Replica Exchange Algorithm Research Articles

Complexation behavior of oppositely charged polyelectrolytes: Effect of charge distribution.

Complexation behavior of oppositely charged polyelectrolytes in a solution is investigated using a combination of computer simulations and experiments, focusing on the influence of polyelectrolyte charge distributions along the chains on the structure of the polyelectrolyte complexes. The simulations are performed using Monte Carlo with the replica-exchange algorithm for three model systems where each system is composed of a mixture of two types of oppositely charged model polyelectrolyte chains (EGEG)5/(KGKG)5, (EEGG)5/(KKGG)5, and (EEGG)5/(KGKG)5, in a solution including explicit solvent molecules. Among the three model systems, only the charge distributions along the chains are not identical. Thermodynamic quantities are calculated as a function of temperature (or ionic strength), and the microscopic structures of complexes are examined. It is found that the three systems have different transition temperatures, and form complexes with different sizes, structures, and densities at a given temperature. Complex microscopic structures with an alternating arrangement of one monolayer of E/K monomers and one monolayer of G monomers, with one bilayer of E and K monomers and one bilayer of G monomers, and with a mixture of monolayer and bilayer of E/K monomers in a box shape and a trilayer of G monomers inside the box are obtained for the three mixture systems, respectively. The experiments are carried out for three systems where each is composed of a mixture of two types of oppositely charged peptide chains. Each peptide chain is composed of Lysine (K) and glycine (G) or glutamate (E) and G, in solution, and the chain length and amino acid sequences, and hence the charge distribution, are precisely controlled, and all of them are identical with those for the corresponding model chain. The complexation behavior and complex structures are characterized through laser light scattering and atomic force microscopy measurements. The order of the apparent weight-averaged molar mass and the order of density of complexes observed from the three experimental systems are qualitatively in agreement with those predicted from the simulations.

Effect of surface corrugation on low temperature phases of adsorbed (p-H2)7: A quantum path integral Monte Carlo study

Abstract By using path integral Monte Carlo simulations coupled to Replica Exchange algorithms, various phases of (p-H2)7 physically adsorbed on a model graphite surface were identified at low temperatures. At T = 0.5 K , the expected superfluid phase was observed for flat and slightly corrugated surfaces. At intermediate and high corrugations, a “supersolid” phase in C7/16 registry and a solid phase in C1/3 registry were observed, respectively. At higher temperatures, the superfluid is converted to a fluid and the “supersolid” to a solid.

Coexistence and interfacial properties of triangle-well fluids

Coexistence and interfacial properties of triangle-well fluids are determined by combining the slab technique and the replica exchange algorithm for different interaction ranges (λ = 1.5, 1.75, 2.0, 2.5, and 3). This is implemented using both Monte Carlo and molecular dynamics methods. We make use of a recently proposed substitution of the hard-core repulsion by a linear function with a large negative slope. This makes possible to gain access through the virial route to thermodynamical properties and to employ widely spread packages such as Gromacs. Coexistence curves of these systems were calculated with both implementations and compared to those previously reported in the literature. A good agreement was found among them. Surface tension data obtained from Monte Carlo and molecular dynamics techniques also show a good agreement.

Sampling the Denatured State of Polypeptides in Water, Urea, and Guanidine Chloride to Strict Equilibrium Conditions with the Help of Massively Parallel Computers.

The denatured state of polypeptides and proteins, stabilized by chemical denaturants like urea and guanidine chloride, displays residual secondary structure when studied by nuclear-magnetic-resonance spectroscopy. However, these experimental techniques are weakly sensitive, and thus molecular-dynamics simulations can be useful to complement the experimental findings. To sample the denatured state, we made use of massively-parallel computers and of a variant of the replica exchange algorithm, in which the different branches, connected with unbiased replicas, favor the formation and disruption of local secondary structure. The algorithm is applied to the second hairpin of GB1 in water, in urea, and in guanidine chloride. We show with the help of different criteria that the simulations converge to equilibrium. It results that urea and guanidine chloride, besides inducing some polyproline-II structure, have different effect on the hairpin. Urea disrupts completely the native region and stabilizes a state which resembles a random coil, while guanidine chloride has a milder effect.

