Skip to main content
SpringerLink
Log in
Book cover

Physical Models of Cell Motility pp 135–197Cite as

Cell Locomotion in One Dimension

  • Chapter
  • First Online:

Part of the book series: Biological and Medical Physics, Biomedical Engineering ((BIOMEDICAL))

Abstract

We overview a sub-class of mathematical models of lamellipodial cell motility on a substrate (crawling) that are based on a projection of a complex intra-cellular dynamics into one dimension. Despite the unavoidable oversimplifications associated with such a representation (loss of flow continuity, neglect of orientational order, misrepresentation of volume control mechanisms, etc.), one-dimensional models are extremely helpful in elucidating the individual roles of the three main active elements of lamellipodial motility: contraction, protrusion and adhesion. Moreover, by shifting the focus from shape to velocity, one-dimensional models reveal in an analytically transparent setting an intricate interplay between these mechanisms involving cooperation and competition.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. M. Abercrombie, Contact inhibition: the phenomenon and its biological implications. Natl. Cancer Inst. Monogr. 26, 249 (1967)

    Google Scholar 

  2. M. Abercrombie, Croonian lecture, 1978 - crawling movement of metazoan cells (English). Proc. R. Soc. Lond. Ser B Biol. Sci. 207(1167), 129–147 (1980)

    Article  ADS  Google Scholar 

  3. Y. Adler, S. Givli, Closing the loop: lamellipodia dynamics from the perspective of front propagation. Phys. Rev. E 88(4), 042708 (2013)

    Google Scholar 

  4. A. Ahmadi, M.C. Marchetti, T.B. Liverpool, Hydrodynamics of isotropic and liquid crystalline active polymer solutions. Phys. Rev. E 74(6), 061913 (2006)

    Google Scholar 

  5. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell, 4th edn. (Garland Science Taylor & Francis Group, New York, 2002)

    Google Scholar 

  6. J. Allard, A. Mogilner, Traveling waves in actin dynamics and cell motility. Curr. Opin. Cell Biol. 25(1), 107–115 (2013)

    Article  Google Scholar 

  7. W. Alt, M. Dembo, Cytoplasm dynamics and cell motion: two-phase flow models. Math. Biosci. 156(12), 207–228 (1999)

    Article  MATH  Google Scholar 

  8. S.J. Altschuler, S.B. Angenent, Y. Wang, L.F. Wu, On the spontaneous emergence of cell polarity. Nature 454(7206), 886–889 (2008)

    Article  ADS  Google Scholar 

  9. J.C. Amazigo, B. Budiansky, G.F. Carrier, Asymptotic analyses of the buckling of imperfect columns on nonlinear elastic foundations. Int. J. Solids Struct. 6(10), 1341–1356 (1970)

    Article  MATH  Google Scholar 

  10. S. Banerjee, M.C. Marchetti, Substrate rigidity deforms and polarizes active gels. Europhys. Lett. 96(2), 28003 (2011)

    Google Scholar 

  11. S. Banerjee, M.C. Marchetti, Contractile stresses in cohesive cell layers on finite-thickness substrates. Phys. Rev. Lett. 109(10), 108101 (2012)

    Google Scholar 

  12. E.L. Barnhart, G.M. Allen, F. Julicher, J.A. Theriot, Bipedal locomotion in crawling cells. Biophys. J. 98(6), 933–942 (2010)

    Article  ADS  Google Scholar 

  13. E.L. Barnhart, K.-C. Lee, K. Keren, A. Mogilner, J.A. Theriot, An adhesion-dependent switch between mechanisms that determine motile cell shape. PLoS Biol. 9(5), e1001059 (2011)

    Google Scholar 

  14. E. Barnhart, K.-C. Lee, G.M. Allen, J.A. Theriot, A. Mogilner, Balance between cell-substrate adhesion and myosin contraction determines the frequency of motility initiation in fish keratocytes. PNAS 112(16), 5045–5050 (2015) published ahead of print April 6, 2015

    Google Scholar 

  15. P. Bell, Cell behavior - a tribute to Abercrombie, in ed. by R. Michael - Bellairs, A. Curtis, G. Dunn (English). Am Scientist 72(1), 88–89 (1984)

    Google Scholar 

  16. R. Bellairs, Michael Abercrombie (1912–1979) (English). Int. J. Dev. Biol. 44(1), 23–28 (2000)

    Google Scholar 

  17. A. Bernheim-Groswasser, J. Prost, C. Sykes, Mechanism of actin-based motility: a dynamic state diagram (English). Biophys. J. 89(2), 1411–1419 (2005)

    Article  Google Scholar 

  18. A. Bershadsky, M. Kozlov, B. Geiger, Adhesion-mediated mechanosensitivity: a time to experiment, and a time to theorize. Curr. Opin. Cell Biol. 18(5), 472–481 (2006)

    Article  Google Scholar 

  19. A. Besser, U.S. Schwarz, Coupling biochemistry and mechanics in cell adhesion: a model for inhomogeneous stress fiber contraction. New J. Phys. 9(11), 425 (2007)

    Google Scholar 

  20. C. Blanch-Mercader, J. Casademunt, Spontaneous motility of actin lamellar fragments. Phys. Rev. Lett. 110(7), 078102 (2013)

