Skip to main content
SpringerLink
Log in

Mathematics of cell motility: have we got its number?

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

Mathematical and computational modeling is rapidly becoming an essential research technique complementing traditional experimental biological methods. However, lack of standard modeling methods, difficulties of translating biological phenomena into mathematical language, and differences in biological and mathematical mentalities continue to hinder the scientific progress. Here we focus on one area—cell motility—characterized by an unusually high modeling activity, largely due to a vast amount of quantitative, biophysical data, ‘modular’ character of motility, and pioneering vision of the area’s experimental leaders. In this review, after brief introduction to biology of cell movements, we discuss quantitative models of actin dynamics, protrusion, adhesion, contraction, and cell shape and movement that made an impact on the process of biological discovery. We also comment on modeling approaches and open questions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abercrombie M., Heaysman J.E. and Pegrum S.M. (1970). The locomotion of fibroblasts in culture. I. Movements of the leading edge. Exp. Cell Res. 59: 393–398

    Google Scholar 

  2. Abraham V.C., Krishnamurthi V., Taylor D.L. and Lanni F. (1999). The actin-based nanomachine at the leading edge of migrating cells. Biophys. J. 77: 1721–1732

    Google Scholar 

  3. Alberts J.B. and Odell G.M. (2004). In silico reconstitution of Listeria propulsion exhibits nano-saltation. PLoS Biol. 2: e412

    Google Scholar 

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

    MATH  Google Scholar 

  5. Asthagiri A.R. and Lauffenburger D.A. (2000). Bioengineering models of cell signaling. Annu. Rev. Biomed. Eng. 2: 31–53

    Google Scholar 

  6. Atilgan E., Wirtz D. and Sun S.X. (2006). Mechanics and dynamics of actin-driven thin membrane protrusions. Biophys. J. 90: 65–76

    Google Scholar 

  7. Bershadsky A.D., Balaban N.Q. and Geiger B. (2003). Adhesion-dependent cell mechanosensitivity. Annu. Rev. Cell Dev. Biol. 19: 677–695

    Google Scholar 

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

    Google Scholar 

  9. Bindschadler M., Osborn E.A., McGrath J.L. Jr. and Dewey C.F. (2004). A mechanistic model of the actin cycle. Biophys. J. 86: 2720–2739

    Google Scholar 

  10. Bohnet S., Ananthakrishnan R., Mogilner A., Meister J.J. and Verkhovsky A.B. (2006). Weak force stalls protrusion at the leading edge of the lamellipodium. Biophys. J. 90: 1810–1820

    Google Scholar 

  11. Bottino D.C. and Fauci L.J. (1998). A computational model of ameboid deformation and locomotion. Eur. Biophys. J. 27: 532–539

    Google Scholar 

  12. Bottino D., Mogilner A., Roberts T., Stewart M. and Oster G. (2002). How nematode sperm crawl. J. Cell Sci. 115: 367–384

    Google Scholar 

  13. Bray D. (2001). Cell Movements: From Molecules to Motility. Garland, New York

    Google Scholar 

  14. Cameron L.A., Giardini P.A., Soo F.S. and Theriot J.A. (2000). Secrets of actin-based motility revealed by a bacterial pathogen. Nat. Rev. Mol. Cell Biol. 1: 110–119

    Google Scholar 

  15. Carlier M.F. and Pantaloni D. (1997). Control of actin dynamics in cell motility. J. Mol. Biol. 269: 459–467

    Google Scholar 

  16. Carlsson A.E. (2001). Growth of branched actin networks against obstacles. Biophys. J. 81: 1907–1923

    Google Scholar 

  17. Carlsson A.E. (2003). Growth velocities of branched actin networks. Biophys. J. 84: 2907–2918

    Google Scholar 

  18. Carlsson A.E. and Sept D. (2004). Mathematical modeling of cell migration. Methods Cell Biol. 84: 911–937

    Google Scholar 

  19. Caron-Lormier G. and Berry H. (2005). Amplification and oscillations in the FAK/Src kinase system during integrin signaling. J. Theor. Biol. 232: 235–248

