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

Ensemble-Based Thermodynamics of the Fuzzy Binding between Intrinsically Disordered Proteins and Small-Molecule Ligands

In contrast to the "lock-and-key" model underlying the long-term success of structural biology and rational drug design, intrinsically disordered proteins (IDPs) exist in an ensemble of highly heterogeneous conformations even after binding with small-molecule ligands. It remains controversial how to characterize the thermodynamics of such fuzzy interactions. Here, we derive an ensemble-based thermodynamic framework to analyze the apparent affinity between IDPs and ligands. It is shown that the apparent affinity is related to the interaction free energy between the individual conformation and ligand in a way similar to Jarzynski's equality in nonequilibrium statistics. The oncoprotein c-Myc is adopted as an example to demonstrate the related properties, for example, the distribution of conformation-ligand interaction free energy, the entropic contribution from the ensemble, the conformation shift under ligand binding, and how to control the error under a limited number of sampled conformations.

Native contact density and nonnative hydrophobic effects in the folding of bacterial immunity proteins

The bacterial colicin-immunity proteins Im7 and Im9 fold by different mechanisms. Experimentally, at pH 7.0 and 10°C, Im7 folds in a three-state manner via an intermediate but Im9 folding is two-state-like. Accordingly, Im7 exhibits a chevron rollover, whereas the chevron arm for Im9 folding is linear. Here we address the biophysical basis of their different behaviors by using native-centric models with and without additional transferrable, sequence-dependent energies. The Im7 chevron rollover is not captured by either a pure native-centric model or a model augmented by nonnative hydrophobic interactions with a uniform strength irrespective of residue type. By contrast, a more realistic nonnative interaction scheme that accounts for the difference in hydrophobicity among residues leads simultaneously to a chevron rollover for Im7 and an essentially linear folding chevron arm for Im9. Hydrophobic residues identified by published experiments to be involved in nonnative interactions during Im7 folding are found to participate in the strongest nonnative contacts in this model. Thus our observations support the experimental perspective that the Im7 folding intermediate is largely underpinned by nonnative interactions involving large hydrophobics. Our simulation suggests further that nonnative effects in Im7 are facilitated by a lower local native contact density relative to that of Im9. In a one-dimensional diffusion picture of Im7 folding with a coordinate- and stability-dependent diffusion coefficient, a significant chevron rollover is consistent with a diffusion coefficient that depends strongly on native stability at the conformational position of the folding intermediate.

In order to fold correctly, a globular protein must avoid being trapped in wrong, i.e., nonnative conformations. Thus a biophysical account of how attractive nonnative interactions are bypassed by some amino acid sequences but not others is key to deciphering protein structure and function. We examine two closely related bacterial immunity proteins, Im7 and Im9, that are experimentally known to fold very differently: Whereas Im9 folds directly, Im7 folds through a mispacked conformational intermediate. A simple model we developed accounts for their intriguingly different folding kinetics in terms of a balance between the density of native-promoting contacts and the hydrophobicity of local amino acid sequences. This emergent principle is extensible to other biomolecular recognition processes.

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

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

Discrete kinetic models from funneled energy landscape simulations

A general method for facilitating the interpretation of computer simulations of protein folding with minimally frustrated energy landscapes is detailed and applied to a designed ankyrin repeat protein (4ANK). In the method, groups of residues are assigned to foldons and these foldons are used to map the conformational space of the protein onto a set of discrete macrobasins. The free energies of the individual macrobasins are then calculated, informing practical kinetic analysis. Two simple assumptions about the universality of the rate for downhill transitions between macrobasins and the natural local connectivity between macrobasins lead to a scheme for predicting overall folding and unfolding rates, generating chevron plots under varying thermodynamic conditions, and inferring dominant kinetic folding pathways. To illustrate the approach, free energies of macrobasins were calculated from biased simulations of a non-additive structure-based model using two structurally motivated foldon definitions at the full and half ankyrin repeat resolutions. The calculated chevrons have features consistent with those measured in stopped flow chemical denaturation experiments. The dominant inferred folding pathway has an “inside-out”, nucleation-propagation like character.

