Exact and efficient calculation of lagrange multipliers in biological polymers with constrained bond lengths and bond angles: Proteins and nucleic acids as example cases

Abstract

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 N c 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(N c ) operations, instead of the usual O(N 3 c ) 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 * Email: garcia.risueno@gmail.com the technique in a series of polyalanine peptides of different lengths using the AMBER molecular dynamics package.

Accelerating Sparse Arithmetic in the Context of Newton’s Method for Small Molecules with Bond Constraints

“…(1) will then contain the result of the linear update (7). In this manner we fused the backward sweep and the linear update.…”

“…to keep their values constant throughout the simulation. Constraining fast degrees of freedom allows for an increase in the time step of the MD simulation, so that larger systems and intervals of real time can be simulated [7].…”

Linearly scaling direct method for accurately inverting sparse banded matrices

“…Proteins, nucleic acids and other biological molecules have an essentially linear topology, which makes it possible to calculate the Lagrange multipliers associated to their constrained internal degrees of freedom by solving banded systems. More explanations on how to impose constraints on molecules and on how to calculate the Lagrange multipliers in biomolecules can be found in [22].…”

“…5) with different numbers of residues (R). See [22] for further information on the way the systems of equations to solve were generated. In our tests, we measured the error as calculated with (39), as well as the execution time of the algorithms.…”

“…An important case for the calculation of Lagrange multipliers deals with molecules with angular constraints. The banded method presented here is suitable to calculate the associated Lagrange multipliers exactly and efficiently [22]. Banded matrix techniques are useful not only in linear systems, but also in linearized ones, which also appear frequently in the literature [4,[6][7][8][9]18].…”