A comparative study of molecular dynamics in Cartesian and in internal coordinates: dynamical instability in the latter caused by nonlinearity of the equations of motion

The stability of a general molecular dynamics (MD) integration scheme is examined for simulations in generalized (internal plus external) coordinates (GCs). An analytic expression is derived for the local error in energy during each integration time step. This shows that the explicit dependence of the mass-matrix on GCs, which makes the system's Lagrange equations of motion nonlinear, causes MD simulations in GCs to be less stable than those in Cartesian coordinates (CCs). In terms of CCs, the corresponding mass-matrix depends only on atomic masses and thus atomistic motion is subject to the linear Newton equations, which makes the system more stable. Also investigated are two MD methods in GCs that utilize nonzero elements of the vibrational spectroscopic B-matrices. One updates positions and velocities in GCs that are iteratively adjusted so as to conform to the velocity Verlet equivalent in GCs. The other updates positions in GCs and velocities in CCs that are adjusted to satisfy the internal constraints of the new constrained WIGGLE MD scheme. The proposed methods are applied to an isolated n-octane molecule and their performances are compared with those of several CCMD schemes. The simulation results are found to be consistent with the analytic stability analysis. Finally, a method is presented for computing nonzero elements of B-matrices for external rotations without imposing the Casimir-Eckart conditions.     

Internal coordinate density of state from molecular dynamics simulation

The vibrational density of states (DoS), calculated from the Fourier transform of the velocity autocorrelation function, provides profound information regarding the structure and dynamic behavior of a system. However, it is often difficult to identify the exact vibrational mode associated with a specific frequency if the DoS is determined based on velocities in Cartesian coordinates. Here, the DoS is determined based on velocities in internal coordinates, calculated from Cartesian atomic velocities using a generalized Wilson's B ‐matrix. The DoS in internal coordinates allows for the correct detection of free dihedral rotations that may be mistaken as hindered rotation in Cartesian DoS. Furthermore, the pronounced enhancement of low frequency modes in Cartesian DoS for macromolecules should be attributed to the coupling of dihedral and angle motions. The internal DoS, thus deconvolutes the internal motions and provides fruitful insights to the dynamic behaviors of a system. © 2015 Wiley Periodicals, Inc.     

New out-of-plane angle and bond angle internal coordinates and related potential energy functions for molecular mechanics and dynamics simulations

With currently used definitions of out-of-plane angle and bond angle internal coordinates, Cartesian derivatives have singularities, at ±π/2 in the former case and π in the latter. If either of these occur during molecular mechanics or dynamics simulations, the forces are not well defined. To avoid such difficulties, we provide new out-of-plane and bond angle coordinates and associated potential energy functions that inherently avoid these singularities. The application of these coordinates is illustrated by ab initio calculations on ammonia, water, and carbon dioxide. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1067–1084, 1999     

Nonlinear shear of entangled polymers from nonequilibrium molecular dynamics

This study aims to use molecular dynamics (MD) simulations of Kremer–Grest (KG) chains to inform future developments of models of entangled polymer dynamics. We perform nonequilibrium MD simulations, under shear flow, for well‐entangled KG chains. We study chains of 512 and 1000 KG beads, corresponding to 8 and 15 entanglements, respectively. We compute the linear rheological properties from equilibrium simulations of the stress autocorrelation and obtain from these data the tube model parameters. Under nonlinear shear flow, we compute the shear viscosity, the first and second normal stress differences, and chain contour length. For chains of 512 monomers, we obtain agreement with the results of Cao and Likhtman (ACS Macro. Lett. 2015, 4, 1376). We also compare our nonlinear results with the Graham, Likhtman and Milner‐McLeish (GLaMM) model. We identify some systematic disagreement that becomes larger for the longer chains. We made a comparison of the transient shear stress maximum from our simulations, two nonlinear models and experiments on a wide range of melts and solutions, including polystyrene (PS), polybutadiene, and styrene–butadiene rubber. This comparison establishes that the PS melt data show markedly different behavior to all other melts and solutions and KG simulations reproduce the PS data more closely than either the GLaMM or Xie and Schweizer models. We discuss the performance of these models against the data and simulations. Finally, by imposing a rapid reversing flow, we produce a method to extract the recoverable strain from MD simulations, valid for sufficiently entangled monodisperse polymers. We explore how the resulting data can probe the melt state just before the reversing flow. © 2019 The Authors. Journal of Polymer Science Part B: Polymer Physics published by Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 1692–1704     

Effect of molecular structure of curing agents on cyclic creep in highly cross-linked epoxy polymers

