Thermodynamics of a diffusional protein folding reaction

The folding reactions of several proteins are well described as diffusional barrier crossing processes, which suggests that they should be analyzed by Kramers' rate theory rather than by transition state theory. For the cold shock protein Bc-Csp from Bacillus caldolyticus, we measured stability and folding kinetics, as well as solvent viscosity as a function of temperature and denaturant concentration. Our analysis indicates that diffusional folding reactions can be treated by transition state theory, provided that the temperature and denaturant dependence of the solvent viscosity is properly accounted for, either at the level of the measured rate constants or of the calculated activation parameters. After viscosity correction the activation barriers for folding become less enthalpic and more entropic. The transition from an enthalpic to an entropic folding barrier with increasing temperature is, however, apparent in the data before and after this correction. It is a consequence of the negative activation heat capacity of refolding, which is independent of solvent viscosity. Bc-Csp and its mesophilic homolog Bs-CspB from Bacillus subtilis differ strongly in stability but show identical enthalpic and entropic barriers to refolding. The increased stability of Bc-Csp originates from additional enthalpic interactions that are established after passage through the activated state. As a consequence, the activation enthalpy of unfolding is increased relative to Bs-CspB.     

Solvent Friction Changes the Folding Pathway of the Tryptophan Zipper TZ2

Because the rate of a diffusional process such as protein folding is controlled by friction encountered along the reaction pathway, the speed of folding is readily tunable through adjustment of solvent viscosity. The precise relationship between solvent viscosity and the rate of diffusion is complex and even conformation-dependent, however, because both solvent friction and protein internal friction contribute to the total reaction friction. The heterogeneity of the reaction friction along the folding pathway may have subtle consequences. For proteins that fold on a multidimensional free-energy surface, an increase in solvent friction may drive a qualitative change in folding trajectory. Our time-resolved experiments on the rapidly and heterogeneously folding β-hairpin TZ2 show a shift in the folding pathway as viscosity increases, even though the energetics of folding is unaltered. We also observe a nonlinear or saturating behavior of the folding relaxation time with rising solvent viscosity, potentially an experimental signature of the shifting pathway for unfolding. Our results show that manipulations of solvent viscosity in folding experiments and simulations may have subtle and unexpected consequences on the folding dynamics being studied.     

Diffusional barrier crossing in a two-state protein folding reaction.

There has been some debate as to whether protein folding involves diffusive chain motions and thus depends on solvent viscosity. The interpretation of folding kinetics in viscous solvents has remained difficult and controversial, in that viscogenic agents affect folding rates not only by increasing solvent viscosity but also by increasing protein stability. By carefully choosing experimental conditions, we can now eliminate the effect on stability and show that the folding dynamics of the cold shock protein CspB are viscosity dependent. Thus Kramers' theory of reaction rates rather than transition state theory should be used to describe this folding reaction.     

Internal friction controls the speed of protein folding from a compact configuration

Several studies have found millisecond protein folding reactions to be controlled by the viscosity of the solvent: Reducing the viscosity allows folding to accelerate. In the limit of very low solvent viscosity, however, one expects a different behavior. Internal interactions, occurring within the solvent-excluded interior of a compact molecule, should impose a solvent-independent upper limit to folding speed once the bulk diffusional motions become sufficiently rapid. Why has this not been observed? We have studied the effect of solvent viscosity on the folding of cytochrome c from a highly compact, late-stage intermediate configuration. Although the folding rate accelerates as the viscosity declines, it tends toward a finite limiting value approximately 10(5) s(-1) as the viscosity tends toward zero. This limiting rate is independent of the cosolutes used to adjust solvent friction. Therefore, interactions within the interior of a compact denatured polypeptide can limit the folding rate, but the limiting time scale is very fast. It is only observable when the solvent-controlled stages of folding are exceedingly rapid or else absent. Interestingly, we find a very strong temperature dependence in these "internal friction"-controlled dynamics, indicating a large energy scale for the interactions that govern reconfiguration within compact, near-native states of a protein.     

Kinetic pathways of beta-hairpin (un)folding in explicit solvent

We examine the dynamical (un)folding pathways of the C-terminal beta-hairpin of protein G-B1 at room temperature in explicit solvent, by employing transition path sampling algorithms. The path ensembles contain information on the folding kinetics, including solvent motion. We determine the transition state ensembles for the two main transitions: 1), the hydrophobic collapse; and 2), the backbone hydrogen bond formation. In both cases the transition state ensembles are characterized by a layer (1) or a strip (2) of water molecules in between the two hairpin strands, supporting the hypothesis of the solvent as lubricant in the folding process. The transition state ensembles do not correspond with saddle points in the equilibrium free-energy landscapes. The kinetic pathways are thus not completely determined by the free-energy landscape. This phenomenon can occur if the order parameters obey different timescales. Using the transition interface sampling technique, we calculate the rate constants for (un)folding and find them in reasonable agreement with experiments, thus supporting the validation of using all-atom force fields to study protein folding.     

