Cell-Based, Continuum and Hybrid Models of Tissue Dynamics

Part of the Lecture Notes in Mathematics book series (LNM, volume 2167)


Movement of amoeboid cells is involved in embryonic development, wound repair, the immune response to bacterial invasion, and tumor formation and metastasis. Individual cells detect extracellular chemical and mechanical signals via membrane receptors, and this initiates signal transduction cascades that produce intracellular signals. These signals control the motile machinery of the cell and thereby determine the spatial localization of contact sites with the substrate and the sites of force-generation needed to produce directed motion. The coordination and control of this complex process of direction sensing, amplification of spatial differences in the signal, assembly of the motile machinery, and control of the attachment to the substratum involves numerous molecules whose spatial distribution serves to distinguish the front from the rear of the cell, and whose temporal expression is tightly controlled. How chemical and mechanical signals are integrated, how spatial differences in signals are produced, and how propulsive and adhesive forces are controlled are issues that are amenable to mathematical modeling. An overview of some approaches to these complex problems is the subject of this chapter.


Actin Filament Fruiting Body Actin Network Stochastic Simulation Algorithm Perfusion Experiment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell, 4th edn. (Garland Science, New York, London, 2002)Google Scholar
  2. 2.
    U. Alon, M.G. Surette, N. Barkai, S. Leibler, Robustness in bacterial chemotaxis. Nature 15, 168–171 (1999)Google Scholar
  3. 3.
    R. Ananthakrishnan, A. Ehrlicher, The forces behind cell movement. Int. J. Biol. Sci. 3 (5), 303–17 (2007)CrossRefGoogle Scholar
  4. 4.
    L. Blanchoin, T.D. Pollard, Hydrolysis of ATP by polymerized actin depends on the bound divalent cation but not profilin. Biochemistry 41 (2), 597–602 (2002)CrossRefGoogle Scholar
  5. 5.
    D. Boal, Mechanics of the Cell (Cambridge University Press, Cambridge, 2002)Google Scholar
  6. 6.
    J.T. Bonner, The Cellular Slime Molds (Princeton University Press, Princeton, NJ, 1967)CrossRefGoogle Scholar
  7. 7.
    J.T. Bonner, A way of following individual cells in the migrating slugs of Dictyostelium discoideum. Proc. Natl. Acad. Sci. 95 (16), 9355–9359 (1998)CrossRefGoogle Scholar
  8. 8.
    D. Bray, Cell Movements: From Molecules to Motility (Garland Publishing, New York, 2001)Google Scholar
  9. 9.
    T. Bretschneider, K. Anderson, M. Ecke, A. Müller-Taubenberger, B. Schroth-Diez, H.C. Ishikawa-Ankerhold, G. Gerisch, The three-dimensional dynamics of actin waves, a model of cytoskeletal self-organization. Biophys. J. 96 (7), 2888–2900 (2009)CrossRefGoogle Scholar
  10. 10.
    J.A. Brzostowski, C.A. Parent, A.R. Kimmel, A Gα-dependent pathway that antagonizes multiple chemoattractant responses that regulate directional cell movement. Genes Dev. 18, 805–15 (2004)CrossRefGoogle Scholar
  11. 11.
    B. Bugyi, M.F. Carlier, Control of actin filament treadmilling in cell motility. Annu. Rev. Biophys. 39, 449–470 (2010)CrossRefGoogle Scholar
  12. 12.
    L. Cai, A.M. Makhov, D.A. Schafer, J.E. Bear, Coronin 1B antagonizes cortactin and remodels arp2/3-containing actin branches in lamellipodia. Cell 134 (5), 828–842 (2008)CrossRefGoogle Scholar
  13. 13.
    Y. Cao, H. Li, L. Petzold, Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. J. Chem. Phys. 121 (9), 4059–67 (2004)CrossRefGoogle Scholar
  14. 14.
    M. Carlier, D. Pantaloni, E. Korn, The effects of Mg2+ at the high-affinity and low-affinity sites on the polymerization of actin and associated ATP hydrolysis. J. Biol. Chem. 261, 10785–10792 (1986)Google Scholar
  15. 15.
    L.B. Case, C.M. Waterman, Adhesive F-actin waves: a novel integrin-mediated adhesion complex coupled to ventral actin polymerization. PLoS One 6 (11), e26631 (2011)Google Scholar
  16. 16.
    M.Y. Chen, R.H. Insall, P.N. Devreotes, Signaling through chemoattracnt receptors in Dictyostelium. Trends Genet. 12 (2), 52–57 (1996)Google Scholar
  17. 17.
