Algorithms for constrained molecular dynamics
kabbab abdelhamid1995, Journal of …
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
In molecular dynamics simulations, the fastest components of the potential eld impose severe restrictions on the stability and hence the speed of computational methods. One possibility for treating this problem is to replace the fastest components with algebraic length constraints. In this paper, the resulting systems of mixed di erential and algebraic equations are studied. Commonly used discretization schemes for constrained Hamiltonian systems are discussed. The form of the nonlinear equations is examined in detail and used to give convergence results for the traditional nonlinear solution technique SHAKE iteration and for a modi cation based on Successive OverRelaxation (SOR). A simple adaptive algorithm for nding the optimal relaxation parameter is presented. Alternative direct methods using sparse matrix techniques are discussed. Numerical results are given for the new techniques, implemented in the molecular modeling software package CHARMM, showing as much as twofold improvement over SHAKE iteration. matrix methods 1. Introduction. In molecular dynamics, the length of timestep for numerically integrating the equations of motion is dictated by the contributions to the force vector which maintain pairs of atoms near some equilibrium distance. The imposition of algebraic constraints that x these lengths removes the associated rapid vibrational modes, enabling the use of longer timesteps without substantially altering important physical characteristics of the motion 1]. Although we treat only length constraints in the present work, constrained techniques are also of interest for conformational search and conformational free energy simulations 2]. In 3] the SHAKE iteration was described for solving the nonlinear equations at each timestep of a constrained version of the Verlet discretization, and a similar scheme was proposed in 4] for use with the RATTLE discretization.
Related papers
Reduced variable molecular dynamics
Journal of Computational Chemistry, 1995
This article describes an extension to previously developed constraint techniques. These enhanced constraint methods will enable the study of large computational chemistry problems that cannot be easily handled with current constrained molecular dynamics (MD) methods. These methods are based on an O( N solution to the constrained equations of motion. The benefits of this approach are that (1) the system constraints are solved exactly at each time step, (2) the solution algorithm is noniterative, (3) the algorithm is recursive and scales as O( N 1, (4) the algorithm is numerically stable, (5) the algorithm is highly amenable to parallel processing, and (6) potentially greater integration step sizes are possible. It is anticipated that application of this methodology will provide a 10-to 100-improvement in the speed of a large molecular trajectory as compared with the time required to run a conventional atomistic unconstrained simulation. It is, therefore, anticipated that this methodology will provide an enabling capacity for pursuing the drug discovery process for large molecular systems. 0 1995 by John Wiley & Sons, Inc. mous impact on our understanding of the structure-function relationships pertinent to the discovery of new molecules with desired properties, such as new pharmaceutical drugs. The use of molecular simulations and the increasing performance of modem computers makes it possible to study the precise physicochemical nature of protein-ligand
LINCS: A linear constraint solver for molecular simulations
Journal of Computational Chemistry, 1997
In this article we present a new LINear Constraint Solver (LINCS) for molecular simulations with bond constraints. The algorithm is inherently stable, as the constraints themselves are reset instead of derivatives of the constraints, thereby eliminating drift. Although the derivation of the algorithm is presented in terms of matrices, no matrix matrix multiplications are needed and only the nonzero matrix elements have to be stored, making the method useful for very large molecules. At the same accuracy, the LINCS algorithm is 3 to 4 times faster than the SHAKE algorithm. Parallelization of the algorithm is straightforward.
General framework for constraints in molecular dynamics simulations
Molecular Physics, 2017
The article presents a theoretical framework for molecular dynamics simulations of complex systems subject to any combination of holonomic and non-holonomic constraints. Using the concept of constrained inverse matrices both the particle accelerations and the associated constraint forces can be determined from given external forces and kinematical conditions. The formalism enables in particular the construction of explicit kinematical conditions which lead to the well-known Nosé-Hoover type equations of motion for the simulation of non-standard molecular dynamics ensembles. Illustrations are given for a few examples and an outline is presented for a numerical implementation of the method.
