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Two-Phase Acto-Cytosolic Fluid Flow in a Moving Keratocyte: A 2D Continuum Model

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Abstract

The F-actin network and cytosol in the lamellipodia of crawling cells flow in a centripetal pattern and spout-like form, respectively. We have numerically studied this two-phase flow in the realistic geometry of a moving keratocyte. Cytosol has been treated as a low viscosity Newtonian fluid flowing through the high viscosity porous medium of F-actin network. Other involved phenomena including myosin activity, adhesion friction, and interphase interaction are also discussed to provide an overall view of this problem. Adopting a two-phase coupled model by myosin concentration, we have found new accurate perspectives of acto-cytosolic flow and pressure fields, myosin distribution, as well as the distribution of effective forces across the lamellipodia of a keratocyte with stationary shape. The order of magnitude method is also used to determine the contribution of forces in the internal dynamics of lamellipodia.

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References

  • Alt W, Dembo M (1999) Cytoplasm dynamics and cell motion: two-phase flow models. Math Biosci 156:207–228

    Article  MATH  Google Scholar 

  • Abraham VC, Krishnamurthi V, Taylor DL, Lanni F (1999) The actin-based nanomachine at the leading edge of migrating cells. Biophys J 77:1721–1732

    Article  Google Scholar 

  • Alexandrova AY, Arnold K, Schaub S, Vasiliev JM, Meister J-J, Bershadsky AD, Verkhovsky AB (2008) Comparative dynamics of retrograde actin flow and focal adhesions: formation of nascent adhesions triggers transition from fast to slow flow. PLoS ONE 3(9):e3234

    Article  Google Scholar 

  • Borisy GG, Svitkina TM (2000) Actin machinery: pushing the envelope. Curr Opin Cell Biol 12:104–112

    Article  Google Scholar 

  • Bottino DC, Fauci LJ (1998) A computational model of amoeboid deformation and locomotion. Eur Biophys J 27:532–539

    Article  Google Scholar 

  • Bendix PM et al (2008) A quantitative analysis of contractility in active cytoskeletal protein networks. Biophys J 94:3126–3136

    Article  Google Scholar 

  • Bindschadler M, Osborn EA Jr, Dewey CF, McGrath JL (2004) A mechanistic model of the actin cycle. Biophys J 86:2720–2739

    Article  Google Scholar 

  • Barnhart EL, Lee K-C, Keren K, Mogilner A, Theriot JA (2011) An adhesion-dependent switch between mechanisms that determine motile cell shape. PLOS Biol 9(5):e1001059

    Article  Google Scholar 

  • Cogan NG, Guy RD (2010) Multiphase flow models of biogels from crawling cells to bacterial biofilms. HFSP J 4:11–25

    Article  Google Scholar 

  • Danuser G, Oldenbourg R (2000) Probing f-actin flow by tracking shape fluctuations of radial bundles in lamellipodia of motile cells. Biophys J 79:191–201

    Article  Google Scholar 

  • Elliott CM, Stinner B, Venkataraman C (2012) Modelling cell motility and chemotaxis with evolving surface finite elements. J R Soc Interface 9:3027–3044

    Article  Google Scholar 

  • Fournier MF, Sauser R, Ambrosi D, Meister J-J, Verkhovsky AB (2010) Force transmission in migrating cells. J Cell Biol 188:287–297

    Article  Google Scholar 

  • Fuhs T, Goegler M, Brunner CA, Wolgemuth CW, Kaes JA (2014) Causes of retrograde flow in fish keratocytes. Cytoskeleton 71:24–35

    Article  Google Scholar 

  • Galbraith CG, Sheetz MP (1999) Keratocytes pull with similar forces on their dorsal and ventral surfaces. J Cell Biol 147:1313–1324

    Article  Google Scholar 

  • George UZ, Stéphanou A, Madzvamuse A (2012) Mathematical modeling and numerical simulations of actin dynamics in the eukaryotic cell. J Math Biol 66:547–593

    Article  Google Scholar 

  • Gardel ML, Sabass B, Ji L, Danuser G, Schwarz US, Waterman CM (2008) Traction stress in focal adhesions correlates biphasically with actin retrograde flow speed. J Cell Biol 183:999–1005

    Article  Google Scholar 

  • Herant M, Dembo M (2010) Cytopede: a three-dimensional tool for modeling cell motility on a flat surface. J Comput Biol 17:1639–1677