Performance of Replica-exchange Wang-landau Sampling for the 2D Ising Model: A Brief Survey

Abstract We report a brief performance study of the replica-exchange Wang-Landau algorithm, a recently proposed parallel realization of Wang-Landau sampling, using the 2D Ising model as a test case. The simulation time is found to scale inversely with the square root of the number of subwindows (and thus number of processors) used to span the global parameter space. We also investigate the “time profiles” for random walkers in different subwindows to complete iterations, which will aid the development of an adaptive load-balancing scheme.

Hydophobic Guided Protein Folding

Protein folding simulations have the potential to help us figure out the problem of protein folding. However, the highly frustrated energy landscapes of protein folding limits the use of these techniques. Hence, many new algorithms are developed each year aiming at improving sampling efficiency, producing results in shorter times. We have developed and tested an algorithm using GPU technology that allows us to fold several proteins within weeks of computational usage.Our work extends on the principles of protein folding through zipping and assembly. In this view, local events such as helix formation happen first (zipping), and non-local ones later (assembly). The assembly step is slow: in order to form native like contacts the residues involved have to first find each other. Our algorithm takes advantage of the fact that proteins form hydrophobic cores to guide the assembly step. We accelerate the process by adding restraints in our simulations that brings pairs of hydrophobic residues together, favoring the formation of native-like contacts. Since not all hydrophobic pairs will lead to native-like contacts we use a Hamiltonian and Temperature Replica exchange algorithm (H,T-REMD), that allows us to modulate the strengths of these restraints. Thus, the restraints that the underlying molecular force field favors are enforced, where as the ones that are not favored are turned off.This method allows us to jump between different sets of contacts easily, sampling many different topologies and limiting the amount of sampling that is required in between. In this way we have been able to get native-like structures for a small set of proteins. During my presentation I will show how the method works and how successful it is in a small set of proteins.

From Atoms to Fullerene: Stochastic Surface Walking Solution for Automated Structure Prediction of Complex Material.

It is of general concern whether the automated structure prediction of unknown material without recourse to any knowledge from experiment is ever possible considering the daunting complexity of potential energy surface (PES) of material. Here we demonstrate that the stochastic surface walking (SSW) method can be a general and promising solution to this ultimate goal, which is applied to assemble carbon fullerenes containing up to 100 atoms (including 60, 70, 76, 78, 80, 84, 90, 96, and 100 atoms) from randomly distributed atoms, a long-standing challenge in global optimization. Combining the SSW method with a parallel replica exchange algorithm, we can locate the global minima (GM) of these large fullerenes efficiently without being trapped in numerous energy-nearly degenerate isomers. Detailed analyses on the SSW trajectories allow us to rationalize how and why the SSW method is able to explore the highly complex PES, which highlights the abilities of SSW method for surmounting the high barrier and the preference of SSW trajectories to the low energy pathways. The work demonstrates that the parallel SSW method is a practical tool for predicting unknown materials.

Calculation of Free Energy Landscape in Multi-Dimensions with Hamiltonian-Exchange Umbrella Sampling on Petascale Supercomputer.

An extremely scalable computational strategy is described for calculations of the potential of mean force (PMF) in multidimensions on massively distributed supercomputers. The approach involves coupling thousands of umbrella sampling (US) simulation windows distributed to cover the space of order parameters with a Hamiltonian molecular dynamics replica-exchange (H-REMD) algorithm to enhance the sampling of each simulation. In the present application, US/H-REMD is carried out in a two-dimensional (2D) space and exchanges are attempted alternatively along the two axes corresponding to the two order parameters. The US/H-REMD strategy is implemented on the basis of parallel/parallel multiple copy protocol at the MPI level, and therefore can fully exploit computing power of large-scale supercomputers. Here the novel technique is illustrated using the leadership supercomputer IBM Blue Gene/P with an application to a typical biomolecular calculation of general interest, namely the binding of calcium ions to the small protein Calbindin D9k. The free energy landscape associated with two order parameters, the distance between the ion and its binding pocket and the root-mean-square deviation (rmsd) of the binding pocket relative the crystal structure, was calculated using the US/H-REMD method. The results are then used to estimate the absolute binding free energy of calcium ion to Calbindin D9k. The tests demonstrate that the 2D US/H-REMD scheme greatly accelerates the configurational sampling of the binding pocket, thereby improving the convergence of the potential of mean force calculation.

