# On a poroviscoelastic model for cell crawling

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## Abstract

In this paper a minimal, one-dimensional, two-phase, viscoelastic, reactive, flow model for a crawling cell is presented. Two-phase models are used with a variety of constitutive assumptions in the literature to model cell motility. We use an upper-convected Maxwell model and demonstrate that even the simplest of two-phase, viscoelastic models displays features relevant to cell motility. We also show care must be exercised in choosing parameters for such models as a poor choice can lead to an ill–posed problem. A stability analysis reveals that the initially stationary, spatially uniform strip of cytoplasm starts to crawl in response to a perturbation which breaks the symmetry of the network volume fraction or network stress. We also demonstrate numerically that there is a steady travelling-wave solution in which the crawling velocity has a bell-shaped dependence on adhesion strength, in agreement with biological observation.

## Keywords

Cell motility Two-phase flow Viscoelastic Cell Adhesion## Mathematics Subject Classification (2000)

92C17 76A10 76T99## Notes

### Acknowledgments

This publication is based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). S.L.W. is grateful for funding from the EPSRC in the form of an Advanced Research Fellowship.

## References

- Alt W, Dembo M (1999) Cytoplasm dynamics and cell motion: two-phase flow models. Math Biosci 156(1–2):207–228CrossRefzbMATHGoogle Scholar
- Alt W, Tranquillo R (1995) Basic morphogenetic system modeling shape changes of migrating cells: how to explain fluctuating lamellipodial dynamics. J Biol Syst 3(4):905–916CrossRefGoogle Scholar
- Alt W, Bock M, Möhl C (2010) Cell mechanics: from single-scale based models to multiscale modeling. In: Chauvière A, Preziosi L, Verdier C (eds) Coupling of cytoplasm and adhesion dynamics determines cell polarization and locomotion. Chapman and Hall/CRC, Boca Raton, London, New York, p 89–132Google Scholar
- Balay S, Gropp WD, Curfman McInnes L, Smith BF (1997) Modern software tools for scientific computing. In: Arge E, Bruaset AM, Langtangen HP (eds) Efficient management of parallelism in object oriented numerical software libraries. Birkhäuser Press, Boston, p 163–202Google Scholar
- Balay S, Brown J, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, Curfman McInnes L, Smith BF, Zhang H (2010) PETSc users manual. anl-95/11—revision 3.1. Argonne National LaboratoryGoogle Scholar
- Balay S, Brown J, Buschelman K, Gropp WD, Kaushik D, Knepley MG, Curfman McInnes L, Smith BF, Zhang H (2011) PETSc Web page. http://www.mcs.anl.gov/petsc.
- Bausch A, Ziemann F, Boulbitch A, Jacobson K, Sackmann E (1998) Local measurements of viscoelastic parameters of adherent cell surfaces by magnetic bead microrheometry. Biophys J 75(4):2038–2049CrossRefGoogle Scholar
- Borm B, Requardt R, Herzog V, Kirfel G (2005) Membrane ruffles in cell migration: indicators of inefficient lamellipodia adhesion and compartments of actin filament reorganization. Exp Cell Res 302(1):83–95CrossRefGoogle Scholar
- Bray D (2001) Cell movements: from molecules to motility. Garland Science, New YorkGoogle Scholar
- Burnette D, Manley S, Sengupta P, Sougrat R, Davidson M, Kachar B, Lippincott-Schwartz J (2011) A role for actin arcs in the leading-edge advance of migrating cells. Nat Cell Biol 13(4):371–382CrossRefGoogle Scholar
- Burton K, Park J, Taylor D (1999) Keratocytes generate traction forces in two phases. Mol Biol Cell 10(11):3745–3769CrossRefGoogle Scholar
- Callan-Jones A, Jülicher F (2011) Hydrodynamics of active permeating gels. New J Phys 13:093027CrossRefGoogle Scholar
- Callan-Jones A, Joanny J, Prost J (2008) Viscous-fingering-like instability of cell fragments. Phys Rev Lett 100(25):258106CrossRefGoogle Scholar
- Cogan N, Guy R (2010) Multiphase flow models of biogels from crawling cells to bacterial biofilms. HFSP J 4(1):11–25CrossRefGoogle Scholar
- Cogan NG, Keener JP (2004) The role of the biofilm matrix in structural development. Math Med Biol 21(2):147–166Google Scholar
- Dembo M, Harlow F (1986) Cell motion, contractile networks, and the physics of interpenetrating reactive flow. Biophys J 50(1):109–121CrossRefGoogle Scholar
