Replica-exchange methods and predictions of helix configurations of membrane proteins

A simulation in generalized ensemble is based on a non-Boltzmann weight factor and performs a random walk in potential energy space, which allows the simulation to avoid getting trapped in states of local-minimum energy states. In this article, we review uses of the generalized-ensemble algorithms. Three well-known methods, namely, multicanonical algorithm (MUCA), simulated tempering (ST) and replica-exchange method (REM), are described first. Both Monte Carlo (MC) and molecular dynamics (MD) versions of the algorithms are given. We then present the results of the application of replica-exchange MC method to the predictions of membrane protein structures.     

Comparison of molecular dynamics simulation methods for amyloid β(1-42) monomers containing D-aspartic acid residues for predicting retention times in chromatography

Molecular dynamics simulations of amyloid β(1-42) containing D-aspartic acid residues were performed using several continuous solvent models to investigate the usefulness of simulation methods for D-amino acid-containing proteins and peptides. Normal molecular dynamics simulations and replica exchange molecular dynamics simulations, which are one of the generalized-ensemble algorithms, were performed. Because the β-structure contents of amyloid β(1-42) peptides obtained by replica exchange molecular dynamics simulations with Onufriev-Bashford-Case generalized Born implicit solvent were qualitatively consistent with experimental data, replica exchange molecular dynamics rather than other methods appeared to be more reasonable for calculations of amyloid β(1-42) containing D-aspartic acid residues. Computational results revealed that peptides with stereoinversion of Asp23 tend to form β-sheet structures by themselves, in contrast to the wild-type peptides that form β-sheet structures only after aggregation. These results are expected to be useful for computational investigations of proteins and peptides such as prediction of retention time of peptides and proteins containing D-aspartic acid residues.     

Computation of elastic constants of solids using molecular simulation: comparison of constant volume and constant pressure ensemble methods

We compute the elastic stiffness tensor of fcc argon at 60 K and 1 bar using molecular simulation tools. Three different methods are investigated: explicit deformations of the simulation box, strain fluctuations at constant pressure and stress fluctuations at constant volume. Statistical ensemble sampling is done using molecular dynamics and Monte Carlo simulations. We observe a good agreement between the different methods and sampling algorithms excepted with molecular dynamics simulations in the (NpT) ensemble. There, we notice a strong dependence of the computed elastic constants with the barostat parameter, whereas molecular dynamics simulations in the (NVT) ensemble are not affected by the thermostat parameter.     

Application of the multiensemble sampling to the equilibrium folding of proteins

MOTIVATION: Conventional Monte Carlo and molecular dynamics simulations of proteins in the canonical ensemble are of little use, because they tend to get trapped in states of energy local minima at low temperatures. One way to surmount this difficulty is to use a non-Boltzmann sampling method in which conformations are sampled upon a general weighting function instead of the conventional Boltzmann weighting function. The multiensemble sampling (MES) method is a non-Boltzmann sampling method that was originally developed to estimate free energy differences between systems with different potential energies and/or at different thermodynamic states. The method has not yet been applied to studies of complex molecular systems such as proteins. RESULTS: MES Monte Carlo simulations of small proteins have been carried out using a united-residue force field. The proteins at several temperatures from the unfolded to the folded states were simulated in a single MC run at a time and their equilibrium thermodynamic properties were calculated correctly. The distributions of sampled conformations clearly indicate that, when going through states of energy local minima, the MES simulation did not get trapped in them but escaped from them so quickly that all the relevant parts of conformation space could be sampled properly. A two-step folding process consisting of a collapse transition followed by a folding transition is observed. This study demonstrates that the use of MES alleviates the multiple-minima problem greatly. AVAILABILITY: Available on request from the authors.     

