Design of quasisymplectic propagators for Langevin dynamics

A vector field splitting approach is discussed for the systematic derivation of numerical propagators for deterministic dynamics. Based on the formalism, a class of numerical integrators for Langevin dynamics are presented for single and multiple time step algorithms.     

Langevin dynamics for rigid bodies of arbitrary shape

We present an algorithm for carrying out Langevin dynamics simulations on complex rigid bodies by incorporating the hydrodynamic resistance tensors for arbitrary shapes into an advanced rotational integration scheme. The integrator gives quantitative agreement with both analytic and approximate hydrodynamic theories for a number of model rigid bodies and works well at reproducing the solute dynamical properties (diffusion constants and orientational relaxation times) obtained from explicitly solvated simulations.     

Langevin dynamics simulation of polymer-assisted virus-like assembly

Starting from a coarse grained representation of the building units of the minute virus of mice and a flexible polyelectrolyte molecule, we have explored the mechanism of assembly into icosahedral structures with the help of Langevin dynamics simulations and the parallel tempering technique. Regular icosahedra with appropriate symmetry form only in a narrow range of temperature and polymer length. Within this region of parameters where successful assembly would proceed, we have systematically investigated the growth kinetics. The assembly of icosahedra is found to follow the classical nucleation and growth mechanism in the absence of the polymer, with the three regimes of nucleation, linear growth, and slowing down in the later stage. The calculated average nucleation time obeys the laws expected from the classical nucleation theory. The linear growth rate is found to obey the laws of secondary nucleation as in the case of lamellar growth in polymer crystallization. The same mechanism is seen in the simulations of the assembly of icosahedra in the presence of the polymer as well. The polymer reduces the nucleation barrier significantly by enhancing the local concentration of subunits via adsorbing them on their backbone. The details of growth in the presence of the polymer are also found to be consistent with the classical nucleation theory, despite the smallness of the assembled structures.     

Long-time overdamped Langevin dynamics of molecular chains

We present a novel algorithm of constrained, overdamped dynamics to study the long-time properties of peptides, proteins, and related molecules. The constraints are applied to an all-atom model of the molecule by projecting out all components of the nonbonding interactions which tend to alter fixed bond lengths and angles. Because the overdamped dynamical equations are first order in time, the constraints are satisfied by inversion of a banded matrix at each timestep, which is computationally efficient. Thermal effects are included through a Langevin noise term in the equation of motion. Because high-frequency components of the motion have been eliminated, the timestep of the algorithm is determined by the nonbonding forces, which are two to three orders of magnitude weaker than the bonding forces. Using polyalanine as a test example, we demonstrate that trajectories simulating a microsecond of motion can be run about 10³ times faster than an equivalent molecular dynamics simulation. © 1994 by John Wiley & Sons, Inc.     

Langevin dynamics in constant pressure extended systems

The advantages of performing Langevin dynamics in extended systems are discussed. A simple Langevin dynamics scheme for producing the canonical ensemble is reviewed, and is then extended to the Hoover ensemble. We show that the resulting equations of motion generate the isobaric-isothermal ensemble. The Parrinello-Rahman ensemble is then discussed and we show that despite the presence of intrinsic probability gradients in this system, a Langevin dynamics approach samples the extended phase space in the correct fashion. The implementation of these methods in the ab initio plane wave density functional theory code CASTEP [M. D. Segall, P. L. D. Lindan, M. J. Probert, C. J. Pickard, P. J. Hasnip, S. J. Clarke, and M. C. Payne, J. Phys.: Condens. Matter 14, 2717 (2003)] is demonstrated.     

Dynamic bond constraints in protein Langevin dynamics

Bond constraint algorithms for molecular dynamics typically take, as the target constraint lengths, the values of the equilibrium bond lengths defined in the potential. In Langevin form, the equations of motion are temperature dependent, which gives the average value for the individual bond lengths a temperature dependence. In addition to this, locally constant force fields can shift the local equilibrium bond lengths. To restore the average bond lengths in constrained integration to their unconstrained values, we suggest changing the constraint length used by popular constraint methods such as RATTLE [H. C. Andersen, J. Comput. Phys. 52, 23 (1983)] at each step. This allows us to more accurately capture the equilibrium bond length changes (with respect to the potential) due to the local equilibration and temperature effects. In addition, the approximations to the unconstrained nonbonded energies are closer using the dynamic constraint method than a traditional fixed constraint algorithm. The mechanism for finding the new constrained lengths involves one extra calculation of the bonded components of the force, and therefore adds O(N) time to the constraint algorithm. Since most molecular dynamics calculations are dominated by the O(N2) nonbonded forces, this new method does not take significantly more time than a fixed constraint algorithm.     

Langevin dynamics simulations of macromolecules on parallel computers

A parallel algorithm is developed that allows efficient Langevin-dynamics simulations of macromolecular coils, which is the usual structure of synthetic polymers in solution and in bulk. Contrary to usual so-called spatial decomposition algorithms, we map the one-dimensional topology of the chain molecule on the parallel computer. The speedup of the algorithm is measured on different multi-processor systems. The reliability of the parallel calculations is shown by comparison with sequential simulations.     