    MathSciNet  Google Scholar 

  20. Charras G.T., Yarrow J.C., Horton M.A., Mahadevan L. and Mitchison T.J. (2005). Non-equilibration of hydrostatic pressure in blebbing cells. Nature 435: 365–369

    Google Scholar 

  21. Charras G.T., Coughlin M., Mitchison T.J. and Mahadevan L. (2008). Life and times of a cellular bleb. Biophys. J. 94: 1836–1853

    Google Scholar 

  22. Chaudhuri O., Parekh S.H. and Fletcher D.A. (2007). Reversible stress softening of actin networks. Nature 445: 295–298

    Google Scholar 

  23. Chicurel M. (2002). Cell biology: cell migration research is on the move. Science 295: 606–609

    Google Scholar 

  24. Choi Y.S., Lee J. and Lui R. (2004). Traveling wave solutions for a one-dimensional crawling nematode sperm cell model. J. Math. Biol. 49: 310–328

    MATH  MathSciNet  Google Scholar 

  25. Condeelis J. (1993). Life at the leading edge: the formation of cell protrusions. Annu. Rev. Cell Biol. 9: 411–444

    Google Scholar 

  26. Dawes A.T. and Edelstein-Keshet L. (2007). Phosphoinositides and Rho proteins spatially regulate actin polymerization to initiate and maintain directed movement in a one-dimensional model of a motile cell. Biophys. J. 92: 744–768

    Google Scholar 

  27. Dembo M., Tuckerman L. and Goad W. (1981). Motion of polymorphonuclear leukocytes: theory of receptor redistribution and the frictional force on a moving cell. Cell Motil. 1: 205–235

    Google Scholar 

  28. Devreotes P. and Janetopoulos C. (2003). Eukaryotic chemotaxis: distinctions between directional sensing and polarization. J. Biol. Chem. 278: 20445–20448

    Google Scholar 

  29. Dickinson R.B. and Purich D.L. (2002). Clamped-filament elongation model for actin-based motors. Biophys. J. 82: 605–617

    Google Scholar 

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

    Google Scholar 

  31. Edelstein-Keshet L. (1988). Mathematical Models in Biology. Random House, New York

    MATH  Google Scholar 

  32. Edelstein-Keshet L. and Ermentrout G.B. (2001). A model for actin-filament length distribution in a lamellipod. J. Math. Biol. 43: 325–355

    MATH  MathSciNet  Google Scholar 

  33. Elson E.L., Felder S.F., Jay P.Y., Kolodney M.S. and Pasternak C. (1999). Forces in cell locomotion. Biochem. Soc. Symp. 65: 299–314

    Google Scholar 

  34. Evans E. (1993). New physical concepts for cell amoeboid motion. Biophys. J. 64: 1306–1322

    Google Scholar 

  35. Footer M.J., Kerssemakers J.W., Theriot J.A. and Dogterom M. (2007). Direct measurement of force generation by actin filament polymerization using an optical trap. Proc. Natl. Acad. Sci. USA 104: 2181–2186

    Google Scholar 

  36. Friedl P. (2004). Prespecification and plasticity: shifting mechanisms of cell migration. Curr. Opin. Cell Biol. 16: 14–23

    Google Scholar 

  37. Friedl P., Borgmann S. and Brocker E.B. (2001). Amoeboid leukocyte crawling through extracellular matrix: lessons from the Dictyostelium paradigm of cell movement. J. Leukoc Biol. 70: 491–509

    Google Scholar 

  38. Gardel M.L., Shin J.H., MacKintosh F.C., Mahadevan L., Matsudaira P.A. and Weitz D.A. (2004). Scaling of F-actin network rheology to probe single filament elasticity and dynamics. Phys. Rev. Lett. 93: 188102

    Google Scholar 

  39. Gerbal F., Chaikin P., Rabin Y. and Prost J. (2000). An elastic analysis of Listeria monocytogenes propulsion. Biophys. J. 79: 2259–2275

    Google Scholar 

  40. Giannone G., Dubin-Thaler B.J., Dobereiner H.G., Kieffer N., Bresnick A.R. and Sheetz M.P. (2004). Periodic lamellipodial contractions correlate with rearward actin waves. Cell 116: 431–443