Molecular weight distribution effects on the structure of strongly adsorbed polymers by Monte Carlo simulation

Monte Carlo simulations are reported to study the structure of polymers adsorbed from solution onto strongly attractive, perfectly smooth substrates. Six systems spanning a range of molecular weight distributions are investigated with a coarse-grained united atom model for freely rotating chains. By employing a global replica exchange algorithm and topology altering Monte Carlo moves, a range of monomer-surface attraction from weak (0.27kT) to strong (4kT) is simultaneously explored. Thus for the first time ever, equilibrium polymer adsorption on highly attractive surfaces is studied, with all adsorbed molecules displaying similar properties and statistics. The architecture of the adsorbed layers, including density profiles, bond orientation order parameters, radii of gyration, and distribution of the adsorbed chain fractions, is shown to be highly dependent on the polydispersity of the polymer phase. The homology of polymer chains, and the ergodicity of states explored by the molecules is in contrast to the metastable, kinetically constrained paradigm of irreversible adsorption. The structure of more monodisperse systems is qualitatively similar to experimental results and theoretical predictions, but result from very different chain conformations and statistics. The polydispersity-dependent behavior is explained in the context of the competition between polymers to make contact with the surface.

Phase Transition in Frustrated Heisenberg Antiferromagnet on a Triangular Lattice with Next-Nearest Neighbor Interactions

We study the critical behavior of three-dimensional antiferromagnet Heisenberg model with nearest-neighbor (J) and next-nearest-neighbor (J1) interactions by the Monte Carlo method using a high-effective replica exchange algorithm. Here is calculated a full set of main static critical exponents for values R =J1/J= 0.0; 0.025; 0.05; 0.075; 0.1; 0.115 using the finite-size scaling theory. A phase diagram of dependency of the critical temperature on a relation between nearest-neighbor and next-nearest-neighbor R is plotted.

Efficient large-scale replica-exchange simulations on production infrastructure

Replica-exchange (RE) algorithms are used to understand physical phenomena--ranging from protein folding dynamics to binding affinity calculations. They represent a class of algorithms that involve a large number of loosely coupled ensembles, and are thus amenable to using distributed resources. We develop a framework for RE that supports different replica pairing (synchronous versus asynchronous) and exchange coordination mechanisms (centralized versus decentralized) and which can use a range of production cyberinfrastructures concurrently. We characterize the performance of both RE algorithms at an unprecedented number of cores employed--the number of replicas and the typical number of cores per replica--on the production distributed infrastructure. We find that the asynchronous algorithms outperform the synchronous algorithms, even though details of the specific implementations are important determinants of performance.

Surface aided replica exchange algorithm applied to the prewetting transition

A novel algorithm, termed surface aided replica exchange (SARE), was introduced in which the fluid–surface interaction was varied in order to generate different replicas in a particular ensemble. Exchange between replicas was allowed with a probability of acceptance obtained by imposing detailed balance. The method was implemented in a modified isothermal-isobaric ensemble that permitted precise characterization of the prewetting line of a simple adsorbed fluid. The prewetting line for each surface was characterized by computing the adsorption of the fluid as a function of pressure, and the wetting temperature estimated for each surface.

Simulated Tempering Distributed Replica Sampling, Virtual Replica Exchange, and Other Generalized-Ensemble Methods for Conformational Sampling.