Kinetic analysis of folding and unfolding the 56 amino acid IgG-binding domain of streptococcal protein G.

The 56 amino acid B domain of protein G (GB) is a stable globular folding unit with no disulfide cross-links. The physical properties of GB offer extraordinary flexibility for evaluating the energetics of the folding reaction. The protein is monomeric and very soluble in both folded and unfolded forms. The folding reaction has been previously examined by differential scanning calorimetry (Alexander et al., 1992) and found to exhibit two-state unfolding behavior over a wide pH range with an unfolding transition near 90 degrees C (GB1) at neutral pH. Here, the kinetics of folding and unfolding two naturally occurring versions of GB have been measured using stopped-flow mixing methods and analyzed according to transition-state theory. GB contains no prolines, and the kinetics of folding and unfolding can be fit to a single, first-order rate constant over the temperature range of 5-35 degrees C. The major thermodynamic changes going from the unfolded state to the transition state are (1) a large decrease in heat capacity (delta Cp), indicating that the transition state is compact and solvent inaccessible relative to the unfolded state; (2) a large loss of entropy; and (3) a small increase in enthalpy. The most surprising feature of the folding of GB compared to that of previously studied proteins is that its folding approximates a rapid diffusion controlled process with little increase in enthalpy going from the unfolded to the transition state.     

The effect of increasing the stability of non-native interactions on the folding landscape of the bacterial immunity protein Im9

How stabilising non-native interactions influence protein folding energy landscapes is currently not well understood: such interactions could speed folding by reducing the conformational search to the native state, or could slow folding by increasing ruggedness. Here, we examine the influence of non-native interactions in the folding process of the bacterial immunity protein Im9, by exploiting our ability to manipulate the stability of the intermediate and rate-limiting transition state (TS) in the folding of this protein by minor alteration of its sequence or changes in solvent conditions. By analysing the properties of these species using Phi-value analysis, and exploration of the structural properties of the TS ensemble using molecular dynamics simulations, we demonstrate the importance of non-native interactions in immunity protein folding and demonstrate that the rate-limiting step involves partial reorganisation of these interactions as the TS ensemble is traversed. Moreover, we show that increasing the contribution to stability made by non-native interactions results in an increase in Phi-values of the TS ensemble without altering its structural properties or solvent-accessible surface area. The data suggest that the immunity proteins fold on multiple, but closely related, micropathways, resulting in a heterogeneous TS ensemble that responds subtly to mutation or changes in the solvent conditions. Thus, altering the relative strength of native and non-native interactions influences the search to the native state by restricting the pathways through the folding energy landscape.     

Molecular basis of cooperativity in protein folding IV. Core: A general cooperative folding model

The cooperative nature of the protein folding process is independent of the characteristic fold and the specific secondary structure attributes of a globular protein. A general folding/unfolding model should, therefore, be based upon structural features that transcend the peculiarities of α-helices, β-sheets, and other structural motifs found in proteins. The studies presented in this paper suggest that a single structural characteristic common to all globular proteins is essential for cooperative folding. The formation of a partly folded state from the native state results in the exposure to solvent of two distinct regions: (1) the portions of the protein that are unfolded; and (2) the “complementary surfaces,” located in the regions of the protein that remain folded. The cooperative character of the folding/unfolding transition is determined largely by the energetics of exposing complementary surface regions to the solvent. By definition, complementary regions are present only in partly folded states; they are absent from the native and unfolded states. An unfavorable free energy lowers the probability of partly folded states and increases the cooperativity of the transition. In this paper we present a mathematical formulation of this behavior and develop a general cooperative folding/unfolding model, termed the “complementary region” (CORE) model. This model successfully reproduces the main properties of folding/unfolding transitions without limiting the number of partly folded states accessible to the protein, thereby permitting a systematic examination of the structural and solvent conditions under which intermediates become populated. It is shown that the CORE model predicts two-state folding/unfolding behavior, even though the two-state character is not assumed in the model. © 1993 Wiley-Liss, Inc.     

Posttransition state desolvation of the hydrophobic core of the src-SH3 protein domain

The folding thermodynamics of the src-SH3 protein domain were characterized under refolding conditions through biased fully atomic molecular dynamics simulations with explicit solvent. The calculated free energy surfaces along several reaction coordinates revealed two barriers. The first, larger barrier was identified as the transition state barrier for folding, associated with the formation of the first hydrophobic sheet of the protein. phi values calculated from structures residing at the transition state barrier agree well with experimental phi values. The microscopic information obtained from our simulations allowed us to unambiguously assign intermediate phi values as the result of multiple folding pathways. The second, smaller barrier occurs later in the folding process and is associated with the cooperative expulsion of water molecules between the hydrophobic sheets of the protein. This posttransition state desolvation barrier cannot be observed through traditional folding experiments, but is found to be critical to the correct packing of the hydrophobic core in the final stages of folding. Hydrogen exchange and NMR experiments are suggested to probe this barrier.     