    Google Scholar 

  41. Gov N.S. and Gopinathan A. (2006). Dynamics of membranes driven by actin polymerization. Biophys. J. 90: 454–469

    Google Scholar 

  42. Gracheva M.E. and Othmer H.G. (2004). A continuum model of motility in ameboid cells. Bull. Math. Biol. 66: 167–193

    MathSciNet  Google Scholar 

  43. Haviv L., Brill-Karniely Y., Mahaffy R., Backouche F., Ben-Shaul A., Pollard T.D. and Bernheim-Groswasser A. (2006). Reconstitution of the transition from lamellipodium to filopodium in a membrane-free system. Proc. Natl. Acad. Sci. USA 103: 4906–4911

    Google Scholar 

  44. Head D.A., Levine A.J. and MacKintosh F.C. (2003). Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks. Phys. Rev. E. 68: 061907

    Google Scholar 

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

    Google Scholar 

  46. Hill T.L. and Kirschner M.W. (1982). Bioenergetics and kinetics of microtubule and actin filament assembly-disassembly. Int. Rev. Cytol. 78: 1–125

    Google Scholar 

  47. Howard, J.: Mechanics of motor proteins and the cytoskeleton, Sunderland (2001)

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

    Google Scholar 

  49. Keller M., Tharmann R., Dichtl M.A., Bausch A.R. and Sackmann E. (2003). Slow filament dynamics and viscoelasticity in entangled and active actin networks. Philos. Trans. A Math. Phys. Eng. Sci. 361: 699–711

    Google Scholar 

  50. Kovar D.R. and Pollard T.D. (2004). Insertional assembly of actin filament barbed ends in association with formins produces piconewton forces. Proc. Natl. Acad. Sci. USA 101: 14725–14730

    Google Scholar 

  51. Kruse K. and Julicher F. (2000). Actively contracting bundles of polar filaments. Phys. Rev. Lett. 85: 1778–1781

    Google Scholar 

  52. Kruse K., Joanny J.F., Julicher F. and Prost J. (2005). Contractility and retrograde flow in lamellipodium motion. Phys. Biol. 3: 130–137

    Google Scholar 

  53. Lacayo, C.I., Pincus, Z., VanDuijn, M.M., Wilson, C.A., Fletcher, D.A., Gertler, F.B., Mogilner, A., Theriot, J.A.: Emergence of large-scale cell morphology and movement from local actin filament growth dynamics, PLoS Biol. (2007) (in press)

  54. Larripa K. and 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

    Google Scholar 

  55. Laurent V.M., Kasas S., Yersin A., Schaffer T.E., Catsicas S., Dietler G., Verkhovsky A.B. and Meister J.J. (2005). Gradient of rigidity in the lamellipodia of migrating cells revealed by atomic force microscopy. Biophys. J. 89: 667–675

    Google Scholar 

  56. Lee J., Ishihara A., Theriot J.A. and Jacobson K. (1993). Principles of locomotion for simple-shaped cells. Nature 362: 167–171

    Google Scholar 

  57. Lee J. and Jacobson K. (1997). The composition and dynamics of cell-substratum adhesions in locomoting fish keratocytes. J. Cell Sci. 110: 2833–2844

    Google Scholar 

  58. Levchenko A. and Iglesias P.A. (2002). Models of eukaryotic gradient sensing: application to chemotaxis of amoebae and neutrophils. Biophys. J. 82: 50–63

    Google Scholar 

  59. Li S., Guan J.L. and Chien S. (2005). Biochemistry and biomechanics of cell motility. Annu. Rev. Biomed. Eng. 7: 105–150

    Google Scholar 

  60. Loisel T.P., Boujemaa R., Pantaloni D. and Carlier M.F. (1999). Reconstitution of actin-based motility of Listeria and Shigella using pure proteins. Nature 401: 613–616

    Google Scholar 

  61. MacKintosh F.C., Kas J. and Janmey P.A. (1995). Elasticity of semiflexible biopolymer networks. Phys. Rev. Lett. 75: 4425–4428

