Protein-Folding Dynamics: Overview of Molecular Simulation Techniques

Temperature control in molecular dynamic simulations of non-equilibrium processes

Thermostats are often used in various condensed matter problems, e.g. when a biological molecule undergoes a transformation in a solution, a crystal surface is irradiated with energetic particles, a crack propagates in a solid upon applied stress, two surfaces slide with respect to each other, an excited local phonon dissipates its energy into a crystal bulk, and so on. In all of these problems, as well as in many others, there is an energy transfer between different local parts of the entire system kept at a constant temperature. Very often, when modelling such processes using molecular dynamics simulations, thermostatting is done using strictly equilibrium approaches serving to describe the NVT ensemble. In this paper we critically discuss the applicability of such approaches to non-equilibrium problems, including those mentioned above, and stress that the correct temperature control can only be achieved if the method is based on the generalized Langevin equation (GLE). Specifically, we emphasize that a meaningful compromise between computational efficiency and a physically appropriate implementation of the NVT thermostat can be achieved, at least for solid state and surface problems, if the so-called stochastic boundary conditions (SBC), recently derived from the GLE (Kantorovich and Rompotis 2008 Phys. Rev. B 78 094305), are used. For SBC, the Langevin thermostat is only applied to the outer part of the simulated fragment of the entire system which borders the surrounding environment (not considered explicitly) serving as a heat bath. This point is illustrated by comparing the performance of the SBC and some of the equilibrium thermostats in two problems: (i) irradiation of the Si(001) surface with an energetic CaF(2) molecule using an ab initio density functional theory based method, and (ii) the tribology of two amorphous SiO(2) surfaces coated with self-assembled monolayers of methyl-terminated hydrocarbon alkoxylsilane molecules using a classical atomistic force field. We discuss the differences in behaviour of these systems due to applied thermostatting, and show that in some cases a qualitatively different physical behaviour of the simulated system can be obtained if an equilibrium thermostat is used.

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.

On the information expressed in enzyme primary structure: lessons from Ribonuclease A

The information expressed in an enzyme's primary structure is investigated. Brownian computations are directed at Ribonuclease A (RNase A) so as to quantify the information at the atom/covalent bond level. The information content and distribution are crucial because the primary structure underpins the molecule's chemical functions. Brownian computation data are illustrated for the native protein, mutants, and sequence isomers. The results identify signature features of the active protein on new information grounds. The same tools offer rapid screening of proteins and polypeptides whereby several examples are illustrated.

Conformational averaging in structural biology: issues, challenges and computational solutions

Most experimental methods in structural biology provide time- and ensemble-averaged signals and, consequently, molecular structures based on such signals often exhibit only idealized, average features. Second, most experimental signals are only indirectly related to real, molecular geometries, and solving a structure typically involves a complicated procedure, which may not always result in a unique solution. To what extent do such conformationally-averaged, non-linear experimental signals and structural models derived from them accurately represent the underlying microscopic reality? Are there some structural motifs that are actually artificially more likely to be "seen" in an experiment simply due to the averaging artifact? Finally, what are the practical consequences of ignoring the averaging effects when it comes to functional and mechanistic implications that we try to glean from experimentally-based structural models? In this review, we critically address the work that has been aimed at studying such questions. We summarize the details of experimental methods typically used in structural biology (most notably nuclear magnetic resonance, X-ray crystallography and different types of spectroscopy), discuss their individual susceptibility to conformational (motional) averaging, and review several theoretical approaches, most importantly molecular dynamics simulations that are increasingly being used to aid experimentalists in interpreting structural biology experiments.

Deconstructing the native state: energy landscapes, function, and dynamics of globular proteins

Proteins are highly complex molecules with features exquisitely selected by nature to carry out essential biological functions. Physical chemistry and polymer physics provide us with the tools needed to make sense of this complexity. Upon translation, many proteins fold to a thermodynamically stable form known as the native state. The native state is not static, but consists of a hierarchy of conformations, that are continuously explored through dynamics. In this review we provide a brief introduction to some of the core concepts required in the discussion of the protein native dynamics using energy landscapes ideas. We first discuss recent works which have challenged the structure-function paradigm by demonstrating function in disordered proteins. Next we examine the hierarchical organization in the energy landscapes using atomistic molecular dynamics simulations and principal component analysis. In particular, the role of direct and water-mediated contacts in sculpting the landscape is elaborated. Another approach to studying the native state ensemble is based on choosing high-resolution order parameters for computing one- or two-dimensional free energy surfaces. We demonstrate that 2D free energy surfaces provide rich thermodynamic and kinetic information about the native state ensemble. Brownian dynamics simulations on such a surface indicate that protein conformational dynamics is weakly activated. Finally, we briefly discuss implicit and coarse-grained protein models and emphasize the solvent role in determining native state structure and dynamics.

