Kies, T. and Gräser, C. (2017) On differentiability of the membranemediated mechanical interaction energy of discretecontinuum membraneparticle models. SFB 1114 Preprint in arXiv:1711.11192 . pp. 122. (Submitted)

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Official URL: https://arxiv.org/abs/1711.11192
Abstract
We consider a discretecontinuum 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 CanhamHelfrich 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 membranemediated mechanical particleparticle 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 membranemediated mechanical particleparticle 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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