Simple two-state protein folding kinetics requires near-levinthal thermodynamic cooperativity

Analytical Formulation and Field-Theoretic Simulation of Sequence-Specific Phase Separation of Protein-Like Heteropolymers with Short- and Long-Spatial-Range Interactions

A theory for sequence-dependent liquid-liquid phase separation (LLPS) of intrinsically disordered proteins (IDPs) in the study of biomolecular condensates is formulated by extending the random phase approximation (RPA) and field-theoretic simulation (FTS) of heteropolymers with spatially long-range Coulomb interactions to include the fundamental effects of short-range, hydrophobic-like interactions between amino acid residues. To this end, short-range effects are modeled by Yukawa interactions between multiple nonelectrostatic charges derived from an eigenvalue decomposition of pairwise residue-residue contact energies. Chain excluded volume is afforded by incompressibility constraints. A mean-field approximation leads to an effective Flory-Huggins χ parameter, which, in conjunction with RPA, accounts for the contact-interaction effects of amino acid composition and the sequence-pattern effects of long-range electrostatics in IDP LLPS, whereas FTS based on the formulation provides full sequence dependence for both short- and long-range interactions. This general approach is illustrated here by applications to variants of a natural IDP in the context of several different amino-acid interaction schemes as well as a set of different model hydrophobic-polar sequences sharing the same composition. Effectiveness of the methodology is verified by coarse-grained explicit-chain molecular dynamics simulations.

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

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

The hydrophobic effect characterises the thermodynamic signature of amyloid fibril growth

Many proteins have the potential to aggregate into amyloid fibrils, protein polymers associated with a wide range of human disorders such as Alzheimer’s and Parkinson’s disease. The thermodynamic stability of amyloid fibrils, in contrast to that of folded proteins, is not well understood: the balance between entropic and enthalpic terms, including the chain entropy and the hydrophobic effect, are poorly characterised. Using a combination of theory, in vitro experiments, simulations of a coarse-grained protein model and meta-data analysis, we delineate the enthalpic and entropic contributions that dominate amyloid fibril elongation. Our prediction of a characteristic temperature-dependent enthalpic signature is confirmed by the performed calorimetric experiments and a meta-analysis over published data. From these results we are able to define the necessary conditions to observe cold denaturation of amyloid fibrils. Overall, we show that amyloid fibril elongation is associated with a negative heat capacity, the magnitude of which correlates closely with the hydrophobic surface area that is buried upon fibril formation, highlighting the importance of hydrophobicity for fibril stability.

Most proteins fold in the cell into stable, compact structures. Nevertheless, many proteins also have the ability to stick together, forming long fibrillar structures that are associated with a wide range of human disorders including Alzheimer’s and Parkinson’s disease. The exact nature of the amyloid-causing stickiness is not well understood, nevertheless amyloid fibrils show some very specific thermodynamic characteristics. Some fibrils even destabilise at low temperatures. In this work we translate hydrophobic theory previously used to model protein folding to fibril formation. We combine this theory with experimental measurements, simulations and meta-data analysis of different types of fibrils. This allowed us to unravel the nature of the stickiness in amyloid fibrils by observing the effect of temperature changes, specifically at low temperatures, on hydrophobicity.

Temperature, Hydrostatic Pressure, and Osmolyte Effects on Liquid-Liquid Phase Separation in Protein Condensates: Physical Chemistry and Biological Implications