Structure and stability of helices in square-well homopolymers

Recently, it has been demonstrated [Magee, Phys. Rev. Lett. 96, 207802 (2006)] that isolated square-well homopolymers can spontaneously break chiral symmetry and "freeze" into helical structures at sufficiently low temperatures. This behavior is interesting because the square-well homopolymer is itself achiral. In this work, we use event-driven molecular dynamics combined with an optimized parallel tempering scheme to study this polymer model over a wide range of parameters. We examine the conditions where the helix structure is stable and determine how the interaction parameters of the polymer govern the details of the helix structure. The width of the square well (proportional to lambda) is found to control the radius of the helix, which decreases with increasing well width until the polymer forms a coiled sphere for sufficiently large wells. The helices are found to be stable for only a "window" of molecular weights. If the polymer is too short, the helix will not form. If the polymer is too long, the helix is no longer the minimum energy structure, and other folded structures will form. The size of this window is governed by the chain stiffness, which in this model is a function of the ratio of the monomer size to the bond length. Outside this window, the polymer still freezes into a locked structure at low temperature; however, unless the chain is sufficiently stiff, this structure will not be unique and is similar to a glassy state.

Atomistic details of the disordered states of KID and pKID. Implications in coupled binding and folding

Intrinsically disordered proteins (IDPs) are a newly recognized class of functional proteins for which a lack of stable tertiary fold is required for function. Because of the heterogeneous and dynamical nature, molecular modeling is necessary to provide the missing details of disordered states of IDP that are crucial for understanding their functions. In particular, generalized Born (GB) implicit solvent, combined with replica exchange (REX), might offer an optimal balance between accuracy and efficiency for modeling IDPs. We carried out extensive REX simulations in an optimized GB force field to characterize the disordered states of a regulatory IDP, KID domain of transcription factor CREB, and its phosphorylated form, pKID. The results revealed that both KID and pKID, though highly disordered on the tertiary level, are compact and mainly occupy a small number of helical substates. Interestingly, although phosphorylation of KID Ser133 leads only to marginal changes in average helicities on the ensemble level, underlying conformational substates differ significantly. In particular, pSer133 appears to restrict the accessible conformational space of the loop region and thus reduces the entropic cost of KID folding upon binding to the KIX domain of CREB-binding protein. Such an expanded role of phosphorylation in the KID:KIX recognition was not previously recognized because of a lack of substantial conformational changes on the ensemble level and inaccessibility of the structural details from experiments. The results also suggest that an implicit solvent-based modeling framework, despite various existing limitations, might be feasible for accurate atomistic simulation of small IDPs in general.

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

The replica exchange (RE) method is increasingly used to improve sampling in molecular dynamics (MD) simulations of biomolecular systems. Recently, we implemented the united-residue UNRES force field for mesoscopic MD. Initial results from UNRES MD simulations show that we are able to simulate folding events that take place in a microsecond or even a millisecond time scale. To speed up the search further, we applied the multiplexing replica exchange molecular dynamics (MREMD) method. The multiplexed variant (MREMD) of the RE method, developed by Rhee and Pande, differs from the original RE method in that several trajectories are run at a given temperature. Each set of trajectories run at a different temperature constitutes a layer. Exchanges are attempted not only within a single layer but also between layers. The code has been parallelized and scales up to 4000 processors. We present a comparison of canonical MD, REMD, and MREMD simulations of protein folding with the UNRES force-field. We demonstrate that the multiplexed procedure increases the power of replica exchange MD considerably and convergence of the thermodynamic quantities is achieved much faster.

Molecular dynamics simulation of the acidic compact state of apomyoglobin from yellowfin tuna