Solid-State NMR-Restrained Ensemble Dynamics of a Membrane Protein in Explicit Membranes

Solid-state NMR has been used to determine the structures of membrane proteins in native-like lipid bilayer environments. Most structure calculations based on solid-state NMR observables are performed using simulated annealing with restrained molecular dynamics and an energy function, where all nonbonded interactions are represented by a single, purely repulsive term with no contributions from van der Waals attractive, electrostatic, or solvation energy. To our knowledge, this is the first application of an ensemble dynamics technique performed in explicit membranes that uses experimental solid-state NMR observables to obtain the refined structure of a membrane protein together with information about its dynamics and its interactions with lipids. Using the membrane-bound form of the fd coat protein as a model membrane protein and its experimental solid-state NMR data, we performed restrained ensemble dynamics simulations with different ensemble sizes in explicit membranes. For comparison, a molecular dynamics simulation of fd coat protein was also performed without any restraints. The average orientation of each protein helix is similar to a structure determined by traditional single-conformer approaches. However, their variations are limited in the resulting ensemble of structures with one or two replicas, as they are under the strong influence of solid-state NMR restraints. Although highly consistent with all solid-state NMR observables, the ensembles of more than two replicas show larger orientational variations similar to those observed in the molecular dynamics simulation without restraints. In particular, in these explicit membrane simulations, Lys⁴⁰, residing at the C-terminal side of the transmembrane helix, is observed to cause local membrane curvature. Therefore, compared to traditional single-conformer approaches in implicit environments, solid-state NMR restrained ensemble simulations in explicit membranes readily characterize not only protein dynamics but also protein-lipid interactions in detail.     

Atomic-level characterization of disordered protein ensembles

The roles of unfolded states of proteins in normal folding and in diseases involving aggregation, as well as the prevalence and regulatory functions of intrinsically disordered proteins, have become increasingly recognized. The structural representation of these disordered states as ensembles of interconverting conformers can therefore provide critical insights. Experimental methods can be used to probe ensemble-averaged structural properties of disordered states and computational approaches generate representative ensembles of conformers using experimental restraints. In particular, NMR and small-angle X-ray scattering provide quantitative data that can readily be incorporated into calculations. These techniques have gleaned structural information about denatured, unfolded and intrinsically disordered proteins. The use of experimental data in different computational approaches, including ensemble molecular dynamics simulations and algorithms that assign populations to pregenerated conformers, has highlighted the presence of both local and long-range structure, and the occurrence of native-like and non-native interactions in unfolded and denatured states. Analysis of the resulting ensembles has suggested important implications of this fluctuating structure for folding, aggregation and binding.     

First-Principles Protein Folding Simulations

Abstract

It is widely believed that the prediction of the three-dimensional structures of proteins from the first principles is impossible. This view is based on the fact that the number of possible structures for each protein is astronomically large. The question is then why a protein folds into its native structure with the proper biological functions in the time scale of milliseconds to minutes, and this is called Levinthal's paradox. In this article I will discuss our strategy for attacking the protein folding problem. Our approach consists of two elements: the inclusion of accurate solvent effects and the development of powerful simulation algorithms that can avoid getting trapped in states of energy local minima. For the former, we discuss several models varying in nature from crude (distance-dependent dielectric function) to rigorous (reference interaction site model). For the latter, we show the effectiveness of Monte Carlo simulated annealing and generalized-ensemble algorithms.     

N-terminal region of human chemokine receptor CXCR3: Structural analysis of CXCR3(1–48) by experimental and computational studies

Our study on the highly charged N-terminal peptide of the human chemokine receptor CXCR3 by spectroscopic methods in solution and by means of molecular dynamics simulations showed that the charge content modulates the intrinsic structural preference of its flexible backbone. Collectively, our findings suggest that the structural organization of a protein should be seen as a part of a continuum in which the ratio between electrostatic and hydrophobic interactions and the intrinsic flexibility are important properties used to optimize the folding. When this ratio changes and the structure is intrinsically flexible, the structural organization of the system moves along the continuum of the possible conformational states. By all this combined information, one can describe the structure of CXCR3(1–48) as an ensemble of conformations. In fact, the peptide shows stretches of negative charges embedded in a flexible sequence which can be used to maximize promiscuous interactions relevant to molecular recognition but globally the peptide appears as a poly-structured globule-like ensemble that is dynamically stabilized by H-bonds. We have approached the study of the most populated ensembles with subset selection to explain our experimental data also by evidencing that the changes into the fraction of charged residues discriminate between dynamically poly-structured states, conceivably because of small free energy barriers existing between the different conformations of CXCR3(1–48). Therefore, the overlap of a highly flexible backbone, negatively charged residues and sites which can be modified by post-translational modifications represent the structural organization that controls the molecular mechanisms underlying the biological functions carried out by CXCR3(1–48).     