Enthalpic barriers dominate the folding and unfolding of the human Cu, Zn superoxide dismutase monomer

The rate-limiting step in the formation of the native dimeric state of human Cu, Zn superoxide dismutase (SOD1) is a very slow monomer folding reaction that governs the lifetime of its unfolded state. Mutations at dozens of sites in SOD1 are known to cause a fatal motor neuron disease, amyotrophic lateral sclerosis, and recent experiments implicate the unfolded state as a source of soluble oligomers and histologically observable aggregates thought to be responsible for toxicity. To determine the thermodynamic properties of the transition state ensemble (TSE) limiting the folding of this high-contact-order β-sandwich motif, we performed a combined thermal/urea denaturation thermodynamic/kinetic analysis. The barriers to folding and unfolding are dominated by the activation enthalpy at 298 K and neutral pH; the activation entropy is favorable and reduces the barrier height for both reactions. The absence of secondary structure formation or large-scale chain collapse prior to crossing the barrier for folding led to the conclusion that dehydration of nonpolar surfaces in the TSE is responsible for the large and positive activation enthalpy. Although the activation entropy favors the folding reaction, the transition from the unfolded state to the native state is entropically disfavored at 298 K. The opposing entropic contributions to the free energies of the TSE and the native state during folding provide insights into structural properties of the TSE. The results also imply a crucial role for water in governing the productive folding reaction and enhancing the propensity for the aggregation of SOD1.

Effect of solvation-related interaction on the low-temperature dynamics of proteins

The effect of solvation-related interaction on the low-temperature dynamics of proteins is studied by taking into account the desolvation barriers in the interactions of native contacts. It is found out that about the folding transition temperature, the protein folds in a cooperative manner, and the water molecules are expelled from the hydrophobic core at the final stage in the folding process. At low temperature, however, the protein would generally be trapped in many metastable conformations with some water molecules frozen inside the protein. The desolvation takes an important role in these processes. The number of frozen water molecules and that of frozen states of proteins are further analyzed with the methods based on principal component analysis (PCA) and the clustering of conformations. It is found out that both the numbers of frozen water molecules and the frozen states of the protein increase quickly below a certain temperature. Especially, the number of frozen states of the protein increases exponentially following the decrease in the temperature, which resembles the basic features of glassy dynamics. Interestingly, it is observed that the freezing of water molecules and that of protein conformations happen at almost the same temperature. This suggests that the solvation-related interaction performs an important role for the low-temperature dynamics of the model protein.

Replica exchange molecular dynamics simulations of coarse-grained proteins in implicit solvent

Current approaches aimed at determining the free energy surface of all-atom medium-size proteins in explicit solvent are slow and are not sufficient to converge to equilibrium properties. To ensure a proper sampling of the configurational space, it is preferable to use reduced representations such as implicit solvent and/or coarse-grained protein models, which are much lighter computationally. Each model must be verified, however, to ensure that it can recover experimental structures and thermodynamics. Here we test the coarse-grained implicit solvent OPEP model with replica exchange molecular dynamics (REMD) on six peptides ranging in length from 10 to 28 residues: two alanine-based peptides, the second beta-hairpin from protein G, the Trp-cage and zinc-finger motif, and a dimer of a coiled coil peptide. We show that REMD-OPEP recovers the proper thermodynamics of the systems studied, with accurate structural description of the beta-hairpin and Trp-cage peptides (within 1-2 A from experiments). The light computational burden of REMD-OPEP, which enables us to generate many hundred nanoseconds at each temperature and fully assess convergence to equilibrium ensemble, opens the door to the determination of the free energy surface of larger proteins and assemblies.

Effect of the ordered water on protein folding: an off-lattice Gō-like model study

Recent experiments and numerical simulations have shown that the water molecules confined on the surfaces of some substrates, including the surfaces of cellular components in tissues and cells, form icelike ordered structures. If a protein folds in an environment with those icelike ordered water molecules, its behavior may be different from that in bulk water. Here, the effect of this ordered water environment on protein folding is studied by using an off-lattice Gō-like model. It is found that the ordered water environment significantly improves the native state stability and greatly speeds up the folding rate of the proteins.