Liquid-liquid phase separation (LLPS) of proteins and other biomolecules play a critical role in the organization of extracellular materials and membrane-less compartmentalization of intra-organismal spaces through the formation of condensates. Structural properties of such mesoscopic droplet-like states were studied by spectroscopy, microscopy, and other biophysical techniques. The temperature dependence of biomolecular LLPS has been studied extensively, indicating that phase-separated condensed states of proteins can be stabilized or destabilized by increasing temperature. In contrast, the physical and biological significance of hydrostatic pressure on LLPS is less appreciated. Summarized here are recent investigations of protein LLPS under pressures up to the kbar-regime. Strikingly, for the cases studied thus far, LLPSs of both globular proteins and intrinsically disordered proteins/regions are typically more sensitive to pressure than the folding of proteins, suggesting that organisms inhabiting the deep sea and sub-seafloor sediments, under pressures up to 1 kbar and beyond, have to mitigate this pressure-sensitivity to avoid unwanted destabilization of their functional biomolecular condensates. Interestingly, we found that trimethylamine-N-oxide (TMAO), an osmolyte upregulated in deep-sea fish, can significantly stabilize protein droplets under pressure, pointing to another adaptive advantage for increased TMAO concentrations in deep-sea organisms besides the osmolyte's stabilizing effect against protein unfolding. As life on Earth might have originated in the deep sea, pressure-dependent LLPS is pertinent to questions regarding prebiotic proto-cells. Herein, we offer a conceptual framework for rationalizing the recent experimental findings and present an outline of the basic thermodynamics of temperature-, pressure-, and osmolyte-dependent LLPS as well as a molecular-level statistical mechanics picture in terms of solvent-mediated interactions and void volumes.

The role of chain-stiffness in lattice protein models: A replica-exchange Wang-Landau study

Using Monte Carlo simulations, we investigate simple, physically motivated extensions to the hydrophobic-polar lattice protein model for the small (46 amino acid) protein Crambin. We use two-dimensional replica-exchange Wang-Landau sampling to study the effects of a bond angle stiffness parameter on the folding and uncover a new step in the collapse process for particular values of this stiffness parameter. A physical interpretation of the folding is developed by analysis of changes in structural quantities, and the free energy landscape is explored. For these special values of stiffness, we find non-degenerate ground states, a property that is consistent with behavior of real proteins, and we use these unique ground states to elucidate the formation of native contacts during the folding process. Through this analysis, we conclude that chain-stiffness is particularly influential in the low energy, low temperature regime of the folding process once the lattice protein has partially collapsed.

On the polymer physics origins of protein folding thermodynamics

A remarkable feature of the spontaneous folding of many small proteins is the striking similarity in the thermodynamics of the folding process. This process is characterized by simple two-state thermodynamics with large and compensating changes in entropy and enthalpy and a funnel-like free energy landscape with a free-energy barrier that varies linearly with temperature. One might attribute the commonality of this two-state folding behavior to features particular to these proteins (e.g., chain length, hydrophobic/hydrophilic balance, attributes of the native state) or one might suspect that this similarity in behavior has a more general polymer-physics origin. Here we show that this behavior is also typical for flexible homopolymer chains with sufficiently short range interactions. Two-state behavior arises from the presence of a low entropy ground (folded) state separated from a set of high entropy disordered (unfolded) states by a free energy barrier. This homopolymer model exhibits a funneled free energy landscape that reveals a complex underlying dynamics involving competition between folding and non-folding pathways. Despite the presence of multiple pathways, this simple physics model gives the robust result of two-state thermodynamics for both the cases of folding from a basin of expanded coil states and from a basin of compact globule states.

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.

Probing the origins of two-state folding

Many protein systems fold in a two-state manner. Random models, however, rarely display two-state kinetics and thus such behavior should not be accepted as a default. While theories for the prevalence of two-state kinetics have been presented, none sufficiently explain the breadth of experimental observations. A model, making minimal assumptions, is introduced that suggests two-state behavior is likely for any system with an overwhelmingly populated native state. We show two-state folding is a natural consequence of such two-state thermodynamics, and is strengthened by increasing the population of the native state. Further, the model exhibits hub-like behavior, with slow interconversions between unfolded states. Despite this, the unfolded state equilibrates quickly relative to the folding time. This apparent paradox is readily understood through this model. Finally, our results compare favorable with measurements of folding rates as a function of chain length and Keq, providing new insight into these relations.

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.

How long does it take to equilibrate the unfolded state of a protein?