A molecular model of the acidic compact state of apomyoglobin (A-state) from yellowfin tuna was obtained using molecular dynamics simulations (MD) by calculating multiple trajectories. To cause partial unfolding within a reasonable amount of CPU time, both an acidic environment (pH 3 and 0.15M NaCl) and a temperature jump to 500 K were needed. Twenty-five acidic structures of apomyoglobin were generated by MD, 10 of them can be clustered by RMSD in an average structure having a common hydrophobic core as was reported for acidic sperm whale apomyoglobin, with shortened helices A,G,E, and H (the helix A appears to be translated along the sequence). Prolonging the MD runs at 500 K did not cause further substantial unfolding, suggesting that the ensemble of generated structures is indicative of a region of the conformational space accessible to the apoprotein at acidic pH corresponding to a local energy minimum. The comparison of experimentally determined values of specific spectroscopic properties of the apomyoglobin in acidic salt conditions with the expected ones on the basis of the MD generated structures shows a reasonable agreement considering the characteristic uncertainties of both experimental and simulation techniques. We used frequency domain fluorometry, acrylamide fluorescence quenching, and fluorescence correlation spectroscopy together with far UV circular dichroism to estimate the helical content, the Stern-Volmer quenching constant and the radius of gyration of the protein. Tuna apomyoglobin is a single tryptophan protein and thus, interpretation of its intrinsic fluorescence is simpler than for other proteins. The high sensitivity of the applied fluorescence techniques enabled experiments to be performed under very dilute conditions, that is, at concentrations of subnanomolar for the FCS measurements and 6 muM for the other fluorescence measurements. As high concentrations of proteins can strongly affect the association equilibrium among partially unfolded states, fluorescence techniques can provide complementary information with respect to other techniques requiring higher sample concentrations, such as NMR. The analysis of exposed hydrophobic regions in each of the MD-generated acidic structures reveals potential candidates involved in the aggregation processes of apomyoglobin in the acidic compact state. Our investigation represents an effective model system for studying amyloid fibril formation found in important diseases that are believed to proceed via aggregation of protein in the molten globule state.

Dinâmica molecular: teoria e aplicações em planejamento de fármacos

Dinâmica Molecular (DM) é uma ferramenta computacional poderosa usada em Química Medicinal para o planejamento racional de fármacos. DM é uma extensão da Mecânica Molecular, onde o comportamento dinâmico de um sistema molecular é simulado através da integração numérica das equações de movimento. Esta técnica tem sido usada extensivamente para auxiliar e complementar o planejamento de novos ligantes de um alvo terapêutico, bem como estimar a sua potência. Este artigo enfoca a teoria básica da DM clássica e suas importantes aplicações no planejamento racional de potenciais compostos bioativos, particularmente compostos com atividade anti-HIV.

Predicting protein fold pattern with functional domain and sequential evolution information

The fold pattern of a protein is one level deeper than its structural classification, and hence is more challenging and complicated for prediction. Many efforts have been made in this regard, but so far all the reported success rates are still under 70%, indicating that it is extremely difficult to enhance the success rate even by 1% or 2%. To address this problem, here a novel approach is proposed that is featured by combining the functional domain information and the sequential evolution information through a fusion ensemble classifier. The predictor thus developed is called PFP-FunDSeqE. Tests were performed for identifying proteins among their 27 fold patterns. Compared with the existing predictors tested by a same stringent benchmark dataset, the new predictor can, for the first time, achieve over 70% success rate. The PFP-FunDSeqE predictor is freely available to the public as a web server at http://www.csbio.sjtu.edu.cn/bioinf/PFP-FunDSeqE/.

Implementations of Nosé-Hoover and Nosé-Poincaré thermostats in mesoscopic dynamic simulations with the united-residue model of a polypeptide chain

Molecular dynamics (MD) simulations generate a canonical ensemble only when integration of the equations of motion is coupled to a thermostat. Three extended phase space thermostats, one version of Nose-Hoover and two versions of Nose-Poincare, are compared with each other and with the Berendsen thermostat and Langevin stochastic dynamics. Implementation of extended phase space thermostats was first tested on a model Lennard-Jones fluid system; subsequently, they were implemented with our physics-based protein united-residue (UNRES) force field MD. The thermostats were also implemented and tested for the multiple-time-step reversible reference system propagator (RESPA). The velocity and temperature distributions were analyzed to confirm that the proper canonical distribution is generated by each simulation. The value of the artificial mass constant, Q, of the thermostat has a large influence on the distribution of the temperatures sampled during UNRES simulations (the velocity distributions were affected only slightly). The numerical stabilities of all three algorithms were compared with each other and with that of microcanonical MD. Both Nose-Poincare thermostats, which are symplectic, were not very stable for both the Lennard-Jones fluid and UNRES MD simulations started from nonequilibrated structures which implies major changes of the potential energy throughout a trajectory. Even though the Nose-Hoover thermostat does not have a canonical symplectic structure, it is the most stable algorithm for UNRES MD simulations. For UNRES with RESPA, the "extended system inside-reference system propagator algorithm" of the RESPA implementation of the Nose-Hoover thermostat was the only stable algorithm, and enabled us to increase the integration time step.