    H. Chen, B.W. Bernstein, J.R. Bamburg, Regulating actin-filament dynamics in vivo. Trends Biochem. Sci. 25 (1), 19–23 (2000). ReviewCrossRefGoogle Scholar
  18. 18.
    Y. Cheng, H.G. Othmer, A model for direction sensing in dictyostelium discoideum: Ras activity and symmetry breaking driven by a gβ γ- mediated, gα2-ric8 – dependent signal transduction network. PLoS Comput. Biol. 12, e1004900 (2016)CrossRefGoogle Scholar
  19. 19.
    C.Y. Chung, S. Funamoto, R.A. Firtel, Signaling pathways controlling cell polarity and chemotaxis. Trends Biochem. Sci. 26 (9), 557–566 (2001). ReviewCrossRefGoogle Scholar
  20. 20.
    J. Condeelis, A. Bresnick, M. Demma, S. Dharmawardhane, R. Eddy, A.L. Hall, R. Sauterer, V. Warren, Mechanisms of amoeboid chemotaxis: an evaluation of the cortical expansion model. Dev. Genet. 11 (5–6), 333–340 (1990)CrossRefGoogle Scholar
  21. 21.
    L.P. Cramer, M. Siebert, T.J. Mitchison, Identification of novel graded polarity actin filament bundles in locomoting heart fibroblasts: implications for the generation of motile force. J. Cell Biol. 136 (6), 1287–1305 (1997)CrossRefGoogle Scholar
  22. 22.
    J.C. Dallon, H.G. Othmer, A discrete cell model with adaptive signalling for aggregation of dictyostelium discoideum. Philos. Trans. R. Soc. Lond. B Biol. Sci. 352 (1351), 391–417 (1997)CrossRefGoogle Scholar
  23. 23.
    J.C. Dallon, H.G. Othmer, A continuum analysis of the chemotactic signal seen by dictyostelium discoideum. J. Theor. Biol. 194 (4), 461–483 (1998)Google Scholar
  24. 24.
    J.C. Dallon, H.G. Othmer, How cellular movement determines the collective force generated by the dictyostelium discoideum slug. J. Theor. Biol. 231, 203–222 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    P. Dancker, L. Hess, Phalloidin reduces the release of inorganic phosphate during actin polymerization. Biochim. Biophys. Acta 1035 (2), 197–200 (1990)CrossRefGoogle Scholar
  26. 26.
    L.A. Davidson, R.E. Keller, Neural tube closure in xenopus laevis involves medial migration, directed protrusive activity, cell intercalation and convergent extension. Development 126 (20), 4547–4556 (1999)Google Scholar
  27. 27.
    P.N. Devreotes, T.L. Steck, Cyclic 3′, 5′ AMP relay in Dictyostelium discoideum II. Requirements for the initiation and termination of the response. J. Cell Biol. 80, 300–309 (1979)Google Scholar
  28. 28.
    P.N. Devreotes, P.L. Derstine, T.L. Steck, Cyclic 3′, 5′ AMP relay in Dictyostelium discoideum I. A technique to monitor responses to controlled stimuli. J. Cell Biol. 80, 291–299 (1979)Google Scholar
  29. 29.
    G. DeYoung, P.B. Monk, H.G. Othmer, Pacemakers in aggregation fields of Dictyostelium disc oideum. Does a single cell suffice? J. Math. Biol. 26, 486–517 (1988)Google Scholar
  30. 30.
    D. Dormann, C.J. Weijer, Propagating chemoattractant waves coordinate periodic cell movement in dictyostelium slugs. Development 128 (22), 4535–4543 (2001)Google Scholar
  31. 31.
    D. Dormann, G. Weijer, C.A. Parent, P.N. Devreotes, C.J. Weijer, Visualizing PI3 kinase-mediated cell-cell signaling during dictyostelium development. Curr. Biol. 12 (14), 1178–1188 (2002)CrossRefGoogle Scholar
  32. 32.
    R.A. Firtel, R. Meili, Dictyostelium: a model for regulated cell movement during morphogenesis. Curr. Opin. Genet. Dev. 10 (4), 421–427 (2000). ReviewCrossRefGoogle Scholar
  33. 33.
    P. Friedl, K. Wolf, Tumour-cell invasion and migration: diversity and escape mechanisms. Nat. Rev. Cancer 3 (5), 362–74 (2003)CrossRefGoogle Scholar
  34. 34.