GPU-accelerated replica exchange molecular simulation on solid–liquid phase transition study of Lennard-Jones fluids

Determining the solid–liquid phase transition point by conventional molecular dynamics (MD) simulations is difficult because of the tendency of the system to get trapped in local minimum energy states at low temperatures and hysteresis during cooling and heating cycles. The replica exchange method, used in performing many MD simulations of the system at different temperature conditions simultaneously and performs exchanges of these temperatures at certain intervals, has been introduced as a tool to overcome this local-minimum problem. However, around the phase transition temperature, a greater number of different temperatures are required to adequately find the phase transition point. In addition, the number of different temperature values increases when treating larger systems resulting in huge computation times. We propose a computational acceleration of the replica exchange MD simulation on graphics processing units (GPUs) in studying first-order solid–liquid phase transitions of Lennard-Jones (LJ) fluids. The phase transition temperature for a 108-atom LJ fluid has been calculated to validate our new code. The result corresponds with that of a previous study using multicanonical ensemble. The computational speed is measured for various GPU-cluster sizes. A peak performance of 196.3 GFlops with one GPU and 8.13 TFlops with 64 GPUs is achieved.     

Phosphorylation-induced conformational changes in a mitogen-activated protein kinase substrate. Implications for tyrosine hydroxylase activation

Mitogen-activated protein (MAP) kinase-mediated phosphorylation of specific residues in tyrosine hydroxylase leads to an increase in enzyme activity. However, the mechanism whereby phosphorylation affects enzyme turnover is not well understood. We used a combination of fluorescence resonance energy transfer (FRET) measurements and molecular dynamics simulations to explore the conformational free energy landscape of a 10-residue MAP kinase substrate found near the N terminus of the enzyme. This region is believed to be part of an autoregulatory sequence that overlies the active site of the enzyme. FRET was used to measure the effect of phosphorylation on the ensemble of peptide conformations, and molecular dynamics simulations generated free energy profiles for both the unphosphorylated and phosphorylated peptides. We demonstrate how FRET transfer efficiencies can be calculated from molecular dynamics simulations. For both the unphosphorylated and phosphorylated peptides, the calculated FRET efficiencies are in excellent agreement with the experimentally determined values. Moreover, the FRET measurements and molecular simulations suggest that phosphorylation causes the peptide backbone to change direction and fold into a compact structure relative to the unphosphorylated state. These results are consistent with a model of enzyme activation where phosphorylation of the MAP kinase substrate causes the N-terminal region to adopt a compact structure away from the active site. The methods we employ provide a general framework for analyzing the accessible conformational states of peptides and small molecules. Therefore, they are expected to be applicable to a variety of different systems.     

Advances in biomolecular simulations: methodology and recent applications

Molecular dynamics simulations are widely used today to tackle problems in biochemistry and molecular biology. In the 25 years since the first simulation of a protein computers have become faster by many orders of magnitude, algorithms and force fields have been improved, and simulations can now be applied to very large systems, such as protein-nucleic acid complexes and multimeric proteins in aqueous solution. In this review we give a general background about molecular dynamics simulations, and then focus on some recent technical advances, with applications to biologically relevant problems.     

Grand canonical ensemble Monte Carlo simulation of the dCpG/proflavine crystal hydrate.

The grand canonical ensemble Monte Carlo molecular simulation method is used to investigate hydration patterns in the crystal hydrate structure of the dCpG/proflavine intercalated complex. The objective of this study is to show by example that the recently advocated grand canonical ensemble simulation is a computationally efficient method for determining the positions of the hydrating water molecules in protein and nucleic acid structures. A detailed molecular simulation convergence analysis and an analogous comparison of the theoretical results with experiments clearly show that the grand ensemble simulations can be far more advantageous than the comparable canonical ensemble simulations.     

An Extended Gibbs Ensemble

Abstract

A general extended Gibbs ensemble, obtained by augmenting the standard Gibbs ensemble by intermediate states in the spirit of the scaled particle method of Nezbeda and Kolafa [Molec. Simul., 5, 391 (1991)], is introduced. The intermediate states span the states with different number of particles in the simulation boxes and facilitate the transfer of particles even in such complex systems as e.g., mixtures of very different components, systems of flexible polymeric molecules, or systems at very high densities. A general formulation of the ensemble is given and two implementations are considered in detail. The method is very general and is exemplified by studying the fluid-fluid coexistence in a dense binary mixture of the hard-sphere and square-well fluids. It is found that its efficiency is about by factor three greater in comparison with the standard Gibbs ensemble simulations.     

