# American Institute of Mathematical Sciences

• Previous Article
Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data
• DCDS Home
• This Issue
• Next Article
Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach
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
##### 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.  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), 26350-26357. Google Scholar [3] A. F. Huxley, Muscular contraction, J. Physiology, 243 (1974), 1-43. Google Scholar [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.  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), 484-501. 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), 1653-1676. 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), 70-83. 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), 4553-4567. 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), 963-980. 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), 271-281. 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), 397-415. 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), 778-790. 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), 926-937. 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), 26350-26357. Google Scholar [3] A. F. Huxley, Muscular contraction, J. Physiology, 243 (1974), 1-43. Google Scholar [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.  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), 484-501. 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), 1653-1676. 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), 70-83. 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), 4553-4567. 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), 963-980. 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), 271-281. 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), 397-415. 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), 778-790. doi: 10.1038/nrm2786.  Google Scholar
 [1] Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271 [2] Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273 [3] Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775 [4] Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4721-4736. doi: 10.3934/dcdsb.2020121 [5] Ruhua Wang, Senjian An, Wanquan Liu, Ling Li. Fixed-point algorithms for inverse of residual rectifier neural networks. Mathematical Foundations of Computing, 2021, 4 (1) : 31-44. doi: 10.3934/mfc.2020024 [6] Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619 [7] Jian Yang. Asymptotic behavior of solutions for competitive models with a free boundary. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3253-3276. doi: 10.3934/dcds.2015.35.3253 [8] Marta García-Huidobro, Raul Manásevich. A three point boundary value problem containing the operator. Conference Publications, 2003, 2003 (Special) : 313-319. doi: 10.3934/proc.2003.2003.313 [9] Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625 [10] Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737 [11] Yuan Wu, Jin Liang, Bei Hu. A free boundary problem for defaultable corporate bond with credit rating migration risk and its asymptotic behavior. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1043-1058. doi: 10.3934/dcdsb.2019207 [12] Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799 [13] Min Chen, Olivier Goubet. Long-time asymptotic behavior of dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 509-528. doi: 10.3934/dcds.2007.17.509 [14] Jiaohui Xu, Tomás Caraballo, José Valero. Asymptotic behavior of nonlocal partial differential equations with long time memory. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021140 [15] Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61 [16] Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709 [17] Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834 [18] John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337 [19] Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485 [20] John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269

2020 Impact Factor: 1.392