How long does it take to equilibrate the unfolded state of a protein? The answer to this question has important implications for our understanding of why many small proteins fold with two state kinetics. When the equilibration within the unfolded state U is much faster than the folding, the folding kinetics will be two state even if there are many folding pathways with different barriers. Yet the mean first passage times (MFPTs) between different regions of the unfolded state can be much longer than the folding time. This seems to imply that the equilibration within U is much slower than the folding. In this communication we resolve this paradox. We present a formula for estimating the time to equilibrate the unfolded state of a protein. We also present a formula for the MFPT to any state within U, which is proportional to the average lifetime of that state divided by the state population. This relation is valid when the equilibration within U is very fast as compared with folding as it often is for small proteins. To illustrate the concepts, we apply the formulas to estimate the time to equilibrate the unfolded state of Trp-cage and MFPTs within the unfolded state based on a Markov State Model using an ultra-long 208 microsecond trajectory of the miniprotein to parameterize the model. The time to equilibrate the unfolded state of Trp-cage is ∼100 ns while the typical MFPTs within U are tens of microseconds or longer.

First-order coil-globule transition driven by vibrational entropy

By shifting the balance between conformational entropy and internal energy, polymers modify their shape under external stimuli, such as changes in temperature. Prominent among such transformations is the coil-globule transition, whereby a polymer can switch from an entropy-dominated coil conformation to a globular one, governed by energy. The nature of the coil-globule transition has remained elusive, with evidence for both continuous and discontinuous transitions, with the two-state behaviour of proteins as an instance of the latter. Theoretical models mostly predict second-order transitions. Here we introduce a model that takes into consideration hitherto neglected features common to any polymer. We show that a first-order phase transition smoothly appears as a function of the model parameters. Our results can relieve part of the conflicts between theory and experiments in the field of protein folding, in the wake of recent studies tracing back the remarkable properties of proteins to basic polymer physics.

Kinetic consequences of native state optimization of surface-exposed electrostatic interactions in the Fyn SH3 domain

Optimization of surface exposed charge-charge interactions in the native state has emerged as an effective means to enhance protein stability; but the effect of electrostatic interactions on the kinetics of protein folding is not well understood. To investigate the kinetic consequences of surface charge optimization, we characterized the folding kinetics of a Fyn SH3 domain variant containing five amino acid substitutions that was computationally designed to optimize surface charge-charge interactions. Our results demonstrate that this optimized Fyn SH3 domain is stabilized primarily through an eight-fold acceleration in the folding rate. Analyses of the constituent single amino acid substitutions indicate that the effects of optimization of charge-charge interactions on folding rate are additive. This is in contrast to the trend seen in folded state stability, and suggests that electrostatic interactions are less specific in the transition state compared to the folded state. Simulations of the transition state using a coarse-grained chain model show that native electrostatic contacts are weakly formed, thereby making the transition state conducive to nonspecific, or even nonnative, electrostatic interactions. Because folding from the unfolded state to the folding transition state for small proteins is accompanied by an increase in charge density, nonspecific electrostatic interactions, that is, generic charge density effects can have a significant contribution to the kinetics of protein folding. Thus, the interpretation of the effects of amino acid substitutions at surface charged positions may be complicated and consideration of only native-state interactions may fail to provide an adequate picture.

Conformational flexibility and the mechanisms of allosteric transitions in topologically similar proteins

Conformational flexibility plays a central role in allosteric transition of proteins. In this paper, we extend the analysis of our previous study [S. Tripathi and J. J. Portman, Proc. Natl. Acad. Sci. U.S.A. 106, 2104 (2009)] to investigate how relatively minor structural changes of the meta-stable states can significantly influence the conformational flexibility and allosteric transition mechanism. We use the allosteric transitions of the domains of calmodulin as an example system to highlight the relationship between the transition mechanism and the inter-residue contacts present in the meta-stable states. In particular, we focus on the origin of transient local unfolding (cracking), a mechanism that can lower free energy barriers of allosteric transitions, in terms of the inter-residue contacts of the meta-stable states and the pattern of local strain that develops during the transition. We find that the magnitude of the local strain in the protein is not the sole factor determining whether a region will ultimately crack during the transition. These results emphasize that the residue interactions found exclusively in one of the two meta-stable states is the key in understanding the mechanism of allosteric conformational change.