Minimizing frustration by folding in an aqueous environment

Although life as we know it evolved in an aqueous medium, the properties of water are not completely understood. In this review, we focus on the role of water in guiding protein folding and stability. Specifically, we discuss the mechanisms of protein folding in an aqueous environment, the effects of water on the folding energy landscape as well as the transition state ensemble, and interactions of water with the folded state. We show that water cannot be viewed as a passive solvent, but rather, plays a very active role in the life of a protein.     

Stabilization of proteins in confined spaces

We present theory showing that confining a protein to a small inert space (a "cage") should stabilize the protein against reversible unfolding. Examples of such spaces might include the pores within chromatography columns, the Anfinsen cage in chaperonins, the interiors of ribosomes, or regions of steric occlusion inside cells. Confinement eliminates some expanded configurations of the unfolded chain, shifting the equilibrium from the unfolded state toward the native state. The partition coefficient for a protein in a confined space is predicted to decrease significantly when the solvent is changed from native to denaturing conditions. Small cages are predicted to increase the stability of the native state by as much as 15 kcal/mol. Confinement may also increase the rates of protein or RNA folding.     

Theory for the folding and stability of globular proteins

Using lattice statistical mechanics, we develop theory to account for the folding of a heteropolymer molecule such as a protein to the globular and soluble state. Folding is assumed to be driven by the association of solvophobic monomers to avoid solvent and opposed by the chain configurational entropy. Theory predicts a phase transition as a function of temperature or solvent character. Molecules that are too short or too long or that have too few solvophobic residues are predicted not to fold. Globular molecules should have a largely solvophobic core, but there is an entropic tendency for some residues to be "out of place", particularly in small molecules. For long chains, molecules comprised of globular domains are predicted to be thermodynamically more stable than spherical molecules. The number of accessible conformations in the globular state is calculated to be an exceedingly small fraction of the number available to the random coil. Previous estimates of this number, which have motivated kinetic theories of folding, err by many tens of orders of magnitude.     

Non-linear rate-equilibrium free energy relationships and Hammond behavior in protein folding

Non-linear rate-equilibrium relationships upon mutation or changes in solvent conditions are frequently observed in protein folding reactions and are usually interpreted in terms of Hammond behavior. Here we first give a general overview over the concept of transition state movements in chemical reactions and discuss its application to protein folding. We then show examples for genuine Hammond behavior and for apparent transition state movements caused by other effects like changes in the rate-limiting step of the folding reaction or ground state effects, i.e. structural changes in either the native state or the unfolded state. These examples show that apparent transition state movements can easily be mistaken for Hammond behavior. We describe experimental tests using self- and cross-interaction parameters to distinguish between structural changes in a single transition state following Hammond behavior and apparent transition state movements caused by other effects.     

Approach for an equation of state for adsorbed protein surfaces.

A surface equation of state for polymer interfaces is obtained by equating the chemical potentials of the solvent in the bulk and surface phases. This equation of state contains the fraction of the surface covered with polymer and the surface activity coefficient as parameters. These parameters may be obtained by measuring the thermodynamic modulus of elasticity. Moreover, if the amount of protein in the interface is known the degree of folding in the surface may be evaluated. The present theory was applied to adsorbed beta-lactoglobulin in surfaces. The results are in agreement with qualitative equations and results from other sources.     

An effective solvent theory connecting the underlying mechanisms of osmolytes and denaturants for protein stability

An all-atom Gō model of Trp-cage protein is simulated using discontinuous molecular dynamics in an explicit minimal solvent, using a single, contact-based interaction energy between protein and solvent particles. An effective denaturant or osmolyte solution can be constructed by making the interaction energy attractive or repulsive. A statistical mechanical equivalence is demonstrated between this effective solvent model and models in which proteins are immersed in solutions consisting of water and osmolytes or denaturants. Analysis of these studies yields the following conclusions: 1), Osmolytes impart extra stability to the protein by reducing the entropy of the unfolded state. 2), Unfolded states in the presence of osmolyte are more collapsed than in water. 3), The folding transition in osmolyte solutions tends to be less cooperative than in water, as determined by the ratio of van 't Hoff to calorimetric enthalpy changes. The decrease in cooperativity arises from an increase in native structure in the unfolded state, and thus a lower thermodynamic barrier at the transition midpoint. 4), Weak denaturants were observed to destabilize small proteins not by lowering the unfolded enthalpy, but primarily by swelling the unfolded state and raising its entropy. However, adding a strong denaturant destabilizes proteins enthalpically. 5), The folding transition in denaturant-containing solutions is more cooperative than in water. 6), Transfer to a concentrated osmolyte solution with purely hard-sphere steric repulsion significantly stabilizes the protein, due to excluded volume interactions not present in the canonical Tanford transfer model. 7), Although a solution with hard-sphere interactions adds a solvation barrier to native contacts, the folding is nevertheless less cooperative for reasons 1–3 above, because a hard-sphere solvent acts as a protecting osmolyte.     

