# American Institute of Mathematical Sciences

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September  2016, 36(9): 4945-4962. doi: 10.3934/dcds.2016014

## Existence and uniqueness of solutions for a model of non-sarcomeric actomyosin bundles

 1 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria 2 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012-1185, United States, United States

Received  August 2015 Revised  October 2015 Published  May 2016

The model for disordered actomyosin bundles recently derived in [6] includes the effects of cross-linking of parallel and anti-parallel actin filaments, their polymerization and depolymerization, and, most importantly, the interaction with the motor protein myosin, which leads to sliding of anti-parallel filaments relative to each other. The model relies on the assumption that actin filaments are short compared to the length of the bundle. It is a two-phase model which treats actin filaments of both orientations separately. It consists of quasi-stationary force balances determining the local velocities of the filament families and of transport equation for the filaments. Two types of initial-boundary value problems are considered, where either the bundle length or the total force on the bundle are prescribed. In the latter case, the bundle length is determined as a free boundary. Local in time existence and uniqueness results are proven. For the problem with given bundle length, a global solution exists for short enough bundles. For small prescribed force, a formal approximation can be computed explicitly, and the bundle length tends to a limiting value.
Citation: Stefanie Hirsch, Dietmar Ölz, Christian Schmeiser. Existence and uniqueness of solutions for a model of non-sarcomeric actomyosin bundles. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4945-4962. doi: 10.3934/dcds.2016014
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