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

TL;DR

This paper proves local existence and uniqueness of solutions for a mathematical model of actomyosin bundles, analyzing conditions for global solutions and the behavior of bundle length under force constraints.

Contribution

It establishes the first rigorous mathematical results on existence and uniqueness for a two-phase actomyosin bundle model with specific boundary conditions.

Findings

Local in time existence and uniqueness of solutions proven.

Global solutions exist for short bundles under certain conditions.

Bundle length tends to a limit for small prescribed force.

Abstract

The model for disordered actomyosin bundles recently derived by Oelz, in the work 'A viscous two-phase model for contractile actomyosin bundles' (Math. Biol., 68 (2013), 1653--1676) 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 equations 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.