Exact and efficient calculation of the derivatives of Lagrange multipliers for molecular dynamics simulations of biological molecules and polymers

Abstract

In the simulation of biological molecules and polymers, it is customary to impose constraints on the fastest degrees of freedom so as to freeze their motion and increase the time step. An evaluation of the corresponding constraint forces must be performed in an efficient manner, otherwise it would create a bottleneck in the calculations. If integrators of a higher order than two (for example, Gear predictor–corrector methods) are used to find the trajectories of atoms, the derivatives of the forces on the atoms with respect to the time—including the derivatives of constraint forces—also need to be calculated. In this letter, we present a method to perform the calculation of the constraint forces (i.e. of the Lagrange multipliers) as well as their time derivatives in an analytic, accurate and efficient manner. This method can be used together with integrators of a higher order than two, going beyond the simplification of uniform acceleration to perform calculations of enhanced accuracy for atomic paths, while keeping a low numerical complexity in the calculation of constraint forces.