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

TL;DR

This paper proves the differentiability of membrane-mediated interaction energies in a coupled membrane-particle model and provides practical derivative formulas to facilitate optimization algorithms for particle configuration analysis.

Contribution

It establishes the differentiability of the interaction potential with respect to particle positions and derives explicit derivative formulas for computational use.

Findings

Interaction potential is differentiable with respect to particle positions.

Explicit derivative formulas are validated against difference quotient approximations.

Gradient flow implemented successfully for optimizing particle configurations.

Abstract

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