Thermodynamics of protein folding: a microscopic view

Statistical thermodynamics provides a powerful theoretical framework for analyzing, understanding and predicting the conformational properties of biomolecules. The central quantity is the potential of mean force or effective energy as a function of conformation, which consists of the intramolecular energy and the solvation free energy. The intramolecular energy can be reasonably described by molecular mechanics-type functions. While the solvation free energy is more difficult to model, useful results can be obtained with simple approximations. Such functions have been used to estimate the intramolecular energy contribution to protein stability and obtain insights into the origin of thermodynamic functions of protein folding, such as the heat capacity. With reasonable decompositions of the various energy terms, one can obtain meaningful values for the contribution of one type of interaction or one chemical group to stability. Future developments will allow the thermodynamic characterization of ever more complex biological processes.     

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.     

Thermodynamics and kinetics of protein folding: an evolutionary perspective

This article appeals to an evolutionary model which postulates that primordial proteins were described by small polypeptide chains which (i) lack disulfide bridges, and (ii) display slow folding rates with multi-state kinetics, to determine relations between structural properties of proteins and their folding kinetics. We parameterize the energy landscape of proteins in terms of thermodynamic activation variables. The model studies evolutionary changes in these thermodynamic parameters, and we invoke relations between these activation variables and structural properties of the protein to predict the following correspondence between protein structure and folding kinetics. 1. Proteins with inter- and intra-chain disulfide bridges: large variability in both folding rates and stability of intermediates, multi-state kinetics. 2. Proteins which lack inter and intra-chain disulfide bridges. 2.1 Single-domain chains: fast folding rates; unstable intermediates; two-state kinetics. 2.2 Multi-domain monomers: intermediate rates; metastable intermediates; multi-state kinetics. 2.3 Multi-domain oligomers: slow rates; metastable intermediates; multi-state kinetics. The evolutionary model thus provides a kinetic characterization of one important subfamily of proteins which we describe by the following properties: Folding dynamics of single-domain proteins which lack disulfide bridges are described by two-state kinetics. Folding rate of this class of proteins is positively correlated with the thermodynamic stability of the folded state.     

Heat-induced unfolding of neocarzinostatin, a small all-beta protein investigated by small-angle X-ray scattering

Neocarzinostatin is an all-beta protein, 113 amino acid residues long, with an immunoglobulin-like fold. Its thermal unfolding has been studied by small-angle X-ray scattering. Preliminary differential scanning calorimetry and fluorescence measurements suggest that the transition is not a simple, two-state transition. The apparent radius of gyration is determined using three different approaches, the validity of which is critically assessed using our experimental data as well as a simple, two-state model. Similarly, each step of data analysis is evaluated and the underlying assumptions plainly stated. The existence of at least one intermediate state is formally demonstrated by a singular value decomposition of the set of scattering patterns. We assume that the pattern of the solution before the onset of the transition is that of the native protein, and that of the solution at the highest temperature is that of the completely unfolded protein. Given these, actually not very restrictive, boundary constraints, a least-squares procedure yields a scattering pattern of the intermediate state. However, this solution is not unique: a whole class of possible solutions is derived by adding to the previous linear combination of the native and completely unfolded states. Varying the initial conditions of the least-squares calculation leads to very similar solutions. Whatever member of the class is considered, the conformation of this intermediate state appears to be weakly structured, probably less than the transition state should be according to some proposals. Finally, we tried and used the classical model of three thermodynamically well-defined states to account for our data. The failure of the simple thermodynamic model suggests that there is more than the single intermediate structure required by singular value decomposition analysis. Formally, there could be several discrete intermediate species at equilibrium, or an ensemble of conformations differently populated according to the temperature. In the latter case, a third state would be a weighted average of all non native and not completely unfolded states of the protein but, since the weights change with temperature, no meaningful curve is likely to be derived by a global analysis using the simple model of three thermodynamically well-defined states.     

Understanding protein folding: small proteins in silico

Recent improvements in methodology and increased computer power now allow atomistic computer simulations of protein folding. We briefly review several advanced Monte Carlo algorithms that have contributed to this development. Details of folding simulations of three designed mini proteins are shown. Adding global translations and rotations has allowed us to handle multiple chains and to simulate the aggregation of six beta-amyloid fragments. In a different line of research we have developed several algorithms to predict local features from sequence. In an outlook we sketch how such biasing could extend the application spectrum of Monte Carlo simulations to structure prediction of larger proteins.     

Hierarchical protein folding pathways: a computational study of protein fragments

We have previously presented a building block folding model. The model postulates that protein folding is a hierarchical top-down process. The basic unit from which a fold is constructed, referred to as a hydrophobic folding unit, is the outcome of combinatorial assembly of a set of "building blocks." Results obtained by the computational cutting procedure yield fragments that are in agreement with those obtained experimentally by limited proteolysis. Here we show that as expected, proteins from the same family give very similar building blocks. However, different proteins can also give building blocks that are similar in structure. In such cases the building blocks differ in sequence, stability, contacts with other building blocks, and in their 3D locations in the protein structure. This result, which we have repeatedly observed in many cases, leads us to conclude that while a building block is influenced by its environment, nevertheless, it can be viewed as a stand-alone unit. For small-sized building blocks existing in multiple conformations, interactions with sister building blocks in the protein will increase the population time of the native conformer. With this conclusion in hand, it is possible to develop an algorithm that predicts the building block assignment of a protein sequence whose structure is unknown. Toward this goal, we have created sequentially nonredundant databases of building block sequences. A protein sequence can be aligned against these, in order to be matched to a set of potential building blocks.     