Insights from coarse-grained Gō models for protein folding and dynamics

Exploring the landscape of large scale conformational changes such as protein folding at atomistic detail poses a considerable computational challenge. Coarse-grained representations of the peptide chain have therefore been developed and over the last decade have proved extremely valuable. These include topology-based Gō models, which constitute a smooth and funnel-like approximation to the folding landscape. We review the many variations of the Gō model that have been employed to yield insight into folding mechanisms. Their success has been interpreted as a consequence of the dominant role of the native topology in folding. The role of local contact density in determining protein dynamics is also discussed and is used to explain the ability of Gō-like models to capture sequence effects in folding and elucidate conformational transitions.

Probing possible downhill folding: native contact topology likely places a significant constraint on the folding cooperativity of proteins with approximately 40 residues

Experiments point to appreciable variations in folding cooperativity among natural proteins with approximately 40 residues, indicating that the behaviors of these proteins are valuable for delineating the contributing factors to cooperative folding. To explore the role of native topology in a protein's propensity to fold cooperatively and how native topology might constrain the degree of cooperativity achievable by a given set of physical interactions, we compared folding/unfolding kinetics simulated using three classes of native-centric C(alpha) chain models with different interaction schemes. The approach was applied to two homologous 45-residue fragments from the peripheral subunit-binding domain family and a 39-residue fragment of the N-terminal domain of ribosomal protein L9. Free-energy profiles as functions of native contact number were computed to assess the heights of thermodynamic barriers to folding. In addition, chevron plots of folding/unfolding rates were constructed as functions of native stability to facilitate comparison with available experimental data. Although common Gō-like models with pairwise Lennard-Jones-type interactions generally fold less cooperatively than real proteins, the rank ordering of cooperativity predicted by these models is consistent with experiment for the proteins investigated, showing increasing folding cooperativity with increasing nonlocality of a protein's native contacts. Models that account for water-expulsion (desolvation) barriers and models with many-body (nonadditive) interactions generally entail higher degrees of folding cooperativity indicated by more linear model chevron plots, but the rank ordering of cooperativity remains unchanged. A robust, experimentally valid rank ordering of model folding cooperativity independent of the multiple native-centric interaction schemes tested here argues that native topology places significant constraints on how cooperatively a protein can fold.

Excluded volume, local structural cooperativity, and the polymer physics of protein folding rates

A coarse-grained variational model is used to investigate the polymer dynamics of barrier crossing for a diverse set of two-state folding proteins. The model gives reliable folding rate predictions provided excluded volume terms that induce minor structural cooperativity are included in the interaction potential. In general, the cooperative folding routes have sharper interfaces between folded and unfolded regions of the folding nucleus and higher free energy barriers. The calculated free energy barriers are strongly correlated with native topology as characterized by contact order. Increasing the rigidity of the folding nucleus changes the local structure of the transition state ensemble nonuniformly across the set of proteins studied. Nevertheless, the calculated prefactors k(0) are found to be relatively uniform across the protein set, with variation in 1/k(0) less than a factor of 5. This direct calculation justifies the common assumption that the prefactor is roughly the same for all small two-state folding proteins. Using the barrier heights obtained from the model and the best-fit monomer relaxation time 30 ns, we find that 1/k(0) approximately 1-5 mus (with average 1/k(0) approximately 4 micros). This model can be extended to study subtle aspects of folding such as the variation of the folding rate with stability or solvent viscosity and the onset of downhill folding.

Distinguishing between cooperative and unimodal downhill protein folding

Conventional cooperative protein folding invokes discrete ensembles of native and denatured state structures in separate free-energy wells. Unimodal noncooperative ("downhill") folding, however, proposes an ensemble of states occupying a single free-energy well for proteins folding at >/=4 x 10(4) s(-1) at 298 K. It is difficult to falsify unimodal mechanisms for such fast folding proteins by standard equilibrium experiments because both cooperative and unimodal mechanisms can present the same time-averaged structural, spectroscopic, and thermodynamic properties when the time scale used for observation is longer than for equilibration. However, kinetics can provide the necessary evidence. Chevron plots with strongly sloping linear refolding arms are very difficult to explain by downhill folding and are a signature for cooperative folding via a transition state ensemble. The folding kinetics of the peripheral subunit binding domain POB and its mutants fit to strongly sloping chevrons at observed rate constants of >6 x 10(4) s(-1) in denaturant solution, extrapolating to 2 x 10(5) s(-1) in water. Protein A, which folds at 10(5) s(-1) at 298 K, also has a well-defined chevron. Single-molecule fluorescence energy transfer experiments on labeled Protein A in the presence of denaturant demonstrated directly bimodal distributions of native and denatured states.

