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

Stefanie  Hirsch; Dietmar Ölz; Christian  Schmeiser

doi:10.3934/dcds.2016014

Article Contents

2016, Volume 36, Issue 9: 4945-4962. Doi: 10.3934/dcds.2016014

This issue Previous Article Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach Next Article Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data

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
2.
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012-1185

Received: July 31, 2015

Revised: September 30, 2015

Published: May 2017

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35Q92; Secondary: 35N30, 92C40.

Citation:

Related Papers

Cited by

References

[1]	A. Carvalho, A. Desai and K. Oegema, Structural memory in the contractile ring makes the duration of cytokinesis independent of cell size, Cell, 137 (2009), 926-937.doi: 10.1016/j.cell.2009.03.021.
[2]	D. S. Courson and R. S. Rock, Actin cross-link assembly and disassembly mechanics for $\alpha$-actinin and fascin, J. Biol. Chemistry, 285 (2010), 26350-26357.
[3]	A. F. Huxley, Muscular contraction, J. Physiology, 243 (1974), 1-43.
[4]	A. Jayo and M. Parsons, Fascin: A key regulator of cytoskeletal dynamics, Int. J. Biochem. Cell Biol., 42 (2010), 1614-1617.doi: 10.1016/j.biocel.2010.06.019.
[5]	V. Milisic and D. Ölz, On the asymptotic regime of a model for friction mediated by transient elastic linkages, J. Math. Pures Appl., 96 (2011), 484-501.doi: 10.1016/j.matpur.2011.03.005.
[6]	D. Ölz, A viscous two-phase model for contractile actomyosin bundles, J. Math. Biol., 68 (2013), 1653-1676.doi: 10.1007/s00285-013-0682-6.
[7]	D. Oelz and A. Mogilner, Actomyosin contraction, aggregation and traveling waves in a treadmilling actin array, Physica-D., 318 (2016), 70-83.doi: 10.1016/j.physd.2015.10.005.
[8]	D. Oelz and A. Mogilner, A drift-diffusion model for molecular motor transport in anisotropic filament bundles, Discrete and Continuous Dynamical Systems - Series A, 36 (2016), 4553-4567.doi: 10.3934/dcds.2016.36.4553.
[9]	D. Oelz, B. Rubinstein and A. Mogilner, Contraction of random actomyosin arrays is enabled by the combined effect of actin treadmilling and crosslinking, Accepted for publication in Biophys. J., (2015).
[10]	D. Oelz and C. Schmeiser, Derivation of a model for symmetric lamellipodia with instantaneous cross-link turnover, Arch. Rational Mech. Anal., 198 (2010), 963-980.doi: 10.1007/s00205-010-0304-z.
[11]	J. V. Small, K. Rottner, I. Kaverina and K. I. Anderson, Assembling an actin cytoskeleton for cell attachment and movement, Biochimica et Biophysica Acta - Molecular Cell Research, 1404 (1998), 271-281.doi: 10.1016/S0167-4889(98)00080-9.
[12]	T. M. Svitkina, A. B. Verkhovsky, K. M. McQuade and G. G. Borisy, Analysis of the actin-myosin II system in fish epidermal keratocytes: mechanism of cell body translocation, J. Cell Biol., 139 (1997), 397-415.doi: 10.1083/jcb.139.2.397.
[13]	M. Vicente-Manzanares, X. Ma, R. S. Adelstein and A. R. Horwitz, Non-muscle myosin II takes centre stage in cell adhesion and migration, Nature Rev. Mol. Cell Biol., 10 (2009), 778-790.doi: 10.1038/nrm2786.