Distance and angular holonomic constraints in molecular simulations

Methods for Classical-Mechanical Molecular Simulation in Chemistry: Achievements, Limitations, Perspectives

More than a half century ago it became feasible to simulate, using classical-mechanical equations of motion, the dynamics of molecular systems on a computer. Since then classical-physical molecular simulation has become an integral part of chemical research. It is widely applied in a variety of branches of chemistry and has significantly contributed to the development of chemical knowledge. It offers understanding and interpretation of experimental results, semiquantitative predictions for measurable and nonmeasurable properties of substances, and allows the calculation of properties of molecular systems under conditions that are experimentally inaccessible. Yet, molecular simulation is built on a number of assumptions, approximations, and simplifications which limit its range of applicability and its accuracy. These concern the potential-energy function used, adequate sampling of the vast statistical-mechanical configurational space of a molecular system and the methods used to compute particular properties of chemical systems from statistical-mechanical ensembles. During the past half century various methodological ideas to improve the efficiency and accuracy of classical-physical molecular simulation have been proposed, investigated, evaluated, implemented in general simulation software or were abandoned. The latter because of fundamental flaws or, while being physically sound, computational inefficiency. Some of these methodological ideas are briefly reviewed and the most effective methods are highlighted. Limitations of classical-physical simulation are discussed and perspectives are sketched.

Comparison of the United- and All-Atom Representations of (Halo)alkanes Based on Two Condensed-Phase Force Fields Optimized against the Same Experimental Data Set

The level of accuracy that can be achieved by a force field is influenced by choices made in the interaction-function representation and in the relevant simulation parameters. These choices, referred to here as functional-form variants (FFVs), include for example the model resolution, the charge-derivation procedure, the van der Waals combination rules, the cutoff distance, and the treatment of the long-range interactions. Ideally, assessing the effect of a given FFV on the intrinsic accuracy of the force-field representation requires that only the specific FFV is changed and that this change is performed at an optimal level of parametrization, a requirement that may prove extremely challenging to achieve in practice. Here, we present a first attempt at such a comparison for one specific FFV, namely the choice of a united-atom (UA) versus an all-atom (AA) resolution in a force field for saturated acyclic (halo)alkanes. Two force-field versions (UA vs AA) are optimized in an automated way using the CombiFF approach against 961 experimental values for the pure-liquid densities ρ_liq and vaporization enthalpies ΔH_vap of 591 compounds. For the AA force field, the torsional and third-neighbor Lennard-Jones parameters are also refined based on quantum-mechanical rotational-energy profiles. The comparison between the UA and AA resolutions is also extended to properties that have not been included as parameterization targets, namely the surface-tension coefficient γ, the isothermal compressibility κ_T, the isobaric thermal-expansion coefficient α_P, the isobaric heat capacity c_P, the static relative dielectric permittivity ϵ, the self-diffusion coefficient D, the shear viscosity η, the hydration free energy ΔG_wat, and the free energy of solvation ΔG_che in cyclohexane. For the target properties ρ_liq and ΔH_vap, the UA and AA resolutions reach very similar levels of accuracy after optimization. For the nine other properties, the AA representation leads to more accurate results in terms of η; comparably accurate results in terms of γ, κ_T, α_P, ϵ, D, and ΔG_che; and less accurate results in terms of c_P and ΔG_wat. This work also represents a first step toward the calibration of a GROMOS-compatible force field at the AA resolution.

On the use of intra-molecular distance and angle constraints to lengthen the time step in molecular and stochastic dynamics simulations of proteins

Computer simulation of proteins in aqueous solution at the atomic level of resolution is still limited in time span and system size due to limited computing power available and thus employs a variety of time‐saving techniques that trade some accuracy against computational effort. An example of such a time‐saving technique is the application of constraints to particular degrees of freedom when integrating Newton's or Langevin's equations of motion in molecular dynamics (MD) or stochastic dynamics (SD) simulations, respectively. The application of bond‐length constraints is standard practice in protein simulations and allows for a lengthening of the time step by a factor of three. Applying recently proposed algorithms to constrain bond angles or dihedral angles, it is investigated, using the protein trypsin inhibitor as test molecule, whether bond angles and dihedral angles involving hydrogen atoms or even stiff proper (torsional) dihedral angles as well as improper ones (maintaining particular tetrahedral or planar geometries) may be constrained without generating too many artificial side effects. Constraining the relative positions of the hydrogen atoms in the protein allows for a lengthening of the time step by a factor of two. Additionally constraining the improper dihedral angles and the stiff proper (torsional) dihedral angles in the protein does not allow for an increase of the MD or SD time step.