    Google Scholar 

  62. Maly I.V. and Borisy G.G. (2001). Self-organization of a propulsive actin network as an evolutionary process. Proc. Natl. Acad. Sci. USA 98: 11324–11329

    Google Scholar 

  63. Marcy Y., Prost J., Carlier M.F. and Sykes C. (2004). Forces generated during actin-based propulsion: a direct measurement by micromanipulation. Proc. Natl. Acad. Sci. USA 20(101): 5992–5997

    Google Scholar 

  64. Maree A.F., Jilkine A., Dawes A., Grieneisen V.A. and Edelstein-Keshet L. (2006). Polarization and movement of keratocytes: a multiscale modelling approach. Bull. Math. Biol. 68: 1169–1211

    Google Scholar 

  65. Mazzag B.M., Tamaresis J.S. and Barakat A.I. (2003). A model for shear stress sensing and transmission in vascular endothelial cells. Biophys. J. 84: 4087–4101

    Google Scholar 

  66. Meinhardt H. (1999). Orientation of chemotactic cells and growth cones: models and mechanisms. J. Cell Sci. 112: 2867–2874

    Google Scholar 

  67. Mitchison T.J. and Cramer L.P. (1996). Actin-based cell motility and cell locomotion. Cell 84: 371–379

    Google Scholar 

  68. Mizuno D., Tardin C., Schmidt C.F. and Mackintosh F.C. (2007). Nonequilibrium mechanics of active cytoskeletal networks. Science 315: 370–373

    Google Scholar 

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

    Google Scholar 

  70. Mogilner A. and Verzi D. (2003). A simple 1-D physical model for the crawling nematode sperm cell. J. Stat. Phys. 110: 1169–1189

    MATH  Google Scholar 

  71. Mogilner A. and Oster G. (2003). Force generation by actin polymerization II: The elastic ratchet and tethered filaments. Biophys. J. 84: 1591–1605

    Google Scholar 

  72. Mogilner A. and Rubinstein B. (2005). The physics of filopodial protrusion. Biophys. J. 89: 782–795

    Google Scholar 

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

    Google Scholar 

  74. Mullins R.D., Heuser J.A. and Pollard T.D. (1998). The interaction of Arp2/3 complex with actin: nucleation, high affinity pointed end capping, and formation of branching networks of filaments. Proc. Natl. Acad. Sci. USA 95: 6181–6186

    Google Scholar 

  75. Narang A., Subramanian K.K. and Lauffenburger D.A. (2001). A mathematical model for chemoattractant gradient sensing based on receptor-regulated membrane phospholipid signaling dynamics. Ann. Biomed. Eng. 29: 677–691

    Google Scholar 

  76. Nicolas A. and Safran S.A. (2006). Limitation of cell adhesion by the elasticity of the extracellular matrix. Biophys. J. 91: 61–73

    Google Scholar 

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

    Google Scholar 

  78. Oliver T., Dembo M. and Jacobson K. (1999). Separation of propulsive and adhesive traction stresses in locomoting keratocytes. J. Cell Biol. 145: 589–604

    Google Scholar 

  79. Oosawa F. and Asakura S. (1962). A theory of linear and helical aggregations of macromolecules. J. Mol. Biol. 4: 10–21

    Google Scholar 

  80. Oster G.F. (1984). On the crawling of cells. J. Embryol. Exp. Morphol. 83: 329–364

    Google Scholar 

  81. Palecek S.P., Loftus J.C., Ginsberg M.H., Lauffenburger D.A. and Horwitz A.F. (1997). Integrin-ligand binding properties govern cell migration speed through cell-substratum adhesiveness. Nature 385: 537–540

    Google Scholar 

  82. Palecek S.P., Horwitz A.F. and Lauffenburger D.A. (1999). Kinetic model for integrin-mediated adhesion release during cell migration. Ann. Biomed. Eng. 27: 219–235

    Google Scholar 

  83. Paluch E., Sykes C. and Gucht J. (2006). Cracking up: symmetry breaking in cellular systems. J. Cell Biol. 175: 687–692

