Application of Multiplexed Replica Exchange Molecular Dynamics to the UNRES Force Field: Tests with alpha and alpha+beta Proteins

Molecular modeling of peptides

This article presents a review of the field of molecular modeling of peptides. The main focus is on atomistic modeling with molecular mechanics potentials. The description of peptide conformations and solvation through potentials is discussed. Several important computer simulation methods are briefly introduced, including molecular dynamics, accelerated sampling approaches such as replica-exchange and metadynamics, free energy simulations and kinetic network models like Milestoning. Examples of recent applications for predictions of structure, kinetics, and interactions of peptides with complex environments are described. The reliability of current simulation methods is analyzed by comparison of computational predictions obtained using different models with each other and with experimental data. A brief discussion of coarse-grained modeling and future directions is also presented.

Important roles of hydrophobic interactions in folding and charge interactions in misfolding of α-helix bundle protein

An enhanced-sampling molecular dynamics simulation is presented to quantitatively demonstrate the important roles of hydrophobic and charge interactions in the folding and misfolding of α-helix bundle protein, respectively.

Folding and self-assembly of a small heterotetramer

Designed miniproteins offer a possibility to study folding and association of protein complexes, both experimentally and in silico. Using replica exchange molecular dynamics and the coarse-grain UNRES force field, we have simulated the folding and self-assembly of the heterotetramer BBAThet1, comparing it with that of the homotetramer BBAT1 which has the same size and ββα-fold. For both proteins, association of the tetramer precedes and facilitates folding of the individual chains.

WeFold: a coopetition for protein structure prediction

The protein structure prediction problem continues to elude scientists. Despite the introduction of many methods, only modest gains were made over the last decade for certain classes of prediction targets. To address this challenge, a social-media based worldwide collaborative effort, named WeFold, was undertaken by 13 labs. During the collaboration, the laboratories were simultaneously competing with each other. Here, we present the first attempt at "coopetition" in scientific research applied to the protein structure prediction and refinement problems. The coopetition was possible by allowing the participating labs to contribute different components of their protein structure prediction pipelines and create new hybrid pipelines that they tested during CASP10. This manuscript describes both successes and areas needing improvement as identified throughout the first WeFold experiment and discusses the efforts that are underway to advance this initiative. A footprint of all contributions and structures are publicly accessible at http://www.wefold.org.

A unified coarse-grained model of biological macromolecules based on mean-field multipole-multipole interactions

A unified coarse-grained model of three major classes of biological molecules—proteins, nucleic acids, and polysaccharides—has been developed. It is based on the observations that the repeated units of biopolymers (peptide groups, nucleic acid bases, sugar rings) are highly polar and their charge distributions can be represented crudely as point multipoles. The model is an extension of the united residue (UNRES) coarse-grained model of proteins developed previously in our laboratory. The respective force fields are defined as the potentials of mean force of biomacromolecules immersed in water, where all degrees of freedom not considered in the model have been averaged out. Reducing the representation to one center per polar interaction site leads to the representation of average site–site interactions as mean-field dipole–dipole interactions. Further expansion of the potentials of mean force of biopolymer chains into Kubo’s cluster-cumulant series leads to the appearance of mean-field dipole–dipole interactions, averaged in the context of local interactions within a biopolymer unit. These mean-field interactions account for the formation of regular structures encountered in biomacromolecules, e.g., α-helices and β-sheets in proteins, double helices in nucleic acids, and helicoidally packed structures in polysaccharides, which enables us to use a greatly reduced number of interacting sites without sacrificing the ability to reproduce the correct architecture. This reduction results in an extension of the simulation timescale by more than four orders of magnitude compared to the all-atom representation. Examples of the performance of the model are presented.