Trotter derived algorithms for molecular dynamics with constraints: Velocity Verlet revisited
Journal of Computational Physics, 2007
On the basis of the molecular dynamics algorithm proposed by Kalibaeva et al. [G. Kalibaeva, M. Ferrario, G. Ciccotti, Mol. Phys. 101 (2003) 765.] for systems with holonomic constraints in isobaric-isothermal ensemble, we discuss a new recursive algorithm which eliminates the inconsistency associated with the double calculation of constraint forces present in RATTLE. The algorithm is tested on bulk water and on a system containing a polymer with a large number of constraints to evaluate the CPU gain with respect to the usual RATTLE algorithm.
New Optimized Algorithms for Molecular Dynamics Simulations
Condensed Matter Physics, 2002
The method of molecular dynamics (MD) is a powerful tool for the prediction and investigation of various phenomena in physics, chemistry and biology. The development of efficient MD algorithms for integration of the equations of motion in classical and quantum many-body systems should therefore impact a lot of fields of fundamental research. In the present study it is shown that most of the existing MD integrators are far from being ideal and further significant improvement in the efficiency of the calculations can be reached. As a result, we propose new optimized algorithms which allow to reduce the numerical uncertainties to a minimum with the same overall computational costs. The optimization is performed within the well recognized decomposition approach and concerns the widely used symplectic Verlet-, Forest-Ruth-, Suzuki-as well as force-gradient-based schemes. It is concluded that the efficiency of the new algorithms can be achieved better with respect to the original integrators in factors from 3 to 1000 for orders from 2 to 12. This conclusion is confirmed in our MD simulations of a Lennard-Jones fluid for a particular case of second-and fourth-order integration schemes.
Correcting for the free energy costs of bond or angle constraints in molecular dynamics simulations
Biochimica et biophysica acta, 2015
Free energy simulations are an important tool in the arsenal of computational biophysics, allowing the calculation of thermodynamic properties of binding or enzymatic reactions. This paper introduces methods to increase the accuracy and precision of free energy calculations by calculating the free energy costs of constraints during post-processing. The primary purpose of employing constraints for these free energy methods is to increase the phase space overlap between ensembles, which is required for accuracy and convergence. The free energy costs of applying or removing constraints are calculated as additional explicit steps in the free energy cycle. The new techniques focus on hard degrees of freedom and use both gradients and Hessian estimation. Enthalpy, vibrational entropy, and Jacobian free energy terms are considered. We demonstrate the utility of this method with simple classical systems involving harmonic and anharmonic oscillators, four-atomic benchmark systems, an alchemi...
Molecular dynamics as a natural solver
1998
A universal character of molecular dynamics (MD) method is discussed. Contrary to the classical area of MD applications in microscopic world investigations, MD simulation of mesoscopic phenomena is considered. Sample results of MD simulations of the Rayleigh-Taylor instability are shown and discussed briefly. To cover the larger time-and-space scale either simplified MD model or more sophisticated particle based algorithms can be used. In the first case MD method can be directly applied as a predictive display in computer animation. In the second, MD code can be a "backbone" of efficient computer realization of such particle based methods as dissipative particle dynamics and smoothed particle hydrodynamics. Applications of MD approach in global optimization problems are discussed also. It is emphasized that inherent parallelism of MD method resulting in efficient realization on MPP systems together with its universal properties makes the method a powerful natural solver.
Novel algorithms for massively parallel, long-term, simulation of molecular dynamics systems
Advances in Engineering Software, 1998
In this paper, a novel algorithm for solution of the constrained equations of motion with application to simulation of the molecular dynamics systems is presented. The algorithm enables the solution of equations of motion with an internal coordinates model wherein the high-frequency oscillations are frozen by explicit inclusion of hard constraints in the system as well as by clustering of atoms and, thus, allowing a much larger time step in the integration. For a molecular system with N clusters, the algorithm achieves the optimal sequential complexity of O(N). However, the main advantage of this new algorithm is its efficiency for massively parallel computation. In fact, this is the first known algorithm that achieves a both time-and processor-optimal parallel solution for the constrained equations of motion, i.e. an optimal computation time of O(logN) by using an optimal number of O(N) processors. In addition to its theoretical significance, this algorithm is also very efficient for practical implementation on the coarse grain MIMD parallel architectures owing to its highly decoupled computational structure. ᭧
Introduction to Molecular Dynamics Simulation
In this chapter a summary is given of the key ingredients necessary to carry out a molecular dynamics simulation, with particular emphasis on macromolecular systems. We discuss the form of the intermolecular potential for molecules composed of atoms, and of non-spherical sub-units, giving examples of how to compute the forces and torques. We also describe some of the MD algorithms in current use. Finally, we briefly refer to the factors that influence the size of systems, and length of runs, that are needed to calculate statistical properties.