    I. Fujiwara, S. Takahashi, H. Tadakuma, S. Ishiwata, Microscopic analysis of polymerization dynamics with individual actin filaments. Nat. Cell. Biol. 4 (9), 666–673 (2002)CrossRefGoogle Scholar
  35. 35.
    C. Gadgil, C.H. Lee, H.G. Othmer, A stochastic analysis of first-order reaction networks. Bull. Math. Biol. 67, 901–946 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    C.W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, Heildeberg, 1983)CrossRefzbMATHGoogle Scholar
  37. 37.
    G. Gerisch, Chemotaxis in dictyostelium. Annu. Rev. Physiol. 44 (1), 535–552 (1982)CrossRefGoogle Scholar
  38. 38.
    G. Gerisch, U. Wick, Intracellular oscillations and release of cyclic AMP from Dictyostelium cells. Biochem. Biophys. Res. Commun. 65, 364–370 (1975)CrossRefGoogle Scholar
  39. 39.
    G. Gerisch, D. Hulser, D. Malchow, U. Wick, Cell communication by periodic cyclic amp pulses. Philos. Trans. R. Soc. Lond. 272, 181–192 (1975)CrossRefGoogle Scholar
  40. 40.
    G. Gerisch, T. Bretschneider, A. Muler-Taubenberger, E. Simmeth, M. Ecke, S. Diez, K. Anderson, Mobile actin clusters and traveling waves in cells recovering from actin depolymerization. Biophys. J. 87 (5), 3493–3503 (2004)CrossRefGoogle Scholar
  41. 41.
    D.T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434 (1976)MathSciNetCrossRefGoogle Scholar
  42. 42.
    D.T. Gillespie, Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81 (25), 2340–2361 (1977)CrossRefGoogle Scholar
  43. 43.
    A.R. Gingle, Critical density for relaying in Dictyostelium discoideum and its relation to phosphodiesterase secretion into the extracellular medium. J. Cell Sci. 20, 1–20 (1976)Google Scholar
  44. 44.
    A.L. Hall, A. Schlein, J. Condeelis, Relationship of pseudopod extension to chemotactic hormone-induced actin polymerization in amoeboid cells. J. Cell Biol. 37 (3), 285–299 (1988)Google Scholar
  45. 45.
    B. Heit, S. Tavener, E. Raharjo, P. Kubes, An intracellular signaling hierarchy determines direction of migration in opposing chemotactic gradients. J. Cell Biol. 159 (1), 91–102 (2002)CrossRefGoogle Scholar
  46. 46.
    J. Hu, H.G. Othmer, A theoretical analysis of filament length fluctuations in actin and other polymers. J. Math. Biol. 63 (6), 1001–1049 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    J. Hu, A. Matzavinos, H.G. Othmer, A theoretical approach to actin filament dynamics. J. Stat. Phys. 128 (1–2), 111–138 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    J. Hu, H.W. Kang, H.G. Othmer, Stochastic analysis of reaction–diffusion processes. Bull. Math. Biol. 76, 854–894 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    M. Iijima, Y.E. Huang, P. Devreotes, Temporal and spatial regulation of chemotaxis. Dev. Cell 3 (4), 469–478 (2002). ReviewCrossRefGoogle Scholar
  50. 50.
    K. Inouye, Measurement of the motive force of the migrating slug of dictyostelium discoideum by a centrifuge method. Protoplasma 121, 171–177 (1984)CrossRefGoogle Scholar
  51. 51.
    K. Inouye, I. Takeuchi, Analytical studies on migrating movement of the pseudoplasmodium of Dictyostelium Discoideum. Protoplasma 99, 289–304 (1979)Google Scholar
  52. 52.
    R.H. Insall, O.D. Weiner, PIP3, PIP2, and cell movement–similar messages, different meanings? Dev. Cell. 1 (6), 743–747 (2001)CrossRefGoogle Scholar
  53. 53.
    C. Janetopoulos, T. Jin, P. Devreotes, Receptor-mediated activation of heterotrimeric G-proteins in living cells. Science 291 (5512), 2408–2411 (2001)CrossRefGoogle Scholar
  54. 54.
    P.A. Janmey, Mechanical properties of cytoskeletal polymers. Curr. Opin. Cell Biol. 2, 4–11 (1991)CrossRefGoogle Scholar
  55. 55.
    T. Jin, N. Zhang, Y. Long, C.A. Parent, P.N. Devreotes, Localization of the G protein β γ complex in living cells during chemotaxis. Science 287 (5455), 1034–1036 (2000)CrossRefGoogle Scholar
  56. 56.