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

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

A toy model of polymer stretching

We present an extremely simplified model of multiple-domain polymer stretching in an atomic force microscopy experiment. We portray each module as a binary set of contacts and decompose the system energy into a harmonic term (the cantilever) and long-range interaction terms inside each domain. Exact equilibrium computations and Monte Carlo simulations qualitatively reproduce the experimental sawtooth pattern of force-extension profiles, corresponding (in our model) to first-order phase transitions. We study the influence of the coupling induced by the cantilever and the pulling speed on the relative heights of the force peaks. The results suggest that the increasing height of the critical force for subsequent unfolding events is an out-of-equilibrium effect due to a finite pulling speed. The dependence of the average unfolding force on the pulling speed is shown to reproduce the experimental logarithmic law.

Criteria for downhill protein folding: Calorimetry, chevron plot, kinetic relaxation, and single-molecule radius of gyration in chain models with subdued degrees of cooperativity

Recent investigations of possible downhill folding of small proteins such as BBL have focused on the thermodynamics of non-two-state, "barrierless" folding/denaturation transitions. Downhill folding is noncooperative and thermodynamically "one-state," a phenomenon underpinned by a unimodal conformational distribution over chain properties such as enthalpy, hydrophobic exposure, and conformational dimension. In contrast, corresponding distributions for cooperative two-state folding are bimodal with well-separated population peaks. Using simplified atomic modeling of a three-helix bundle-in a scheme that accounts for hydrophobic interactions and hydrogen bonding-and coarse-grained C(alpha) models of four real proteins with various degrees of cooperativity, we evaluate the effectiveness of several observables at defining the underlying distribution. Bimodal distributions generally lead to sharper transitions, with a higher heat capacity peak at the transition midpoint, compared with unimodal distributions. However, the observation of a sigmoidal transition is not a reliable criterion for two-state behavior, and the heat capacity baselines, used to determine the van't Hoff and calorimetric enthalpies of the transition, can introduce ambiguity. Interestingly we find that, if the distribution of the single-molecule radius of gyration were available, it would permit discrimination between unimodal and bimodal underlying distributions. We investigate kinetic implications of thermodynamic noncooperativity using Langevin dynamics. Despite substantial chevron rollovers, the relaxation of the models considered is essentially single-exponential over an extended range of native stabilities. Consistent with experiments, significant deviations from single-exponential behavior occur only under strongly folding conditions.

Testing simplified proteins models of the hPin1 WW domain

The WW domain of the human Pin1 protein for its simple topology and large amount of experimental data is an ideal candidate to assess theoretical approaches to protein folding. The purpose of this work is to compare the reliability of the chemically based Sorenson/Head-Gordon (SHG) model and a standard native centric model in reproducing, through molecular dynamics simulations, some of the well known features of the folding transition of this small domain. Our results show that the Gō model correctly reproduces the cooperative, two-state, folding mechanism of the WW-domain, while the SHG model predicts a transition occurring in two stages: a collapse, followed by a structural rearrangement. The lack of a cooperative folding in the SHG simulations appears to be related to the nonfunnel shape of the energy landscape featuring a partitioning of the native valley in subbasins corresponding to different chain chiralities. However, the SHG approach remains more reliable in estimating the phi-values with respect to Gō-like description. This may suggest that the WW-domain folding process is stirred by energetic and topological factors as well, and it highlights the better suitability of chemically based models in simulating mutations.

WATER MEDIATION IN PROTEIN FOLDING AND MOLECULAR RECOGNITION

Water is essential for life in many ways, and without it biomolecules might no longer truly be biomolecules. In particular, water is important to the structure, stability, dynamics, and function of biological macromolecules. In protein folding, water mediates the collapse of the chain and the search for the native topology through a funneled energy landscape. Water actively participates in molecular recognition by mediating the interactions between binding partners and contributes to either enthalpic or entropic stabilization. Accordingly, water must be included in recognition and structure prediction codes to capture specificity. Thus water should not be treated as an inert environment, but rather as an integral and active component of biomolecular systems, where it has both dynamic and structural roles. Focusing on water sheds light on the physics and function of biological machinery and self-assembly and may advance our understanding of the natural design of proteins and nucleic acids.