- Dembo M, Harlow F, Alt W (1984) Cell surface dynamics: concepts and models. In: Perelson AS, DeLisi C, Wiegel FW (eds) The biophysics of cell surface motility. Marcel Dekker, New York, p 495–542Google Scholar
- Doubrovinski K, Kruse K (2007) Self-organization of treadmilling filaments. Phys Rev Lett 99(22):228104CrossRefGoogle Scholar
- Doubrovinski K, Kruse K (2008) Cytoskeletal waves in the absence of molecular motors. Eur Phys Lett 83:18003CrossRefGoogle Scholar
- Doubrovinski K, Kruse K (2010) Self-organization in systems of treadmilling filaments. Eur Phys J E 31(1):95–104CrossRefGoogle Scholar
- Doubrovinski K, Kruse K (2011) Cell motility resulting from spontaneous polymerization waves. Phys Rev Lett 107(25):258103CrossRefGoogle Scholar
- Du X, Doubrovinski K, Osterfield M (2012) Self-organized cell motility from motor-filament interactions. Biophys J 102:1738–1745CrossRefGoogle Scholar
- Eriksson K (1996) Computational differential equations. Cambridge University Press, CambridgezbMATHGoogle Scholar
- Giannone G, Dubin-Thaler B, Döbereiner H, Kieffer N, Bresnick A, Sheetz M (2004) Periodic lamellipodial contractions correlate with rearward actin waves. Cell 116(3):431–443CrossRefGoogle Scholar
- Gracheva M, Othmer H (2004) A continuum model of motility in ameboid cells. Bull Math Biol 66(1):167–193CrossRefMathSciNetGoogle Scholar
- Hanein D, Horwitz A (2012) The structure of cell-matrix adhesions: the new frontier. Curr Opin Cell Biol 24:134–140CrossRefGoogle Scholar
- Herant M, Marganski W, Dembo M (2003) The mechanics of neutrophils: synthetic modeling of three experiments. Biophys J 84(5):3389–3413CrossRefGoogle Scholar
- Herant M, Heinrich V, Dembo M (2006) Mechanics of neutrophil phagocytosis: experiments and quantitative models. J Cell Sci 119:1903–1913Google Scholar
- Hinz B, Alt W, Johnen C, Herzog V, Kaiser H (1999) Quantifying lamella dynamics of cultured cells by SACED, a new computer-assisted motion analysis. Exp Cell Res 251(1):234–243CrossRefGoogle Scholar
- Hodge N, Papadopoulos P (2012) Continuum modeling and numerical simulation of cell motility. J Math Biol 64:1253–1279CrossRefMathSciNetzbMATHGoogle Scholar
- Joanny J, Jülicher F, Kruse K, Prost J (2007) Hydrodynamic theory for multi-component active polar gels. New J Phys 9:422CrossRefGoogle Scholar
- Jülicher F, Kruse K, Prost J, Joanny J (2007) Active behavior of the cytoskeleton. Phys Rep 449:3–28CrossRefMathSciNetGoogle Scholar
- Keren K, Yam P, Kinkhabwala A, Mogilner A, Theriot J (2009) Intracellular fluid flow in rapidly moving cells. Nat Cell Biol 11(10):1219–1224CrossRefGoogle Scholar
- Kimpton L, Whiteley J, Waters S, King J, Oliver J (2012) Multiple travelling-wave solutions in a minimal model for cell motility. Math Med Biol Adv Access. doi: 10.1093/imammb/dqs023 (published July 11, 2012)
- King J, Oliver J (2005) Thin-film modelling of poroviscous free surface flows. Eur J Appl Math 16(04):519–553CrossRefMathSciNetzbMATHGoogle Scholar
- Knapp D, Barocas V, Moon A, Yoo K, Petzold L, Tranquillo R (1997) Rheology of reconstituted type I collagen gel in confined compression. J Rheol 41(5):971–993CrossRefGoogle Scholar
- Kole T, Tseng Y, Jiang I, Katz J, Wirtz D (2005) Intracellular mechanics of migrating fibroblasts. Mol Biol Cell 16:328–338CrossRefGoogle Scholar
- Kruse K, Joanny J, Jülicher F, Prost J, Sekimoto K (2005) Generic theory of active polar gels: a paradigm for cytoskeletal dynamics. Eur Phys J E 16:5–16CrossRefGoogle Scholar
- Kuusela E, Alt W (2009) Continuum model of cell adhesion and migration. J Math Biol 58(1):135–161CrossRefMathSciNetzbMATHGoogle 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(1):113–123CrossRefGoogle Scholar
- Lee J, Leonard M, Oliver T, Ishihara A, Jacobson K (1994) Traction forces generated by locomoting keratocytes. J Cell Biol 127(6):1957–1964CrossRefGoogle Scholar
- Levayer R, Lecuit T (2011) Biomechanical regulation of contractility: spatial control and dynamics. Trends Cell Biol 22(2):61–81CrossRefGoogle Scholar
- Moeendarbary E, Valon L, Fritzsche M, Harris A, Moulding D, Thrasher A, Stride E, Mahadevan L, Charras G (2013) The cytoplasm of living cells behaves as a poroelastic material. Nat Mater 12(3):253–261CrossRefGoogle Scholar
- Mofrad M (2009) Rheology of the cytoskeleton. Annu Rev Fluid Mech 41:433–453CrossRefGoogle Scholar