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Revised Backbone-Virtual-Bond-Angle Potentials to Treat the l- and d-Amino Acid Residues in the Coarse-Grained United Residue (UNRES) Force Field

Continuing our effort to introduce d-amino-acid residues in the united residue (UNRES) force field developed in our laboratory, in this work the Cα ··· Cα ··· Cα backbone-virtual-bond-valence-angle (θ) potentials for systems containing d-amino-acid residues have been developed. The potentials were determined by integrating the combined energy surfaces of all possible triplets of terminally blocked glycine, alanine, and proline obtained with ab initio molecular quantum mechanics at the MP2/6-31G(d,p) level to calculate the corresponding potentials of mean force (PMFs). Subsequently, analytical expressions were fitted to the PMFs to give the virtual-bond-valence potentials to be used in UNRES. Alanine represented all types of amino-acid residues except glycine and proline. The blocking groups were either the N-acetyl and N',N'-dimethyl or N-acetyl and pyrrolidyl group, depending on whether the residue next in sequence was an alanine-type or a proline residue. A total of 126 potentials (63 symmetry-unrelated potentials for each set of terminally blocking groups) were determined. Together with the torsional, double-torsional, and side-chain-rotamer potentials for polypeptide chains containing d-amino-acid residues determined in our earlier work (Sieradzan et al. J. Chem. Theory Comput., 2012, 8, 4746), the new virtual-bond-angle (θ) potentials now constitute the complete set of physics-based potentials with which to run coarse-grained simulations of systems containing d-amino-acid residues. The ability of the extended UNRES force field to reproduce thermodynamics of polypeptide systems with d-amino-acid residues was tested by comparing the experimentally measured and the calculated free energies of helix formation of model KLALKLALxxLKLALKLA peptides, where x denotes any d- or l- amino-acid residue. The obtained results demonstrate that the UNRES force field with the new potentials reproduce the changes of free energies of helix formation upon d-substitution but overestimate the free energies of helix formation. To test the ability of UNRES with the new potentials to reproduce the structures of polypeptides with d-amino-acid residues, an ab initio replica-exchange folding simulation of thurincin H from Bacillus thuringiensis, which has d-amino-acid residues in the sequence, was carried out. UNRES was able to locate the native α-helical hairpin structure as the dominant structure even though no native sulfide-carbon bonds were present in the simulation.

The LA loop as an important regulatory element of the HtrA (DegP) protease from Escherichia coli: structural and functional studies

Bacterial HtrAs are serine proteases engaged in extracytoplasmic protein quality control and are required for the virulence of several pathogenic species. The proteolytic activity of HtrA (DegP) from Escherichia coli, a model prokaryotic HtrA, is stimulated by stressful conditions; the regulation of this process is mediated by the LA, LD, L1, L2, and L3 loops. The precise mechanism of action of the LA loop is not known due to a lack of data concerning its three-dimensional structure as well as its mode of interaction with other regulatory elements. To address these issues we generated a theoretical model of the three-dimensional structure of the LA loop as per the resting state of HtrA and subsequently verified its correctness experimentally. We identified intra- and intersubunit contacts that formed with the LA loops; these played an important role in maintaining HtrA in its inactive conformation. The most significant proved to be the hydrophobic interactions connecting the LA loops of the hexamer and polar contacts between the LA' (the LA loop on an opposite subunit) and L1 loops on opposite subunits. Disturbance of these interactions caused the stimulation of HtrA proteolytic activity. We also demonstrated that LA loops contribute to the preservation of the integrity of the HtrA oligomer and to the stability of the monomer. The model presented in this work explains the regulatory role of the LA loop well; it should also be applicable to numerous Enterobacteriaceae pathogenic species as the amino acid sequences of the members of this bacterial family are highly conserved.

SSThread: Template-free protein structure prediction by threading pairs of contacting secondary structures followed by assembly of overlapping pairs

Acquiring the three-dimensional structure of a protein from its amino acid sequence alone, despite a great deal of work and significant progress on the subject, is still an unsolved problem. SSThread, a new template-free algorithm is described here that consists of making several predictions of contacting pairs of α-helices and β-strands derived from a database of experimental structures using a knowledge-based potential, secondary structure prediction, and contact map prediction followed by assembly of overlapping pair predictions to create an ensemble of core structure predictions whose loops are then predicted. In a set of seven CASP10 targets SSThread outperformed the two leading methods for two targets each. The targets were all β-strand containing structures and most of them have a high relative contact order which demonstrates the advantages of SSThread. The primary bottlenecks based on sets of 74 and 21 test cases are the pair prediction and loop prediction stages.