    Google Scholar 

  21. D. Boal, Mechanics of the Cell (Cambridge University Press, Cambridge, 2002)

    Google Scholar 

  22. J.S. Bois, F. Jülicher, S.W. Grill, Pattern formation in active fluids. Phys. Rev. Lett. 106(2), 028103 (2011)

    Google Scholar 

  23. D. Bray, Cell Movements: From Molecules to Motility, 2nd edn. (Garland Science, New York, 2000)

    Google Scholar 

  24. C.P. Broedersz, F.C. MacKintosh, Modeling semiflexible polymer networks. Rev. Mod. Phys. 86(3), 995–1036 (2014)

    Article  ADS  Google Scholar 

  25. J. Brugués, J. Casademunt, Self-organization and cooperativity of weakly coupled molecular motors under unequal loading. Phys. Rev. Lett. 102(11), 118104 (2009)

    Google Scholar 

  26. C. Brunner, A. Ehrlicher, B. Kohlstrunk, D. Knebel, J. Ks, M. Goegler, Cell migration through small gaps. Eur. Biophys. J. 35(8), 713–719 (2006). doi:10.1007/s00249-006-0079-1

    Article  Google Scholar 

  27. E. Caglioti, P.L. Lions, C. Marchioro, M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143, 501–525 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. A.C. Callan-Jones, F. Julicher, Hydrodynamics of active permeating gels. New J. Phys. 13(9), 093027 (2011). http://stacks.iop.org/1367-2630/13/i=9/a=093027

    Google Scholar 

  29. A. Callan-Jones, R. Voituriez, Active gel model of amoeboid cell motility. New J. Phys. 15(2), 025022 (2013)

    Google Scholar 

  30. A.C. Callan-Jones, J.-F. Joanny, J. Prost, Viscous-fingering-like instability of cell fragments. Phys. Rev. Lett. 100(25), 258106 (2008)

    Google Scholar 

  31. V. Calvez, N. Meunier, R. Voituriez, A one-dimensional Keller-Segel equation with a drift issued from the boundary. C. R. Math. 348(1112),  629–634 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. O. Campas, L. Mahadevan, J.-F. Joanny, Actin network growth under load (English). Biophys. J. 102(5), 1049–1058 (2012)

    Article  ADS  Google Scholar 

  33. A.E. Carlsson, Mechanisms of cell propulsion by active stresses. New J. Phys. 13(7), 073009 (2011)

    Google Scholar 

  34. A.E. Carlsson, D. Sept, Mathematical modeling of cell migration (English), in Biophysical Tools for Biologists: Vol 1 In Vitro Techniques. Methods in Cell Biology, vol. 84 (Elsevier Academic Press Inc, San Diego, 2008), pp. 911+

    Google Scholar 

  35. J. Carr, R.L. Pego, Metastable patterns in solutions of ut = e2 uxx - f(u). Commun. Pure Appl. Math. 42(5), 523–576 (1989)

    Article  MATH  Google Scholar 

  36. C. Chen, C. Lin, On the symmetry of blowup solutions to a mean field equation. Ann. Inst. Henri Poincare (C) Non Linear Anal. 18(3), 271–296 (2001) [Elsevier]

    Google Scholar 

  37. D.T.N. Chen, Q. Wen, P.A. Janmey, J.C. Crocker, A.G. Yodh, Rheology of soft materials, in Annual Review of Condensed Matter Physics, ed. by J.S. Langer, vol. 1 (2010), pp. 301–322. http://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-070909-104120

    Google Scholar 

  38. B. Cleuren, C. Van den Broeck, Brownian motion with absolute negative mobility (English) Phys. Rev. E 67(5), Part 2 (2003)

    Google Scholar 

  39. B.N. Cox, D.W. Smith, On strain and stress in living cells. J. Mech. Phys. Solids 71, 239–252 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  40. G. Csucs, K. Quirin, G. Danuser, Locomotion of fish epidermal keratocytes on spatially selective adhesion patterns. Cell Motil. Cytoskeleton 64(11), 856–867 (2007)

    Article  Google Scholar 

  41. A.T. Dawes, G.B. Ermentrout, E.N. Cytrynbaum, L. Edelstein-Keshet, Actin filament branching and protrusion velocity in a simple 1D model of a motile cell. J. Theor. Biol. 242(2), 265–279 (2006)

    Article  MathSciNet  Google Scholar 

  42. S.R. De Groot, P. Mazur, Non-Equilibrium Thermodynamics (Courier Dover Publications, New York, 2013)

    MATH  Google Scholar 

  43. V.S. Deshpande, M. Mrksich, R.M. McMeeking, A.G. Evans, A bio-mechanical model for coupling cell contractility with focal adhesion formation (English). J. Mech. Phys. Solids 56(4), 1484–1510 (2008)

    Article  ADS  MATH  Google Scholar 

  44. P. DiMilla, K. Barbee, D. Lauffenburger, Mathematical-model for the effects of adhesion and mechanics on cell-migration speed (English). Biophys. J. 60(1), 15–37 (1991)