RNA folding: conformational statistics, folding kinetics, and ion electrostatics

RNA folding is a remarkably complex problem that involves ion-mediated electrostatic interaction, conformational entropy, base pairing and stacking, and noncanonical interactions. During the past decade, results from a variety of experimental and theoretical studies pointed to (a) the potential ion correlation effect in Mg2+-RNA interactions, (b) the rugged energy landscapes and multistate RNA folding kinetics even for small RNA systems such as hairpins and pseudoknots, (c) the intraloop interactions and sequence-dependent loop free energy, and (d) the strong nonadditivity of chain entropy in RNA pseudoknot and other tertiary folds. Several related issues, which have not been thoroughly resolved, require combined approaches with thermodynamic and kinetic experiments, statistical mechanical modeling, and all-atom computer simulations.

Computer simulations of the refolding of sperm whale apomyoglobin from high-temperature denaturated state

The refolding mechanism of apomyoglobin (apoMb) subsequent to high-temperature unfolding has been examined using computer simulations with atomic level detail. The folding of this protein has been extensively studied experimentally, providing a large database of folding parameters which can be probed using simulations. In the present study, 4-folding trajectories of apoMb were computed starting from coiled structures. A crystal structure of sperm whale myoglobin taken from the Protein Data Bank was used to construct the final native conformation by removal of the heme group followed by energy optimization. The initial unfolded conformations were obtained from high-temperature molecular dynamics simulations. Room-temperature refolding trajectories at neutral pH were obtained using the stochastic difference equation in length algorithm. The folding trajectories were compared with experimental results and two previous molecular dynamics studies at low pH. In contrast to the previous simulations, an extended intermediate with large helical content was not observed. In the present study, a structural collapse occurs without formation of helices or native contacts. Once the protein structure is more compact (radius of gyration<18 A) secondary and tertiary structures appear. These results suggest that apoMb follows a different folding pathway after high-temperature denaturation.

Studying the unfolding kinetics of proteins under pressure using long molecular dynamic simulation runs

The usefulness of computational methods such as molecular dynamics simulation has been extensively established for studying systems in equilibrium. Nevertheless, its application to complex non-equilibrium biological processes such as protein unfolding has been generally regarded as producing results which cannot be interpreted straightforwardly. In the present study, we present results for the kinetics of unfolding of apomyoglobin, based on the analysis of long simulation runs of this protein in solution at 3 kbar (1 atm = 1.01325, bar = 101,325 Pa). We hereby demonstrate that the analysis of the data collected within a simulated time span of 0.18 mus suffices for producing results, which coincide remarkably with the available unfolding kinetics experimental data. This not only validates molecular dynamics simulation as a valuable alternative for studying non-equilibrium processes, but also enables a detailed analysis of the actual structural mechanism which underlies the unfolding process of proteins under elusive denaturing conditions such as high pressure.

Simulation of Top7-CFr: a transient helix extension guides folding

Protein structures often feature beta-sheets in which adjacent beta-strands have large sequence separation. How the folding process orchestrates the formation and correct arrangement of these strands is not comprehensively understood. Particularly challenging are proteins in which beta-strands at the N and C termini are neighbors in a beta-sheet. The N-terminal beta-strand is synthesized early on, but it can not bind to the C terminus before the chain is fully synthesized. During this time, there is a danger that the beta-strand at the N terminus interacts with nearby molecules, leading to potentially harmful aggregates of incompletely folded proteins. Simulations of the C-terminal fragment of Top7 show that this risk of misfolding and aggregation can be avoided by a "caching" mechanism that relies on the "chameleon" behavior of certain segments.

Molecular hydrophobic attraction and ion-specific effects studied by molecular dynamics

Much is written about "hydrophobic forces" that act between solvated molecules and nonpolar interfaces, but it is not always clear what causes these forces and whether they should be labeled as hydrophobic. Hydrophobic effects roughly fall in two classes, those that are influenced by the addition of salt and those that are not. Bubble adsorption and cavitation effects plague experiments and simulations of interacting extended hydrophobic surfaces and lead to a strong, almost irreversible attraction that has little or no dependence on salt type and concentration. In this paper, we are concerned with hydrophobic interactions between single molecules and extended surfaces and try to elucidate the relation to electrostatic and ion-specific effects. For these nanoscopic hydrophobic forces, bubbles and cavitation effects play only a minor role and even if present cause no equilibration problems. In specific, we study the forced desorption of peptides from nonpolar interfaces by means of molecular dynamics simulations and determine the adsorption potential of mean force. The simulation results for peptides compare well with corresponding AFM experiments. An analysis of the various contributions to the total peptide-surface interactions shows that structural effects of water as well as van der Waals interactions between surface and peptide are important. Hofmeister ion effects are studied by separately determining the effective interaction of various ions with hydrophobic surfaces. An extension of the Poisson-Boltzmann equation that includes the ion-specific potential of mean force yields surface potentials, interfacial tensions, and effective interactions between hydrophobic surfaces. There, we also analyze the energetic contributions to the potential of mean force and find that the most important factor determining ion-specific adsorption at hydrophobic surfaces can best be described as surface-modified ion hydration.