Improvement of the treatment of loop structures in the UNRES force field by inclusion of coupling between backbone- and side-chain-local conformational states

The UNited RESidue (UNRES) coarse-grained model of polypeptide chains, developed in our laboratory, enables us to carry out millisecond-scale molecular-dynamics simulations of large proteins effectively. It performs well in ab initio predictions of protein structure, as demonstrated in the last Community Wide Experiment on the Critical Assessment of Techniques for Protein Structure Prediction (CASP10). However, the resolution of the simulated structure is too coarse, especially in loop regions, which results from insufficient specificity of the model of local interactions. To improve the representation of local interactions, in this work we introduced new side-chain-backbone correlation potentials, derived from a statistical analysis of loop regions of 4585 proteins. To obtain sufficient statistics, we reduced the set of amino-acid-residue types to five groups, derived in our earlier work on structurally optimized reduced alphabets, based on a statistical analysis of the properties of amino-acid structures. The new correlation potentials are expressed as one-dimensional Fourier series in the virtual-bond-dihedral angles involving side-chain centroids. The weight of these new terms was determined by a trial-and-error method, in which Multiplexed Replica Exchange Molecular Dynamics (MREMD) simulations were run on selected test proteins. The best average root-mean-square deviations (RMSDs) of the calculated structures from the experimental structures below the folding-transition temperatures were obtained with the weight of the new side-chain-backbone correlation potentials equal to 0.57. The resulting conformational ensembles were analyzed in detail by using the Weighted Histogram Analysis Method (WHAM) and Ward's minimum-variance clustering. This analysis showed that the RMSDs from the experimental structures dropped by 0.5 Å on average, compared to simulations without the new terms, and the deviation of individual residues in the loop region of the computed structures from their counterparts in the experimental structures (after optimum superposition of the calculated and experimental structure) decreased by up to 8 Å. Consequently, the new terms improve the representation of local structure.

Lessons from application of the UNRES force field to predictions of structures of CASP10 targets

The performance of the physics-based protocol, whose main component is the United Residue (UNRES) physics-based coarse-grained force field, developed in our laboratory for the prediction of protein structure from amino acid sequence, is illustrated. Candidate models are selected, based on probabilities of the conformational families determined by multiplexed replica-exchange simulations, from the 10th Community Wide Experiment on the Critical Assessment of Techniques for Protein Structure Prediction (CASP10). For target T0663, classified as a new fold, which consists of two domains homologous to those of known proteins, UNRES predicted the correct symmetry of packing, in which the domains are rotated with respect to each other by 180° in the experimental structure. By contrast, models obtained by knowledge-based methods, in which each domain is modeled very accurately but not rotated, resulted in incorrect packing. Two UNRES models of this target were featured by the assessors. Correct domain packing was also predicted by UNRES for the homologous target T0644, which has a similar structure to that of T0663, except that the two domains are not rotated. Predictions for two other targets, T0668 and T0684_D2, are among the best ones by global distance test score. These results suggest that our physics-based method has substantial predictive power. In particular, it has the ability to predict domain-domain orientations, which is a significant advance in the state of the art.

Electrostatic-Consistent Coarse-Grained Potentials for Molecular Simulations of Proteins

We present a new generation of coarse-grained (CG) potentials that account for a simplified electrostatic description of soluble proteins. The treatment of permanent electrostatic dipoles of the backbone and polar side-chains allows to simulate proteins, preserving an excellent structural and dynamic agreement with respective reference structures and all-atom molecular dynamics simulations. Moreover, multiprotein complexes can be well described maintaining their molecular interfaces thanks to the ability of this scheme to better describe the actual electrostatics at a CG level of resolution. An efficient and robust heuristic algorithm based on particle swarm optimization is used for the derivation of CG parameters via a force-matching procedure. The ability of this protocol to deal with high dimensional search spaces suggests that the extension of this optimization procedure to larger data sets may lead to the generation of a fully transferable CG force field. At the present stage, these electrostatic-consistent CG potentials are easily and efficiently parametrized, show a good degree of transferability, and can be used to simulate soluble proteins or, more interestingly, large macromolecular assemblies for which long all-atom simulations may not be easily affordable.

Mean-field interactions between nucleic-acid-base dipoles can drive the formation of a double helix

A proposed coarse-grained model of nucleic acids demonstrates that average interactions between base dipoles, together with chain connectivity and excluded-volume interactions, are sufficient to form double-helical structures of DNA and RNA molecules. Additionally, local interactions determine helix handedness and direction of strand packing. This result, and earlier research on reduced protein models, suggests that mean-field multipole-multipole interactions are the principal factors responsible for the formation of regular structure of biomolecules.