Langevin dynamics simulations of polymer translocation through nanopores

We investigate the dynamics of polymer translocation through a nanopore using two-dimensional Langevin dynamics simulations. In the absence of an external driving force, we consider a polymer which is initially placed in the middle of the pore and study the escape time tau(e) required for the polymer to completely exit the pore on either side. The distribution of the escape times is wide and has a long tail. We find that tau(e) scales with the chain length N as tau(e) approximately N(1+2nu), where nu is the Flory exponent. For driven translocation, we concentrate on the influence of the friction coefficient xi, the driving force E, and the length of the chain N on the translocation time tau, which is defined as the time duration between the first monomer entering the pore and the last monomer leaving the pore. For strong driving forces, the distribution of translocation times is symmetric and narrow without a long tail and tau approximately E(-1). The influence of xi depends on the ratio between the driving and frictional forces. For intermediate xi, we find a crossover scaling for tau with N from tau approximately N(2nu) for relatively short chains to tau approximately N(1+nu) for longer chains. However, for higher xi, only tau approximately N(1+nu) is observed even for short chains, and there is no crossover behavior. This result can be explained by the fact that increasing xi increases the Rouse relaxation time of the chain, in which case even relatively short chains have no time to relax during translocation. Our results are in good agreement with previous simulations based on the fluctuating bond lattice model of polymers at intermediate friction values, but reveal additional features of dependency on friction.     

Semi-classical generalized Langevin equation for equilibrium and nonequilibrium molecular dynamics simulation

Molecular dynamics (MD) simulation based on Langevin equation has been widely used in the study of structural, thermal properties of matter in different phases. Normally, the atomic dynamics are described by classical equations of motion and the effect of the environment is taken into account through the fluctuating and frictional forces. Generally, the nuclear quantum effects and their coupling to other degrees of freedom are difficult to include in an efficient way. This could be a serious limitation on its application to the study of dynamical properties of materials made from light elements, in the presence of external driving electrical or thermal fields. One example of such system is single molecule dynamics on metal surface, an important system that has received intense study in surface science. In this review, we summarize recent effort in extending the Langevin MD to include nuclear quantum effect and their coupling to flowing electrical current. We discuss its applications in the study of adsorbate dynamics on metal surface, current-induced dynamics in molecular junctions, and quantum thermal transport between different reservoirs.     

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.     

Comparison between self-guided Langevin dynamics and molecular dynamics simulations for structure refinement of protein loop conformations

This article presents a comparative analysis of two replica‐exchange simulation methods for the structure refinement of protein loop conformations, starting from low‐resolution predictions. The methods are self‐guided Langevin dynamics (SGLD) and molecular dynamics (MD) with a Nosé–Hoover thermostat. We investigated a small dataset of 8‐ and 12‐residue loops, with the shorter loops placed initially from a coarse‐grained lattice model and the longer loops from an enumeration assembly method (the Loopy program). The CHARMM22 + CMAP force field with a generalized Born implicit solvent model (molecular‐surface parameterized GBSW2) was used to explore conformational space. We also assessed two empirical scoring methods to detect nativelike conformations from decoys: the all‐atom distance‐scaled ideal‐gas reference state (DFIRE‐AA) statistical potential and the Rosetta energy function. Among the eight‐residue loop targets, SGLD out performed MD in all cases, with a median of 0.48 Å reduction in global root‐mean‐square deviation (RMSD) of the loop backbone coordinates from the native structure. Among the more challenging 12‐residue loop targets, SGLD improved the prediction accuracy over MD by a median of 1.31 Å, representing a substantial improvement. The overall median RMSD for SGLD simulations of 12‐residue loops was 0.91 Å, yielding refinement of a median 2.70 Å from initial loop placement. Results from DFIRE‐AA and the Rosetta model applied to rescoring conformations failed to improve the overall detection calculated from the CHARMM force field. We illustrate the advantage of SGLD over the MD simulation model by presenting potential‐energy landscapes for several loop predictions. Our results demonstrate that SGLD significantly outperforms traditional MD in the generation and populating of nativelike loop conformations and that the CHARMM force field performs comparably to other empirical force fields in identifying these conformations from the resulting ensembles. Published 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Langevin dynamics simulations of ds-DNA translocation through synthetic nanopores

We have implemented a coarse-grained model to study voltage-driven as-DNA translocation through nanopores located in synthetic membranes. The simulated trajectory of the DNA through the nanopores was calculated using Langevin dynamics. We present the results based on more than 120,000 individual translocations. We are particularly interested in this work in probing the physical basis of various experimentally observed--yet poorly understood--phenomena. Notably, we observe in our simulations the formation of ds-DNA hairpins, widely suspected to be the basis for quantized blockage. We study the translocation time, a measurable quantity crucially important in polyelectrolyte characterization, as a function of hairpin vertex location along the polymer backbone, finding that this behavior can be tuned to some degree by simulation parameters. We also study the voltage dependence of the tendency of hairpins to serve as the initiators of translocation events. Surprisingly, we find that the resulting probability depends vitally upon whether the events counted are ultimately successful or not. Further details lead us to propose that failed attempts in experimental translocation studies may be more common--and deceptive--than is generally recognized. We find the time taken by successful single file translocations to be directly proportional to the ratio of chain length to the applied voltage. Finally, we address a common yet puzzling phenomenon in translocation experiments: translocation events in which the current through the pore is highly, yet incompletely, blocked. We present the findings that offer a new explanation for such events.     

