The folding thermodynamics and kinetics of crambin using an all-atom Monte Carlo simulation

We present a novel Monte Carlo simulation of protein folding, in which all heavy atoms are represented as interacting hard spheres. This model includes all degrees of freedom relevant to folding, all side-chain and backbone torsions, and uses a Go potential. In this study, we focus on the 46 residue alpha/beta protein crambin and two of its structural components, the helix and helix hairpin. For a wide range of temperatures, we recorded multiple folding events of these three structures from random coils to native conformations that differ by less than 1 A C(alpha) dRMS from their crystal structure coordinates. The thermodynamics and kinetic mechanism of the helix-coil transition obtained from our simulation shows excellent agreement with currently available experimental and molecular dynamics data. Based on insights obtained from folding its smaller structural components, a possible folding mechanism for crambin is proposed. We observed that the folding occurs via a cooperative, first order-like process, and that many folding pathways to the native state exist. One particular sequence of events constitutes a "fast-folding" pathway where kinetic traps are avoided. At very low temperatures, a kinetic trap arising from the incorrect packing of side-chains was observed. These results demonstrate that folding to the native state can be observed in a reasonable amount of time on desktop computers even when an all-atom representation is used, provided the energetics sufficiently stabilize the native state.

Mode-coupling theory for heteropolymers

We study the Langevin dynamics of a heteropolymer by means of a mode-coupling approximation scheme, giving rise to a set of coupled integro-differential equations relating the response and correlation functions. The analysis shows that there is a regime at low temperature characterized by out-of-equilibrium dynamics, with violation of time-translational invariance and of the fluctuation-dissipation theorem. The onset of aging dynamics at low temperatures gives insight into the nature of the slow dynamics of a disordered polymer. We also introduce a renormalization-group treatment of our mode-coupling equations, which supports our analysis, and might be applicable to other systems.

Different circular permutations produced different folding nuclei in proteins: a computational study

There have been many studies about the effect of circular permutation on the transition state/folding nucleus of proteins, with sometimes conflicting conclusions from different proteins and permutations. To clarify this important issue, we have studied two circular permutations of a lattice protein model with side-chains. Both permuted sequences have essentially the same native state as the original (wild-type) sequence. Circular permutant 1 cuts at the folding nucleus of the wild-type sequence. As a result, the permutant has a drastically different nucleus and folds more slowly than wild-type. In contrast, circular permutant 2 involves an incision at a site unstructured in the wild-type transition state, and the wild-type nucleus is largely retained in the permutant. In addition, permutant 2 displays both two-state and multi-state folding, with a native-like intermediate state occasionally populated. Neither the wild-type nor permutant 1 has a similar intermediate, and both fold in an apparently two-state manner. Surprisingly, permutant 2 folds at a rate identical with that of the wild-type. The intermediate in permutant 2 is stabilised by native and non-native interactions, and cannot be classified simply as on or off-pathway. So we advise caution in attributing experimental data to on or off-pathway intermediates. Finally, our work illuminates the results on alpha-spectrin SH3, chymotrypsin inhibitor 2 and beta-lactoglobulin, and supports a key assumption in the experimental efforts to locate potential nucleation sites of real proteins via circular permutations.

Differential stabilization of two hydrophobic cores in the transition state of the villin 14T folding reaction

We report the distribution of hydrophobic core contacts during the folding reaction transition state for villin 14T, a small 126-residue protein domain. The solution structure of villin 14T contains a central beta-sheet with two flanking hydrophobic cores; transition states for this protein topology have not been previously studied. Villin 14T has no disulfide bonds or cis-proline residues in its native state; it folds reversibly, and in an apparently two-state manner under some conditions. To map the hydrophobic core contacts in the transition state, 27 point mutations were generated at positions spread throughout the two hydrophobic cores. After each point mutation, comparison of the change in folding kinetics with the equilibrium destabilization indicates whether the site of mutation is stabilized in the transition state. The results show that the folding nucleus, or the sub-region with the strongest transition state contacts, is located in one of the two hydrophobic cores (the predominantly aliphatic core). The other hydrophobic core, which is mostly aromatic, makes much weaker contacts in the transition state. This work is the first transition state mapping for a protein with multiple major hydrophobic cores in a single folding unit; the hydrophobic cores cannot be separated into individual folding subdomains. The stabilization of only one hydrophobic core in the transition state illustrates that hydrophobic core formation is not intrinsically capable of nucleating folding, but must also involve the right specific interactions or topological factors in order to be kinetically important.