High concentrations of viscogens decrease the protein folding rate constant by prematurely collapsing the coil

In several studies, viscogenic osmolytes have been suggested to decrease the folding rate constant of polypeptides by slowing their motion through the solvent. Here, we show that osmolytes may slow protein folding by prematurely collapsing the coil. At low or moderate concentrations of osmolytes (<30%), folding of the two-state protein CI2 becomes faster with increasing osmolyte concentrations, suggesting that the kinetics are governed by protein stability. However, at higher concentrations of osmolyte, the coil collapses in the dead-time of the refolding experiment, causing a dramatic drop in the folding rate. The collapsed state is non-native and appears to be different for different osmolytes.     

Protein dynamics: From the native to the unfolded state and back again

Simulations to study protein unfolding and folding were performed. The unfolding simulations make use of molecular dynamics and treat an atomic model of barnase in aqueous solvent. The cooperative nature of the unfolding transition and the important role of water are described. The folding simulations are based on a bead model of the protein on a cubic lattice. It is shown for the 27-mer model that a large energy gap between the lowest energy (native) state and the excited states is a necessary and sufficient condition for fast folding.     

Common Structural Transitions in Explicit-Solvent Simulations of Villin Headpiece Folding

Molecular dynamics simulations of protein folding can provide very high-resolution data on the folding process; however, due to computational challenges most studies of protein folding have been limited to small peptides, or made use of approximations such as Gō potentials or implicit solvent models. We have performed a set of molecular dynamics simulations totaling >50 μs on the villin headpiece subdomain, one of the most stable and fastest-folding naturally occurring proteins, in explicit solvent. We find that the wild-type villin headpiece reliably folds to a native conformation on timescales similar to experimentally observed folding, but that a fast folding double-norleucine mutant shows significantly more heterogeneous behavior. Along with other recent simulation studies, we note the occurrence of nonnative structures intermediates, which may yield a nativelike signal in the fluorescence measurements typically used to study villin folding. Based on the wild-type simulations, we propose alternative approaches to measure the formation of the native state.     

Mapping the folding pathway of an immunoglobulin domain: structural detail from Phi value analysis and movement of the transition state

BACKGROUND: Do proteins that have the same structure fold by the same pathway even when they are unrelated in sequence? To address this question, we are comparing the folding of a number of different immunoglobulin-like proteins. Here, we present a detailed protein engineering phi value analysis of the folding pathway of TI I27, an immunoglobulin domain from human cardiac titin. RESULTS: TI I27 folds rapidly via a kinetic intermediate that is destabilized by most mutations. The transition state for folding is remarkably native-like in terms of solvent accessibility. We use phi value analysis to map this transition state and show that it is highly structured; only a few residues close to the N-terminal region of the protein remain completely unfolded. Interestingly, most mutations cause the transition state to become less native-like. This anti-Hammond behavior can be used as a novel means of obtaining additional structural information about the transition state. CONCLUSIONS: The residues that are involved in nucleating the folding of TI I27 are structurally equivalent to the residues that form the folding nucleus in an evolutionary unrelated fibronectin type III protein. These residues form part of the common structural core of Ig-like domains. The data support the hypothesis that interactions essential for defining the structure of these beta sandwich proteins are also important in nucleation of folding.     

Ten-microsecond molecular dynamics simulation of a fast-folding WW domain

All-atom molecular dynamics (MD) simulations of protein folding allow analysis of the folding process at an unprecedented level of detail. Unfortunately, such simulations have not yet reached their full potential both due to difficulties in sufficiently sampling the microsecond timescales needed for folding, and because the force field used may yield neither the correct dynamical sequence of events nor the folded structure. The ongoing study of protein folding through computational methods thus requires both improvements in the performance of molecular dynamics programs to make longer timescales accessible, and testing of force fields in the context of folding simulations. We report a ten-microsecond simulation of an incipient downhill-folding WW domain mutant along with measurement of a molecular time and activated folding time of 1.5 microseconds and 13.3 microseconds, respectively. The protein simulated in explicit solvent exhibits several metastable states with incorrect topology and does not assume the native state during the present simulations.