Analytical methods for structural ensembles and dynamics of intrinsically disordered proteins

Intrinsically disordered proteins, proteins that do not have a well-defined three-dimensional structure, make up a significant proportion of our proteome and are particularly prevalent in signaling and regulation. Although their importance has been realized for two decades, there is a lack of high-resolution experimental data. Molecular dynamics simulations have been crucial in reaching our current understanding of the dynamical structural ensemble sampled by intrinsically disordered proteins. In this review, we discuss enhanced sampling simulation methods that are particularly suitable to characterize the structural ensemble, along with examples of applications and limitations. The dynamics within the ensemble can be rigorously analyzed using Markov state models. We discuss recent developments that make Markov state modeling a viable approach for studying intrinsically disordered proteins. Finally, we briefly discuss challenges and future directions when applying molecular dynamics simulations to study intrinsically disordered proteins.     

Combined monte carlo and molecular dynamics simulation of fully hydrated dioleyl and palmitoyl-oleyl phosphatidylcholine lipid bilayers

We have applied a new equilibration procedure for the atomic level simulation of a hydrated lipid bilayer to hydrated bilayers of dioleyl-phosphatidylcholine (DOPC) and palmitoyl-oleyl phosphatidylcholine (POPC). The procedure consists of alternating molecular dynamics trajectory calculations in a constant surface tension and temperature ensemble with configurational bias Monte Carlo moves to different regions of the configuration space of the bilayer in a constant volume and temperature ensemble. The procedure is applied to bilayers of 128 molecules of POPC with 4628 water molecules, and 128 molecules of DOPC with 4825 water molecules. Progress toward equilibration is almost three times as fast in central processing unit (CPU) time compared with a purely molecular dynamics (MD) simulation. Equilibration is complete, as judged by the lack of energy drift in 200-ps runs of continuous MD. After the equilibrium state was reached, as determined by agreement between the simulation volume per lipid molecule with experiment, continuous MD was run in an ensemble in which the lateral area was restrained to fluctuate about a mean value and a pressure of 1 atm applied normal to the bilayer surface. Three separate continuous MD runs, 200 ps in duration each, separated by 10,000 CBMC steps, were carried out for each system. Properties of the systems were calculated and averaged over the three separate runs. Results of the simulations are presented and compared with experimental data and with other recent simulations of POPC and DOPC. Analysis of the hydration environment in the headgroups supports a mechanism by which unsaturation contributes to reduced transition temperatures. In this view, the relatively horizontal orientation of the unsaturated bond increases the area per lipid, resulting in increased water penetration between the headgroups. As a result the headgroup-headgroup interactions are attenuated and shielded, and this contributes to the lowered transition temperature.     

Using robotics to fold proteins and dock ligands

The problems of protein folding and ligand docking have been explored largely using molecular dynamics or Monte Carlo methods. These methods are very compute intensive because they often explore a much wider range of energies, conformations and time than necessary. In addition, Monte Carlo methods often get trapped in local minima. We initially showed that robotic motion planning permitted one to determine the energy of binding and dissociation of ligands from protein binding sites (Singh et al., 1999). The robotic motion planning method maps complicated three-dimensional conformational states into a much simpler, but higher dimensional space in which conformational rearrangements can be represented as linear paths. The dimensionality of the conformation space is of the same order as the number of degrees of conformational freedom in three-dimensional space. We were able to determine the relative energy of association and dissociation of a ligand to a protein by calculating the energetics of interaction for a few thousand conformational states in the vicinity of the protein and choosing the best path from the roadmap. More recently, we have applied roadmap planning to the problem of protein folding (Apaydin et al., 2002a). We represented multiple conformations of a protein as nodes in a compact graph with the edges representing the probability of moving between neighboring states. Instead of using Monte Carlo simulation to simulate thousands of possible paths through various conformational states, we were able to use Markov methods to calculate the steady state occupancy of each conformation, needing to calculate the energy of each conformation only once. We referred to this Markov method of representing multiple conformations and transitions as stochastic roadmap simulation or SRS. We demonstrated that the distribution of conformational states calculated with exhaustive Monte Carlo simulations asymptotically approached the Markov steady state if the same Boltzman energy distribution was used in both methods. SRS permits one to calculate contributions from all possible paths simultaneously with far fewer energy calculations than Monte Carlo or molecular dynamics methods. The SRS method also permits one to represent multiple unfolded starting states and multiple, near-native, folded states and all possible paths between them simultaneously. The SRS method is also independent of the function used to calculate the energy of the various conformational states. In a paper to be presented at this conference (Apaydin et al., 2002b) we have also applied SRS to ligand docking in which we calculate the dynamics of ligand-protein association and dissociation in the region of various binding sites on a number of proteins. SRS permits us to determine the relative times of association to and dissociation from various catalytic and non-catalytic binding sites on protein surfaces. Instead of just following the best path in a roadmap, we can calculate the contribution of all the possible binding or dissociation paths and their relative probabilities and energies simultaneously.     