LIN: A new algorithm to simulate the dynamics of biomolecules by combining implicit-integration and normal mode techniques

A central goal in molecular dynamics simulations is increasing the integration time-step to allow the capturing of biomolecular motion on biochemically interesting time frames. We previously made a step in that direction by developing the Langevin/implicit–Euler scheme. Here, we present a modified Langevin/implicit–Euler formulation for molecular dynamics. The new method still maintains the major advantage of the original scheme, namely, stability over a wide range of time-steps. However, it substantially reduces the damping effect of the high-frequency modes inherent in the original implicit scheme. The new formulation involves separation of the solution into two components, one of which is solved exactly using normal-mode techniques, the other of which is solved by implicit numerical integration. In this way, the high-frequency and fast-varying components are well resolved in the analytic solution component, while the remaining components of the motion are obtained by a large time-step integration phase. Full details of the new scheme are presented, accompanied by illustrative examples for a simple pendulum system. An application to liquid butane demonstrates stability of the simulations at time-steps up to 50 fs, still with activation of the high-frequency modes. © John Wiley & Sons, Inc.     

The Langevin Hull: Constant pressure and temperature dynamics for non-periodic systems

We have developed a new isobaric-isothermal (NPT) algorithm which applies an external pressure to the facets comprising the convex hull surrounding the system. A Langevin thermostat is also applied to the facets to mimic contact with an external heat bath. This new method, the "Langevin Hull", can handle heterogeneous mixtures of materials with different compressibilities. These systems are problematic for traditional affine transform methods. The Langevin Hull does not suffer from the edge effects of boundary potential methods, and allows realistic treatment of both external pressure and thermal conductivity due to the presence of an implicit solvent. We apply this method to several different systems including bare metal nanoparticles, nanoparticles in an explicit solvent, as well as clusters of liquid water. The predicted mechanical properties of these systems are in good agreement with experimental data and previous simulation work.     

Kinetic partitioning mechanism as a unifying theme in the folding of biomolecules

We describe a unified approach to describe the kinetics of protein and RNA folding. The underlying conceptual basis for this framework relies on the notion that biomolecules are topologically frustrated due to their polymeric nature and due to the presence of conflicting energies. As a result, the free energy surface (FES) has, in addition to the native basin of attraction (NBA), several competing basins of attraction. A rough FES results in direct and indirect pathways to the NBA, i.e., a kinetic partitioning mechanism (KPM). The KPM leads to a foldability principle according to which fast folding sequences are characterized by the folding transition temperature T _F being close to the collapse transition temperature T _θ, at which a transition from the random coil to the compact structure takes place. Biomolecules with T _θ ≈ T _F , such as small proteins and tRNAs, are expected to fold rapidly with two-state kinetics. Estimates for the multiple time scales in KPM are also given. We show that experiments on proteins and RNA can be understood semi-quantitatively in terms of the KPM. Received: 14 January 1997 / Accepted: 23 January 1997     

Entropy and enthalpy of polyelectrolyte complexation: Langevin dynamics simulations

We report a systematic study by Langevin dynamics simulation on the energetics of complexation between two oppositely charged polyelectrolytes of same charge density in dilute solutions of a good solvent with counterions and salt ions explicitly included. The enthalpy of polyelectrolyte complexation is quantified by comparisons of the Coulomb energy before and after complexation. The entropy of polyelectrolyte complexation is determined directly from simulations and compared with that from a mean-field lattice model explicitly accounting for counterion adsorption. At weak Coulomb interaction strengths, e.g., in solvents of high dielectric constant or with weakly charged polyelectrolytes, complexation is driven by a negative enthalpy due to electrostatic attraction between two oppositely charged chains, with counterion release entropy playing only a subsidiary role. In the strong interaction regime, complexation is driven by a large counterion release entropy and opposed by a positive enthalpy change. The addition of salt reduces the enthalpy of polyelectrolyte complexation by screening electrostatic interaction at all Coulomb interaction strengths. The counterion release entropy also decreases in the presence of salt, but the reduction only becomes significant at higher Coulomb interaction strengths. More significantly, in the range of Coulomb interaction strengths appropriate for highly charged polymers in aqueous solutions, complexation enthalpy depends weakly on salt concentration and counterion release entropy exhibits a large variation as a function of salt concentration. Our study quantitatively establishes that polyelectrolyte complexation in highly charged Coulomb systems is of entropic origin.