Proteins with selected sequences: a heteropolymeric study

Protein sequences are expected not to be random but selected in order to form a stable native structure that is kinetically accessible. Therefore our model contains a selective temperature in sequence space (see [S. Ramanathan and E. Shakhnovich, Phys. Rev. E 50, 1303 (1994)] ) to optimize the sequence for the target conformation statistically. Replica calculations, which go beyond quadratic approximations in the field-theoretical Hamiltonian, are presented. A phase diagram indicating the temperatures and selective temperatures at which transitions to a frozen globule, i.e., the native state, occur is obtained. It is shown that going beyond the quadratic approximation in the field Hamiltonian is very important, since it results in a significant change of the phase diagram. Moreover, we suggest that a one-step replica permutation symmetry scheme is sufficient to solve the model. In addition to this we present a result for the sequence correlation function along the chain in the case of a short-ranged potential between the monomers. A correlation function between monomers that form a contact in the native state is given depending on the temperature and the interaction parameter.

Comparison of two optimization methods to derive energy parameters for protein folding: perceptron and Z score

Two methods were proposed recently to derive energy parameters from known native protein conformations and corresponding sets of decoys. One is based on finding, by means of a perceptron learning scheme, energy parameters such that the native conformations have lower energies than the decoys. The second method maximizes the difference between the native energy and the average energy of the decoys, measured in terms of the width of the decoys' energy distribution (Z-score). Whereas the perceptron method is sensitive mainly to "outlier" (i.e., extremal) decoys, the Z-score optimization is governed by the high density regions in decoy-space. We compare the two methods by deriving contact energies for two very different sets of decoys: the first obtained for model lattice proteins and the second by threading. We find that the potentials derived by the two methods are of similar quality and fairly closely related. This finding indicates that standard, naturally occurring sets of decoys are distributed in a way that yields robust energy parameters (that are quite insensitive to the particular method used to derive them). The main practical implication of this finding is that it is not necessary to fine-tune the potential search method to the particular set of decoys used.

Statistical significance of protein structure prediction by threading

In this study, we estimate the statistical significance of structure prediction by threading. We introduce a single parameter epsilon that serves as a universal measure determining the probability that the best alignment is indeed a native-like analog. Parameter epsilon takes into account both length and composition of the query sequence and the number of decoys in threading simulation. It can be computed directly from the query sequence and potential of interactions, eliminating the need for sequence reshuffling and realignment. Although our theoretical analysis is general, here we compare its predictions with the results of gapless threading. Finally we estimate the number of decoys from which the native structure can be found by existing potentials of interactions. We discuss how this analysis can be extended to determine the optimal gap penalties for any sequence-structure alignment (threading) method, thus optimizing it to maximum possible performance.

What can disulfide bonds tell us about protein energetics, function and folding: simulations and bioninformatics analysis

We study the impact of disulfide bonds on protein stability and folding. Using lattice model simulations, we show that formation of a disulfide bond stabilizes a protein to an extent that depends on the distance along the chain between linked cysteine residues. However, the impact of disulfide bonds on folding kinetics varies broadly, from acceleration when disulfides are introduced in or close to the folding nucleus, to slowing when disulfides are introduced outside the nucleus. Having established the effect of disulfide bonds on stability, we study the correlation between the number of disulfide bonds and the composition of certain amino acid classes with the goal to use it as a statistical probe into factors that contribute to stability of proteins. We find that the number of disulfides is negatively correlated with aliphatic hydrophobic but not aromatic content. It is surprising that we observe a strong correlation of disulfide content with polar (Q,S,T,N) amino acid content and a strong negative correlation with charged (E,D,K,R) content. These findings provide insights into factors that determine protein stability and principles of protein design as well as possible relations of disulfide bonds and protein function.