    Article  Google Scholar 

  45. E. Doedel, A. Champneys, F. Dercole, T. Fairgrieve, Y.A. Kuznetsov, B. Oldeman, R. Paffenroth, B Sandstede, X. Wang, C. Zhang, AUTO-07p, in Continuation and Bifurcation Software for Ordinary Differential Equations (2007). Available at http://cmvl.cs.concordia.ca/auto

  46. K. Doubrovinski, K. Kruse, Self-organization of treadmilling filaments. Phys. Rev. Lett. 99(22), 228104 (2007)

    Google Scholar 

  47. K. Doubrovinski, K. Kruse, Self-organization in systems of treadmilling filaments (English). Eur. Phys. J. E 31(1), 95–104 (2010)

    Article  Google Scholar 

  48. K. Doubrovinski, K. Kruse, Cell motility resulting from spontaneous polymerization waves. Phys. Rev. Lett. 107(25), 258103 (2011)

    Google Scholar 

  49. X. Du, K. Doubrovinski, M. Osterfield, Self-organized cell motility from motor-filament interactions. Biophys. J. 102(8), 1738–1745 (2012)

    Article  ADS  Google Scholar 

  50. J. Étienne, D. Mitrossilis, J. Fouchard, N. Bufi, P. Durand-Smet, A. Asnacios, Collective dynamics of actomyosin cortex endow cells with intrinsic mechanosensing properties. arXiv preprint. arXiv:1407.2765, accepted in PNAS (2014)

    Google Scholar 

  51. B. Finlayson, L. Scriven, Convective instability by active stress. Proc. R. Soc. Lond. A. Math. Phys. Sci. 310(1501), 183–219 (1969)

    Article  ADS  MATH  Google Scholar 

  52. H. Gao, J. Qian, B. Chen, Probing mechanical principles of focal contacts in cell-matrix adhesion with a coupled stochastic-elastic modelling framework. J. R. Soc. Interface 8(62), 1217–1232 (2011)

    Article  Google Scholar 

  53. M.L. Gardel, B. Sabass, L. Ji, G. Danuser, U.S. Schwarz, C.M. Waterman, Traction stress in focal adhesions correlates biphasically with actin retrograde flow speed. J. Cell Biol. 183(6), 999–1005 (2008)

    Article  Google Scholar 

  54. M.L. Gardel, I.C. Schneider, Y. Aratyn-Schaus, C.M. Waterman, Mechanical integration of actin and adhesion dynamics in cell migration, in Annual Review of Cell and Developmental Biology, ed. by R. Schekman, L. Goldstein, R. Lehmann, vol. 26. (2010), pp. 315–333. http://www.annualreviews.org/doi/abs/10.1146/annurev.cellbio.011209.122036

  55. L. Giomi, A. DeSimone, Spontaneous division and motility in active nematic droplets. Phys. Rev. Lett. 112(14), 147802 (2014)

    Google Scholar 

  56. F. Gladiali, M. Grossi, H. Ohtsuka, T. Suzuki, Morse indices of multiple blow-up solutions to the two-dimensional Gel’fand problem. arXiv preprint arXiv:1210.1373 (2012)

    Google Scholar 

  57. P. Haenggi, F. Marchesoni, S. Savel’ev, G. Schmid, Asymmetry in shape causing absolute negative mobility (English). Phys. Rev. E 82(4), Part 1 (2010)

    Google Scholar 

  58. R.J. Hawkins, R. Voituriez, Mechanisms of cell motion in confined geometries. Math. Model. Nat. Phenom. 5(1), 84–105 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  59. R.J. Hawkins, O. Bénichou, M. Piel, R. Voituriez, Rebuilding cytoskeleton roads: Active-transport-induced polarization of cells. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(4), 040903+ (2009)

    Google Scholar 

  60. R.J. Hawkins, M. Piel, G. Faure-Andre, A.M. Lennon-Dumenil, J.F. Joanny, J. Prost, R. Voituriez, Pushing off the walls: a mechanism of cell motility in confinement (English). Phys. Rev. Lett. 102(5), 058103 (2009)

    Google Scholar 

  61. R.J. Hawkins, R. Poincloux, O. Benichou, M. Piel, P. Chavrier, R. Voituriez, Spontaneous contractility-mediated cortical flow generates cell migration in three-dimensional environments. Biophys. J. 101(5), 1041–1045 (2011)

    Article  ADS  Google Scholar 

  62. C.A. Heckman, Contact inhibition revisited. J. Cellular Physiol. 220(3), 574–575 (2009)

    Article  Google Scholar 

  63. M. Herant, M. Dembo, Form and function in cell motility: from fibroblasts to keratocytes. Biophys. J. 98(8), 1408–1417 (2010)

    Article  ADS  Google Scholar 

  64. N. Hodge, P. Papadopoulos, Continuum modeling and numerical simulation of cell motility. J. Math. Biol. 64(7), 1253–1279 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  65. B.D. Hoffman, J.C. Crocker, Cell mechanics: dissecting the physical responses of cells to force. Annu. Rev. Biomed. Eng. 11, 259–288 (2009)

    Article  Google Scholar 

  66. J. Howard, Mechanics of Motor Proteins and the Cytoskeleton (Sinauer Associates, Sunderland, 2001)

    Google Scholar 

  67. J. Howard, S.W. Grill, J.S. Bois, Turing’s next steps: the mechanochemical basis of morphogenesis. Nat. Rev. Mol. Cell Biol. 12(6), 392–398 (2011)