Ratcheting behavior of highly–cross-linked epoxy polymers was investigated considering the effect of molecular structure of curing agents by molecular dynamics simulations. Cyclic loading–unloading simulations at two different frequencies were conducted using atomistic models for epoxies cured by aliphatic and aromatic curing agents, triethylenetetramine (TETA) and diethyltoluenediamine (DETDA), respectively. Different ratcheting strain evolutions, dihedral angle stress accumulations, and stiffness variations were observed during the cyclic deformation simulations depending on the molecular structure of curing agents. The epoxy cured by DETDA exhibited a more rapid increase of ratcheting strain and a decrease of the stiffness toward the loading direction. Structural analyses were carried out by observing the orientation order parameter of the monomers, radius of gyration, and free volume evolution to understand the ratcheting strain behaviors and stiffness variations at atomistic scale. The structural analyses revealed that irreversible dihedral angle transitions near the benzene ring of the curing agent DETDA were responsible for low ratcheting resistance and stiffness degradation during the cyclic deformations. Whereas, the aliphatic curing agent TETA, which does not exhibit any stress possession by the irreversible dihedral angle change, was revealed to be advantageous for the ratcheting resistance and stiffness variation of the epoxy polymers.     

Morphological studies of blends of conjugated polymers and acceptor molecules using langevin dynamics simulations

We use coarse‐grained Langevin dynamics simulations of blends of generic conjugated polymers and acceptor molecules to show how architecture (e.g., side chains, backbone flexibility of oligomers) and the pair‐wise interactions between the constituents of the blend affect morphology and phase transition. Alkyl side chains on the conjugated oligomer backbones shift the liquid crystal (LC) transition temperature from that of bare conjugated backbones and the direction of the shift depends on backbone–backbone interactions. Rigid backbones and constrained side chains cause a layer‐by‐layer morphology of conjugated polymers and amorphous acceptors, whereas flexible backbones and unconstrained side chains facilitate highly ordered acceptor arrangement. Strong backbone–backbone attraction shifts LC transition to higher temperatures than weak backbone–backbone attraction, and strong acceptor–acceptor attraction increases acceptor aggregation. Pure macro‐phase separated domains form when all pair‐wise interactions in the blend are strongly attractive, whereas interconnected domains form at intermediate acceptor–acceptor attraction and strong polymer–polymer attractions. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2013     

Numerical examination of the extended phase‐space volume‐preserving integrator by the Nosé‐Hoover molecular dynamics equations

This article illustrates practical applications to molecular dynamics simulations of the recently developed numerical integrators [Phys Rev E 2006, 73, 026703] for ordinary differential equations. This method consists of extending any set of ordinary differential equations in order to define a time invariant function, and then use the techniques of divergence‐free solvable decomposition and symmetric composition to obtain volume‐preserving integrators in the extended phase space. Here, we have developed the technique by constructing multiple extended‐variable formalism in order to enhance the handling in actual simulation, and by constituting higher order integrators to obtain further accuracies. Using these integrators, we perform constant temperature molecular dynamics simulations of liquid water, liquid argon and peptide in liquid water droplet. The temperature control is obtained through an extended version of the Nosé‐Hoover equations. Analyzing the effects of the simulation conditions including time step length, initial values, boundary conditions, and equation parameters, we investigate local accuracy, global accuracy, computational cost, and sensitivity along with the sampling validity. According to the results of these simulations, we show that the volume‐preserving integrators developed by the current method are more effective than traditional integrators that lack the volume‐preserving property. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Computation of internal coordinates,derivatives, and gradient expressions: torsion and improper torsion

Laplacians and gradient dot products are required for the recently developed internal coordinate quantum Monte Carlo method. New formulas are presented for these quantities for torsion and improper torsion angles. The Laplacians can also be used to economize calculation of sets of second derivatives used in molecular mechanics and other methods. Formulas for torsion angle gradient dot products and Laplacians, and completely new formulas for improper torsion, are presented. In addition, calculations of cos τ and sin τ, some suitable for energy subroutines and others for force subroutines, are shown. Finally, in a related development, several sets of conditions for three atom linearity or four atom planarity involving internal coordinate derivatives are reported. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 553–561, 2000     

Application of torsion angle molecular dynamics for efficient sampling of protein conformations