Hiking in the energy landscape in sequence space: a bumpy road to good folders

With the help of a simple 20-letter lattice model of heteropolymers, we investigated the energy landscape in the space of designed good-folder sequences. Low-energy sequences form clusters, interconnected via neutral networks, in the space of sequences. Residues that play a key role in the foldability of the chain and in the stability of the native state are highly conserved, even among the chains belonging to different clusters. If, according to the interaction matrix, some strong attractive interactions are almost degenerate (i.e., they can be realized by more than one type of amino acid contacts), sequence clusters group into a few superclusters. Sequences belonging to different superclusters are dissimilar, displaying very small ( approximately 10%) similarity, and residues in key sites are, as a rule, not conserved. Similar behavior is observed in the analysis of real protein sequences.

Kinetics, thermodynamics and evolution of non-native interactions in a protein folding nucleus

A lattice model with side chains was used to investigate protein folding with computer simulations. In this model, we rigorously demonstrate the existence of a specific folding nucleus. This nucleus contains specific interactions not present in the native state that, when weakened, slow folding but do not change protein stability. Such a decoupling of folding kinetics from thermodynamics has been observed experimentally for real proteins. From our results, we conclude that specific non-native interactions in the transition state would give rise to straight phi-values that are negative or larger than unity. Furthermore, we demonstrate that residue Ile 34 in src SH3, which has been shown to be kinetically, but not thermodynamically, important, is universally conserved in proteins with the SH3 fold. This is a clear example of evolution optimizing the folding rate of a protein independent of its stability and function.

Analysis of knowledge-based protein-ligand potentials using a self-consistent method

We propose a self-consistent approach to analyze knowledge-based atom-atom potentials used to calculate protein-ligand binding energies. Ligands complexed to actual protein structures were first built using the SMoG growth procedure (DeWitte & Shakhnovich, 1996) with a chosen input potential. These model protein-ligand complexes were used to construct databases from which knowledge-based protein-ligand potentials were derived. We then tested several different modifications to such potentials and evaluated their performance on their ability to reconstruct the input potential using the statistical information available from a database composed of model complexes. Our data indicate that the most significant improvement resulted from properly accounting for the following key issues when estimating the reference state: (1) the presence of significant nonenergetic effects that influence the contact frequencies and (2) the presence of correlations in contact patterns due to chemical structure. The most successful procedure was applied to derive an atom-atom potential for real protein-ligand complexes. Despite the simplicity of the model (pairwise contact potential with a single interaction distance), the derived binding free energies showed a statistically significant correlation (approximately 0.65) with experimental binding scores for a diverse set of complexes.

Identifying the protein folding nucleus using molecular dynamics

Molecular dynamics simulations of folding in an off-lattice protein model reveal a nucleation scenario, in which a few well-defined contacts are formed with high probability in the transition state ensemble of conformations. Their appearance determines folding cooperativity and drives the model protein into its folded conformation. Amino acid residues participating in those contacts may serve as "accelerator pedals" used by molecular evolution to control protein folding rate.

Modeling the kinetics of acylation of insulin using a recursive method for solving the systems of coupled differential equations

This paper describes a theoretical method for solving systems of coupled differential equations that describe the kinetics of complicated reaction networks in which a molecule having multiple reaction sites reacts irreversibly with multiple equivalents of a ligand (reagent). The members of the network differ in the number of equivalents of reagent that have reacted, and in the patterns of sites of reaction. A recursive algorithm generates series, asymptotic, and average solutions describing this kinetic scheme. This method was validated by successfully simulating the experimental data for the kinetics of acylation of insulin.