    Article  Google Scholar 

  68. E.-M. Hur, I.H. Yang, D.-H. Kim, J. Byun, W.-L. Xu, P.R. Nicovich, R. Cheong, A. Levchenko, N. Thakor, F.-Q. Zhou, et al., Engineering neuronal growth cones to promote axon regeneration over inhibitory molecules. Proc. Natl. Acad. Sci. 108(12), 5057–5062 (2011)

    Article  ADS  Google Scholar 

  69. H. Jiang, S.X. Sun, Cellular pressure and volume regulation and implications for cell mechanics. Biophys. J. 105(3), 609–619 (2013)

    Article  ADS  Google Scholar 

  70. A. Jilkine, L. Edelstein-Keshet, A comparison of mathematical models for polarization of single eukaryotic cells in response to guided cues. PLoS Comput. Biol. 7(4), e1001121 (2011)

    Google Scholar 

  71. J.-F. Joanny, J. Prost, Constructing tools for the description of cell dynamics, in Biological Physics: Poincare Seminar 2009, ed. by B. Duplantier, V. Rivasseau. Progress in Mathematical Physics, vol. 60. 12th Poincare Seminar on Biological Physics, Inst Henri Poincare, Paris, Jan 31, 2009. Commissariat Energie Atomique, Div Sci Matiere; Daniel Iagolnitzer Fdn; Ecole Polytechnique (2011), pp. 1–32

    Google Scholar 

  72. J. Joanny, F. Julicher, J. Prost, Motion of an adhesive gel in a swelling gradient: a mechanism for cell locomotion (English). Phys. Rev. Lett. 90(16), 168102 (2003)

    Google Scholar 

  73. J.F. Joanny, F. Julicher, K. Kruse, J. Prost, Hydrodynamic theory for multi-component active polar gels (English). New J. Phys. 9(11), 422 (2007)

    Google Scholar 

  74. K. John, P. Peyla, K. Kassner, J. Prost, C. Misbah, Nonlinear study of symmetry breaking in actin gels: implications for cellular motility. Phys. Rev. Lett. 100(6), 068101 (2008)

    Google Scholar 

  75. F. Julicher, K. Kruse, J. Prost, J.-F. Joanny, Active behavior of the cytoskeleton (English). Phys. Rep. Rev. Section Phys. Lett. 449(1–3), 3–28 (2007)

    Google Scholar 

  76. H. Keller, A.D. Zadeh, P. Eggli, Localised depletion of polymerised actin at the front of Walker carcinosarcoma cells increases the speed of locomotion. Cell Motil. Cytoskeleton 53(3), 189–202 (2002)

    Article  Google Scholar 

  77. L. Kimpton, J. Whiteley, S. Waters, J. Oliver, On a poroviscoelastic model for cell crawling. J. Math. Biol. 70(1), 133–171 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  78. W. Koiter, Current Trends in the Theory of Buckling (Springer, Berlin, 1976)

    Book  Google Scholar 

  79. K. Kruse, F. Julicher, Actively contracting bundles of polar filaments. Phys. Rev. Lett. 85(8), 1778–1781 (2000)

    Article  ADS  Google Scholar 

  80. K. Kruse, F. Jülicher, Self-organization and mechanical properties of active filament bundles. Phys. Rev. E 67(5), 051913 (2003)

    Google Scholar 

  81. K. Kruse, A. Zumdieck, F. Jlicher, Continuum theory of contractile fibres. Europhys. Lett. 64(5), 716 (2003)

    Google Scholar 

  82. K. Kruse, J. Joanny, F. Julicher, J. Prost, K. Sekimoto, Generic theory of active polar gels: a paradigm for cytoskeletal dynamics (English). Eur. Phys. J. E 16(1), 5–16 (2005)

    Article  Google Scholar 

  83. K. Kruse, J.F. Joanny, F. Julicher, J. Prost, Contractility and retrograde flow in lamellipodium motion (English). Phys. Biol. 3(2), 130–137 (2006)

    Article  ADS  Google Scholar 

  84. O.M. Lancaster, B. Baum, Shaping up to divide: coordinating actin and microtubule cytoskeletal remodelling during mitosis. Semin. Cell Dev. Biol. 34(0), 109–115 (2014)

    Article  Google Scholar 

  85. O.M. Lancaster, M.L. Berre, A. Dimitracopoulos, D. Bonazzi, E. Zlotek- Zlotkiewicz, R. Picone, T. Duke, M. Piel, B. Baum, Mitotic rounding alters cell geometry to ensure efficient bipolar spindle formation. Dev. Cell 25(3), 270–283 (2013)

    Google Scholar 

  86. K. Larripa, A. Mogilner, Transport of a 1D viscoelastic actin-myosin strip of gel as a model of a crawling cell (English). Physica A-Stat. Mech. Appl. 372(1) (2006). Workshop on Common Trends in Traffic Systems, Kanpur, Feb 08–10, 2006, pp. 113–123

    Google Scholar 

  87. E. Lauga, T.R. Powers, The hydrodynamics of swimming microorganisms (English). Rep. Progr. Phys. 72(9), 096601 (2009)

