact enumeration of approximately 10(5) possible conformations, with two adjustable parameters in the partition function, and either the fraction of native contacts (Q) or the number of ordered residues (P) as reaction coordinates. The variable-barrier model provides an excellent fit to the data and yields a barrier height at the folding temperature ranging from 0.4 to 1.1 kcal mol(-1), while the Ising-like model provides a less good fit and yields barrier heights of 2.3 +/- 0.1 kcal mol(-1) and 2.1 +/- 0.1 kcal mol(-1) for the Q and P reaction coordinates, respectively. In both models, the barrier to folding increases with increasing temperature. Assuming a sufficiently large activation energy for diffusion on the free-energy surfaces, both models are consistent with the observation of a temperature-independent folding rate in previously published laser temperature-jump experiments. Analysis of this kinetic data, using an approximate form for the pre-exponential factor of Kramers theory and the 70 ns relaxation time for the fast phase that precedes the unfolding/refolding relaxation to determine the diffusion coefficient, results in a barrier height of 1.6 +/- 0.3 kcal mol-1 for an unspecified reaction coordinate. Although no independent test of the validity of the H, Q, or P reaction coordinates is given, the barrier-height estimates obtained with the three reaction coordinates are in quite good agreement with the value derived from a Kramers analysis of the kinetics that makes no assumptions about the reaction coordinate. However, the higher estimates obtained using Q or P appear more consistent with the finding of barrier-crossing kinetics of a villin mutant that folds in 700 ns, corresponding to a 1.3 kcal mol-1 reduction in the folding barrier relative to wild-type. All of the results suggest that the free-energy barrier to folding is sufficiently low that it should be possible to engineer this protein or find solution conditions that would eliminate the barrier to create the "downhill" folding scenario of Wolynes and Onuchic.