Free energy landscape of protein folding in water: explicit vs. implicit solvent

The Generalized Born (GB) continuum solvent model is arguably the most widely used implicit solvent model in protein folding and protein structure prediction simulations; however, it still remains an open question on how well the model behaves in these large-scale simulations. The current study uses the beta-hairpin from C-terminus of protein G as an example to explore the folding free energy landscape with various GB models, and the results are compared to the explicit solvent simulations and experiments. All free energy landscapes are obtained from extensive conformation space sampling with a highly parallel replica exchange method. Because solvation model parameters are strongly coupled with force fields, five different force field/solvation model combinations are examined and compared in this study, namely the explicit solvent model: OPLSAA/SPC model, and the implicit solvent models: OPLSAA/SGB (Surface GB), AMBER94/GBSA (GB with Solvent Accessible Surface Area), AMBER96/GBSA, and AMBER99/GBSA. Surprisingly, we find that the free energy landscapes from implicit solvent models are quite different from that of the explicit solvent model. Except for AMBER96/GBSA, all other implicit solvent models find the lowest free energy state not the native state. All implicit solvent models show erroneous salt-bridge effects between charged residues, particularly in OPLSAA/SGB model, where the overly strong salt-bridge effect results in an overweighting of a non-native structure with one hydrophobic residue F52 expelled from the hydrophobic core in order to make better salt bridges. On the other hand, both AMBER94/GBSA and AMBER99/GBSA models turn the beta-hairpin in to an alpha-helix, and the alpha-helical content is much higher than the previously reported alpha-helices in an explicit solvent simulation with AMBER94 (AMBER94/TIP3P). Only AMBER96/GBSA shows a reasonable free energy landscape with the lowest free energy structure the native one despite an erroneous salt-bridge between D47 and K50. Detailed results on free energy contour maps, lowest free energy structures, distribution of native contacts, alpha-helical content during the folding process, NOE comparison with NMR, and temperature dependences are reported and discussed for all five models.     

Complexity of protein folding

It is believed that the native folded three-dimensional conformation of a protein is its lowest free energy state, or one of its lowest. It is shown here that both a two-and three-dimensional mathematical model describing the folding process as a free energy minimization problems is NP-hard. This means that the problem belongs to a large set of computational problems, assumed to be very hard (“conditionally intractable”). Some of the possible ramifications of this results are speculated upon.     

N-glycosylation induced changes in tau protein dynamics reveal its role in tau misfolding and aggregation: A microsecond long molecular dynamics study

Various posttranslational modifications like hyperphosphorylation, O-GlcNAcylation, and acetylation have been attributed to induce the abnormal folding in tau protein. Recent in vitro studies revealed the possible involvement of N-glycosylation of tau protein in the abnormal folding and tau aggregation. Hence, in this study, we performed a microsecond long all atom molecular dynamics simulation to gain insights into the effects of N-glycosylation on Asn-359 residue which forms part of the microtubule binding region. Trajectory analysis of the stimulations coupled with essential dynamics and free energy landscape analysis suggested that tau, in its N-glycosylated form tends to exist in a largely folded conformation having high beta sheet propensity as compared to unmodified tau which exists in a large extended form with very less beta sheet propensity. Residue interaction network analysis of the lowest energy conformations further revealed that Phe378 and Lys353 are the functionally important residues in the peptide which helped in initiating the folding process and Phe378, Lys347, and Lys370 helped to maintain the stability of the protein in the folded state.     

Water as a conformational editor in protein folding

As molecules approach one another in aqueous solution, desolvation free energy barriers to association are encountered. Experiments suggest these (de)solvation effects contribute to the free energy barriers separating the folded and unfolded states of protein molecules. To explore their influence on the energy landscapes of protein folding reactions, we have incorporated desolvation barriers into a semi-realistic, off-lattice protein model that uses a simplified physico-chemical force-field determined solely by the sequence of amino acids. Monte Carlo sampling techniques were used to study the effects on the thermodynamics and kinetics of folding of a number of systems, diverse in structure and sequence. In each case, desolvation barriers increase the stability of the native conformation and the cooperativity of the major folding/unfolding transition. The folding times of these systems are reduced significantly upon inclusion of desolvation barriers, demonstrating that the particulate nature of the solvent engenders a more defined route to the native fold.     