Folding myoglobin within a sol-gel glass: protein folding constrained to a small volume

The unfolding and refolding reaction of myoglobin was examined in solution and within a porous silica sol-gel glass. The sol-gel pores constrain the protein to a volume that is the same size and shape as the folded native state accompanied by a few layers of water solvation. Denaturants such as low pH buffers can be diffused through the gel pores to the protein to initiate unfolding and refolding. Acid-induced unfolding was hindered by the steric constraints imposed by the gel pores such that more denaturing conditions were required within the gel than in solution to create the unfolded state. No new folding intermediates were observed. Refolding of myoglobin was not complete in millimolar pH 7 buffer alone. Addition of 25% glycerol to the pH 7 buffer resulted in nearly complete refolding, and the use of 1 M phosphate buffer resulted in complete refolding. The role of this cosolvent and salt in disrupting the ordered water surrounding the protein within the gel is discussed in light of the Hofmeister series and entropic trapping via a diminished hydrophobic effect within the gel. These results are consistent with the premises of folding models in which secondary and tertiary structures are considered to form within a compact conformation of the protein backbone.     

Thermodynamics of BPTI folding.

A calorimetric study of the basic pancreatic trypsin inhibitor (BPTI) has been performed using the new generation of the adiabatic scanning microcalorimeters, operating in an extended temperature range of 5-130 degrees C. Precise measurements of the heat capacities of the native and unfolded states of BPTI show that the heat capacity change upon unfolding strongly depends on temperature; its value is maximal at about 50 degrees C and diminishes as the temperature is increased. The temperature dependencies of the enthalpy and entropy changes upon BPTI unfolding were found to be similar to those normally observed for other small globular proteins. The stability of BPTI has been correlated with its structure.     

Protein modeling with reduced representation: statistical potentials and protein folding mechanism

A high resolution reduced model of proteins is used in Monte Carlo dynamics studies of the folding mechanism of a small globular protein, the B1 immunoglobulin-binding domain of streptococcal protein G. It is shown that in order to reproduce the physics of the folding transition, the united atom based model requires a set of knowledge-based potentials mimicking the short-range conformational propensities and protein-like chain stiffness, a model of directional and cooperative hydrogen bonds, and properly designed knowledge-based potentials of the long-range interactions between the side groups. The folding of the model protein is cooperative and very fast. In a single trajectory, a number of folding/unfolding cycles were observed. Typically, the folding process is initiated by assembly of a native-like structure of the C-terminal hairpin. In the next stage the rest of the four-ribbon beta-sheet folds. The slowest step of this pathway is the assembly of the central helix on the scaffold of the beta-sheet.     

A Gaussian statistical mechanical model for the equilibrium thermodynamics of barnase folding

Given an all non-hydrogen-atom potential function that implicitly includes solvation effects, it is possible to adjust its parameters to favor the correct native structure for several proteins over decoys produced by ungapped threading. It is also possible to further train it to reproduce the experimental free energy of unfolding in aqueous solution at 298 K for wild-type barnase and 66 mutants. For this, the native state is represented by the crystal structure at a single energy level with a calculated low degeneracy; the denatured state is represented by the extended conformation and a high calculated degeneracy. The same two-state model can be extended to account for the stability of all 67 sequences toward urea denaturation at 298 K by building in a solvation term that depends on urea concentration. With the addition of one more parameter set to give the correct heat capacity of unfolded barnase in solution, it is possible to approximate the experimental thermodynamics of barnase thermal denaturation: melting temperature, width of thermal transition, deltaG, deltaH, deltaS, and deltaCp. This requires a novel sort of statistical mechanical model where the two states each have a Gaussian density of microscopic state distribution as a function of energy.     

Computer simulations of protein folding with a small number of distance restraints

A high coordination lattice model was used to represent the protein chain. Lattice points correspond to amino-acid side groups. A complicated force field was designed in order to reproduce a protein-like behavior of the chain. Long-distance tertiary restraints were also introduced into the model. The Replica Exchange Monte Carlo method was applied to find the lowest energy states of the folded chain and to solve the problem of multiple minima. In this method, a set of replicas of the model chain was simulated independently in different temperatures with the exchanges of replicas allowed. The model chains, which consisted of up to 100 residues, were folded to structures whose root-mean-square deviation (RMSD) from their native state was between 2.5 and 5 A. Introduction of restrain based on the positions of the backbone hydrogen atoms led to an improvement in the number of successful simulation runs. A small improvement (about 0.5 A) was also achieved in the RMSD of the folds. The proposed method can be used for the refinement of structures determined experimentally from NMR data.     