    H.W. Kang, L. Zheng, H.G. Othmer, A new method for choosing the computational cell in stochastic reaction–diffusion systems. J. Math. Biol. 60, 1017–1099 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    H.W. Kang, L. Zheng, H.G. Othmer, A new method for choosing the computational cell in stochastic reaction-diffusion systems. J. Math. Biol. 60, 1017–1099 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    V. Khamviwath, J. Hu, H.G. Othmer, A continuum model of actin waves in dictyostelium discoideum. PloS One 8 (5), e64272 (2013)Google Scholar
  59. 59.
    A.R. Kimmel, C.A. Parent, The signal to move: D. discoideum go orienteering. Science 300 (5625), 1525–1527 (2003)Google Scholar
  60. 60.
    J.R. Kuhn, T.D. Pollard, Real-time measurements of actin filament polymerization by total internal reflection fluorescence microscopy. Biophys. J. 88 (2), 1387–1402 (2005)CrossRefGoogle Scholar
  61. 61.
    D.A. Lauffenburger, A.F. Horwitz, Cell migration: a physically integrated molecular process. Cell 84, 359–369 (1996)CrossRefGoogle Scholar
  62. 62.
    A. Levchenko, P.A. Iglesias, Models of eukaryotic gradient sensing: application to chemotaxis of amoebae and neutrophils. Biophys. J. 82 (1 Pt 1), 50–63 (2002)CrossRefGoogle Scholar
  63. 63.
    L. Limozin, M. Barmann, E. Sackmann, On the organization of self-assembled actin networks in giant vesicles. Eur. Phys. J E 10 (4), 319–330 (2003)CrossRefGoogle Scholar
  64. 64.
    F.C. MacKintosh, Theoretical models of viscoelasticity of actin solutions and the actin cortex. Biol. Bull. 194 (3), 351–353 (1998). No abstract availableGoogle Scholar
  65. 65.
    A. Matzavinos, H.G. Othmer, A stochastic analysis of actin polymerization in the presence of twinfilin and gelsolin. J. Theor. Biol. 249 (4), 723–736 (2007)MathSciNetCrossRefGoogle Scholar
  66. 66.
    J.L. McGrath, E.A. Osborn, Y.S. Tardy, C.F. Dewey Jr, J.H. Hartwig, Regulation of the actin cycle in vivo by actin filament severing. Proc. Natl. Acad. Sci. USA 97 (12), 6532–7 (2000)CrossRefGoogle Scholar
  67. 67.
    S. McLaughlin, J. Wang, A. Gambhir, D. Murray, PIP2 and proteins: interactions, organization and information flow. Annu. Rev. Biophys. Biomol. Struct. 31, 151–175 (2002). ReviewCrossRefGoogle Scholar
  68. 68.
    R. Meili, C. Ellsworth, S. Lee, T.B. Reddy, H. Ma, R.A. Firtel, Chemoattractant-mediated transient activation and membrane localization of akt/PKB is required for efficient chemotaxis to cAMP in dictyostelium. EMBO J. 18 (8), 2092–2105 (1999)CrossRefGoogle Scholar
  69. 69.
    H. Meinhardt, Orientation of chemotactic cells and growth cones: models and mechanisms. J. Cell Sci. 17 (17), 2867–2874 (1999)Google Scholar
  70. 70.
    R. Melki, S. Fievez, M.F. Carlier, Continuous monitoring of pi release following nucleotide hydrolysis in actin or tubulin assembly using 2-amino-6-mercapto-7-methylpurine ribonucleoside and purine-nucleoside phosphorylase as an enzyme-linked assay. Biochemistry 35 (37), 12038–45 (1996)CrossRefGoogle Scholar
  71. 71.
    T.J. Mitchison, L.P. Cramer, Actin-based cell motility and cell locomotion. Cell 3, 371–379 (1996). Review. No abstract availableGoogle Scholar
  72. 72.
    F. Oosawa, S. Asakura, Thermodynamics of the Polymerization of Protein (Academic, New York, 1975)Google Scholar
  73. 73.
    H.G. Othmer, A graph-theoretic analysis of chemical reaction networks (1979). Lecture Notes, Rutgers University. Available at
  74. 74.
    H.G. Othmer, P. Schaap, Oscillatory cAMP signaling in the development of dictyostelium discoideum. Comments Theor. Biol. 5, 175–282 (1998)Google Scholar
  75. 75.