Effects of surface tethering on protein folding mechanisms

The folding mechanisms of proteins are increasingly being probed through single-molecule experiments in which the protein is immobilized on a surface. Nevertheless, a clear understanding of how the surface might affect folding, and whether or not it changes folding from its bulk behavior, is lacking. In this work, we use molecular dynamics simulations of a model beta-barrel protein tethered to a surface to systematically investigate how the surface impacts folding. In the bulk, this protein folds in a three-state manner through a compact intermediate state, and its transition state (TS) has a well formed hydrophobic core. Upon tethering, we find that folding rates and stability are impacted differently by the surface, with dependencies on both the length and location of the tether. Significant changes in folding times are observed for tether points that do not alter the folding temperature. Tethering also locally enhances the formation of structure for residues proximal to the tether point. We find that neither the folding mechanism nor the TS of this protein are altered if the tether is in a fully structured or completely unstructured region of the TS. By contrast, tethering in a partially structured region of the TS leads to dramatic changes. For one such tether point, the intermediate present in bulk folding is eliminated, leading to a two-state folding process with a heterogeneous, highly unstructured TS ensemble. These results have implications for both the design of single-molecule experiments and biotechnological applications of tethered proteins.

Energy barriers, cooperativity, and hidden intermediates in the folding of small proteins

Current theoretical views of the folding process of small proteins (< approximately 100 amino acids) postulate that the landscape of potential mean force (PMF) for the formation of the native state has a funnel shape and that the free energy barrier to folding arises from the chain configurational entropy only. However, recent theoretical studies on the formation of hydrophobic clusters with explicit water suggest that a barrier should exist on the PMF of folding, consistent with the fact that protein folding generally involves a large positive activation enthalpy at room temperature. In addition, high-resolution structural studies of the hidden partially unfolded intermediates have revealed the existence of non-native interactions, suggesting that the correction of the non-native interactions during folding should also lead to barriers on PMF. To explore the effect of a PMF barrier on the folding behavior of proteins, we modified Zwanzig's model for protein folding with an uphill landscape of PMF for the formation of transition states. We found that the modified model for short peptide segments can satisfy the thermodynamic and kinetic criteria for an apparently two-state folding. Since the Levinthal paradox can be solved by a stepwise folding of short peptide segments, a landscape of PMF with a locally uphill search for the transition state and cooperative stabilization of folding intermediates/native state is able to explain the available experimental results for small proteins. We speculate that the existence of cooperative hidden folding intermediates in small proteins could be the consequence of the highly specific structures of the native state, which are selected by evolution to perform specific functions and fold in a biologically meaningful time scale.

Solvation and desolvation effects in protein folding: native flexibility, kinetic cooperativity and enthalpic barriers under isostability conditions

As different parts of a protein chain approach one another during folding, they are expected to encounter desolvation barriers before optimal packing is achieved. This impediment originates from the water molecule's finite size, which entails a net energetic cost for water exclusion when the formation of compensating close intraprotein contacts is not yet complete. Based on recent advances, we extend our exploration of these microscopic elementary desolvation barriers' roles in the emergence of generic properties of protein folding. Using continuum Gō-like C(alpha) chain models of chymotrypsin inhibitor 2 (CI2) and barnase as examples, we underscore that elementary desolvation barriers between a protein's constituent groups can significantly reduce native conformational fluctuations relative to model predictions that neglected these barriers. An increasing height of elementary desolvation barriers leads to thermodynamically more cooperative folding/unfolding transitions (i.e., higher overall empirical folding barriers) and higher degrees of kinetic cooperativity as manifested by more linear rate-stability relationships under constant temperature. Applying a spatially non-uniform thermodynamic parametrization we recently introduced for the pairwise C(alpha) potentials of mean force, the present barnase model further illustrates that desolvation is a probable physical underpinning for the experimentally observed high intrinsic enthalpic folding barrier under isostability conditions.

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

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