    Article  MathSciNet  Google Scholar 

  • Herant M, Marganski WA, Dembo M (2003) The mechanics of neutrophils: synthetic modeling of three experiments. Biophys J 84:3389–3413

    Article  Google Scholar 

  • Ji L, Lim J, Danuser G (2008) Fluctuations of intracellular forces during cell protrusion. Nat Cell Biol 10:1393–1400

    Article  Google Scholar 

  • Kuusela E, Alt W (2009) Continuum model of cell adhesion and migration. J Math Biol 58:135–161

    Article  MATH  MathSciNet  Google Scholar 

  • Kaverina I, Krylyshkina O, Small JV (2002) Regulation of substrate adhesion dynamics during cell motility. Int J Biochem Cell Biol 34:746–761

    Article  Google Scholar 

  • Kirfel G, Rigort A, Borm B, Herzog V (2004) Cell migration: mechanisms of rear detachment and the formation of migration tracks. Eur J Cell Biol 83:717–724

    Article  Google Scholar 

  • Keren K, Yam PT, Kinkhabwala A, Mogilner A, Theriot JA (2009) Intracellular fluid flow in rapidly moving cells. Nat Cell Biol 11:1219–1224

    Article  Google Scholar 

  • Kole TP, Tseng Y, Jiang I, Katz JL, Wirtz D (2005) Intracellular mechanics of migrating fibroblasts. Mol Biol Cell 16:328–338

    Article  Google Scholar 

  • Keren K, Pincus Z, Allen GM, Barnhart EL, Marriott G, Mogilner A, Theriot JA (2008) Mechanism of shape determination in motile cells. Nature 453:475–480

    Article  Google Scholar 

  • Larripa K, Mogilner A (2006) Transport of a 1D viscoelastic actin-myosin strip of gel as a model of a crawling cell. Phys A 372:113–123

    Article  Google Scholar 

  • Li Y, Bhimalapuram P, Dinner AR (2010) Model for how retrograde actin flow regulates adhesion traction stresses. J Phys Condens Matter 22(19):194113

    Article  Google Scholar 

  • Mogilner A (2006) On the edge: modeling protrusion. Curr Opin Cell Biol 18:32–39

    Article  Google Scholar 

  • Madzvamuse A, George UZ (2013) The moving gridfinite element method applied to cell movement and deformation. Finite Elem Anal Des 74:76–92

    Article  MATH  MathSciNet  Google Scholar 

  • Neilson M, Mackenzie JA, Webb SD, Insall RH (2011) Modelling cell movement and chemotaxis using pseudopod-based feedback. SIAM J Sci 33(3):1035–1057

    Article  MATH  MathSciNet  Google Scholar 

  • Ofer N, Mogilner A, Keren K (2011) Actin disassembly clock determines shape and speed of lamellipodial fragments. PNAS 108:20394–20399

    Article  Google Scholar 

  • Palmer JS, Boyce MC (2008) Constitutive modeling of the stress–strain behavior of f-actin filament networks. Acta Biomater 4:597–612

    Article  Google Scholar 

  • Paluch E, Sykes C, Prost J, Bornens M (2006) Dynamic modes of the cortical actomyosin gel during cell locomotion and division. Trends Cell Biol 16:5–10

    Article  Google Scholar 

  • Panorchan P, Lee JSH, Kole TP, Tseng Y, Wirtz D (2006) Microrheology and ROCK signaling of human endothelial cells embedded in a 3D matrix. Biophys J 91:3499–3507

    Article  Google Scholar 

  • Rottner K, Stradal TEB (2011) Actin dynamics and turnover in cell motility. Curr Opin Cell Biol 23:569–578

    Article  Google Scholar 

  • Ryan GL, Petroccia HM, Watanabe N, Vavylonis D (2012) Excitable actin dynamics in lamellipodial protrusion and retraction. Biophys J 102:1493–1502

    Article  Google Scholar 

  • Rubinstein B, Fournier MF, Jacobson K, Verkhovsky AB, Mogilner A (2009) Actin–myosin viscoelastic flow in the keratocyte lamellipod. Biophys J 97:1853–1863

    Article  Google Scholar 

  • Sambeth R, Baumgaertner A (2001) Locomotion of a two-dimensional keratocyte model. J Biol Syst 9:201–219

    Article  Google Scholar 

  • Sastry SK, Burridge K (2000) Focal adhesions: a nexus for intracellular signaling and cytoskeletal dynamics. Exp Cell Res 261:25–36