A one-dimensional reaction coordinate for identification of transition states from explicit solvent P(fold)-like calculations

A properly identified transition state ensemble (TSE) in a molecular dynamics (MD) simulation can reveal a tremendous amount about how a protein folds and offer a point of comparison to experimentally derived Phi(F) values, which reflect the degree of structure in these transient states. In one such method of TSE identification, dubbed P(fold), MD simulations of individual protein structures taken from an unfolding trajectory are used to directly assess an input structure's probability of folding before unfolding, and P(fold) is, by definition, 0.5 for the TSE. Other, less computationally intensive methods, such as multidimensional scaling (MDS) of the pairwise root mean-squared deviation (RMSD) matrix of the conformations sampled in a thermal unfolding trajectory, have also been used to identify the TSE. Identification of the TSE is made from the original MD simulation without the need to run further simulations. Here we present a P(fold)-like study and describe methods for identification of the TSE through the derivation of a high fidelity, bounded, one-dimensional reaction coordinate for protein folding. These methods are applied to the engrailed homeodomain. The TSE identified by this approach is essentially identical to the TSE identified previously by MDS of the pairwise RMSD matrix. However, the cost of performing P(fold), or even our reduced P(fold)-like calculations, is at least 36,000 times greater than the MDS method.     

Bayesian Parameter Inference and Model Selection by Population Annealing in Systems Biology

Parameter inference and model selection are very important for mathematical modeling in systems biology. Bayesian statistics can be used to conduct both parameter inference and model selection. Especially, the framework named approximate Bayesian computation is often used for parameter inference and model selection in systems biology. However, Monte Carlo methods needs to be used to compute Bayesian posterior distributions. In addition, the posterior distributions of parameters are sometimes almost uniform or very similar to their prior distributions. In such cases, it is difficult to choose one specific value of parameter with high credibility as the representative value of the distribution. To overcome the problems, we introduced one of the population Monte Carlo algorithms, population annealing. Although population annealing is usually used in statistical mechanics, we showed that population annealing can be used to compute Bayesian posterior distributions in the approximate Bayesian computation framework. To deal with un-identifiability of the representative values of parameters, we proposed to run the simulations with the parameter ensemble sampled from the posterior distribution, named “posterior parameter ensemble”. We showed that population annealing is an efficient and convenient algorithm to generate posterior parameter ensemble. We also showed that the simulations with the posterior parameter ensemble can, not only reproduce the data used for parameter inference, but also capture and predict the data which was not used for parameter inference. Lastly, we introduced the marginal likelihood in the approximate Bayesian computation framework for Bayesian model selection. We showed that population annealing enables us to compute the marginal likelihood in the approximate Bayesian computation framework and conduct model selection depending on the Bayes factor.     

New version of Monte Carlo expanded ensemble method for precise calculations of free energy difference

A new version of Monte Carlo (MC) expanded ensemble (EE) method is proposed for the calculations of free energy difference (FED) between two different systems with close values of the free energy. In order to check the method the FED between simple model systems (fluid of hard spheres and freely jointed polymer chain of hard spheres) was calculated. The free energy of the mentioned above systems was also calculated by a standard MC EE method in order to compare the results of two simulations. It was shown that the accuracy of a new algorithm is the same as of a standard one. At the same time new version of EE allows us to obtain FED between two systems having quite different structures, but similar free energies, during one simulation run.