    Google Scholar 

  88. Y. Lin, A model of cell motility leading to biphasic dependence of transport speed on adhesive strength. J. Mech. Phys. Solids 58(4), 502–514 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  89. Y. Lin, M. Inamdar, L. Freund, The competition between Brownian motion and adhesion in soft materials. J. Mech. Phys. Solids 56(1), 241–250 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  90. Z. Liu, L. A. van Grunsven, E. Van Rossen, B. Schroyen, J.-P. Timmermans, A. Geerts, H. Reynaert, Blebbistatin inhibits contraction and accelerates migration in mouse hepatic stellate cells. Br. J. Pharmacol. 159(2), 304–315 (2010)

    Article  Google Scholar 

  91. J. Löber, F. Ziebert, I.S. Aranson, Modeling crawling cell movement on soft engineered substrates. Soft Matter 10(9), 1365–1373 (2014)

    Article  ADS  Google Scholar 

  92. M.L. Lombardi, D.A. Knecht, M. Dembo, J. Lee, Traction force microscopy in Dictyostelium reveals distinct roles for myosin II motor and actin-crosslinking activity in polarized cell movement. J. Cell Sci. 120(Pt 9), 1624–1634 (2007)

    Article  Google Scholar 

  93. A.J. Loosley, J.X. Tang, Stick-slip motion and elastic coupling in crawling cells. Phys. Rev. E 86(3), Part 1 (2012)

    Google Scholar 

  94. L. Machura, M. Kostur, P. Talkner, J. Luczka, P. Haenggi, Absolute negative mobility induced by thermal equilibrium fluctuations (English). Phys. Rev. Lett. 98(4), 040601 (2007)

    Google Scholar 

  95. M.C. Marchetti, J.-F. Joanny, S. Ramaswamy, T.B. Liverpool, J. Prost, M. Rao, R. Aditi Simha, Soft active matter. (2012). ArXiv e-prints. arXiv:1207.2929 [cond-mat.soft]

    Google Scholar 

  96. M. Marchetti, J. Joanny, S. Ramaswamy, T. Liverpool, J. Prost, M. Rao, R.A. Simha, Hydrodynamics of soft active matter. Rev. Mod. Phys. 85(3), 1143 (2013)

    Google Scholar 

  97. M. Mayer, M. Depken, J.S. Bois, F. Julicher, S.W. Grill, Anisotropies in cortical tension reveal the physical basis of polarizing cortical flows. Nature 467(7315), 617–U150 (2010)

    Article  ADS  Google Scholar 

  98. G. Meurant, A review on the inverse of symmetrical tridiagonal and block tridiagonal matrices (English). SIAM J. Matrix Anal. Appl. 13(3), 707–728 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  99. Q. Mi, D. Swigon, B. Riviere, S. Cetin, Y. Vodovotz, D.J. Hackam, One-dimensional elastic continuum model of enterocyte layer migration (English). Biophys. J. 93(11), 3745–3752 (2007)

    Article  ADS  Google Scholar 

  100. S. Mikhlin, Linear Integral Equations (Hindustan Publishing Corp., DELHI (India) 1960)

    MATH  Google Scholar 

  101. M.R.K. Mofrad, Rheology of the cytoskeleton. Ann. Rev. Fluid Mech. 41, 433–453 (2009)

    Article  ADS  MATH  Google Scholar 

  102. A. Mogilner, Mathematics of cell motility: have we got its number? (English). J. Math. Biol. 58(1–2) (2009). International Conference on Industrial and Applied Mathematics (Zurich, 2007), pp. 105–134

    Google Scholar 

  103. A. Mogilner, L. Edelstein-Keshet, Regulation of actin dynamics in rapidly moving cells: a quantitative analysis (English). Biophys. J. 83(3), 1237–1258 (2002)

    Article  Google Scholar 

  104. Y. Mori, A. Jilkine, L. Edelstein-Keshet, Wave-pinning and cell polarity from a bistable reaction-diffusion system. Biophys. J. 94(9), 3684–3697 (2008)

    Article  ADS  Google Scholar 

  105. C. Nelson, Paluch, Ewa K, Nelson, Celeste M, Biais, Nicolas, Fabry, Ben, Moeller, Jens, Pruitt, Beth L, Wollnik, Carina, Kudryasheva, Galina and Rehfeldt, Florian, Federle, Walter, Mechanotransduction: use the force (s). BMC Biol., BioMed Central Ltd 13(1), 47 (2015). http://www.biomedcentral.com/1741-7007/13/47

  106. L. Nirenberg, Topics in Nonlinear Functional Analysis, vol. 6 (American Mathematical Society, Providence, RI, 1974)

    MATH  Google Scholar 

  107. I.L. Novak, B.M. Slepchenko, A. Mogilner, L.M. Loew, Cooperativity between cell contractility and adhesion. Phys. Rev. Lett. 93(26), 268109 (2004)

    Google Scholar 

  108. H. Ohtsuka, A concentration phenomenon around a shrinking hole for solutions of mean field equations. Osaka J. Math. 39, 395–407 (2002)