蛋白质折叠机理的理论研究

简要综述了近年来蛋白质折叠机理的理论研究。首先回顾了蛋白质折叠理论的发展历程,然后对折叠中间体的研究现状作了较详细的介绍。同时,对折叠机理理论研究中的几种理论模型和模拟算法作了细致评述,分析了其现状和存在的问题。最后,总结和讨论了折叠机理理论研究的现存问题及研究热点,并展望了该领域研究的发展趋势。     

The use of folding patterns in a multisolution approach of the all-beta protein three-dimensional structure. A beta-roll structure is predicted in the retroviral glycoprotein

An automatic macromolecular modelling package of unknown protein structures was developed using the intimate correlation which appears between the observed X-ray structures and their associated predicted folding patterns. The method can be considered as a generalization of both the combinatorial [1] and the template identification [2,3] approaches which were proposed some years ago, and provide a fast way of selecting 'structural motifs' to build new proteins. As an illustration, the tertiary fold of the all-beta-domain of the retroviral outermembrane glycoprotein is proposed.     

Thermodynamics of a beta-hairpin structure: evidence for cooperative formation of folding nucleus

To elucidate early nucleation stages in protein folding, multi-probed thermodynamic characterization was applied to the beta-hairpin structural formation of G-peptide, which is a C-terminal fragment of the B1 domain of streptococcal protein G. The segment corresponding to the sequence of G-peptide is believed to act as a nucleus during the folding process of the B1 domain. In spite of the broad thermal transition of G-peptide, nuclear magnetic resonance (NMR) melting measurements combined with our original analytical theory enabled us to obtain the thermodynamic properties of the beta-hairpin formation with considerable accuracy. Additionally, all the thermodynamic properties determined by every NMR probe on both the main-chain and the side-chains were quite similar, and also comparable to the values that were independently determined by calorimetric analysis of G-peptide. These results demonstrate that G-peptide folds cooperatively throughout the molecule. In other words, the formation of the beta-hairpin is interpreted as the fashion of a first-order phase transition between two states without any distinguishable intermediates. This cooperative formation of the short linear peptide consisting of only 16 residues provides insight into not only the first folding events of the B1 domain, but also the general principles of proteins in terms of structural hierarchy, stability and folding mechanism.     

Brownian dynamics simulation of protein folding: a study of the diffusion-collision model

The dynamic aspects of protein folding are described by a series of diffusion-collision steps involving structural units (microdomains) of various sizes that combine to form the protein in its native state. A method is introduced for obtaining the rate constants for the basic diffusion-collision step by use of Brownian dynamics. The method is applied to an investigation of the folding dynamics of two α-helices connected by a flexible (random-coil) polypeptide chain. The results of this full three-dimensional treatment are compared with simplified model calculations for the diffusion-collision step. Of particular interest are the nature of the collision dynamics and the role of the intervening peptide chain.     

Thermodynamics, kinetics, and salt dependence of folding of YopM, a large leucine-rich repeat protein

Small globular proteins have many contacts between residues that are distant in primary sequence. These contacts create a complex network between sequence-distant segments of secondary structure, which may be expected to promote the cooperative folding of globular proteins. Although repeat proteins, which are composed of tandem modular units, lack sequence-distant contacts, several of considerable length have been shown to undergo cooperative two-state folding. To explore the limits of cooperativity in repeat proteins, we have studied the unfolding of YopM, a leucine-rich repeat (LRR) protein of over 400 residues. Despite its large size and modular architecture (15 repeats), YopM equilibrium unfolding is highly cooperative, and shows a very strong dependence on the concentration of urea. In contrast, kinetic studies of YopM folding indicate a mechanism that includes one or more transient intermediates. The urea dependence of the folding and unfolding rates suggests a relatively small transition state ensemble. As with the urea dependence, we have found an extreme dependence of the free energy of unfolding on the concentration of salt. This salt dependence likely results from general screening of a large number of unfavorable columbic interactions in the folded state, rather than from specific cation binding.     

Thermodynamics of ion-induced RNA folding in the hammerhead ribozyme: an isothermal titration calorimetric study

The hammerhead ribozyme undergoes a well-defined two-stage conformational folding process, induced by the binding of magnesium ions. In this study, we have used isothermal titration calorimetry to analyze the thermodynamics of magnesium binding and magnesium ion-induced folding of the ribozyme. Binding to the natural sequence ribozyme is strongly exothermic and can be analyzed in terms of sequential interaction at two sites with association constants K(A) = 480 and 2840 M(-1). Sequence variants of the hammerhead RNA give very different isothermal titration curves. An A14G variant that cannot undergo ion-induced folding exhibits endothermic binding. By contrast, a deoxyribose G5 variant that can undergo only the first of the two folding transitions gives a complex titration curve. However, despite these differences the ITC data for all three species can be analyzed in terms of the sequential binding of magnesium ions at two sites. While the binding affinities are all in the region of 10(3) M(-1), corresponding to free energies of Delta G degrees = -3.5 to -4 kcal mol(-1), the enthalpic and entropic contributions show much greater variation. The ITC experiments are in good agreement with earlier conformational studies of the folding of the ion-induced folding of the hammerhead ribozyme.