Coarse-grained models for protein folding and aggregation

Coarse-grained models for protein folding and aggregation are used to explore large dimension scales and timescales that are inaccessible to all-atom models in explicit aqueous solution. Combined with enhanced configuration search methods, these simplified models with various levels of granularity offer the possibility to determine equilibrium structures, compare folding kinetics and thermodynamics with experiments for single proteins and understand the dynamic assembly of amyloid proteins leading to neurodegenerative diseases. I shall describe recent progress in developing such models, and discuss their potentials and limitations in probing the folding and misfolding of proteins with computer simulations.

Replica exchanging self-guided Langevin dynamics for efficient and accurate conformational sampling

This work presents a replica exchanging self-guided Langevin dynamics (RXSGLD) simulation method for efficient conformational searching and sampling. Unlike temperature-based replica exchanging simulations, which use high temperatures to accelerate conformational motion, this method uses self-guided Langevin dynamics (SGLD) to enhance conformational searching without the need to elevate temperatures. A RXSGLD simulation includes a series of SGLD simulations, with simulation conditions differing in the guiding effect and/or temperature. These simulation conditions are called stages and the base stage is one with no guiding effect. Replicas of a simulation system are simulated at the stages and are exchanged according to the replica exchanging probability derived from the SGLD partition function. Because SGLD causes less perturbation on conformational distribution than high temperatures, exchanges between SGLD stages have much higher probabilities than those between different temperatures. Therefore, RXSGLD simulations have higher conformational searching ability than temperature based replica exchange simulations. Through three example systems, we demonstrate that RXSGLD can generate target canonical ensemble distribution at the base stage and achieve accelerated conformational searching. Especially for large systems, RXSGLD has remarkable advantages in terms of replica exchange efficiency, conformational searching ability, and system size extensiveness.

Coexistence of phases in a protein heterodimer

A heterodimer consisting of two or more different kinds of proteins can display an enormous number of distinct molecular architectures. The conformational entropy is an essential ingredient in the Helmholtz free energy and, consequently, these heterodimers can have a very complex phase structure. Here, it is proposed that there is a state of proteins, in which the different components of a heterodimer exist in different phases. For this purpose, the structures in the protein data bank (PDB) have been analyzed, with radius of gyration as the order parameter. Two major classes of heterodimers with their protein components coexisting in different phases have been identified. An example is the PDB structure 3DXC. This is a transcriptionally active dimer. One of the components is an isoform of the intra-cellular domain of the Alzheimer-disease related amyloid precursor protein (AICD), and the other is a nuclear multidomain adaptor protein in the Fe65 family. It is concluded from the radius of gyration that neither of the two components in this dimer is in its own collapsed phase, corresponding to a biologically active protein. The UNRES energy function has been utilized to confirm that, if the two components are separated from each other, each of them collapses. The results presented in this work show that heterodimers whose protein components coexist in different phases, can have intriguing physical properties with potentially important biological consequences.

Folding and self-assembly of a small protein complex

The synthetic homotetrameric ββα (BBAT1) protein possesses a stable quaternary structure with a ββα fold. Because of its small size (a total of 84 residues), the homotetramer is an excellent model system with which to study the self-assembly and protein-protein interactions. We find from replica exchange molecular dynamics simulations with the coarse-grain UNRES force field that the folding and association pathway consists of three well-separated steps, where that association to a tetramer precedes and facilitates folding of the four chains. At room temperature the tetramer exists in an ensemble of diverse structures. The crystal structure becomes energetically favored only when the molecule is put in a dense and crystal-like environment. The observed picture of folding promoted by association may mirror the mechanism according to which intrinsically unfolded proteins assume their functional structure.

Folding helical proteins in explicit solvent using dihedral-biased tempering

Using a single-trajectory-based tempering method with a high-temperature dihedral bias, we repeatedly folded four helical proteins [α(3)D (PDB ID: 2A3D, 73 residues), α(3)W (1LQ7, 67 residues), Fap1-NR(α) (2KUB, 81 residues) and S-836 (2JUA, 102 residues)] and some of the mutants in explicit solvent within several microseconds. The lowest root-mean-square deviations of backbone atoms from the experimentally determined structures were 1.9, 1.4, 1.0, and 2.1 Å, respectively. Cluster analyses of folding trajectories showed the native conformation usually occupied the most populated cluster. The simulation protocol can be applied to large-scale simulations of other helical proteins on commonly accessible computing platforms.