    MathSciNet  MATH  Google Scholar 

  109. J. Oliver, J. King, K. McKinlay, P. Brown, D. Grant, C. Scotchford, J. Wood, Thin-film theories for two-phase reactive flow models of active cell motion. Math. Med. Biol. 22(1), 53–98 (2005)

    Article  MATH  Google Scholar 

  110. J.G. Orlandi, C. Blanch-Mercader, J. Brugués, J. Casademunt, Cooperativity of self-organized Brownian motors pulling on soft cargoes. Phys. Rev. E 82(6), 061903 (2010)

    Google Scholar 

  111. A. Pathak, V.S. Deshpande, R.M. McMeeking, A.G. Evans, The simulation of stress fibre and focal adhesion development in cells on patterned substrates (English). J. R. Soc. Interface 5(22), 507–524 (2008)

    Article  Google Scholar 

  112. B. Perthame, Growth, Reaction, Movement and Diffusion from Biology. Lecture Notes, University Paris 6 (2012)

    Google Scholar 

  113. C. Peskin, G. Odell, G. Oster, Cellular motions and thermal fluctuations - the Brownian ratchet (English). Biophys. J. 65(1), 316–324 (1993)

    Article  ADS  Google Scholar 

  114. R. Poincloux, O. Collin, F. Lizárraga, M. Romao, M. Debray, M. Piel, P. Chavrier, Contractility of the cell rear drives invasion of breast tumor cells in 3D Matrigel. Proc. Natl. Acad. Sci. U.S.A. 108(5), 1943–1948 (2011)

    Article  ADS  Google Scholar 

  115. M. Prass, K. Jacobson, A. Mogilner, M. Radmacher, Direct measurement of the lamellipodial protrusive force in a migrating cell. J. Cell Biol. 174, 767–772 (2006)

    Article  Google Scholar 

  116. R.H. Pritchard, Y.Y. Shery Huang, E.M. Terentjev, Mechanics of biological networks: from the cell cytoskeleton to connective tissue. Soft Matter 10(12), 1864–1884 (2014)

    Article  ADS  Google Scholar 

  117. J. Prost, C. Barbetta, J.-F. Joanny, Dynamical control of the shape and size of Stereocilia and Microvilli. Biophys. J. 93(4), 1124–1133 (2007)

    Article  ADS  Google Scholar 

  118. J. Prost, F. Jülicher, J. Joanny, Active gel physics. Nat. Phys. 11(2), 111–117 (2015)

    Article  Google Scholar 

  119. E.M. Purcell, Life at low Reynolds number. Am. J. Phys. 45(1), 3–11 (1977)

    Article  ADS  Google Scholar 

  120. S. Rafelski, J. Theriot, Crawling toward a unified model of cell motility: spatial and temporal regulation of actin dynamics (English). Ann. Rev. Biochem. 73, 209–239 (2004)

    Article  Google Scholar 

  121. J. Ranft, J. Prost, F. Jülicher, J.-F. Joanny, Tissue dynamics with permeation. Eur. Phys. J. E. Soft Matter 35, 9723 (2012)

    Article  Google Scholar 

  122. P. Recho, L. Truskinovsky, Asymmetry between pushing and pulling for crawling cells. Phys. Rev. E 87(2), 022720 (2013)

    Google Scholar 

  123. P. Recho, L. Truskinovsky, Maximum velocity of self-propulsion for an active segment. Math. Mech. Solids (2015). http://mms.sagepub.com/content/early/2015/07/06/1081286515588675.abstract

  124. P. Recho, T. Putelat, L. Truskinovsky, Contraction-driven cell motility. Phys. Rev. Lett. 111(10), 108102 (2013)

    Google Scholar 

  125. P. Recho, J.-F. Joanny, L. Truskinovsky, Optimality of contraction-driven crawling. Phys. Rev. Lett. 112(21), 218101 (2014)

    Google Scholar 

  126. P. Recho, T. Putelat, L. Truskinovsky, Mechanics of motility initiation and motility arrest in crawling cells. J. Mech. Phys. Solids 84, 469–505 (2015). Doi:http://dx.doi.org/10.1016/j.jmps.2015.08.006. Available at http://www.sciencedirect.com/science/article/pii/S0022509615300612

    Google Scholar 

  127. A.J. Ridley, M.A. Schwartz, K. Burridge, R.A. Firtel, M.H. Ginsberg, G. Borisy, J.T. Parsons, A.R. Horwitz, Cell migration: integrating signals from front to back. Science 302(5651), 1704–1709 (2003). eprint: http://www.sciencemag.org/content/302/5651/1704.full.pdf

    Google Scholar 

  128. T. Risler, Cytoskeleton and cell motility. (2011). arXiv:1105.2423 [physics.bio-ph]

    Google Scholar 

  129. W. Ronan, V.S. Deshpande, R.M. McMeeking, J.P. McGarry, Cellular contractility and substrate elasticity: a numerical investigation of the actin cytoskeleton and cell adhesion. Biomech. Model. Mechanobiol. 13(2), 417–435 (2014)

    Article  Google Scholar 

  130. A. Ros, R. Eichhorn, J. Regtmeier, T. Duong, P. Reimann, D. Anselmetti, Brownian motion - absolute negative particle mobility (English). Nature 436(7053), 928 (2005)