A small tripeptide AFA undergoes two state cooperative conformational transitions: implications for conformational biases in unfolded states

It is important to understand the conformational biases that are present in unfolded states to understand protein folding. In this context, it is surprising that even a short tripeptide like AFA samples folded/ordered conformation as demonstrated recently by NMR experiments of the peptide in aqueous solution at 280 K. In this paper, we present molecular dynamics simulation of the peptide in explicit water using OPLS-AA/L all-atom force field. The results are in overall agreement with NMR results and provide some further insights. The peptide samples turn and extended conformational forms corresponding to minima in free energy landscape. Frequent transitions between the minima are observed due to modest free energy barriers. The turn conformation seems to be stabilized by hydrophobic interactions and possibly by bridging water molecules between backbone donors and acceptors. Thus the peptide does not sample conformations randomly, but samples well defined conformations. The peptide served as a model for folding-unfolding equilibrium in the context of peptide folding. Further, implications for drug design are also discussed.     

Equilibrium folding and unfolding pathways for a model protein

The folding–unfolding process of reduced bovine pancreatic trypsin inhibitor was investigated with an idealized model employing approximate free energies. The protein is regarded to consist of only C^α and C^β atoms. The backbone dihedral angles are the only conformational variables and are permitted to take discrete values at every 10°. Intraresidue energies consist of two terms: an empirical part taken from the observed frequency distributions of (?,ψ) and an additional favorable energy assigned to the native conformation of each residue. Interresidue interactions are simplified by assuming that there is an attractive energy operative only between residue pairs in close contact in the native structure. A total of 230,000 molecular conformations, with no atomic overlaps, ranging from the native state to the denatured state, are randomly generated by changing the sampling bias. Each conformation is classified according to its conformational energy, F; a conformational entropy, S(F) is estimated for each value of F from the number of samples. The dependence of S(F) on energy reveals that the folding–unfolding transition for this idealized model is an “all-or-none” type; this is attributable to the specific long-range interactions. Interresidue contact probabilities, averaged over samples representing various stages of folding, serve to characterize folding intermediates. Most probable equilibrium pathways for the folding–unfolding transition are constructed by connecting conformationally similar intermediates. The specific details obtained for bovine pancreatic trypsin inhibitor are as follows: (1) Folding begins with the appearance of nativelike medium-range contacts at a β-turn and at the α-helix. (2) These grow to include the native pair of interacting β-strands. This state includes intact regular secondary conformations, as well as the interstrand sheet contacts, and corresponds to an activated state with the highest free energy on the pathway. (3) Additional native long-range contacts are completely formed either toward the amino terminus or toward the carboxyl terminus. (4) In a final step, the missing contacts appear. Although these folding pathways for this model are not consistent with experimental reports, it does indicate multiple folding pathways. The method is general and can be applied to any set of calculated conformational energies and furthermore permits investigation of gross folding features.     

Structure of a partially unfolded form of Escherichia coli dihydrofolate reductase provides insight into its folding pathway

Proteins frequently fold via folding intermediates that correspond to local minima on the conformational energy landscape. Probing the structure of the partially unfolded forms in equilibrium under native conditions can provide insight into the properties of folding intermediates. To elucidate the structures of folding intermediates of Escherichia coli dihydrofolate reductase (DHFR), we investigated transient partial unfolding of DHFR under native conditions. We probed the structure of a high‐energy conformation susceptible to proteolysis (cleavable form) using native‐state proteolysis. The free energy for unfolding to the cleavable form is clearly less than that for global unfolding. The dependence of the free energy on urea concentration (m‐value) also confirmed that the cleavable form is a partially unfolded form. By assessing the effect of mutations on the stability of the partially unfolded form, we found that native contacts in a hydrophobic cluster formed by the F‐G and Met‐20 loops on one face of the central β‐sheet are mostly lost in the partially unfolded form. Also, the folded region of the partially unfolded form is likely to have some degree of structural heterogeneity. The structure of the partially unfolded form is fully consistent with spectroscopic properties of the near‐native kinetic intermediate observed in previous folding studies of DHFR. The findings suggest that the last step of the folding of DHFR involves organization in the structure of two large loops, the F‐G and Met‐20 loops, which is coupled with compaction of the rest of the protein.     