Simulation of the opening and closing of Hsp70 chaperones by coarse-grained molecular dynamics

Heat-shock proteins 70 (Hsp70s) are key molecular chaperones which assist in the folding and refolding/disaggregation of proteins. Hsp70s, which consist of a nucleotide-binding domain (NBD, consisting of NBD-I and NBD-II subdomains) and a substrate-binding domain [SBD, further split into the β-sheet (SBD-β) and α-helical (SBD-α) subdomains], occur in two major conformations having (a) a closed SBD, in which the SBD and NBD domains do not interact, (b) an open SBD, in which SBD-α interacts with NBD-I and SBD-β interacts with the top parts of NBD-I and NBD-II. In the SBD-closed conformation, SBD is bound to a substrate protein, with release occurring after transition to the open conformation. While the transition from the closed to the open conformation is triggered efficiently by binding of adenosine triphosphate (ATP) to the NBD, it also occurs, although less frequently, in the absence of ATP. The reverse transition occurs after ATP hydrolysis. Here, we report canonical and multiplexed replica exchange simulations of the conformational dynamics of Hsp70s using a coarse-grained molecular dynamics approach with the UNRES force field. The simulations were run in the following three modes: (i) with the two halves of the NBD unrestrained relative to each other, (ii) with the two halves of the NBD restrained in an "open" geometry as in the SBD-closed form of DnaK (2KHO), and (iii) the two halves of NBD restrained in a "closed" geometry as in known experimental structures of ATP-bound NBD forms of Hsp70. Open conformations, in which the SBD interacted strongly with the NBD, formed spontaneously during all simulations; the number of transitions was largest in simulations carried out with the "closed" NBD domain, and smallest in those carried out with the "open" NBD domain; this observation is in agreement with the experimentally-observed influence of ATP-binding on the transition of Hsp70's from the SBD-closed to the SBD-open form. Two kinds of open conformations were observed: one in which SBD-α interacts with NBD-I and SBD-β interacts with the top parts of NBD-I and NBD-II (as observed in the structures of nucleotide exchange factors), and another one in which this interaction pattern is swapped. A third type of motion, in which SBD-α binds to NBD without dissociating from SBD-β was also observed. It was found that the first stage of interdomain communication (approach of SBD-β, to NBD) is coupled with the rotation of the long axes of NBD-I and NBD-II towards each other. To the best of our knowledge, this is the first successful simulation of the full transition of an Hsp70 from the SBD-closed to the SBD-open conformation.

Effects of mutation, truncation, and temperature on the folding kinetics of a WW domain

The purpose of this work is to show how mutation, truncation, and change of temperature can influence the folding kinetics of a protein. This is accomplished by principal component analysis of molecular-dynamics-generated folding trajectories of the triple β-strand WW domain from formin binding protein 28 (FBP28) (Protein Data Bank ID: 1E0L) and its full-size, and singly- and doubly-truncated mutants at temperatures below and very close to the melting point. The reasons for biphasic folding kinetics [i.e., coexistence of slow (three-state) and fast (two-state) phases], including the involvement of a solvent-exposed hydrophobic cluster and another delocalized hydrophobic core in the folding kinetics, are discussed. New folding pathways are identified in free-energy landscapes determined in terms of principal components for full-size mutants. Three-state folding is found to be a main mechanism for folding the FBP28 WW domain and most of the full-size and truncated mutants. The results from the theoretical analysis are compared to those from experiment. Agreements and discrepancies between the theoretical and experimental results are discussed. Because of its importance in understanding protein kinetics and function, the diffusive mechanism by which the FBP28 WW domain and its full-size and truncated mutants explore their conformational space is examined in terms of the mean-square displacement and principal component analysis eigenvalue spectrum analyses. Subdiffusive behavior is observed for all studied systems.

Respice, adspice, and prospice

The title, "Look to the past, Look to the present, and Look to the future," the motto of City College of New York, expresses how my family life and education led me to an academic career in physical chemistry and ultimately to a study of proteins. The economic depression of the 1930s left a lasting impression on my outlook and career aspirations. With fortunate experiences at several stages in my life, I was able to participate in the great adventure of the last half of the twentieth century: the revolution in biology that advanced the field of protein chemistry to so great an extent. The future is bright and limitless, with greater understanding of biology yet to come.