    Google Scholar 

  131. B. Rubinstein, M.F. Fournier, K. Jacobson, A.B. Verkhovsky, A. Mogilner, Actin-myosin viscoelastic flow in the keratocyte lamellipod (English). Biophys. J. 97(7), 1853–1863 (2009)

    Article  ADS  Google Scholar 

  132. D. Saintillan, M.J. Shelley, Emergence of coherent structures and large-scale flows in motile suspensions. J. R. Soc. Interface 9(68), 571–585 (2012)

    Article  Google Scholar 

  133. G. Salbreux, J. Prost, J.F. Joanny, Hydrodynamics of cellular cortical flows and the formation of contractile rings. Phys. Rev. Lett. 103(5), 058102 (2009)

    Google Scholar 

  134. S. Sankararaman, S. Ramaswamy, Instabilities and waves in thin films of living fluids. Phys. Rev. Lett. 102(11), 118107 (2009)

    Google Scholar 

  135. C.H. Schreiber, M. Stewart, T. Duke, Simulation of cell motility that reproduces the force-velocity relationship (English). Proc. Natl. Acad. Sci. U.S.A. 107(20), 9141–9146 (2010)

    Article  ADS  Google Scholar 

  136. U.S. Schwarz, M.L. Gardel, United we stand - integrating the actin cytoskeleton and cell-matrix adhesions in cellular mechanotransduction. J. Cell Science 125(13), 3051–3060 (2012)

    Article  Google Scholar 

  137. U. Schwarz, S. Safran, Elastic interactions of cells. Phys. Rev. Lett. 88(4), 048102 (2002)

    Google Scholar 

  138. U. Schwarz, S. Safran, Physics of adherent cells. Rev. Mod. Phys. 85(3), 1327–1381 (2013)

    Article  ADS  Google Scholar 

  139. J. Sedzinski, M. Biro, A. Oswald, J.-Y. Tinevez, G. Salbreux, E. Paluch, Polar actomyosin contractility destabilizes the position of the cytokinetic furrow. Nature 476(7361), 462–466 (2011)

    Article  ADS  Google Scholar 

  140. T. Senba, T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis. Adv. Math. Sci. Appl. 10(1), 191–224 (2000)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  142. M. Sheetz, J. Sable, H. Dobereiner, Continuous membrane-cytoskeleton adhesion requires continuous accommodation to lipid and cytoskeleton dynamics. Ann. Rev. Biophys. Biomol. Struct. 35, 417–434 (2006)

    Article  Google Scholar 

  143. M. Shutova, C. Yang, J.M. Vasiliev, T. Svitkina, Functions of non-muscle myosin II in assembly of the cellular contractile system. PLoS One 7(7), e40814–e40814 (2012)

    Article  ADS  Google Scholar 

  144. R. Simha, S. Ramaswamy, Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles. Phys. Rev. Lett. 89(5), 058101_1–058101_4 (2002)

    Google Scholar 

  145. M.P. Stewart, J. Helenius, Y. Toyoda, S.P. Ramanathan, D.J. Muller, A.A. Hyman, Hydrostatic pressure and the actomyosin cortex drive mitotic cell rounding. Nature 469(7329), 226–230 (2011)

    Article  ADS  Google Scholar 

  146. T. Stossel, On the crawling of animal-cells (English). Science 260(5111), 1086–1094 (1993)

    Article  ADS  Google Scholar 

  147. K.M. Stroka, H. Jiang, S.-H. Chen, Z. Tong, D. Wirtz, S.X. Sun, K. Konstantopoulos, Water permeation drives tumor cell migration in confined microenvironments. Cell 157(3), 611–623 (2014)

    Article  Google Scholar 

  148. M. Struwe, G. Tarantello, On multivortex solutions in Chern-Simons gauge theory. Bollettino della Unione Matematica Italiana-B 1, 109–122 (1998)

    MathSciNet  MATH  Google Scholar 

  149. K. Tawada, K. Sekimoto, Protein friction exerted by motor enzymes through a weak-binding interaction. J. Theor. Biol. 150(2), 193–200 (1991)

    Article  Google Scholar 

  150. T. Thoresen, M. Lenz, M.L. Gardel, Reconstitution of contractile actomyosin bundles. Biophys. J. 100(11), 2698–2705 (2011)

    Article  ADS  Google Scholar 

  151. J.-Y. Tinevez, U. Schulze, G. Salbreux, J. Roensch, J.-F. Joanny, E. Paluch, Role of cortical tension in bleb growth. Proc. Natl. Acad. Sci. 106(44), 18581–18586 (2009)

    Article  ADS  Google Scholar 

  152. E. Tjhung, A. Tiribocchi, D. Marenduzzo, M. Cates, A minimal physical model captures the shapes of crawling cells. Nat. Commun. 6 (2015)

    Google Scholar 

  153. E. Tjhung, D. Marenduzzo, M.E. Cates, Spontaneous symmetry breaking in active droplets provides a generic route to motility. Proc. Natl. Acad. Sci. U.S.A. 109(31), 12381–12386 (2012)

    Article  ADS  Google Scholar 

  154. P.G. Torres, K. Doubrovinski, K. Kruse, Filament turnover stabilizes contractile cytoskeletal structures. Europhys. Lett. 91(6), 68003 (2010)