- Mogilner A (2009) Mathematics of cell motility: have we got its number? J Math Biol 58(1):105–134CrossRefMathSciNetzbMATHGoogle Scholar
- Mogilner A, Marland E, Bottino D (2001) A minimal model of locomotion applied to the steady gliding movement of fish keratocyte cells. In: Maini P, Othmer H (eds) Mathematical models for biological pattern formation, the IMA volumes in mathematics and its applications, vol 121. Springer, New York, pp 269–293CrossRefGoogle Scholar
- Ohsumi T, Flaherty J, Evans M, Barocas V (2008) Three-dimensional simulation of anisotropic cell-driven collagen gel compaction. Biomech Model Mechanobiol 7(1):53–62CrossRefGoogle Scholar
- Oliver J, King J, McKinlay K, Brown P, Grant D, Scotchford C, Wood J (2005) Thin-film theories for two-phase reactive flow models of active cell motion. Math Med Biol 22(1):53CrossRefzbMATHGoogle Scholar
- Palecek SP, Loftus JC, Ginsberg MH, Lauffenburger DA, Horwitz AF (1997) Integrin-ligand binding properties govern cell migration speed through cell-substratum adhesiveness. Nature 385:537–540CrossRefGoogle Scholar
- Pollard T, Borisy G (2003) Cellular motility driven by assembly and disassembly of actin filaments. Cell 112(4):453–465CrossRefGoogle Scholar
- Ridley A (2011) Life at the leading edge. Cell 145(7):1012–1022CrossRefGoogle Scholar
- Rottner K, Stradal T (2011) Actin dynamics and turnover in cell motility. Curr Opin Cell Biol 23:569–578CrossRefGoogle Scholar
- Rubinstein B, Jacobson K, Mogilner A (2005) Multiscale two-dimensional modeling of a motile simple-shaped cell. Multiscale Model Simul 3(2):413–439CrossRefMathSciNetzbMATHGoogle Scholar
- Rubinstein B, Fournier M, Jacobson K, Verkhovsky A, Mogilner A (2009) Actin-myosin viscoelastic flow in the keratocyte lamellipod. Biophys J 97(7):1853–1863CrossRefGoogle Scholar
- Sakamoto Y, Prudhomme S, Zaman M (2011) Viscoelastic gel-strip model for the simulation of migrating cells. Ann Biomed Eng 39(11):2735–2749CrossRefGoogle Scholar
- Sarvestani A, Jabbari E (2009) Analysis of cell locomotion on ligand gradient substrates. Biotechnol Bioeng 103(2):424–429CrossRefGoogle Scholar
- Schaub S, Bohnet S, Laurent V, Meister J, Verkhovsky A (2007) Comparative maps of motion and assembly of filamentous actin and myosin II in migrating cells. Mol Biol Cell 18(10):3723–3732CrossRefGoogle Scholar
- Shao D, Levine H, Rappel W (2012) Coupling actin flow, adhesion, and morphology in a computational cell motility model. PNAS 109(18):6851–6856CrossRefGoogle Scholar
- Svitkina T, Verkhovsky A, McQuade K, Borisy G (1997) Analysis of the actin-myosin II system in fish epidermal keratocytes: mechanism of cell body translocation. J Cell Biol 139(2):397–415CrossRefGoogle Scholar
- Vallotton P, Gupton S, Waterman-Storer C, Danuser G (2004) Simultaneous mapping of filamentous actin flow and turnover in migrating cells by quantitative fluorescent speckle microscopy. PNAS 101(26):9660–9665CrossRefGoogle Scholar
- Vallotton P, Danuser G, Bohnet S, Meister J, Verkhovsky A (2005) Tracking retrograde flow in keratocytes: news from the front. Mol Biol Cell 16(3):1223–1231CrossRefGoogle Scholar
- Verkhovsky A, Svitkina T, Borisy G (1999) Self-polarization and directional motility of cytoplasm. Curr Biol 9(1):11–20CrossRefGoogle Scholar
- Wolgemuth C, Stajic J, Mogilner A (2011) Redundant mechanisms for stable cell locomotion revealed by minimal models. Biophys J 101(3):545–553CrossRefGoogle 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):98103CrossRefGoogle Scholar
- Wright G, Guy R, Du J, Fogelson A (2011) A high-resolution finite-difference method for simulating two-fluid, viscoelastic gel dynamics. J Non-Newtonian Fluid Mech 166(19):1137–1157CrossRefzbMATHGoogle Scholar
- Yamaoka H, Matsushita S, Shimada Y, Adachi T (2012) Multiscale modeling and mechanics of filamentous actin cytoskeleton. Biomech Model Mechanobiol 11:291–302CrossRefGoogle Scholar
- Zajac M, Dacanay B, Mohler W, Wolgemuth C (2008) Depolymerization-driven flow in nematode spermatozoa relates crawling speed to size and shape. Biophys J 94(10):3810–3823CrossRefGoogle Scholar
- Ziebert F, Swaminathan S, Aranson I (2012) Model for self-polarization and motility of keratocyte fragments. J R Soc Interface 9:1084–1092CrossRefGoogle Scholar