Coarse-grained force field: general folding theory

We review the coarse-grained UNited RESidue (UNRES) force field for the simulations of protein structure and dynamics, which is being developed in our laboratory over the last several years. UNRES is a physics-based force field, the prototype of which is defined as a potential of mean force of polypeptide chains in water, where all the degrees of freedom except the coordinates of α-carbon atoms and side-chain centers have been integrated out. We describe the initial implementation of UNRES to protein-structure prediction formulated as a search for the global minimum of the potential-energy function and its subsequent molecular dynamics and extensions of molecular-dynamics implementation, which enabled us to study protein-folding pathways and thermodynamics, as well as to reformulate the protein-structure prediction problem as a search for the conformational ensemble with the lowest free energy at temperatures below the folding-transition temperature. Applications of UNRES to study biological problems are also described.

Mechanism of fiber assembly: treatment of Aβ peptide aggregation with a coarse-grained united-residue force field

The growth mechanism of β-amyloid (Aβ) peptide fibrils was studied by a physics-based coarse-grained united-residue model and molecular dynamics (MD) simulations. To identify the mechanism of monomer addition to an Aβ(1-40) fibril, we placed an unstructured monomer at a distance of 20 Å from a fibril template and allowed it to interact freely with the latter. The monomer was not biased towards fibril conformation by either the force field or the MD algorithm. With the use of a coarse-grained model with replica-exchange molecular dynamics, a longer timescale was accessible, making it possible to observe how the monomers probe different binding modes during their search for the fibril conformation. Although different assembly pathways were seen, they all follow a dock-lock mechanism with two distinct locking stages, consistent with experimental data on fibril elongation. Whereas these experiments have not been able to characterize the conformations populating the different stages, we have been able to describe these different stages explicitly by following free monomers as they dock onto a fibril template and to adopt the fibril conformation (i.e., we describe fibril elongation step by step at the molecular level). During the first stage of the assembly ("docking"), the monomer tries different conformations. After docking, the monomer is locked into the fibril through two different locking stages. In the first stage, the monomer forms hydrogen bonds with the fibril template along one of the strands in a two-stranded β-hairpin; in the second stage, hydrogen bonds are formed along the second strand, locking the monomer into the fibril structure. The data reveal a free-energy barrier separating the two locking stages. The importance of hydrophobic interactions and hydrogen bonds in the stability of the Aβ fibril structure was examined by carrying out additional canonical MD simulations of oligomers with different numbers of chains (4-16 chains), with the fibril structure as the initial conformation. The data confirm that the structures are stabilized largely by hydrophobic interactions and show that intermolecular hydrogen bonds are highly stable and contribute to the stability of the oligomers as well.

Evidence, from simulations, of a single state with residual native structure at the thermal denaturation midpoint of a small globular protein

The folding of the B-domain of staphylococcal protein A has been studied by coarse-grained canonical and multiplexed replica-exchange molecular dynamics simulations with the UNRES force field in a broad range of temperatures (270 K < or = T < or = 350 K). In canonical simulations, the folding was found to occur either directly to the native state or through kinetic traps, mainly the topological mirror image of the native three-helix bundle. The latter folding scenario was observed more frequently at low temperatures. With increase of temperature, the frequency of the transitions between the folded and misfolded/unfolded states increased and the folded state became more diffuse with conformations exhibiting increased root-mean-square deviations from the experimental structure (from about 4 A at T = 300 K to 8.7 A at T = 325 K). An analysis of the equilibrium conformational ensemble determined from multiplexed replica exchange simulations at the folding-transition temperature (T(f) = 325 K) showed that the conformational ensemble at this temperature is a collection of conformations with residual secondary structures, which possess native or near-native clusters of nonpolar residues in place, and not a 50-50% mixture of fully folded and fully unfolded conformations. These findings contradict the quasi-chemical picture of two- or multistate protein folding, which assumes an equilibrium between the folded, unfolded, and intermediate states, with equilibrium shifting with temperature but with the native conformations remaining essentially unchanged. Our results also suggest that long-range hydrophobic contacts are the essential factor to keep the structure of a protein thermally stable.