Hydrophobic interactions in the formation of secondary structures in small peptides

Effects of the attractive and repulsive parts of hydrophobic interactions on α helices and β sheets in small peptides are investigated using a simple atomic potential. Typically, a physical spatial range of attraction tends to favor β sheets, but α helices would be favored if the attractive range were more extended. We also found that desolvation barriers favor β sheets in collapsed conformations of polyalanine, polyvaline, polyleucine, and three fragments of amyloid peptides tested in this study. Our results provide insight into the multifaceted role of hydrophobicity in secondary structure formation, including the α to β transitions in certain amyloid peptides.

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

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

The contribution of entropy, enthalpy, and hydrophobic desolvation to cooperativity in repeat-protein folding

Cooperativity is a defining feature of protein folding, but its thermodynamic and structural origins are not completely understood. By constructing consensus ankyrin repeat protein arrays that have nearly identical sequences, we quantify cooperativity by resolving stability into intrinsic and interfacial components. Heteronuclear NMR and CD spectroscopy show that these constructs adopt ankyrin repeat structures. Applying a one-dimensional Ising model to a series of constructs chosen to maximize information content in unfolding transitions, we quantify stabilities of the terminal capping repeats, and resolve the effects of denaturant into intrinsic and interfacial components. Reversible thermal denaturation resolves interfacial and intrinsic free energies into enthalpic, entropic, and heat capacity terms. Intrinsic folding is entropically disfavored, whereas interfacial interaction is entropically favored and attends a decrease in heat capacity. These results suggest that helix formation and backbone ordering occurs upon intrinsic folding, whereas hydrophobic desolvation occurs upon interfacial interaction, contributing to cooperativity.

The folding of single domain proteins--have we reached a consensus?

Rather than stressing the most recent advances in the field, this review highlights the fundamental topics where disagreement remains and where adequate experimental data are lacking. These topics include properties of the denatured state and the role of residual structure, the nature of the fundamental steps and barriers, the extent of pathway heterogeneity and non-native interactions, recent comparisons between theory and experiment, and finally, dynamical properties of the folding reaction.

All-or-none proteinlike folding transition of a flexible homopolymer chain

Here we report a first-order all-or-none transition from an expanded coil to a compact crystallite for a flexible polymer chain. Wang-Landau sampling is used to construct the complete density of states for square-well chains up to length 256. Analysis within both the microcanonical and canonical ensembles shows a direct freezing transition for finite length chains with sufficiently short-range interactions. This type of transition is a distinctive feature of "one-step" protein folding and our findings demonstrate that a simple homopolymer model can exhibit protein-folding thermodynamics.

Relationship between protein folding thermodynamics and the energy landscape

The origin of protein folding thermodynamics is examined in terms of the energy landscape, employing an off-lattice protein model with scaled non-native attractions, which is continuously tunable between a Go-like model and a highly frustrated system. Extensive statistical temperature molecular dynamics simulations, combined with inherent structure analysis, reveal the intimate connection between the global geometric properties of the energy landscape and the statistical temperature. The basin depth of the energy landscape is shown to play a key role in the first-order-like characteristics of the statistical temperature, which are easily identified by the squared modulus of the potential energy gradient in the microcanonical ensemble.

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

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

Determination of the structure of the energy landscape for coarse-grained off-lattice models of folding heteropolymers

The structure of energy landscapes for a minimalist coarse-grained off-lattice protein model is presented to investigate the folding behaviors of heteropolymers. The obtained energy landscapes serve as a useful tool for visualization of the funnel-like structure of a considered system in the configuration space. Despite the simplicity of the model, the knowledge of the free-energy landscape enables us to show different folding characteristics known from real proteins and synthetic peptides, such as two-state folding and metastability.