Molecular dynamics simulations: advances and applications
Advances and Applications in Bioinformatics and Chemistry, 2015
Molecular dynamics simulations have evolved into a mature technique that can be used effectively to understand macromolecular structure-to-function relationships. Present simulation times are close to biologically relevant ones. Information gathered about the dynamic properties of macromolecules is rich enough to shift the usual paradigm of structural bioinformatics from studying single structures to analyze conformational ensembles. Here, we describe the foundations of molecular dynamics and the improvements made in the direction of getting such ensemble. Specific application of the technique to three main issues (allosteric regulation, docking, and structure refinement) is discussed.
Related papers
A parallel molecular dynamics simulation scheme for a molecular system with bond constraints in NPT ensemble
Computer Physics Communications, 2006
Equations of motion based on an atomic group scaling scheme are described for a molecular system with bond constraints. The NPT ensemble extended system method is employed along with a numerical integration scheme using an operator technique. For parallelization of the integration scheme, a domain decomposition scheme is employed based on a group of atoms which share common constraints. This decomposition scheme fits well into the integration scheme and involves no extra inter-processor communication during the SHAKE/RATTLE procedures. An example is given for a solvated protein system containing a total of 23 558 atoms on 64 processors.
A test set for molecular dynamics algorithms
Computational Methods for …, 2002
This article describes a collection of model problems for aiding numerical analysts, code developers and others in the design of computational methods for molecular dynamics (MD) simulation. Common types of calculations and desirable features of algorithms are surveyed, and these are used to guide selection of representative models. By including essential features of certain classes of molecular systems, but otherwise limiting the physical and quantitative details, it is hoped that the test set can help to facilitate cross-disciplinary algorithm and code development e orts.
Editorial - Constraints: From physical principles to molecular simulations and beyond
The European Physical Journal Special Topics, 2011
A method of incorporating rate constants as kinetic constraints in molecular dynamics simulations
Proceedings of the National Academy of Sciences, 2020
From the point of view of statistical mechanics, a full characterization of a molecular system requires an accurate determination of its possible states, their populations, and the respective interconversion rates. Toward this goal, well-established methods increase the accuracy of molecular dynamics simulations by incorporating experimental information about states using structural restraints and about populations using thermodynamics restraints. However, it is still unclear how to include experimental knowledge about interconversion rates. Here, we introduce a method of imposing known rate constants as constraints in molecular dynamics simulations, which is based on a combination of the maximum-entropy and maximum-caliber principles. Starting from an existing ensemble of trajectories, obtained from either molecular dynamics or enhanced trajectory sampling, this method provides a minimally perturbed path distribution consistent with the kinetic constraints, as well as modified free...
Reactive Molecular Dynamics: Numerical Methods and Algorithmic Techniques
2009
Modeling atomic and molecular systems requires computation-intensive quantum mechanical methods such as, but not limited to, density functional theory (DFT) . These methods have been successful in predicting various properties of chemical systems at atomistic detail. Due to the inherent nonlocality of quantum mechanics, the scalability of these methods ranges from O(N 3 ) to O(N 7 ) depending on the method used and approximations involved. This significantly limits the size of simulated systems to a few thousands of atoms, even on large scale parallel platforms. On the other hand, classical approximations of quantum systems, although computationally (relatively) easy to implement, yield simpler models that lack essential chemical properties such as reactivity and charge transfer. The recent work of van Duin et al overcomes the limitations of classical molecular dynamics approximations by carefully incorporating limited nonlocality (to mimic quantum behavior) through empirical bond order potential. This reactive molecular dynamics method, called ReaxFF, achieves essential quantum properties, while retaining computational simplicity of classical molecular dynamics, to a large extent.