Generalized-ensemble algorithms in temperature space have become popular tools to enhance conformational sampling in biomolecular simulations. A random walk in temperature leads to a corresponding random walk in potential energy, which can be used to cross over energetic barriers and overcome the problem of quasi-nonergodicity. In this paper, we introduce two novel methods: simulated tempering distributed replica sampling (STDR) and virtual replica exchange (VREX). These methods are designed to address the practical issues inherent in the replica exchange (RE), simulated tempering (ST), and serial replica exchange (SREM) algorithms. RE requires a large, dedicated, and homogeneous cluster of CPUs to function efficiently when applied to complex systems. ST and SREM both have the drawback of requiring extensive initial simulations, possibly adaptive, for the calculation of weight factors or potential energy distribution functions. STDR and VREX alleviate the need for lengthy initial simulations, and for synchronization and extensive communication between replicas. Both methods are therefore suitable for distributed or heterogeneous computing platforms. We perform an objective comparison of all five algorithms in terms of both implementation issues and sampling efficiency. We use disordered peptides in explicit water as test systems, for a total simulation time of over 42 μs. Efficiency is defined in terms of both structural convergence and temperature diffusion, and we show that these definitions of efficiency are in fact correlated. Importantly, we find that ST-based methods exhibit faster temperature diffusion and correspondingly faster convergence of structural properties compared to RE-based methods. Within the RE-based methods, VREX is superior to both SREM and RE. On the basis of our observations, we conclude that ST is ideal for simple systems, while STDR is well-suited for complex systems.

Monte carlo folding of trans‐membrane helical peptides in an implicit generalized Born membrane

An efficient Monte Carlo (MC) algorithm using concerted backbone rotations is combined with a recently developed implicit membrane model to simulate the folding of the hydrophobic transmembrane domain M2TM of the M2 protein from influenza A virus and Sarcolipin at atomic resolution. The implicit membrane environment is based on generalized Born theory and has been calibrated against experimental data. The MC sampling has previously been used to fold several small polypeptides and been shown to be equivalent to molecular dynamics (MD). In combination with a replica exchange algorithm, M2TM is found to form continuous membrane spanning helical conformations for low temperature replicas. Sarcolipin is only partially helical, in agreement with the experimental NMR structures in lipid bilayers and detergent micelles. Higher temperature replicas exhibit a rapidly decreasing helicity, in agreement with expected thermodynamic behavior. To exclude the possibility of an erroneous helical bias in the simulations, the model is tested by sampling a synthetic Alanine-rich polypeptide of known helicity. The results demonstrate there is no overstabilization of helical conformations, indicating that the implicit model captures the essential components of the native membrane environment for M2TM and Sarcolipin.

Monte Carlo vs Molecular Dynamics for All-Atom Polypeptide Folding Simulations

An efficient Monte Carlo (MC) algorithm including concerted rotations is directly compared to molecular dynamics (MD) in all-atom statistical mechanics folding simulations of small polypeptides. The previously reported algorithm "concerted rotations with flexible bond angles" (CRA) has been shown to successfully locate the native state of small polypeptides. In this study, the folding of three small polypeptides (trpzip2/H1/Trp-cage) is investigated using MC and MD, for a combined sampling time of approximately 10(11) MC configurations and 8 micros, respectively. Both methods successfully locate the experimentally determined native states of the three systems, but they do so at different speed, with 2-2.5 times faster folding of the MC runs. The comparison reveals that thermodynamic and dynamic properties can reliably be obtained by both and that results from folding simulations do not depend on the algorithm used. Similar to previous comparisons of MC and MD, it is found that one MD integration step of 2 fs corresponds to one MC scan, revealing the good sampling of MC. The simplicity and efficiency of the MC method will enable its future use in folding studies involving larger systems and the combination with replica exchange algorithms.

Spin-glass behavior of the antiferromagnetic Ising model on a scale-free network

Antiferromagnetic Ising spins on the scale-free Barabasi-Albert network are studied via the Monte Carlo method. Using the replica exchange algorithm, we calculate the temperature dependence of various physical quantities of interest including the overlap and the Binder parameters. We observe a transition between a paramagnetic phase and a spin glass phase and estimate the critical temperature for the phase transition to be $T \sim 4.0(1)$ in units of $J/k_B$, where $J$ is the coupling strength between spins and $k_B$ is the Boltzmann constant. Using the scaling behaviour of the Binder parameter, we estimate the scaling exponent to be $\nu \sim 1.10(2)$.

