On differentiability of the membrane-mediated mechanical interaction energy of discrete–continuum membrane–particle models

We consider a discrete-continuum model of biomembrane with embedded particles. While the membrane is represented by continuous surface, particles are described rigid discrete objects which free to move and rotate in lateral direction. For we linearized Canham-Helfrich energy functional height slope boundary conditions imposed on particle boundaries resulting coupled minimization problem for shape positions. When considering energetically optimal each position obtain reduced that models implicitly given interaction potential membrane-mediated mechanical particle-particle interactions. show this differentiable respect positions orientations. Furthermore derive fully practical representation derivative only terms well defined derivatives membrane. This opens door application algorithms computation minimizers system further investigation interaction. The results illustrated numerical examples comparing explicit formula difference quotient approximations. furthermore demonstrate derived implement gradient flow approximation configurations.