A method to apply bond-angle constraints in molecular dynamics simulations

An algorithm to apply bond-angle constraints in molecular dynamics simulations of macromolecules or molecular liquids is presented. It uses Cartesian coordinates and determines the Lagrange multipliers required for maintaining the constraints iteratively. It constitutes an alternative to the use of only distance constraints (DCs) between particles to maintain a particular geometry. DCs are unsuitable to maintain particular, for example, linear or flat, geometries of molecules. The proposed algorithm can easily handle bond-length, bond-angle, and dihedral-angle constraints simultaneously, as when calculating a potential of mean force along a dihedral-angle degree of freedom.

Photoinduced Dynamics with Constrained Vibrational Motion: FrozeNM Algorithm

Ab initio molecular dynamics (AIMD) simulation, analyzed in terms of vibrational normal modes, is a widely used technique that facilitates understanding of complex structural motions and coupling between electronic and nuclear degrees of freedom. Usually, only a subset of vibrations is directly involved in the process of interest. The impact of these vibrations can be evaluated by performing AIMD simulations by selectively freezing certain motions. Herein, we present frozen normal mode (FrozeNM), a new algorithm to apply normal-mode constraints in AIMD simulations, as implemented in the nonadiabatic excited state molecular dynamics code. We further illustrate its capacity by analyzing the impact of normal-mode constraints on the photoinduced energy transfer between polyphenylene ethynylene dendrimer building blocks. Our results show that the electronic relaxation can be significantly slowed down by freezing a well-selected small subset of active normal modes characterized by their contributions in the direction of energy transfer. The application of these constraints reduces the nonadiabatic coupling between electronic excited states during the entire dynamical simulations. Furthermore, we validate reduced dimensionality models by freezing all the vibrations, except a few active modes. Altogether, we consider FrozeNM as a useful tool that can be broadly used to underpin the role of vibrational motion in a studied process and to formulate reduced models that describe essential physical phenomena.

Algorithms to apply dihedral-angle constraints in molecular or stochastic dynamics simulations

Various algorithms to apply dihedral-angle constraints in molecular dynamics or stochastic dynamics simulations of molecular systems are presented, investigated, and tested. They use Cartesian coordinates and determine the Lagrangian multipliers necessary for maintaining the constraints iteratively. The most suitable algorithm to maintain a dihedral-angle constraint is numerically compared to the alternative to use distance constraints to this end. It can easily be used to obtain a potential of mean force along a dihedral-angle coordinate.

Approximating net interactions among rigid domains

Many physical simulations aim at evaluating the net interaction between two rigid bodies, resulting from the cumulative effect of pairwise interactions between their constituents. This is manifested particularly in biomolecular applications such as hierarchical protein folding instances where the interaction between almost rigid domains directly influences the folding pathway, the interaction between macromolecules for drug design purposes, self-assembly of nanoparticles for drug design and drug delivery, and design of smart materials and bio-sensors. In general, the brute force approach requires quadratic (in terms of the number of particles) number of pairwise evaluation operations for any relative pose of the two bodies, unless simplifying assumptions lead to a collapse of the computational complexity. We propose to approximate the pairwise interaction function using a linear predictor function, in which the basis functions have separated forms, i.e. the variables that describe local geometries of the two rigid bodies and the ones that reflect the relative pose between them are split in each basis function. Doing so replaces the quadratic number of interaction evaluations for each relative pose with a one-time quadratic computation of a set of characteristic parameters at a preprocessing step, plus constant number of pose function evaluations at each pose, where this constant is determined by the required accuracy of approximation as well as the efficiency of the used approximation method. We will show that the standard deviation of the error for the net interaction is linearly (in terms of number of particles) proportional to the regression error, if the regression errors are from a normal distribution. Our results show that proper balance of the tradeoff between accuracy and speed-up yields an approximation which is computationally superior to other existing methods while maintaining reasonable precision.