    E. Palsson, H.G. Othmer, A model for individual and collective cell movement in dictyostelium discoideum. Proc. Natl. Acad. Sci. 97, 11448–11453 (2000)Google Scholar
  76. 76.
    C.A. Parent, P.N. Devreotes, A cell’s sense of direction. Science 284 (5415), 765–770 (1999). ReviewCrossRefGoogle Scholar
  77. 77.
    E. Pate, H.G. Othmer, Differentiation, cell sorting and proportion regulation in the slug stage of Dictyostelium discoideum. J. Theor. Biol. 118, 301–319 (1986)Google Scholar
  78. 78.
    M. Pineda, C. Weijer, R. Eftimie, Modelling cell movement, cell differentiation, cell sorting and proportion regulation in dictyostelium discoideum aggregations. J. Theor. Biol. 370, 135–150 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    T.D. Pollard, Regulation of actin filament assembly by arp2/3 complex and formins. Annu. Rev. Biophys. Biomol. Struct. 36, 451–477 (2007)CrossRefGoogle Scholar
  80. 80.
    T.D. Pollard, L. Blanchoin, R.D. Mullins, Molecular mechanisms controlling actin filament dynamics in nonmuscle cells. Annu. Rev. Biophys. Biomol. Struct. 29 (1), 545–76 (2000)CrossRefGoogle Scholar
  81. 81.
    A.Y. Pollitt, R.H. Insall, WASP and SCAR/WAVE proteins: the drivers of actin assembly. J. Cell Sci. 122 (Pt 15), 2575–2578 (2009). doi: 10.1242/jcs.023879 CrossRefGoogle Scholar
  82. 82.
    R.K. Raman, Y. Hashimoto, M.H. Cohen, A. Robertson, Differentiation for aggregation in the cellular slime molds: the emergence of autonomously signalling cells in Dictyostelium discoideum. J. Cell. Sci. 21, 243–259 (1976)Google Scholar
  83. 83.
    D. Raucher, T. Stauffer, W. Chen, K. Shen, S. Guo, J.D. York, M.P. Sheetz, T. Meyer, Phosphatidylinositol 4,5-bisphosphate functions as a second messenger that regulates cytoskeleton-plasma membrane adhesion. Cell 100 (2), 221–228 (2000)CrossRefGoogle Scholar
  84. 84.
    J. Rosenblatt, P. Peluso, T.J. Mitchison, The bulk of unpolymerized actin in xenopus egg extracts is ATP-bound. Mol. Biol. Cell 6 (2), 227–36 (1995)CrossRefGoogle Scholar
  85. 85.
    K.R. Sanft, H.G. Othmer, Constant-complexity stochastic simulation algorithm with optimal binning. J. Chem. Phys. 143 (8), 074108 (2015)Google Scholar
  86. 86.
    P. Schaap, Evolutionary crossroads in developmental biology: dictyostelium discoideum. Development 138 (3), 387–396 (2011)CrossRefGoogle Scholar
  87. 87.
    D. Sept, J. Xu, T.D. Pollard, J.A. McCammon, Annealing accounts for the length of actin filaments formed by spontaneous polymerization. Biophys. J. 77 (6), 2911–2919 (1999)CrossRefGoogle Scholar
  88. 88.
    M.P. Sheetz, D. Felsenfeld, C.G. Galbraith, D. Choquet, Cell migration as a five-step cycle. Biochem. Soc. Symp. 65, 233–43 (1999)Google Scholar
  89. 89.
    F. Siegert, C.J. Weijer, Analysis of optical density wave propagation and cell movement in the cellular slime mould Dictyostelium discoideum 49, 224–232 (1991)Google Scholar
  90. 90.
    J.V. Small, Microfilament-based motility in non-muscle cells. Curr. Opin. Cell Biol. 1, 75–79 (1989)CrossRefGoogle Scholar
  91. 91.
    J.V. Small, T. Stradal, E. Vignal, K. Rottner, The lamellipodium: where motility begins. Trends Cell Biol. 12 (3), 112–120 (2002). ReviewCrossRefGoogle Scholar
  92. 92.
    D.R. Soll, The use of computers in understanding how animal cells crawl, in International Review of Cytology, vol. 163, ed. by K.W. Jeon, J. Jarvik (Academic, New York, 1995), pp. 43–104Google Scholar
  93. 93.