We investigate the application of torsion angle molecular dynamics (TAMD) to augment conformational sampling of peptides and proteins. Interesting conformational changes in proteins mainly involve torsional degrees of freedom. Carrying out molecular dynamics in torsion space does not only explicitly sample the most relevant degrees of freedom, but also allows larger integration time steps with elimination of the bond and angle degrees of freedom. However, the covalent geometry needs to be fixed during internal coordinate dynamics, which can introduce severe distortions to the underlying potential surface in the extensively parameterized modern Cartesian-based protein force fields. A "projection" approach (Katritch et al. J Comput Chem 2003, 24, 254-265) is extended to construct an accurate internal coordinate force field (ICFF) from a source Cartesian force field. Torsion crossterm corrections constructed from local molecular fragments, together with softened van der Waals and electrostatic interactions, are used to recover the potential surface and incorporate implicit bond and angle flexibility. MD simulations of dipeptide models demonstrate that full flexibility in both the backbone phi/psi and side chain chi1 angles are virtually restored. The efficacy of TAMD in enhancing conformational sampling is then further examined by folding simulations of small peptides and refinement experiments of protein NMR structures. The results show that an increase of several fold in conformational sampling efficiency can be reliably achieved. The current study also reveals some complicated intrinsic properties of internal coordinate dynamics, beyond energy conservation, that can limit the maximum size of the integration time step and thus the achievable gain in sampling efficiency.     

Parameterization of electrostatic interactions for molecular dynamics simulations of heterocyclic polymers

The paper focuses on the problem of electrostatic interactions in molecular dynamics simulations of thermal properties of heterocyclic polymers. The study focuses on three thermoplastic polyimides synthesized on the basis of 1,3‐bis‐(3′,4‐dicarboxyphenoxy)benzene (dianhydride R) and three diamines: 4,4′‐bis‐(4″‐aminophenoxy) diphenylsulfone (diamine BAPS), 4,4′‐bis‐(4″‐aminophenoxy) biphenyl (diamine BAPB), and 4,4′‐bis‐(4''‐aminophenoxy) diphenyloxide (diamine BAPO). In the molecular dynamics simulations these polyimides were described by the Gromos53a5 force field. To parameterize the electrostatic interactions four methods of calculating the partial atomic charges were chosen: B3LYP/6–31G*(Mulliken), AM1(Mulliken), HF/6–31G*(Mulliken), and HF/6–31G*(ChelpG). As our parameterization is targeted to reproduce thermal properties of the thermoplastic polyimides, the choice of proper partial charges was finalized on a basis of the closest match between computational and experimental data for the thermal expansion coefficients of the polyimides below glass transition temperatures. Our finding clearly show that the best agreement with experimental data is achieved with the Mulliken partial atomic charges calculated by the Hartree‐Fock method with 6–31G* basis set. Furthermore, in addition to the thermal expansion coefficients this set of partial atomic charges predicts an experimentally observed relationship between glass transition temperatures of the three polyimides under study: . A mechanism behind the change in thermal properties upon the change in the chemical structure in considered polyimides may be related to an additional spatial ordering of sulfone groups due to dipole‐dipole interactions. Overall, the modified force‐field is proved to be suitable for accurate prediction of thermal properties of thermoplastic polyimides and can serve as a basis for building up atomistic theoretical models for describing other heterocyclic polymers in bulk. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 912–923     

A fast SHAKE algorithm to solve distance constraint equations for small molecules in molecular dynamics simulations

A common method for the application of distance constraints in molecular simulations employing Cartesian coordinates is the SHAKE procedure for determining the Lagrange multipliers regarding the constraints. This method relies on the linearization and decoupling of the equations governing the atomic coordinate resetting corresponding to each constraint in a molecule, and is thus iterative. In the present study, we consider an alternative method, M‐SHAKE, which solves the coupled equations simultaneously by matrix inversion. The performances of the two methods are compared in simulations of the pure solvents water, dimethyl sulfoxide, and chloroform. It is concluded that M‐SHAKE is significantly faster than SHAKE when either (1) the molecules contain few distance constraints (solvent), or (2) when a high level of accuracy is required in the application of the constraints. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 501–508, 2001     

Essential dynamics/factor analysis for the interpretation of molecular dynamics trajectories

Subject of this work is the analysis of molecular dynamics (MD) trajectories of neurophysins I (NPI) and II (NPII) and their complexes with the neurophyseal nonapeptide hormones oxytocin (OT) and vasopresssin (VP), respectively, simulated in water. NPs serve in the neurosecretory granules as carrier proteins for the hormones before their release to the blood. The starting data consisted of two pairs of different trajectories for each of the (NPII/VP)2 and (NPI/OT)2 heterotetramers and two more trajectories for the NPII2 and NPI2 homodimers (six trajectories in total). Using essential dynamics which, to our judgement, is equivalent to factor analysis, we found that only about 10 degrees of freedom per trajectory are necessary and sufficient to describe in full the motions relevant for the function of the protein. This is consistent with these motions to explain about 90% of the total variance of the system. These principal degrees of freedom represent slow anharmonic motional modes, clearly pointing at distinguished mobility of the atoms involved in the protein's functionality.     