[When and how can homologs overcome errors in the energy estimates and make the 3D structure prediction possible]

One still cannot predict the 3D fold of a protein from its amino acid sequence, mainly because of errors in the energy estimates underlying the prediction. However, a recently developed theory [1] shows that having a set of homologs (i.e., the chains with equal, in despite of numerous mutations, 3D folds) one can average the potential of each interaction over the homologs and thus predict the common 3D fold of protein family even when a correct fold prediction for an individual sequence is impossible because the energies are known only approximately. This theoretical conclusion has been verified by simulation of the energy spectra of simplified models of protein chains [2], and the further investigation of these simplified models shows that their true "native" fold can be found by folding of the chain where each interaction potential is averaged over the homologs. In conclusion, the applicability of the "homolog-averaging" approach is tested by recognition of real protein 3D structures. Both the gapless threading of sequences onto the known protein folds [3] and the more practically important gapped threading (which allows to consider not only the known 3D structures, but the more or less similar to them folds as well) shows a significant increase in selectivity of the native chain fold recognition.

Evidence for the role of PrP(C) helix 1 in the hydrophilic seeding of prion aggregates

Prions are mammalian proteins (PrPs) with a unique pathogenic property: a nonendogenous isoform PrP(Sc) can catalyze conversion of the endogenous PrP(C) isoform into additional PrP(Sc). In this work, we demonstrate that PrP(C) helix 1 has certain properties (hydrophilicity, charge distribution) that make it unique among all naturally occurring alpha-helices, and which are indicative of a highly specific model of prion infectivity. The beta-nucleation model proposes that PrP(Sc) is an aggregate with a hydrophilic core, consisting of a beta-sheet-like arrangement of constituent helix 1 components. It is suggested by using structural arguments, and confirmed by using CHARMM energy calculations, that aggregate formation from two PrP(C) molecules is highly unfavorable, but the addition of chains to an existing aggregate is favorable. The beta-nucleation model is shown to be consistent with the prion species-barrier, as well as with infectivity data. Sequence analysis of all known protein structures indicates that PrP is uniquely suited to beta-nucleation, in contrast to the many proteins that readily form less favorable (often nonspecific) hydrophobic aggregates.

Universally conserved positions in protein folds: reading evolutionary signals about stability, folding kinetics and function

Here, we provide an analysis of molecular evolution of five of the most populated protein folds: immunoglobulin fold, oligonucleotide-binding fold, Rossman fold, alpha/beta plait, and TIM barrels. In order to distinguish between "historic", functional and structural reasons for amino acid conservations, we consider proteins that acquire the same fold and have no evident sequence homology. For each fold we identify positions that are conserved within each individual family and coincide when non-homologous proteins are structurally superimposed. As a baseline for statistical assessment we use the conservatism expected based on the solvent accessibility. The analysis is based on a new concept of "conservatism-of-conservatism". This approach allows us to identify the structural features that are stabilized in all proteins having a given fold, despite the fact that actual interactions that provide such stabilization may vary from protein to protein. Comparison with experimental data on thermodynamics, folding kinetics and function of the proteins reveals that such universally conserved clusters correspond to either: (i) super-sites (common location of active site in proteins having common tertiary structures but not function) or (ii) folding nuclei whose stability is an important determinant of folding rate, or both (in the case of Rossman fold). The analysis also helps to clarify the relation between folding and function that is apparent for some folds.

Discrete molecular dynamics studies of the folding of a protein-like model

BACKGROUND

Many attempts have been made to resolve in time the folding of model proteins in computer simulations. Different computational approaches have emerged. Some of these approaches suffer from insensitivity to the geometrical properties of the proteins (lattice models), whereas others are computationally heavy (traditional molecular dynamics).

RESULTS

We used the recently proposed approach of Zhou and Karplus to study the folding of a protein model based on the discrete time molecular dynamics algorithm. We show that this algorithm resolves with respect to time the folding <--> unfolding transition. In addition, we demonstrate the ability to study the core of the model protein.

CONCLUSIONS

The algorithm along with the model of interresidue interactions can serve as a tool for studying the thermodynamics and kinetics of protein models.