    P.A. Spiro, J.S. Parkinson, H.G. Othmer, A model of excitation and adaptation in bacterial chemotaxis. Proc. Natl. Acad. Sci. 94 (14), 7263–7268 (1997)CrossRefGoogle Scholar
  94. 94.
    T.M. Svitkina, G.G. Borisy, Arp2/3 complex and actin depolymerizing factor/cofilin in dendritic organization and treadmilling of actin filament array in lamellipodia. J. Cell Sci. 145(5), 1009–26 (1999)CrossRefGoogle Scholar
  95. 95.
    J. Swanson, D.L. Taylor, Local and spatially coordinated movements in D ictyostelium discoideum amoedae during chemotaxis. Cell 28, 225–232 (1982)CrossRefGoogle Scholar
  96. 96.
    Y. Tang, H.G. Othmer, A G protein-based model of adaptation in Dictyostelium discoideum. Math. Biosci. 120 (1), 25–76 (1994)Google Scholar
  97. 97.
    Y. Tang, H.G. Othmer, Excitation, oscillations and wave propagation in a G-protein-based model of signal transduction in Dictyostelium discoideum. Philos. Trans. R. Soc. Lond. B Biol. Sci. 349 (1328), 179–95 (1995)CrossRefGoogle Scholar
  98. 98.
    J.A. Theriot, T.J. Mitchison, Actin microfilament dynamics in locomoting cells. Nature 352 (6331), 126–131 (1991)CrossRefGoogle Scholar
  99. 99.
    A. Tikhonov, Systems of differential equations containing small parameters multiplying derivatives. Math. Sb. 31, 575–586 (1952)Google Scholar
  100. 100.
    P.H. Vardy, L.R. Fisher, E. Smith, K.L. Williams, Traction proteins in the extracellular matrix of Dictyostelium discoideum slugs. Nature 320 (6062), 526–529 (1986)CrossRefGoogle Scholar
  101. 101.
    B. Varnum, K.B. Edwards, D.R. Soll, Dictyostelium amebae alter motility differently in response to increasing versus decreasing temporal gradients of cAMP. J Cell Biol. 101, 1–5 (1985)CrossRefGoogle Scholar
  102. 102.
    B. Vasiev, C.J. Weijer, Modelling of dictyostelium discoideum slug migration. J. Theor. Biol. 223, 347–59 (2003)MathSciNetCrossRefGoogle Scholar
  103. 103.
    J.B. Wallingford, L.A. Niswander, G.M. Shaw, R.H. Finnell, The continuing challenge of understanding, preventing, and treating neural tube defects. Science 339 (6123), 1222002 (2013)Google Scholar
  104. 104.
    C.J. Weijer, Signalling during dictyostelium development, in Dictyostelids (Springer, Berlin, 2013), pp. 49–70Google Scholar
  105. 105.
    D. Wessels, J. Murray, D.R. Soll, Behavior of Dictyostelium amoebae is regulated primarily by the temporal dynamic of the natural cAMP wave. Cell Motil. Cytoskeleton 23 (2), 145–156 (1992)CrossRefGoogle Scholar
  106. 106.
    L. Wolpert, R. Beddington, T. Jessel, P. Lawrence, E. Meyerowitz, J. Smith, Principles of Development (Oxford University Press, Oxford, 2002)Google Scholar
  107. 107.
    B. Wurster, K. Schubiger, U. Wick, G. Gerisch, Cyclic GMP in Dictyostelium discoideum: oscillations and pulses in response to folic acid and cyclic AMP signals. FEBS Lett. 76, 141–144 (1977)CrossRefGoogle Scholar
  108. 108.
    X. Xin, H.G. Othmer, A trimer of dimers- based model for the chemotactic signal transduction network in bacterial chemotaxis. Bull. Math. Biol. 74, 2339–2382 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  109. 109.
    X.S. Xu, A. Kuspa, D. Fuller, W.F. Loomis, D.A. Knecht, Cell-cell adhesion prevents mutant cells lacking myosin II from penetrating aggregation streams of dictyostelium. Dev. Biol. 175 (2), 218–226 (1996)CrossRefGoogle Scholar
  110. 110.
    J. Xu, Y. Tseng, D. Wirtz, Strain hardening of actin filament networks. regulation by the dynamic cross-linking protein alpha-actinin. J. Biol. Chem. 275 (46), 35886–35892 (2000)Google Scholar
  111. 111.
    S.H. Zigmond, Recent quantitative studies of actin filament turnover during locomotion. Cell Motil. Cytoskeleton 25, 309–316 (1993)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

Personalised recommendations