Gradient SHAKE: An improved method for constrained energy minimization in macromolecular simulations
Journal of Computational Chemistry, 1995
Current macromolecular energy minimization algorithms become inefficient and prone to failure when bond length constraints are imposed. They are required to relieve steric stresses in biomolecules prior to a molecular dynamics simulation. Unfortunately, the latter often require constraints, leading to difficulties in initiating trajectories from unconstrained energy minima. This difficulty was overcome by requiring that the components of the energy gradient vanish along the constrained bonds. The modified energy minimization algorithm converges to a lower energy in a fewer number of iterations and is more robust than current implementations. The method has been successfully applied to the Dickerson DNA dodecamer, CGCGAA'ITCGCG. 0 1995 by John Wiley & Sons, Inc. techniques' are routinely used in X-ray diffraction studies of proteins and DNA to adjust bond lengths, bond angles, etc. so that they conform to known stereochemical values. They have been used extensively in the parameterization of molecular mechanics force fields's3 and in molecular dynambiomolecules, in which the stresses in the molecule lar structure and function. Gradient minimization ics (MD) and Monte Carlo (MC) studies of generally must be prior to the commencement of the simulation^^,^ and in which, for large
A Hamiltonian-Based Algorithm for Equilibrium Molecular Dynamics Simulation at Constant Chemical Potential
2006
The objective of the research described in this thesis is to determine whether dilating mass can produce an equilibrium molecular dynamics algorithm for rigorous constant chemical potential simulation. The hypothesis is tested by developing an equilibrium molecular dynamics algorithm for the grand ensemble (constant chemical potential, volume and energy ensemble or μVE ensemble) following a methodical procedure developed by Keffer et al. [1] and running simulations on possibly a μVE ensemble. A novel concept for a chemicostat controller is described. An equation for the instantaneous chemical potential is not available, thus a property, called the instantaneous partial specific Hamiltonian, that is related to the chemical potential was defined. The Hamiltonian for the μVE ensemble was formulated and from this the equations of motion were derived. The derivation of the algorithm for the integration scheme-single time scale reversible reference system propagator (rRESPA) is presented. We were able to simulate successfully a stable algorithm (i.e., the chemicostat controller functions properly, driving the system to the set point product of the partial specific Hamiltonian and mass), and show an equivalence of the change in mass and the change in number of particles with respect to the change in potential energy. The methodical procedure for algorithm development has great potential for extending the μVE ensemble algorithm to rigorous grand canonical ensemble EMD simulations. N *
A separating framework for increasing the timestep in molecular dynamics
Computer Simulation of Biomolecular Systems, 1997
A new algorithm for rigid body molecular dynamics
Chemical Physics, 2006
The molecular dynamics of a completely rigid molecule is described in terms of external coordinates, namely translations and rotations, and a new algorithm is proposed, which is faster than other known methods and satisfies the constraints up to a desired accuracy. The procedure dispenses with the adoption of Lagrange multipliers and it is derived from an expression previously proposed for the motion of a semirigid molecule, when constraints are imposed to any selected number of intramolecular parameters. The latter need not to be specified for a rigid body but cannot be altogether ignored since it is necessary to guarantee that internal and external coordinates form a complete set of independent variables. This requirement is met by the familiar Eckart-Sayvetz conditions which provide with an iterative procedure for the evaluation, through symmetric orthogonalization, of a matrix of rotation. It turns out that only a first approximation of this matrix is necessary, therefore a final algorithm is proposed, based on the definition of infinitesimal angles of rotation about the mass center.
Introduction fo the molecular dynamic simulation
In this chapter a summary is given of the key ingredients necessary to carry out a molecular dynamics simulation, with particular emphasis on macromolecular systems. We discuss the form of the intermolecular potential for molecules composed of atoms, and of non-spherical sub-units, giving examples of how to compute the forces and torques. We also describe some of the MD algorithms in current use. Finally, we briefly refer to the factors that influence the size of systems, and length of runs, that are needed to calculate statistical properties.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.