Repository: Freie Universität Berlin, Math Department

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

Kies, T. and Gräser, C. (2017) On differentiability of the membrane-mediated mechanical interaction energy of discrete-continuum membrane-particle models. SFB 1114 Preprint in arXiv:1711.11192 . pp. 1-22. (Submitted)

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Official URL: https://arxiv.org/abs/1711.11192

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.

Item Type:Article
Subjects:Mathematical and Computer Sciences > Mathematics > Numerical Analysis
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics
ID Code:2208
Deposited By: Ekaterina Engel
Deposited On:14 Feb 2018 15:50
Last Modified:05 Mar 2018 08:42

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