基于HP模型的蛋白质折叠问题的研究

基于蛋白质二维HP模型提出改进的遗传算法对真实蛋白质进行计算机折叠模拟。结果显示疏水能量函数最小值的蛋白质构象对应含疏水核心的稳定结构,疏水作用在蛋白质折叠中起主要作用。研究表明二维HP模型在蛋白质折叠研究中是可行的和有效的并为进一步揭示蛋白质折叠机理提供重要参考信息。     

Analysis of the free energy in a stochastic RNA secondary structure model

There are two custom ways for predicting RNA secondary structures: minimizing the free energy of a conformation according to a thermodynamic model and maximizing the probability of a folding according to a stochastic model. In most cases, stochastic grammars are used for the latter alternative applying the maximum likelihood principle for determining a grammar's probabilities. In this paper, building on such a stochastic model, we will analyze the expected minimum free energy of an RNA molecule according to Turner's energy rules. Even if the parameters of our grammar are chosen with respect to structural properties of native molecules only (and therefore, independent of molecules' free energy), we prove formulae for the expected minimum free energy and the corresponding variance as functions of the molecule's size which perfectly fit the native behavior of free energies. This gives proof for a high quality of our stochastic model making it a handy tool for further investigations. In fact, the stochastic model for RNA secondary structures presented in this work has, for example, been used as the basis of a new algorithm for the (nonuniform) generation of random RNA secondary structures.     

Modeling disordered regions in proteins using Rosetta

Protein structure prediction methods such as Rosetta search for the lowest energy conformation of the polypeptide chain. However, the experimentally observed native state is at a minimum of the free energy, rather than the energy. The neglect of the missing configurational entropy contribution to the free energy can be partially justified by the assumption that the entropies of alternative folded states, while very much less than unfolded states, are not too different from one another, and hence can be to a first approximation neglected when searching for the lowest free energy state. The shortcomings of current structure prediction methods may be due in part to the breakdown of this assumption. Particularly problematic are proteins with significant disordered regions which do not populate single low energy conformations even in the native state. We describe two approaches within the Rosetta structure modeling methodology for treating such regions. The first does not require advance knowledge of the regions likely to be disordered; instead these are identified by minimizing a simple free energy function used previously to model protein folding landscapes and transition states. In this model, residues can be either completely ordered or completely disordered; they are considered disordered if the gain in entropy outweighs the loss of favorable energetic interactions with the rest of the protein chain. The second approach requires identification in advance of the disordered regions either from sequence alone using for example the DISOPRED server or from experimental data such as NMR chemical shifts. During Rosetta structure prediction calculations the disordered regions make only unfavorable repulsive contributions to the total energy. We find that the second approach has greater practical utility and illustrate this with examples from de novo structure prediction, NMR structure calculation, and comparative modeling.     

Reliable protein folding on complex energy landscapes: the free energy reaction path

A theoretical framework is developed to study the dynamics of protein folding. The key insight is that the search for the native protein conformation is influenced by the rate r at which external parameters, such as temperature, chemical denaturant, or pH, are adjusted to induce folding. A theory based on this insight predicts that 1), proteins with complex energy landscapes can fold reliably to their native state; 2), reliable folding can occur as an equilibrium or out-of-equilibrium process; and 3), reliable folding only occurs when the rate r is below a limiting value, which can be calculated from measurements of the free energy. We test these predictions against numerical simulations of model proteins with a single energy scale.     

An empirical approach to protein conformation stability and flexibility

Experimental measurements of disulfide bond stability at various stages of protein folding are considered in terms of the effective concentrations of the thiol groups relative to each other; values of up to 10⁷M are observed, so that intramolecular interactions within the interior of a protein are much more stable, and provide greater stability to the folded conformation, than those on the surface or in a flexible segment. Intramolecular interactions can have substantially lower free energies than intermolecular, for solely entropic reasons; this implies that polar interactions, such as hydrogen bonds and salt bridges, can provide net stabilization to a folded conformation, in spite of the unfolded protein having intermolecular interactions with the solvent. These considerations can account for the lower free energy and enthalpy of the folded state and are useful for considering protein flexibility.     

