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.     

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 model of the temperature and hydration dependence of protein vibrational dynamics

The modification of internal vibrational modes in a protein due to intraprotein anharmonicity and solvation effects is determined by performing molecular dynamics (MD) simulations of myoglobin, analyzing them using a Langevin model of the vibrational dynamics and comparing the Langevin results to a harmonic, normal mode model of the protein in vacuum. The diagonal and off-diagonal Langevin friction matrix elements, which model the roughness of the vibrational potential energy surfaces, are determined together with the vibrational potentials of mean force from the MD trajectories at 120 K and 300 K in vacuum and in solution. The frictional properties are found to be describable using simple phenomenological functions of the mode frequency, the accessible surface area, and the intraprotein interaction (the displacement vector overlap of any given mode with the other modes in the protein). The frictional damping of a vibrational mode in vacuum is found to be directly proportional to the intraprotein interaction of the mode, whereas in solution, the friction is proportional to the accessible surface area of the mode. In vacuum, the MD frequencies are lower than those of the normal modes, indicating intramolecular anharmonic broadening of the associated potential energy surfaces. Solvation has the opposite effect, increasing the large-amplitude vibrational frequencies relative to in vacuum and thus vibrationally confining the protein atoms. Frictional damping of the low-frequency modes is highly frequency dependent. In contrast to the damping effect of the solvent, the vibrational frequency increase due to solvation is relatively temperature independent, indicating that it is primarily a structural effect. The MD-derived vibrational dynamic structure factor and density of states are well reproduced by a model in which the Langevin friction and potential of mean force parameters are applied to the harmonic normal modes.     

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 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 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 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.     

Normal mode partitioning of Langevin dynamics for biomolecules

We propose a novel normal mode multiple time stepping Langevin dynamics integrator called NML. The aim is to approximate the kinetics or thermodynamics of a biomolecule by a reduced model based on a normal mode decomposition of the dynamical space. Our basis set uses the eigenvectors of a mass reweighted Hessian matrix calculated with a biomolecular force field. This particular choice has the advantage of an ordering according to the eigenvalues, which have a physical meaning of being the square of the mode frequency. Low frequency eigenvalues correspond to more collective motions, whereas the highest frequency eigenvalues are the limiting factor for the stability of the integrator. In NML, the higher frequency modes are overdamped and relaxed near their energy minimum while respecting the subspace of low frequency dynamical modes. Our numerical results confirm that both sampling and rates are conserved for an implicitly solvated alanine dipeptide model, with only 30% of the modes propagated, when compared to the full model. For implicitly solvated systems, NML gives a twofold improvement in efficiency over plain Langevin dynamics for sampling a small 22 atom (alanine dipeptide) model and in excess of an order of magnitude for sampling an 882 atom (bovine pancreatic trypsin inhibitor) model, with good scaling with system size subject to the number of modes propagated. NML has been implemented in the open source software PROTOMOL.     

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.     

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.     

Secondary structure sensitivity of hydrogen bond lifetime dynamics in the protein hydration layer

The heterogeneous nature of a protein surface plays an essential role in its biological activity and molecular recognition, and this role is mediated at least partly through the surrounding water molecules. We have performed atomistic molecular dynamics simulations of an aqueous solution of HP-36 to investigate the correlation between the dynamics of the hydration layer water molecules and the lifetimes of protein-water hydrogen bonds. The nonexponential hydrogen bond lifetime correlation functions have been analyzed by using the formalism of Luzar and Chandler, which allowed identification of the quasi-bound states in the surface and quantification of the dynamic equilibrium between quasi-bound and free water molecules in terms of time-dependent rate of interconversion. It is noticed that, irrespective of the structural heterogeneity of different segments of the protein, namely the three alpha-helices, the positively charged amino acid residues form longer-lived hydrogen bonds with water. The overall relaxation behavior of protein-water hydrogen bonds is found to differ significantly among the three helices of the protein. Study of water number density fluctuation reveals that the hydration layer of helix-3 is much less rigid, which can be correlated with faster structural relaxation of the hydrogen bonds between its residues and water. This also agrees excellently with faster translational and rotational motions of water near helix-3, and hence the lower rigidity of its hydration layer. The lower rigidity of the helix-3 hydration layer also correlates well with the biological activity of the protein, as several of the active-site residues of HP-36 are located in helix-3.     

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.     

Improved configuration space sampling: Langevin dynamics with alternative mobility

We present a new and efficient method for determining optimal configurations of a large number (N) of interacting particles. We use a coarse-grained stochastic Langevin equation in the overdamped limit to describe the dynamics of this system and replace the standard mobility by an effective space dependent inverse Hessian correlation matrix. Due to the analogy of the drift term in the Langevin equation and the update scheme in Newton's method, we expect accelerated dynamics or improved convergence in the convex part of the potential energy surface Phi. The stochastic noise term, however, is not only essential for proper thermodynamic sampling but also allows the system to access transition states in the concave parts of Phi. We employ a Broyden-Fletcher-Goldfarb-Shannon method for updating the local mobility matrix. Quantitative analysis for one and two dimensional systems shows that the new method is indeed more efficient than standard methods with constant effective friction. Due to the construction, our effective mobility adapts high values/low friction in configurations which are less optimal and low values/high friction in configurations that are more optimal.     

Langevin dynamics of molecules with internal rigid fragments in the harmonic regime

An approximation scheme is developed to compute Brownian motion according to the Langevin equation for a molecular system moving in a harmonic force field (corresponding to a quadratic potential energy surface) and characterized by one or more rigid internal fragments. This scheme, which relies on elements of the rotation translation block (RTB) method for computing vibrational normal modes of large molecules developed by Sanejouand and co-workers [Biopolymers 34, 759 (1994); Proteins: Struct., Funct., Genet. 41, 1 (2000)], provides a natural and efficient way to freeze out the small amplitude, high frequency motions within each rigid fragment. The number of dynamical degrees of freedom in the problem is thereby reduced, often dramatically. To illustrate the method, the relaxation kinetics of the small membrane-bound ion channel protein gramicidin-A, subjected to an externally imposed impulse, is computed. The results obtained from all-atom dynamics are compared to those obtained using the RTB-Langevin dynamics approximation (treating eight indole moieties as internal rigid fragments): good agreement between the two treatments is found.     

A Langevin dynamics study of mobile filler particles in phase-separating binary systems

The dynamics of phase separation in a simple binary mixture containing mobile filler particles that are preferentially wet by one of the two components is investigated systematically via Langevin simulations in two dimensions. We found that while the filler particles reduce the growth rate of spinodal decomposition, the domain growth remains essentially identical to that of the pure binary mixture. The growth rate diminishes as either the filler particles concentration is increased or their diffusivity is decreased.