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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

Stefanie Hirsch 1, , Dietmar Ölz 2, and  Christian Schmeiser 2,

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.
Keywords: fixed-point methods, (free) boundary value problem, asymptotic methods, Actomyosin bundles, long time behavior..
Mathematics Subject Classification: Primary: 35Q92; Secondary: 35N30, 92C4.
Citation: Stefanie Hirsch, Dietmar Ölz, Christian Schmeiser. Existence and uniqueness of solutions for a model of non-sarcomeric actomyosin bundles. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4945-4962. doi: 10.3934/dcds.2016014
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.  doi: 10.1016/j.cell.2009.03.021.  Google Scholar

[2]

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[3]

A. F. Huxley, Muscular contraction,, J. Physiology, 243 (1974), 1.   Google Scholar

[4]

A. Jayo and M. Parsons, Fascin: A key regulator of cytoskeletal dynamics,, Int. J. Biochem. Cell Biol., 42 (2010), 1614.  doi: 10.1016/j.biocel.2010.06.019.  Google Scholar

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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.  doi: 10.1016/j.matpur.2011.03.005.  Google Scholar

[6]

D. Ölz, A viscous two-phase model for contractile actomyosin bundles,, J. Math. Biol., 68 (2013), 1653.  doi: 10.1007/s00285-013-0682-6.  Google Scholar

[7]

D. Oelz and A. Mogilner, Actomyosin contraction, aggregation and traveling waves in a treadmilling actin array,, Physica-D., 318 (2016), 70.  doi: 10.1016/j.physd.2015.10.005.  Google Scholar

[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.  doi: 10.3934/dcds.2016.36.4553.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s00205-010-0304-z.  Google Scholar

[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.  doi: 10.1016/S0167-4889(98)00080-9.  Google Scholar

[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.  doi: 10.1083/jcb.139.2.397.  Google Scholar

[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.  doi: 10.1038/nrm2786.  Google Scholar

show all references

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.  doi: 10.1016/j.cell.2009.03.021.  Google Scholar

[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.   Google Scholar

[3]

A. F. Huxley, Muscular contraction,, J. Physiology, 243 (1974), 1.   Google Scholar

[4]

A. Jayo and M. Parsons, Fascin: A key regulator of cytoskeletal dynamics,, Int. J. Biochem. Cell Biol., 42 (2010), 1614.  doi: 10.1016/j.biocel.2010.06.019.  Google Scholar

[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.  doi: 10.1016/j.matpur.2011.03.005.  Google Scholar

[6]

D. Ölz, A viscous two-phase model for contractile actomyosin bundles,, J. Math. Biol., 68 (2013), 1653.  doi: 10.1007/s00285-013-0682-6.  Google Scholar

[7]

D. Oelz and A. Mogilner, Actomyosin contraction, aggregation and traveling waves in a treadmilling actin array,, Physica-D., 318 (2016), 70.  doi: 10.1016/j.physd.2015.10.005.  Google Scholar

[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.  doi: 10.3934/dcds.2016.36.4553.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s00205-010-0304-z.  Google Scholar

[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.  doi: 10.1016/S0167-4889(98)00080-9.  Google Scholar

[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.  doi: 10.1083/jcb.139.2.397.  Google Scholar

[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.  doi: 10.1038/nrm2786.  Google Scholar

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