1,431 research outputs found
Exact and efficient calculation of Lagrange multipliers in constrained biological polymers: Proteins and nucleic acids as example cases
In order to accelerate molecular dynamics simulations, it is very common to
impose holonomic constraints on their hardest degrees of freedom. In this way,
the time step used to integrate the equations of motion can be increased, thus
allowing, in principle, to reach longer total simulation times. The imposition
of such constraints results in an aditional set of Nc equations (the equations
of constraint) and unknowns (their associated Lagrange multipliers), that must
be solved in one way or another at each time step of the dynamics. In this work
it is shown that, due to the essentially linear structure of typical biological
polymers, such as nucleic acids or proteins, the algebraic equations that need
to be solved involve a matrix which is banded if the constraints are indexed in
a clever way. This allows to obtain the Lagrange multipliers through a
non-iterative procedure, which can be considered exact up to machine precision,
and which takes O(Nc) operations, instead of the usual O(Nc3) for generic
molecular systems. We develop the formalism, and describe the appropriate
indexing for a number of model molecules and also for alkanes, proteins and
DNA. Finally, we provide a numerical example of the technique in a series of
polyalanine peptides of different lengths using the AMBER molecular dynamics
package.Comment: 29 pages, 10 figures, 1 tabl
Discontinuous Molecular Dynamics for Semi-Flexible and Rigid Bodies
A general framework for performing event-driven simulations of systems with
semi-flexible or rigid bodies interacting under impulsive torques and forces is
outlined. Two different approaches are presented. In the first, the dynamics
and interaction rules are derived from Lagrangian mechanics in the presence of
constraints. This approach is most suitable when the body is composed of
relatively few point masses or is semi-flexible. In the second method, the
equations of rigid bodies are used to derive explicit analytical expressions
for the free evolution of arbitrary rigid molecules and to construct a simple
scheme for computing interaction rules. Efficient algorithms for the search for
the times of interaction events are designed in this context, and the handling
of missed interaction events is discussed.Comment: 16 pages, double column revte
Non-iterative and exact method for constraining particles in a linear geometry
We present a practical numerical method for evaluating the Lagrange
multipliers necessary for maintaining a constrained linear geometry of
particles in dynamical simulations. The method involves no iterations, and is
limited in accuracy only by the numerical methods for solving small systems of
linear equations. As a result of the non-iterative and exact (within numerical
accuracy) nature of the procedure there is no drift in the constrained
geometry, and the method is therefore readily applied to molecular dynamics
simulations of, e.g., rigid linear molecules or materials of non-spherical
grains. We illustrate the approach through implementation in the commonly used
second-order velocity explicit Verlet method.Comment: 12 pages, 2 figure
Accurate and efficient constrained molecular dynamics of polymers using Newton's method and special purpose code
In molecular dynamics simulations we can often increase the time step by imposing constraints on bond lengths and bond angles. This allows us to extend the length of the time interval and therefore the range of physical phenomena that we can afford to simulate. We examine the existing algorithms and software for solving nonlinear constraint equations in parallel and we explain why it is necessary to advance the state-of-the-art. We present ILVES-PC, a new algorithm for imposing bond constraints on proteins accurately and efficiently. It solves the same system of differential algebraic equations as the celebrated SHAKE algorithm, but ILVES-PC solves the nonlinear constraint equations using Newton’s method rather than the nonlinear Gauss-Seidel method. Moreover, ILVES-PC solves the necessary linear systems using a specialized linear solver that exploits the structure of the protein. ILVES-PC can rapidly solve constraint equations as accurately as the hardware will allow. The run-time of ILVES-PC is proportional to the number of constraints. We have integrated ILVES-PC into GROMACS and simulated proteins of different sizes. Compared with SHAKE, we have achieved speedups of up to 4.9× in single-threaded executions and up to 76× in shared-memory multi-threaded executions. Moreover, ILVES-PC is more accurate than P-LINCS algorithm. Our work is a proof-of-concept of the utility of software designed specifically for the simulation of polymers
A penalty function method for constrained molecular dynamics
We propose a penalty-function method for constrained molecular dynamic simulation by defining a quadratic penalty function for the constraints. The simulation with such a method can be done by using a conventional, unconstrained solver only with the penalty parameter increased in an appropriate manner as the simulation proceeds. More specifically, we scale the constraints with their force constants when forming the penalty terms. The resulting force function can then be viewed as a smooth continuation of the original force field as the penalty parameter increases. The penalty function method is easy to implement and costs less than a Lagrange multiplier method, which requires the solution of a nonlinear system of equations in every time step. We have first implemented a penalty function method in CHARMM and applied it to protein Bovine Pancreatic Trypsin Inhibitor (BPTI). We compared the simulation results with Verlet and Shake, and found that the penalty function method had high correlations with Shake and outperformed Verlet. In particular, the RMSD fluctuations of backbone and non-backbone atoms and the velocity auto correlations of Ca atoms of the protein calculated by the penalty function method agreed well with those by Shake. We have also tested the method on a group of argon clusters constrained with a set of inter-atomic distances in their global energy minimum states. The results showed that the method was able to impose the constraints effectively and the clusters tended to converge to their energy minima more rapidly than not confined by the constraints
A patch that imparts unconditional stability to certain explicit integrators for SDEs
This paper proposes a simple strategy to simulate stochastic differential
equations (SDE) arising in constant temperature molecular dynamics. The main
idea is to patch an explicit integrator with Metropolis accept or reject steps.
The resulting `Metropolized integrator' preserves the SDE's equilibrium
distribution and is pathwise accurate on finite time intervals. As a corollary
the integrator can be used to estimate finite-time dynamical properties along
an infinitely long solution. The paper explains how to implement the patch
(even in the presence of multiple-time-stepsizes and holonomic constraints),
how it scales with system size, and how much overhead it requires. We test the
integrator on a Lennard-Jones cluster of particles and `dumbbells' at constant
temperature.Comment: 29 pages, 5 figure
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