A method for molecular dynamics on curved surfaces

Dynamics simulations of constrained particles can greatly aid in understanding the temporal and spatial evolution of biological processes such as lateral transport along membranes and self-assembly of viruses. Most theoretical efforts in the field of diffusive transport have focused on solving the diffusion equation on curved surfaces, for which it is not tractable to incorporate particle interactions even though these play a crucial role in crowded systems. We show here that it is possible to take such interactions into account by combining standard constraint algorithms with the classical velocity Verlet scheme to perform molecular dynamics simulations of particles constrained to an arbitrarily curved surface. Furthermore, unlike Brownian dynamics schemes in local coordinates, our method is based on Cartesian coordinates, allowing for the reuse of many other standard tools without modifications, including parallelization through domain decomposition. We show that by applying the schemes to the Langevin equation for various surfaces, we obtain confined Brownian motion, which has direct applications to many biological and physical problems. Finally we present two practical examples that highlight the applicability of the method: 1) the influence of crowding and shape on the lateral diffusion of proteins in curved membranes; and 2) the self-assembly of a coarse-grained virus capsid protein model.