Lattice model of equilibrium polymerization. VII. Understanding the role of "cooperativity" in self-assembly

Cooperativity is an emergent many-body phenomenon related to the degree to which elementary entities (particles, molecules, organisms) collectively interact to form larger scale structures. From the standpoint of a formal mean field description of chemical reactions, the cooperativity index m, describing the number of elements involved in this structural self-organization, is the order of the reaction. Thus, m for molecular self-assembly is the number of molecules in the final organized structure, e.g., spherical micelles. Although cooperativity is crucial for regulating the thermodynamics and dynamics of self-assembly, there is a limited understanding of this aspect of self-assembly. We analyze the cooperativity by calculating essential thermodynamic properties of the classical mth order reaction model of self-assembly (FAm model), including universal scaling functions describing the temperature and concentration dependence of the order parameter and average cluster size. The competition between self-assembly and phase separation is also described. We demonstrate that a sequential model of thermally activated equilibrium polymerization can quantitatively be related to the FAm model. Our analysis indicates that the essential requirement for "cooperative" self-assembly is the introduction of constraints (often nonlocal) acting on the individual assembly events to regulate the thermodynamic free energy landscape and, thus, the thermodynamic sharpness of the assembly transition. An effective value of m is defined for general self-assembly transitions, and we find a general tendency for self-assembly to become a true phase transition as m-->infinity. Finally, various quantitative measures of self-assembly cooperativity are discussed in order to identify experimental signatures of cooperativity in self-assembling systems and to provide a reliable metric for the degree of transition cooperativity.

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

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

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

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

Lattice simulations of cotranslational folding of single domain proteins

We use lattice protein models and Monte Carlo simulations to study cotranslational folding of small single domain proteins. We show that the assembly of native structure begins during late extrusion stages, but final formation of native state occurs during de novo folding, when all residues are extruded. There are three main results in our study. First, for the sequences displaying two-state refolding mechanism de novo cotranslational folding pathway differs from that sampled in in vitro refolding. The change in folding pathways is due to partial assembly of native interactions during extrusion that results in different starting conditions for in vitro refolding and for de novo cotranslational folding. For small single domain proteins cotranslational folding is slower than in vitro refolding, but is generally fast enough to be completed before the release from a ribosome. Second, we found that until final stages of biosynthesis cotranslational folding is essentially equilibrium. This observation is explained by low stability of structured states for partially extruded chains. Finally, our data suggest that the proteins, which refold in vitro slowly via intermediates, complete their de novo folding after the release from a ribosome. Comparison of our lattice cotranslational simulations with recent experimental and computational studies is discussed.

The helix-coil transition revisited

In this article, we perform a dynamic Monte Carlo simulation study of the helix-coil transition by using a bond-fluctuation lattice model. The results of the simulations are compared with those predicted by the Zimm-Bragg statistical thermodynamic theory with propagation and nucleation parameters determined from simulation data. The Zimm-Bragg theory provides a satisfactory description of the helix-coil transition of a homopolypeptide chain of 32 residues (N = 32). For such a medium-length chain, however, the analytical equation based on a widely-used large-N approximation to the Zimm-Bragg theory is not suitable to predict the average length of helical blocks at low temperatures when helicity is high. We propose an analytical large-eigenvalue (lambda) approximation. The new equation yields a significantly improved agreement on the average helix-block length with the original Zimm-Bragg theory for both medium and long chain lengths in the entire temperature range. Nevertheless, even the original Zimm-Bragg theory does not provide an accurate description of helix-coil transition for longer chains. We assume that the single-residue nucleation of helix formation as suggested in the original Zimm-Bragg model might be responsible for this deviation. A mechanism of nucleation by a short helical block is proposed by us and provides a significantly improved agreement with our simulation data.