    Article  Google Scholar 

  • Shao D, Levine H, Rappel W-J (2012) Coupling actin flow, adhesion, and morphology in a computational cell motility model. PNAS 109:6851–6856

    Article  Google Scholar 

  • Shao D, Rappel W-J, Levine H (2010) Computational model for cell morphodynamics. Phys Rev Lett 105(10):108104

    Article  Google Scholar 

  • Shemesh T, Bershadsky AD, Kozlov MM (2012) Physical model for self-organization of actin cytoskeleton and adhesion complexes at the cell front. Biophys J 102:1746–1756

    Article  Google Scholar 

  • Small JV, Herzog M, Anderson K (1995) Actin filament organization in the fish keratocyte lamellipodium. J Cell Biol 129:1275–1286

    Article  Google Scholar 

  • Stéphanou A, Chaplain MAJ, Tracqui P (2004) A mathematical model for the dynamics of large membrane deformations of Isolated fibroblasts. Bull Math Biol 66:1119–1154

    Article  MathSciNet  Google Scholar 

  • Stéphanou A, Mylona E, Chaplain M, Tracqui P (2008) A computational model of cell migration coupling the growth of focal adhesions with oscillatory cell protrusions. J Theor Biol 253:701–716

    Article  Google Scholar 

  • Svitkina TM, Verkhovsky AB, McQuade KM, Borisy GG (1997) Analysis of the actin-myosin II system in fish epidermal keratocytes: mechanism of cell body translocation. J Cell Biol 139:397–415

    Article  Google Scholar 

  • Schaub S, Bohnet S, Laurent VM, Meister J-J, Verkhovsky AB (2007) Comparative maps of motion and assembly of filamentous actin and myosin II in migrating cell. Mol Biol Cell 18:3723–3732

    Article  Google Scholar 

  • Shemesh T, Verkhovsky AB, Svitkina TM, Bershadsky AD, Kozlov MM (2009) Role of focal adhesions and mechanical stresses in the formation and progression of lamellum interface. Biophys J 97:1254–1264

    Article  Google Scholar 

  • Tharmann R, Claessens MMAE, Bausch AR (2007) Viscoelasticity of isotropically cross-linked actin networks. Phys Rev Lett 98(8):088103-1–088103-4

  • Taber LA, Shi Y, Yang L, Bayly PV (2011) A poroelastic model for cell crawling including mechanical coupling between cytoskeleton contraction and actin polymerization. J Mech Mater Struct 6:569–589

    Article  Google Scholar 

  • Verkhovsky AB, Svitkina TM, Borisy GG (1998) Network contraction model for cell translocation and retrograde flow. Biochem Soc Symp 65:207–222

    Google Scholar 

  • Verkhovsky AB, Svitkina TM, Borisy GG (1999) Self-polarization and directional motility of cytoplasm. Curr Biol 9:11–20

    Article  Google Scholar 

  • Wolgemuth CW (2005) Lamellipodial contractions during crawling and spreading. Biophys J 89:1643–1649

    Article  Google Scholar 

  • Wong HC, Tang WC (2011) Finite element analysis of the effects of focal adhesion mechanical properties and substrate stiffness on cell migration. J Biomech 44:1046–1050

    Article  Google Scholar 

  • Wottawah F, Schinkinger S, Lincoln B, Ananthakrishnan R, Romeyke M, Guck J, Käs J (2005) Optical rheology of biological cells. Phys Rev Lett 94(9):098103

    Article  Google Scholar 

  • Wilson CA, Tsuchida MA, Allen GM, Barnhart EL, Applegate KT, Yam PT, Ji L, Keren K, Danuser G, Theriot JA (2010) Myosin II contributes to cell-scale actin network treadmilling through network disassembly. Nature 465:373–377

    Article  Google Scholar 

  • Ziebert F, Swaminathan S, Aranson IS (2011) Model for self-polarization and motility of keratocyte fragments. J R Soc Interface 9(70):1084–1092

    Article  Google Scholar 

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Correspondence to B. Firoozabadi.

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Nikmaneshi, M.R., Firoozabadi, B. & Saidi, M.S. Two-Phase Acto-Cytosolic Fluid Flow in a Moving Keratocyte: A 2D Continuum Model. Bull Math Biol 77, 1813–1832 (2015). https://doi.org/10.1007/s11538-015-0105-5

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  • DOI: https://doi.org/10.1007/s11538-015-0105-5

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