Statistical mechanics of protein folding by exhaustive enumeration

It is hard to construct theories for the folding of globular proteins because they are large and complicated molecules having enormous numbers of nonnative conformations and having native states that are complicated to describe. Statistical mechanical theories of protein folding are constructed around major simplifying assumptions about the energy as a function of conformation and/or simplifications of the representation of the polypeptide chain, such as one point per residue on a cubic lattice. It is not clear how the results of these theories are affected by their various simplifications. Here we take a very different simplification approach where the chain is accurately represented and the energy of each conformation is calculated by a not unreasonable empirical function. However, the set of amino acid sequences and allowed conformations is so restricted that it becomes computationally feasible to examine them all. Hence we are able to calculate melting curves for thermal denaturation as well as the detailed kinetic pathway of refolding. Such calculations are based on a novel representation of the conformations as points in an abstract 12-dimensional Euclidean conformation space. Fast folding sequences have relatively high melting temperatures, native structures with relatively low energies, small kinetic barriers between local minima, and relatively many conformations in the global energy minimum's watershed. In contrast to other folding theories, these models show no necessary relationship between fast folding and an overall funnel shape to the energy surface, or a large energy gap between the native and the lowest nonnative structure, or the depth of the native energy minimum compared to the roughness of the energy landscape. Proteins 32:425–437, 1998. © 1998 Wiley-Liss, Inc.     

一种在格点模型上模拟蛋白质折叠结构的优化算法

蛋白质折叠问题是生物信息学中一个经典的多项式复杂程度的非确定性(non-deterministic polynomial,NP)难度问题.势能曲面变平法(ELP)是一种启发式的全局优化算法.通过对ELP方法中的直方图函数提出一种新的更新机制,并将基于贪心策略的初始构象的产生,基于牵引移动的邻域搜索策略与ELP方法相结合,为面心立方体(FCC)格点模型的蛋白质折叠问题提出一种改进的势能曲面变平(ELP+)算法.采用文献中9条常用序列作为测试集.对于每条序列,ELP+算法均能找到与文献中的算法所得到的最低能量相等或更低的能量.实验结果表明,ELP+算法是求解FCC格点模型的蛋白质折叠问题的一种有效算法.     

Study of the compact denatured state of a protein by molecular dynamics simulation]

The native state can be considered as a unique conformation of the protein molecule with the lowest free energy of residue contacts. In this case, all other conformations correspond to the denatured state. The degree of their compactness varies significantly. Under folding conditions, the compact denatured state rather than the random coil is in equilibrium with native protein. The balance between the main forces of protein folding, the solvophobic interactions and conformational entropy, suggests that some properties of the compact denatured state are close to those of native protein, whereas other properties are close to those of the random coil. To investigate the molecular structure of the compact denatured state, the method of molecular dynamics simulation seems to be very useful.     

The highly cooperative folding of small naturally occurring proteins is likely the result of natural selection

To illuminate the evolutionary pressure acting on the folding free energy landscapes of naturally occurring proteins, we have systematically characterized the folding free energy landscape of Top7, a computationally designed protein lacking an evolutionary history. Stopped-flow kinetics, circular dichroism, and NMR experiments reveal that there are at least three distinct phases in the folding of Top7, that a nonnative conformation is stable at equilibrium, and that multiple fragments of Top7 are stable in isolation. These results indicate that the folding of Top7 is significantly less cooperative than the folding of similarly sized naturally occurring proteins, suggesting that the cooperative folding and smooth free energy landscapes observed for small naturally occurring proteins are not general properties of polypeptide chains that fold to unique stable structures but are instead a product of natural selection.     

Polar and nonpolar atomic environments in the protein core: Implications for folding and binding

Hydrophobic interactions are believed to play an important role in protein folding and stability. Semi-empirical attempts to estimate these interactions are usually based on a model of solvation, whose contribution to the stability of proteins is assumed to be proportional to the surface area buried upon folding. Here we propose an extension of this idea by defining an environment free energy that characterizes the environment of each atom of the protein, including solvent, polar or nonpolar atoms of the same protein or of another molecule that interacts with the protein. In our model, the difference of this environment free energy between the folded state and the unfolded (extended) state of a protein is shown to be proportional to the area buried by nonpolar atoms upon folding. General properties of this environment free energy are derived from statistical studies on a database of 82 well-refined protein structures. This free energy is shown to be able to discriminate misfolded from correct structural models, to provide an estimate of the stabilization due to oligomerization, and to predict the stability of mutants in which hydrophobic residues have been substituted by site-directed mutagenesis, provided that no large structural modifications occur. © 1994 Wiley-Liss, Inc.     