Statistical mechanics of helix bundles using a dynamic programming approach

Despite much study, biomolecule folding cooperativity is not well understood. There are quantitative models for helix-coil transitions and for coil-to-globule transitions, but no accurate models yet treat both chain collapse and secondary structure formation together. We develop here a dynamic programming approach to statistical mechanical partition functions of foldamer chain molecules. We call it the ascending levels model. We apply it to helix-coil and helix-bundle folding and cooperativity. For 14- to 50-mer Baldwin peptides, the model gives good predictions for the heat capacity and helicity versus temperature and urea. The model also gives good fits for the denaturation of Oas's three-helix bundle B domain of protein A (F13W*) and synthetic protein alpha3C by temperature and guanidine. The model predicts the conformational distributions. It shows that these proteins fold with transitions that are two-state, although the transitions in the Baldwin helices are nearly higher order. The model shows that the recently developed three-helix bundle polypeptoids of Lee et al. fold anti-cooperatively, with a predicted value of DeltaHvH/DeltaHcal = 0.72. The model also predicts that two-helix bundles are unstable in proteins but stable in peptoids. Our dynamic programming approach provides a general way to explore cooperativity in complex foldable polymers.

Identification of characteristic protein folding channels in a coarse-grained hydrophobic-polar peptide model

Folding channels and free-energy landscapes of hydrophobic-polar heteropolymers are discussed on the basis of a minimalistic off-lattice coarse-grained model. We investigate how rearrangements of hydrophobic and polar monomers in a heteropolymer sequence lead to completely different folding behaviors. Studying three exemplified sequences with the same content of hydrophobic and polar residues, we can reproduce within this simple model two-state folding, folding through intermediates, as well as metastability.

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.

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

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

Exploring the relationship between funneled energy landscapes and two-state protein folding

It should take an astronomical time span for unfolded protein chains to find their native state based on an unguided conformational random search. The experimental observation that folding is fast can be rationalized by assuming that protein energy landscapes are sloped towards the native state minimum, such that rapid folding can proceed from virtually any point in conformational space. Folding transitions often exhibit two-state behavior, involving extensively disordered and highly structured conformers as the only two observable kinetic species. This study employs a simple Brownian dynamics model of "protein particles" moving in a spherically symmetrical potential. As expected, the presence of an overall slope towards the native state minimum is an effective means to speed up folding. However, the two-state nature of the transition is eradicated if a significant energetic bias extends too far into the non-native conformational space. The breakdown of two-state cooperativity under these conditions is caused by a continuous conformational drift of the unfolded proteins. Ideal two-state behavior can only be maintained on surfaces exhibiting large regions that are energetically flat, a result that is supported by other recent data in the literature (Kaya and Chan, Proteins: Struct Funct Genet 2003;52:510-523). Rapid two-state folding requires energy landscapes exhibiting the following features: (i) A large region in conformational space that is energetically flat, thus allowing for a significant degree of random sampling, such that unfolded proteins can retain a random coil structure; (ii) a trapping area that is strongly sloped towards the native state minimum.

On the Nonequilibrium Thermodynamics of Large Departures from Butler−Volmer Behavior

Large deviations from the behavior predicted by the Butler-Volmer equation of electrochemistry are accounted for using mesoscopic nonequilibrium thermodynamics. The nonequilibrium thermodynamic hypotheses are extended to include velocity space and cope with imperfect reactant transport leading to departures from Butler-Volmer behavior. This results in a modified Butler-Volmer equation in good agreement with experimental data. The distinct advantages of the method and its applicability to analyze other systems are briefly discussed.

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.