Geometrical features of the protein folding mechanism are a robust property of the energy landscape: a detailed investigation of several reduced models

The concept of a funneled energy landscape and the principle of minimal frustration are the theoretical foundation justifying the applicability of structure-based models. In simulations, a protein is commonly reduced to a C(alpha)-bead representation. These simulations are sufficient to predict the geometrical features of the folding mechanism observed experimentally utilizing a concise formulation of the Hamiltonian with low computational costs. Toward a better understanding of the interplay between energetic and geometrical features in folding, the side chain is now explicitly included in the simulations. The simplest choice is the addition of C(beta)-beads at the center-of-mass position of the side chains. While one varies the energetic parameters of the model, the geometric aspects of the folding mechanism remain robust for a broad range of parameters. Energetic properties like folding barriers and protein stability are sensitive to the details of simulations. This robustness to geometry and sensitivity to energetic properties provide flexibility in choosing different parameters to represent changes in sequences, environments, stability or folding rate effects. Therefore, minimal frustration and the funnel concept guarantee that the geometrical features are robust properties of the folding landscape, while mutations and/or changes in the environment easily influence energy-dependent properties like folding rates or stability.

Computational techniques for efficient conformational sampling of proteins

In this review, we summarize the computational methods for sampling the conformational space of biomacromolecules. We discuss the methods applicable to find only lowest energy conformations (global minimization of the potential-energy function) and to generate canonical ensembles (canonical Monte Carlo method and canonical molecular dynamics method and their extensions). Special attention is devoted to the use of coarse-grained models that enable simulations to be enhanced by several orders of magnitude.

Coarse-grained models of protein folding: toy models or predictive tools?

Coarse-grained models are emerging as a practical alternative to all-atom simulations for the characterization of protein folding mechanisms over long time scales. While a decade ago minimalist toy models were mainly designed to test general hypotheses on the principles regulating protein folding, the latest coarse-grained models are increasingly realistic and can be used to characterize quantitatively the detailed folding mechanism of specific proteins. The ability of such models to reproduce the essential features of folding dynamics suggests that each single atomic degree of freedom is not by itself particularly relevant to folding and supports a statistical mechanical approach to characterize folding transitions. When combined with more refined models and with experimental studies, the systematic investigation of protein systems and complexes using coarse-grained models can advance our theoretical understanding of the actual organizing principles that emerge from the complex network of interactions among protein atomic constituents.

Conformational analysis of L-prolines in water

The results of the ring conformational analysis of L-proline, N-acetyl-L-proline, and trans-4-hydroxy-L-proline by NMR combined with calculations using density functional theory (DFT) and molecular dynamics (MD) are reported. Accurate values of 1H-1H J-couplings in water and other solvents have been determined. Using a two-site equilibrium model, the Cgamma-endo conformer of L-proline in water has been identified as intermediate between gammaTdelta [twist(Cgamma-endo, Cdelta-exo)] and gammaE [envelope(Cgamma-endo)] and the Cgamma-exo conformer as betaTgamma. Both conformers were equally populated at room temperature. The N-acetyl [cis-rotamer gammaTbeta(80%)/gammaE(20%) and trans-rotamer gammaTbeta(61%)/gammaE(39%)] and 4-hydroxy (gammaEpsilon) derivatives showed significant changes in both the population and the geometries of the preferred ring conformers. The combination of NMR predicted populations with the DFT B3LYP/6-311+G(2d,p)/IEFPCM calculations proved successful, resulting in fairly accurate predictions of J-couplings. Simulations using MD were mostly in favor of the two-site equilibrium model between Cgamma-endo and Cgamma-exo conformers, similar to that used for the analysis of NMR J-couplings. Various force fields examined for MD simulations failed to reproduce the ring conformational geometries and populations of L-proline in water accurately, while significantly better agreement with NMR was found for trans-N-acetyl-L-proline using GROMOS and AMBER force fields.