  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.