Determination of side-chain-rotamer and side-chain and backbone virtual-bond-stretching potentials of mean force from AM1 energy surfaces of terminally-blocked amino-acid residues, for coarse-grained simulations of protein structure and folding. I. The method

In this and the accompanying article, we report the development of new physics-based side-chain-rotamer and virtual-bond-deformation potentials which now replace the respective statistical potentials used so far in our physics-based united-reside UNRES force field for large-scale simulations of protein structure and dynamics. In this article, we describe the methodology for determining the corresponding potentials of mean force (PMF's) from the energy surfaces of terminally-blocked amino-acid residues calculated with the AM1 quantum-mechanical semiempirical method. The approach is based on minimization of the AM1 energy for fixed values of the angles lambda for rotation of the peptide groups about the C(alpha)...C(alpha) virtual bonds, and for fixed values of the side-chain dihedral angles chi, which formed a multidimensional grid. A harmonic-approximation approach was developed to extrapolate from the energy at a given grid point to other points of the conformational space to compute the respective contributions to the PMF. To test the applicability of the harmonic approximation, the rotamer PMF's of alanine and valine obtained with this approach have been compared with those obtained by using a Metropolis Monte Carlo method. The PMF surfaces computed with the harmonic approximation are more rugged and have more pronounced minima than the MC-calculated surfaces but the harmonic-approximation- and MC-calculated PMF values are linearly correlated. The potentials derived with the harmonic approximation are, therefore, appropriate for UNRES for which the weights (scaling factors) of the energy terms are determined by force-field optimization for foldability.

Determination of side-chain-rotamer and side-chain and backbone virtual-bond-stretching potentials of mean force from AM1 energy surfaces of terminally-blocked amino-acid residues, for coarse-grained simulations of protein structure and folding. II. Results, comparison with statistical potentials, and implementation in the UNRES force field

Using the harmonic-approximation approach of the accompanying article and AM1 energy surfaces of terminally blocked amino-acid residues, we determined physics-based side-chain rotamer potentials and the side-chain virtual-bond-deformation potentials of 19 natural amino-acid residues with side chains. The potentials were approximated by analytical formulas and implemented in the UNRES mesoscopic dynamics program. For comparison, the corresponding statistical potentials were determined from 19,682 high-resolution protein structures. The low free-energy region of both the AM1-derived and the statistical potentials is determined by the valence geometry and the L-chirality, and its size increases with side-chain flexibility and decreases with increasing virtual-bond-angle theta. The differences between the free energies of rotamers are greater for the AM1-derived potentials compared with the statistical potentials and, for alanine and other residues with small side chains, a region corresponding to the C(ax)(7) conformation has remarkably low free-energy for the AM1-derived potentials, as opposed to the statistical potentials. These differences probably result from the interactions between neighboring residues and indicate the need for introduction of cooperative terms accounting for the coupling between side-chain rotamer and backbone interactions. Both AM1-derived and statistical virtual-bond-deformation potentials are multimodal for flexible side chains and are topologically similar; however, the regions of minima of the statistical potentials are much narrower, which probably results from imposing restraints in structure determination. The force field with the new potentials was preliminarily optimized using the FBP WW domain (1E0L) and the engrailed homeodomain (1ENH) as training proteins and assessed to be reasonably transferable.

Investigation of protein folding by coarse-grained molecular dynamics with the UNRES force field

Coarse-grained molecular dynamics simulations offer a dramatic extension of the time-scale of simulations compared to all-atom approaches. In this article, we describe the use of the physics-based united-residue (UNRES) force field, developed in our laboratory, in protein-structure simulations. We demonstrate that this force field offers about a 4000-times extension of the simulation time scale; this feature arises both from averaging out the fast-moving degrees of freedom and reduction of the cost of energy and force calculations compared to all-atom approaches with explicit solvent. With massively parallel computers, microsecond folding simulation times of proteins containing about 1000 residues can be obtained in days. A straightforward application of canonical UNRES/MD simulations, demonstrated with the example of the N-terminal part of the B-domain of staphylococcal protein A (PDB code: 1BDD, a three-alpha-helix bundle), discerns the folding mechanism and determines kinetic parameters by parallel simulations of several hundred or more trajectories. Use of generalized-ensemble techniques, of which the multiplexed replica exchange method proved to be the most effective, enables us to compute thermodynamics of folding and carry out fully physics-based prediction of protein structure, in which the predicted structure is determined as a mean over the most populated ensemble below the folding-transition temperature. By using principal component analysis of the UNRES folding trajectories of the formin-binding protein WW domain (PDB code: 1E0L; a three-stranded antiparallel beta-sheet) and 1BDD, we identified representative structures along the folding pathways and demonstrated that only a few (low-indexed) principal components can capture the main structural features of a protein-folding trajectory; the potentials of mean force calculated along these essential modes exhibit multiple minima, as opposed to those along the remaining modes that are unimodal. In addition, a comparison between the structures that are representative of the minima in the free-energy profile along the essential collective coordinates of protein folding (computed by principal component analysis) and the free-energy profile projected along the virtual-bond dihedral angles gamma of the backbone revealed the key residues involved in the transitions between the different basins of the folding free-energy profile, in agreement with existing experimental data for 1E0L .