Entropy reduction effect imposed by hydrogen bond formation on protein folding cooperativity: evidence from a hydrophobic minimalist model

Conformational restrictions imposed by hydrogen bond formation during protein folding are investigated by Monte Carlo simulations of a non-native-centric, two-dimensional, hydrophobic model in which the formation of favorable contacts is coupled to an effective reduction in lattice coordination. This scheme is intended to mimic the requirement that polar backbone groups of real proteins must form hydrogen bonds concomitantly to their burial inside the apolar protein core. In addition to the square lattice, with z=3 conformations per monomer, we use extensions in which diagonal step vectors are allowed, resulting in z=5 and z=7. Thermodynamics are governed by the hydrophobic energy function, according to which hydrophobic monomers tend to make contacts unspecifically while the reverse is true for hydrophilic monomers, with the additional restriction that only contacts between monomers adopting one of zh<z local conformations contribute to the energy, where zh is the number of local conformations assumed to be compatible with hydrogen bond formation. The folding transition abruptness and van't Hoff-to-calorimetric-enthalpy ratio are found to increase dramatically by this simple and physically motivated mechanism. The observed increase in folding cooperativity is correlated to an increase in the convexity of the underlying microcanonical conformational entropy as a function of energy. Preliminary simulations in three dimensions, even though using a smaller relative reduction in lattice effective coordination zh/z=4/5, display a slight increase in cooperativity for a hydrophobic model of 40 monomers and a more pronounced increase in cooperativity for a native-centric Go-model with the same native conformation, suggesting that this purely entropic effect is not an artifact of dimensionality and is likely to be of fundamental importance in the theoretical understanding of folding cooperativity.

Kinetics of Loop Formation and Breakage in the Denatured State of Iso-1-cytochrome c

The earliest events in protein folding involve the formation of simple loops. Observing the rates of loop closure under denaturing conditions can provide direct insight into the relative probability and sequence determinants for formation of loops of different sizes. The persistence of these initial contacts is equally important for efficient folding, so measurement of rates of loop breakage under denaturing conditions is also essential. We have used stopped-flow and continuous-flow methods to measure the rates of histidine-heme loop formation and breakage in the denatured state of iso-1-cytochrome c (in the presence of 3 M guanidine HCl). The data indicate that the mechanism for forming loops is a two-step process, the first step being the deprotonation of the histidine, and the second step being the binding of the histidine to the heme. This mechanism makes it possible to extract both the rate constants of formation, k(f), and breakage, k(b), of loops from the pH dependence of the observed rate constant, k(obs). To determine the dependence of k(f) and k(b) on loop size, we have carried out kinetic measurements for seven single surface histidine variants of iso-1-cytochrome c. A scaling factor (the dependence of k(f) on log[loop size]) of approximately -1.8 is observed for loop formation, similar to that observed in other systems. The magnitude of k(b) varies from 30 s(-1) to 300 s(-1), indicating that the stability of different loops varies considerably. The implications of the kinetics of loop formation and breakage in the denatured state for the mechanism of protein folding are discussed.

Protein flexibility prediction by an all-atom mean-field statistical theory

We extended a mean-field model to proteins with all atomic detail. The all-atom mean-field model was used to calculate the dynamic and thermodynamic properties of a three-helix bundle fragment of Staphylococcal protein A (Protein Data Bank [PDB] ID 1BDD) and alpha-spectrin SH3 domain protein (PDB ID 1SHG). We show that a model with all-atomic detail provides a significantly more accurate prediction of flexibility of residues in proteins than does a coarse-grained residue-level model. The accuracy of flexibility prediction is further confirmed by application of the method to 18 additional proteins with the largest size of 224 residues.

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

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

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

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

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

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

Multicanonical study of coarse-grained off-lattice models for folding heteropolymers

We have performed multicanonical simulations of hydrophobic-hydrophilic heteropolymers with two simple effective, coarse-grained off-lattice models to study the influence of specific interactions in the models on conformational transitions of selected sequences with 20 monomers. Another aspect of the investigation was the comparison with the purely hydrophobic homopolymer and the study of general conformational properties induced by the "disorder" in the sequence of a heteropolymer. Furthermore, we applied an optimization algorithm to sequences with up to 55 monomers and compared the global-energy minimum found with lowest-energy states identified within the multicanonical simulation. This was used to find out how reliable the multicanonical method samples the free-energy landscape, in particular for low temperatures.

Native geometry and the dynamics of protein folding

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