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 focussed 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 combine standard constraint algorithms with the classical velocity Verlet scheme to perform molecular dynamics simulations of particles constrained to an arbitrarily curved surface, in which such interactions can be taken into account. 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 parallelisation through domain decomposition. We show that by applying the schemes to the Langevin equation for various surfaces, confined Brownian motion is obtained, which has direct applications to many biological and physical problems. Finally we present two practical examples that highlight the applicability of the method: (i) the influence of crowding and shape on the lateral diffusion of proteins in curved membranes and (ii) the self-assembly of a coarse-grained virus capsid protein model.