    Google Scholar 

  84. Parekh S.H., Chaudhuri O., Theriot J.A. and Fletcher D.A. (2005). Loading history determines the velocity of actin-network growth. Nat. Cell Biol. 7: 1219–1223

    Google Scholar 

  85. Peskin C.S., Odell G.M. and Oster G.F. (1993). Cellular motions and thermal fluctuations: the Brownian ratchet. Biophys. J. 65: 316–324

    Google Scholar 

  86. Pollard T.D. (1986). Rate constants for the reactions of ATP- and ADP-actin with the ends of actin filaments. J. Cell Biol. 103: 2747–2754

    Google Scholar 

  87. Pollard T.D. (2003). The cytoskeleton, cellular motility and the reductionist agenda. Nature 422: 741–745

    Google Scholar 

  88. Pollard T.D. and Borisy G.G. (2003). Cellular motility driven by assembly and disassembly of actin filaments. Cell 112: 453–465

    Google Scholar 

  89. Pollard D. and Earnshaw W.C. (2007). Cell Biology. Elsevier, New York

    Google Scholar 

  90. Ponti A., Machacek M., Gupton S.L., Waterman-Storer C.M. and Danuser G. (2004). Two distinct actin networks drive the protrusion of migrating cells. Science 305: 1782–1786

    Google Scholar 

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

    Google Scholar 

  92. Preziosi L. (2006). Hybrid and multiscale modelling. J. Math. Biol. 53: 977–978

    MATH  Google Scholar 

  93. Rafelski S.M. and Theriot J.A. (2004). Crawling toward a unified model of cell mobility: spatial and temporal regulation of actin dynamics. Annu. Rev. Biochem. 73: 209–239

    Google Scholar 

  94. Rappel W.J., Thomas P.J., Levine H. and Loomis W.F. (2002). Establishing direction during chemotaxis in eukaryotic cells. Biophys. J. 83: 1361–1367

    Google Scholar 

  95. Rubinstein B., Jacobson K. and Mogilner A. (2005). Multiscale two-dimensional modeling of a motile simple-shaped cell. SIAM J. MMS. 3: 413–439

    MATH  MathSciNet  Google Scholar 

  96. Sambeth R. and Baumgaertner A. (2001). Autocatalytic polymerization generates persistent random walk of crawling cells. Phys. Rev. Lett. 86: 5196–5199

    Google Scholar 

  97. Satulovsky J., Lui R. and Wang Y.-L. (2008). Exploring the control circuit of cell migration by mathematical modeling. Biophys. J. 94: 3671–3683

    Google Scholar 

  98. Satyanarayana S.V. and Baumgaertner A. (2004). Shape and motility of a model cell: a computational study. J. Chem. Phys. 121: 4255–4265

    Google Scholar 

  99. Schaus T.E., Taylor E.W. and Borisy G.G. (2007). Self-organization of actin filament orientation in the dendritic-nucleation/array-treadmilling model. Proc. Natl. Acad. Sci. USA 104: 7086–7091

    Google Scholar 

  100. Schaus, T.E., Borisy, G.G.: Performance of a population of independent filaments in lamellipodial protrusion. Biophys. J. (2008) (in press)

  101. Schwarz U.S., Erdmann T. and Bischofs I.B. (2006). Focal adhesions as mechanosensors: the two-spring model. Biosystems 83: 225–232

    Google Scholar 

  102. Shemesh T., Geiger B., Bershadsky A.D. and Kozlov M.M. (2005). Focal adhesions as mechanosensors: a physical mechanism. Proc. Natl. Acad. Sci. USA 102: 12383–12388

    Google Scholar 

  103. Shenoy V.B., Tambe D.T., Prasad A. and Theriot J.A. (2007). A kinematic description of the trajectories of Listeria monocytogenes propelled by actin comet tails. Proc. Natl. Acad. Sci. USA 104: 8229–8234

    Google Scholar 

  104. Small J.V. (1994). Lamellipodia architecture: actin filament turnover and the lateral flow of actin filaments during motility. Semin. Cell Biol. 5: 157–163

