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

Carsten Gräser,Tobias Kies

doi:10.4171/ifb/461

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.

Highlights

Particles embedded into membranes are commonly expected to be crucial for various biological processes involving the shaping of the membrane
This paper considered a typical model for membrane-mediated particle interactions where the membrane is described as a continuous surface and where the particles are treated as discrete entities that couple to the membrane through certain constraints
Based on methods from shape calculus and the implicit function theorem we were able to give a proof for the differentiability of the interaction energy

Summary

Introduction

Particles embedded into membranes are commonly expected to be crucial for various biological processes involving the shaping of the membrane. To overcome the severe length scale limitations of such approaches, alternative modeling techniques representing the membrane as continuous surface that minimizes an elastic energy have been developed [3, 16] Using such an models it was shown that there are long-range interactions between particles that are predominantly membrane-mediated [10]. In this paper we consider a discrete–continuum model where the membrane is modeled as a continuous graph minimizing a linearized Canham–Helfrich bending energy and where an arbitrary amount of particles are embedded into the membrane These particles are modeled as discrete entities which are coupled to the membrane through certain boundary conditions. The overall system’s energy given fixed boundary conditions and an optimal membrane shape can be written as a function of the particle positions, which we call the interaction energy.

Membrane and particle model

Interaction energy

Differentiation of the reduced interaction energy

Numerical Examples

Conclusion

A dx where