Two methods for molecular dynamics on curved surfaces

Lateral diffusion along membranes is an important transport mecha-nism in biology. Dynamical simulations of this transport can greatly aid in understanding biological processes where this diffusion plays a role. Brownian dynamics simulations in local coordinates are one pos-sibility, but we show here that it is also possible to combine constraint algorithms with a velocity Verlet scheme to perform molecular dy-namics simulations of particles confined on arbitrary time-independent curved surfaces. The main advantage is that this method is based on Cartesian coordinates instead of local coordinates, allowing the reuse of many other standard tools, including parallelisation through do-main decomposition, without adapting those to local coordinates. Of the two constraint algorithms we considered, RATTLE is more com-putationally efficient and easier to implement, while the symmetric projection method has slightly better energy conservation. By ap-plying the schemes to the Langevin equation, Brownian motion on various curved surfaces can be modeled, which can be applied directly to many biological and physical problems. As showcase of that, we use it to model a crystal growing on a sphere.