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

To accelerate molecular dynamics simulations, it is common to impose holonomic constraints on the hardest degrees of freedom. In this way, the time step used to integrate the equations of motion can be increased, thereby allowing 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), whose solution is closely related to any algorithm implementing the constraints in Euclidean coordinates. In this work, it is shown that, due to the essentially linear structure of typical biological polymers the algebraic equations that need to be solved involve a matrix which is not only sparse, but also banded if the constraints are indexed in a skilful way. This allows the Lagrange multipliers to be obtained through a noniterative procedure, which can be considered exact up to machine precision, and which takes O(N(c)) operations, instead of the usual O(N c3) for generic molecular systems. We develop the formalism, and describe the appropriate indexing for a number of model molecules. Finally, we provide a numerical example of the technique in a series of polyalanine peptides of different lengths. Although a use of the Lagrange multipliers without any modification in the solution of the underlying ordinary differential equations yields unstable integration algorithms, the central role of these quantities makes their efficient calculation useful for the improvement of methods that correctly enforce the exact satisfaction of the constraints at each time step. We provide several examples of this.