    Google Scholar 

  155. X. Trepat, M.R. Wasserman, T.E. Angelini, E. Millet, D.A. Weitz, J.P. Butler, J.J. Fredberg, Physical forces during collective cell migration. Nat. Phys. 5(6), 426–430 (2009)

    Article  Google Scholar 

  156. H. Turlier, B. Audoly, J. Prost, J.-F. Joanny, Furrow constriction in animal cell cytokinesis. Biophys. J. 106(1), 114–123 (2014)

    Article  ADS  Google Scholar 

  157. B. Vanderlei, J.J. Feng, L. Edelstein-Keshet, A computational model of cell polarization and motility coupling mechanics and biochemistry. Multiscale Model. Simul. 9(4), 1420–1443 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  158. S.R.K. Vedula, M.C. Leong, T.L. Lai, P. Hersen, A.J. Kabla, C.T. Lim, B. Ladoux, Emerging modes of collective cell migration induced by geometrical constraints. Proc. Natl. Acad. Sci. U.S.A. 109, 12974–12979 (2012)

    Article  ADS  Google Scholar 

  159. A. Verkhovsky, T. Svitkina, G. Borisy, Self-polarization and directional motility of cytoplasm. Curr. Biol. 9(1), 11–20 (1999)

    Article  Google Scholar 

  160. M. Vicente-Manzanares, D. Webb, A. Horwitz, Cell migration at a glance (English). J. Cell Sci. 118(21), 4917–4919 (2005)

    Article  Google Scholar 

  161. M. Vicente-Manzanares, X. Ma, R.S. Adelstein, A.R. Horwitz, Non-muscle myosin II takes centre stage in cell adhesion and migration. Nat. Rev. Mol. Cell Biol. 10(11), 778–790 (2009)

    Article  Google Scholar 

  162. Y. Wang, E. Botvinick, Y. Zhao, M. Berns, S Usami, R. Tsien, S Chien, Visualizing the mechanical activation of Src. Nature 434(7036), 1040–1045 (2005)

    Google Scholar 

  163. Q. Wang, X. Yang, D. Adalsteinsson, T.C. Elston, K. Jacobson, M. Kapustina, M.G. Forest, Computational and modeling strategies for cell motility, in Biological and Medical Physics, Biomedical Engineering, ed. by N.V. Dokholyan (Springer, Heidelberg, 2012), pp. 257–296

    Google Scholar 

  164. H. Wolfenson, Y.I. Henis, B. Geiger, A.D. Bershadsky, The heel and toe of the cell’s foot: a multifaceted approach for understanding the structure and dynamics of focal adhesions. Cell Motil. Cytoskeleton 66(11), 1017–1029 (2009)

    Article  Google Scholar 

  165. C. Wolgemuth, Lamellipodial contractions during crawling and spreading. Biophys. J. 89(3), 1643–1649 (2005)

    Article  ADS  Google Scholar 

  166. C.W. Wolgemuth, J. Stajic, A. Mogilner, Redundant mechanisms for stable cell locomotion revealed by minimal models. Biophys. J. 101(3), 545–553 (2011)

    Article  ADS  Google Scholar 

  167. P.T. Yam, C.A. Wilson, L. Ji, B. Hebert, E.L. Barnhart, N.A. Dye, P.W. Wiseman, G. Danuser, J.A. Theriot, Actin-myosin network reorganization breaks symmetry at the cell rear to spontaneously initiate polarized cell motility. J. Cell Biol. 178(7), 1207–1221 (2007)

    Article  Google Scholar 

  168. F. Ziebert, I.S. Aranson, Effects of adhesion dynamics and substrate compliance on the shape and motility of crawling cells. PloS One 8(5), e64511 (2013)

    Google Scholar 

  169. F. Ziebert, S. Swaminathan, I.S. Aranson, Model for self-polarization and motility of keratocyte fragments. J. R. Soc. Interface 9(70), 1084–1092 (2012)

    Article  Google Scholar 

  170. J. Zimmermann, C. Brunner, M. Enculescu, M. Goegler, A. Ehrlicher, J. Kaes, M. Falcke, Actin filament elasticity and retrograde flow shape the force-velocity relation of motile cells. Biophys. J. 102(2), 287–295 (2012)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

Considerable part of this research was conducted in collaboration with J-F. Joanny and T. Putelat. We thank F. Alouges, D. Ambrosi, O. du Roure, J. Etienne, G. Geymonnat, A. Grosberg, K. Kruse, and C. Verdier for helpful discussions.

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX26GG, UK

    Pierre Recho

  2. LMS, CNRS-UMR 7649, École Polytechnique, Route de Saclay, 91128, Palaiseau, France

    Lev Truskinovsky

Authors
  1. Pierre Recho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lev Truskinovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre Recho .

Editor information

Editors and Affiliations

  1. Argonne National Laboratory, Argonne, Illinois, USA

    Igor S. Aranson

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Recho, P., Truskinovsky, L. (2016). Cell Locomotion in One Dimension. In: Aranson, I. (eds) Physical Models of Cell Motility. Biological and Medical Physics, Biomedical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-24448-8_4

Download citation

Publish with us

Policies and ethics

Search

Navigation