Implementation of molecular dynamics and its extensions with the coarse-grained UNRES force field on massively parallel systems; towards millisecond-scale simulations of protein structure, dynamics, and thermodynamics

We report the implementation of our united-residue UNRES force field for simulations of protein structure and dynamics with massively parallel architectures. In addition to coarse-grained parallelism already implemented in our previous work, in which each conformation was treated by a different task, we introduce a fine-grained level in which energy and gradient evaluation are split between several tasks. The Message Passing Interface (MPI) libraries have been utilized to construct the parallel code. The parallel performance of the code has been tested on a professional Beowulf cluster (Xeon Quad Core), a Cray XT3 supercomputer, and two IBM BlueGene/P supercomputers with canonical and replica-exchange molecular dynamics. With IBM BlueGene/P, about 50 % efficiency and 120-fold speed-up of the fine-grained part was achieved for a single trajectory of a 767-residue protein with use of 256 processors/trajectory. Because of averaging over the fast degrees of freedom, UNRES provides an effective 1000-fold speed-up compared to the experimental time scale and, therefore, enables us to effectively carry out millisecond-scale simulations of proteins with 500 and more amino-acid residues in days of wall-clock time.

An improved functional form for the temperature scaling factors of the components of the mesoscopic UNRES force field for simulations of protein structure and dynamics

Coarse-grained or mesoscopic models of proteins and the corresponding force fields are of great importance because they enable us to reduce the folding simulation time by several orders of magnitude compared to the all-atom approach and, consequently, reach the millisecond time scale of simulations. In the coarse-grained UNRES model for simulations of protein structure and dynamics, developed by our group, each amino acid residue is represented by a united side chain and a united peptide group located in the middle between the two neighboring alpha-carbon atoms, which assist only in the definition of the geometry. The prototype of the UNRES force field has been defined as a potential of mean force or restricted free-energy function corresponding to averaging out the degrees of freedom not present in the coarse-grained representation, which has further been approximated by a truncated Kubo cumulant series to enable us to derive analytical expressions for the corresponding terms. This force field should depend on temperature, and in its simplest form, a term corresponding to the cumulant of order n should be multiplied by f(n) = 1/T(n-1). The temperature dependence has been introduced in recent work ( J. Phys. Chem. B , 2007 , 111 , 260 - 285 ), and in order to prevent too steep a variation with temperature, the factors at the nth order cumulant terms were assumed to have a form f(n) = ln[exp(1) + exp(-1)]/ln{exp[(T/T(0))(n-1)] + exp[-(T/T(0))(n-1)]}, where T(0) = 300 K is the reference temperature. In this work, we have introduced a modified scaling factor f(n) = ln[exp(c) + exp(-c)]/ln{exp[c(T/T(0))(n-1)] + exp[-c(T/T(0))(n-1)]}, where c is an adjustable parameter, and determined c by fitting the analytical approximation of the temperature dependence of the virtual bond torsional term corresponding to rotation about the C(alpha)...C(alpha) virtual bond in terminally blocked dialanine to the respective potential of mean force calculated from the MP2/6-31G(d, p) ab initio energy surfaces of terminally blocked alanine (Ac-Ala-NHMe) and, independently, by optimizing it to obtain a sharp heat capacity curve and the lowest ensemble-averaged root-mean-square deviation over the C(alpha) atoms of 1GAB used as a training protein. Both approaches gave consistent results, and c = 1.4 has been selected as the optimal value of this parameter. The force field with the new temperature scaling factors has been optimized using 1GAB as the training protein. The new force field has been tested on a series of medium size alpha-helical proteins and found to perform better than that with the original temperature scaling factors.