Replica Exchange and Multicanonical Algorithms with the Coarse-Grained United-Residue (UNRES) Force Field

Three algorithms, namely a Replica Exchange method (REM), a Replica Exchange Multicanonical method (REMUCA), and Replica Exchange Multicanonical with Replica Exchange (REMUCAREM), were implemented with the coarse-grained united-residue force field (UNRES) in both Monte Carlo and Molecular Dynamics versions. The MD algorithms use the constant-temperature Berendsen thermostat, with the velocity Verlet algorithm and variable time step. The algorithms were applied to one peptide (20 residues of Alanine with free ends; ala(20)) and two small proteins, namely an α-helical protein of 46 residues (the B-domain of the staphylococal protein A; 1BDD), and an α+β-protein of 48 residues (the E. Coli Mltd Lysm Domain; 1E0G). Calculated thermodynamic averages, such as canonical average energy and heat capacity, are in good agreement among all simulations for poly-L-alanine, showing that the algorithms were implemented correctly, and that all three algorithms are equally effective for small systems. For protein A, all algorithms performed reasonably well, although some variability in the calculated results was observed whereas, for a more complicated α+β-protein (1E0G), only Replica Exchange was capable of producing reliable statistics for calculating thermodynamic quantities. Finally, from the Replica Exchange molecular dynamics results, we calculated free energy maps as functions of RMSD and radius of gyration for different temperatures. The free energy calculations show correct folding behavior for poly-L-alanine and protein A while, for 1E0G, the native structure had the lowest free energy only at very low temperatures. Hence, the entropy contribution for 1E0G is larger than that for protein A at the same temperature. A larger contribution from entropy means that there are more accessible conformations at a given temperature, making it more difficult to obtain an efficient coverage of conformational space to obtain reliable thermodynamic properties. At the same temperature, ala(20) has the smallest entropy contribution, followed by protein A, and then by 1E0G.

The Thermodynamics of Folding of a β Hairpin Peptide Probed Through Replica Exchange Molecular Dynamics Simulations

Peptides that possess a well defined native state are ideal model systems to study the folding of proteins. They possess many of the complexities of larger proteins, yet their small size renders their study computationally tractable. Recent advances in sampling techniques, including replica exchange molecular dynamics, now permit a full characterization of the thermodynamics of folding of small peptides. These simulations not only yield insight into the folding of larger proteins, but equally importantly, they allow, through comparison with experiment, an objective test of the accuracy of force fields, water models and of different numerical schemes for dealing with electrostatic interactions. In this account, we present a molecular dynamics simulation of a small β-hairpin peptide using the replica exchange algorithm and illustrate how this enhanced sampling scheme enables a thorough characterization of the native and unfolded states, and sheds new light into its folding mechanism.

Worm algorithm for two-dimensional spin glasses

A worm algorithm is proposed for the two-dimensional spin glasses. The method is based on a low-temperature expansion of the partition function. The low-temperature configurations of the spin glass on square lattice can be viewed as strings connecting pairs of frustrated plaquettes. The worm algorithm directly manipulates these strings. It is shown that the worm algorithm is efficient, particularly if free boundary conditions are used. We obtain accurate low-temperature specific heat data consistent with a form c approximately T-2 exp [-2J/( kB T) ] , where T is temperature and J is coupling constant, for the two-dimensional +/- J spin glass.

Lattice models of peptide aggregation: Evaluation of conformational search algorithms

We present a series of conformational search calculations on the aggregation of short peptide fragments that form fibrils similar to those seen in many protein mis-folding diseases. The proteins were represented by a face-centered cubic lattice model with the conformational energies calculated using the Miyazawa-Jernigan potential. The searches were performed using algorithms based on the Metropolis Monte Carlo method, including simulated annealing and replica exchange. We also present the results of searches using the tabu search method, an algorithm that has been used for many optimization problems, but has rarely been used in protein conformational searches. The replica exchange algorithm consistently found more stable structures then the other algorithms, and was particularly effective for the octamers and larger systems.