The sequence TGAAKAVALVL from glyceraldehyde-3-phosphate dehydrogenase displays structural ambivalence and interconverts between alpha-helical and beta-hairpin conformations mediated by collapsed conformational states.

The peptide TGAAKAVALVL from glyceraldehyde-3-phosphate dehydrogenase adopts a helical conformation in the crystal structure and is a site for two hydrated helical segments, which are thought to be helical folding intermediates. Overlapping sequences of four to five residues from the peptide, sample both helical and strand conformations in known protein structures, which are dissimilar to glyceraldehyde-3-phosphate dehydrogenase suggesting that the peptide may have a structural ambivalence. Molecular dynamics simulations of the peptide sequence performed for a total simulation time of 1.2 micros, starting from the various initial conformations using GROMOS96 force field under NVT conditions, show that the peptide samples a large number of conformational forms with transitions from alpha-helix to beta-hairpin and vice versa. The peptide, therefore, displays a structural ambivalence. The mechanism from alpha-helix to beta-hairpin transition and vice versa reveals that the compact bends and turns conformational forms mediate such conformational transitions. These compact structures including helices and hairpins have similar hydrophobic radius of gyration (Rgh) values suggesting that similar hydrophobic interactions govern these conformational forms. The distribution of conformational energies is Gaussian with helix sampling lowest energy followed by the hairpins and coil. The lowest potential energy of the full helix may enable the peptide to take up helical conformation in the crystal structure of the glyceraldehyde-3-phosphate dehydrogenase, even though the peptide has a preference for hairpin too. The relevance of folding and unfolding events observed in our simulations to hydrophobic collapse model of protein folding are discussed.     

Determination of the conformation of folding initiation sites in proteins by computer simulation

Experimental evidence and theoretical models both suggest that protein folding begins by specific short regions of the polypeptide chain intermittently assuming conformations close to their final ones. The independent folding properties and small size of these folding initiation sites make them suitable subjects for computational methods aimed at deriving structure from sequence. We have used a torsion space Monte Carlo procedure together with an all-atom free energy function to investigate the folding of a set of such sites. The free energy function is derived by a potential of mean force analysis of experimental protein structures. The most important contributions to the total free energy are the local main chain electrostatics, main chain hydrogen bonds, and the burial of nonpolar area. Six proposed independent folding units and four control peptides 11–14 residues long have been investigated. Thirty Monte Carlo simulations were performed on each peptide, starting from different random conformations. Five of the six folding units adopted conformations close to the experimental ones in some of the runs. None of the controls did so, as expected. The generated conformations which are close to the experimental ones have among the lowest free energies encountered, although some less native like low free energy conformations were also found. The effectiveness of the method on these peptides, which have a wide variety of experimental conformations, is encouraging in two ways: First, it provides independent evidence that these regions of the sequences are able to adopt native like conformations early in folding, and therefore are most probably key components of the folding pathways. Second, it demonstrates that available simulation methods and free energy functions are able to produce reasonably accurate structures. Extensions of the methods to the folding of larger portions of proteins are suggested. © 1995 Wiley-Liss, Inc.     

Evidence for sequential barriers and obligatory intermediates in apparent two-state protein folding

Many small proteins fold fast and without detectable intermediates. This is frequently taken as evidence against the importance of partially folded states, which often transiently accumulate during folding of larger proteins. To get insight into the properties of free energy barriers in protein folding we analyzed experimental data from 23 proteins that were reported to show non-linear activation free-energy relationships. These non-linearities are generally interpreted in terms of broad transition barrier regions with a large number of energetically similar states. Our results argue against the presence of a single broad barrier region. They rather indicate that the non-linearities are caused by sequential folding pathways with consecutive distinct barriers and a few obligatory high-energy intermediates. In contrast to a broad barrier model the sequential model gives a consistent picture of the folding barriers for different variants of the same protein and when folding of a single protein is analyzed under different solvent conditions. The sequential model is also able to explain changes from linear to non-linear free energy relationships and from apparent two-state folding to folding through populated intermediates upon single point mutations or changes in the experimental conditions. These results suggest that the apparent discrepancy between two-state and multi-state folding originates in the relative stability of the intermediates, which argues for the importance of partially folded states in protein folding.