A Lagrangian formulation for interacting particles on a deformable medium

Abstract

Many problems in softmatter and membrane biophysics, such as finding the equilibrium configurations of protein clusters on cell-membranes, the highly defect ridden structure of Gag polyproteins in immature HIV capsids, and the unusual fluid-like state of Archaeal viruses can be understood as systems of interacting particles (typically representing proteins or protein capsomers) on deformable surfaces. The coupled interactions between the particles and the underlying elastic medium to which they are constrained pose significant computational challenges. Existing methods often employ expensive constraints to anchor particles to the surface or artificially restrict the particles movement yielding spurious equilibrium states. In this work, a new method is presented where the particles positions are parameterized in the reference configuration. This pull-back operation allows the particles to move freely on the surface without the need of additional constraints. Furthermore, the mapping necessary to describe the particles positions in a discretized setting, e.g., in a finite element mesh, can be done on the reference undeformed configuration at a reduced computational cost. The proposed method is applied to particles moving on manifolds in one and two dimensions to study its convergence characteristics and error when compared to derived analytical and semi-analytical solutions. Finally, the newly developed method is applied to generalized Tammes problems of packing interacting point particles on a deformable shell.