CHARMM force field and molecular dynamics simulations of protonated polyethylenimine

As a gene delivery vector, polyethylenimine (PEI) shows one of the highest transfection efficiencies, while effectively protecting DNA from enzyme degradation. The distinctive charge pattern of protonated PEI is widely considered responsible for fundamental process such as DNA condensation into PEI/DNA polyplexes (which are able to enter cells via endocytosis), proton sponge effect (which triggers the release of polyplexes from endosome), and release of DNA from polyplexes (to be further processed inside the nucleus). Our investigations are largely motivated by the crucial need for a realistic molecular mechanics force field (FF) for PEI, and, accordingly, we focus on two major issues: (1) development of a new atomistic (CHARMM) FF for PEI in different protonation states, rigorously derived from high‐quality ab initio calculations performed on model polymers, and (2) molecular dynamics investigations of solvated PEI, providing a detailed picture of the dynamic structuring thereof in dependence on their size and protonation state. The modeled PEI chains are essentially described in terms of gyration radius, end‐to‐end distance, persistence length, radial distribution functions, coordination numbers, and diffusion coefficients. They turn out to be more rigid than in other computational studies and we find diffusion coefficients in fair agreement with experimental data. The developed atomistic FF proves adequate for the realistic modeling of the size and protonation behavior of linear PEI, either as individual chains or composing polyplexes. © 2017 Wiley Periodicals, Inc.     

On the use of projection operators in molecular dynamics

Symmetry coordinates are easily constructed when operating in mass-weighted Cartesian space. Such coordinates can be used to block diagonalize automatically the secular equation occurring in the harmonic approximation of dynamics of large molecules. Examples of the usefulness of the method are given.     

Explicit ion, implicit water solvation for molecular dynamics of nucleic acids and highly charged molecules

An explicit ion, implicit water solvent model for molecular dynamics was developed and tested with DNA and RNA simulations. The implicit water model uses the finite difference Poisson (FDP) model with the smooth permittivity method implemented in the OpenEye ZAP libraries. Explicit counter-ions, co-ions, and nucleic acid were treated with a Langevin dynamics molecular dynamics algorithm. Ion electrostatics is treated within the FDP model when close to the solute, and by the Coulombic model when far from the solute. The two zone model reduces computation time, but retains an accurate treatment of the ion atmosphere electrostatics near the solute. Ion compositions can be set to reproduce specific ionic strengths. The entire ion/water treatment is interfaced with the molecular dynamics package CHARMM. Using the CHARMM-ZAPI software combination, the implicit solvent model was tested on A and B form duplex DNA, and tetraloop RNA, producing stable simulations with structures remaining close to experiment. The model also reproduced the A to B duplex DNA transition. The effect of ionic strength, and the structure of the counterion atmosphere around B form duplex DNA were also examined.     

Constraint dynamics algorithm for simulation of semiflexible macromolecules

Semiflexible models are often used to study macromolecules containing stable structural elements. Based on rigid body dynamics, we developed a rigid fragment constraint dynamics algorithm for the simulation of semiflexible macromolecules. Stable structural elements are treated as rigid fragments. Rigid fragment constraints, defined as combinations of distance constraints and position constraints, are introduced to limit internal molecular motion to the required mode. The constraint forces are solved separately for each rigid fragment constraint and iteratively until all constraint conditions are satisfied within a given tolerance at each time step, as is done for the bond length constraint in the SHAKE algorithm. The orientation of a rigid fragment is represented by the quaternion parameters, and both translation and rotation are solved by the leap-frog formulation. We tested the algorithm with molecular dynamics simulations of a series of peptides and a small protein. The computation cost for the constraints is roughly proportional to the size of the molecule. In the microcanonical ensemble simulation of polyvalines, the total energy was conserved satisfactorily with time steps as large as 20 fs. A helix folding simulation of a synthetic peptide was carried out to show the efficiency of the algorithm in a conformational search. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1555–1566, 1998     

Effect of internal viscosity on Brownian dynamics of DNA molecules in shear flow

The results of Brownian dynamics simulations of a single DNA molecule in shear flow are presented taking into account the effect of internal viscosity. The dissipative mechanism of internal viscosity is proved necessary in the research of DNA dynamics. A stochastic model is derived on the basis of the balance equation for forces acting on the chain. The Euler method is applied to the solution of the model. The extensions of DNA molecules for different Weissenberg numbers are analyzed. Comparison with the experimental results available in the literature is carried out to estimate the contribution of the effect of internal viscosity.