    Google Scholar 

  105. Stossel T.P. (1993). On the crawling of animal cells. Science 260: 1086–1094

    Google Scholar 

  106. Stukalin E.B. and Kolomeisky A.B. (2006). ATP hydrolysis stimulates large length fluctuations in single actin filaments. Biophys. J. 90: 2673–2685

    Google Scholar 

  107. Svitkina T.M., Verkhovsky A.B., McQuade K.M. and Borisy G.G. (1997). Analysis of the actin-myosin II system in fish epidermal keratocytes: mechanism of cell body translocation. J. Cell Biol. 139: 397–415

    Google Scholar 

  108. Vallotton P., Danuser G., Bohnet S., Meister J.J. and Verkhovsky A.B. (2005). Tracking retrograde flow in keratocytes: news from the front. Mol. Biol. Cell. 16: 1223–1231

    Google Scholar 

  109. van der Gucht, J., Paluch, E., Plastino, J., Sykes, C. (2005). Stress release drives symmetry breaking for actin-based movement. Proc. Natl. Acad. Sci. USA 102: 7847–7852

    Google Scholar 

  110. van Oudenaarden, A., Theriot, J.A. (1999). Cooperative symmetry-breaking by actin polymerization in a model for cell motility. Nat. Cell Biol. 1: 493–499

    Google Scholar 

  111. Vavylonis D., Yang Q. and O’Shaughnessy B. (2005). Actin polymerization kinetics, cap structure, and fluctuations. Proc. Natl. Acad. Sci. USA 102: 8543–8548

    Google Scholar 

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

    Google Scholar 

  113. Vicente-Manzanares M., Webb D.J. and Horwitz A.R. (2005). Cell migration at a glance. J. Cell Sci. 118: 4917–4919

    Google Scholar 

  114. Vignjevic D., Yarar D., Welch M.D., Peloquin J., Svitkina T. and Borisy G.G. (2003). Formation of filopodia- like bundles in vitro from a dendritic network. J. Cell Biol. 160: 951–962

    Google Scholar 

  115. Voituriez R., Joanny J.F. and Prost J. (2006). Generic phase diagram of active polar films. Phys. Rev. Lett. 96: 028102

    Google Scholar 

  116. Ward M.D. and Hammer D.A. (1994). Focal contact assembly through cytoskeletal polymerization: steady state analysis. J. Math. Biol. 32: 677–704

    MATH  Google Scholar 

  117. Webb D.J., Parsons J.T. and Horwitz A.F. (2002). Adhesion assembly, disassembly and turnover in migrating cells—over and over and over again. Nat. Cell Biol. 4: E97–E100

    Google Scholar 

  118. Wegner A. (1976). Head to tail polymerization of actin. J. Mol. Biol. 108: 139–150

    Google Scholar 

  119. Wolgemuth C.W., Mogilner A. and Oster G. (2004). The hydration dynamics of polyelectrolyte gels with applications to cell motility and drug delivery. Eur. Biophys. J. 33: 146–158

    Google Scholar 

  120. Wolgemuth C.W. (2005). Lamellipodial contractions during crawling and spreading. Biophys. J. 89: 1643–1649

    Google Scholar 

  121. Zaman M.H., Kamm R.D., Matsudaira P. and Lauffenburger D.A. (2005). Computational model for cell migration in three-dimensional matrices. Biophys. J. 89: 1389–1397

    Google Scholar 

  122. Zamir E. and Geiger B. (2001). Molecular complexity and dynamics of cell-matrix adhesions. J. Cell Sci. 114: 3583–3590

    Google Scholar 

  123. Zhu C. and Skalak R. (1988). A continuum model of protrusion of pseudopod in leukocytes. Biophys. J. 54: 1115–1137

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Neurobiology, Physiology and Behavior, University of California, Davis, CA, 95618, USA

    Alex Mogilner

  2. Department of Mathematics, University of California, Davis, CA, 95618, USA

    Alex Mogilner

Authors
  1. Alex Mogilner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alex Mogilner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mogilner, A. Mathematics of cell motility: have we got its number?. J. Math. Biol. 58, 105–134 (2009). https://doi.org/10.1007/s00285-008-0182-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-008-0182-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation