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Multicell Simulations of Development and Disease Using the CompuCell3D Simulation Environment

  • Maciej H. Swat
  • Susan D. Hester
  • Ariel I. Balter
  • Randy W. Heiland
  • Benjamin L. Zaitlen
  • James A. Glazier
Protocol
Part of the Methods in Molecular Biology book series (MIMB, volume 500)

Summary

Mathematical modeling and computer simulation have become crucial to biological fields from genomics to ecology. However, multicell, tissue-level simulations of development and disease have lagged behind other areas because they are mathematically more complex and lack easy-to-use software tools that allow building and running in silico experiments without requiring in-depth knowledge of programming. This tutorial introduces Glazier—Graner—Hogeweg (GGH) multicell simulations and CompuCell3D, a simulation framework that allows users to build, test, and run GGH simulations.

Keywords

Glazier—Graner—Hogeweg model GGH CompuCell3D Mitosis Cell growth Cell sorting Chemotaxis Multicell modeling Tissue-level modeling Developmental biology Computational biology 

Notes

Acknowledgments

We gratefully acknowledge support from the National Institutes of Health, National Institute of General Medical Sciences, grants 1R01 GM077138–01A1 and 1R01 GM076692-01, and the Office of Vice President for Research, the College of Arts and Sciences, the Pervasive Technologies Laboratories and the Biocomplexity Institute at Indiana University. Indiana University's University Information Technology Services provided time on their BigRed clusters for simulation execution. Early versions of CompuCell and CompuCell3D were developed at the University of Notre Dame by J.A.G., Dr. Mark Alber and Dr. Jesus Izaguirre and collaborators with the support of National Science Foundation, Division of Integrative Biology, grant IBN-00836563. Since the primary home of CompuCell3D moved to Indiana University in 2004, the Notre Dame team have continued to provide important support for its development.

References

  1. 1.
    Bassingthwaighte, J. B. (2000) Strategies for the Physiome project. Ann. Biomed. Eng. 28, 1043–1058.PubMedCrossRefGoogle Scholar
  2. 2.
    Merks, R. M. H., Newman, S. A., and Glazier, J. A. (2004) Cell-oriented modeling of in vitro capillary development. Lect. Notes Comp. Sci. 3305, 425–434.CrossRefGoogle Scholar
  3. 3.
    Turing, A. M. (1953) The chemical basis of morphogenesis. Philos. Trans. R. Soc. B 237, 37–72.CrossRefGoogle Scholar
  4. 4.
    Merks, R. M. H. and Glazier, J. A. (2005) A cell-centered approach to developmental biology. Phys. A 352, 113–130.CrossRefGoogle Scholar
  5. 5.
    Dormann, S. and Deutsch, A. (2002) Modeling of self-organized avascular tumor growth with a hybrid cellular automaton. In Silico Biol. 2, 1–14.Google Scholar
  6. 6.
    dos Reis, A. N., Mombach, J. C. M., Walter, M., and de Avila, L. F. (2003) The interplay between cell adhesion and environment rigidity in the morphology of tumors. Phys. A 322, 546–554.CrossRefGoogle Scholar
  7. 7.
    Drasdo, D. and Hohme, S. (2003) Individual-based approaches to birth and death in avascular tumors. Math. Comput. Model. 37, 1163–1175.CrossRefGoogle Scholar
  8. 8.
    Holm, E. A., Glazier, J. A., Srolovitz, D. J., and Grest, G. S. (1991) Effects of lattice anisotropy and temperature on domain growth in the two-dimensional Potts model. Phys. Rev. A 43, 2662–2669.PubMedCrossRefGoogle Scholar
  9. 9.
    Turner, S. and Sherratt, J. A. (2002) Intercellular adhesion and cancer invasion: A discrete simulation using the extended Potts model. J. Theor. Biol. 216, 85–100.PubMedCrossRefGoogle Scholar
  10. 10.
    Drasdo, D. and Forgacs, G. (2000) Modeling the interplay of generic and genetic mechanisms in cleavage, blastulation, and gastrulation. Dev. Dynam. 219, 182–191.CrossRefGoogle Scholar
  11. 11.
    Drasdo, D., Kree, R., and McCaskill, J. S. (1995) Monte-Carlo approach to tissue-cell populations. Phys. Rev. E 52, 6635–6657.CrossRefGoogle Scholar
  12. 12.
    Longo, D., Peirce, S. M., Skalak, T. C., Davidson, L., Marsden, M., and Dzamba, B. (2004) Multicellular computer simulation of morphogenesis: Blastocoel roof thinning and matrix assembly in Xenopus laevis. Dev. Biol. 271, 210–222.PubMedCrossRefGoogle Scholar
  13. 13.
    Collier, J. R., Monk, N. A. M., Maini, P. K., and Lewis, J. H. (1996) Pattern formation by lateral inhibition with feedback: A mathematical model of Delta-Notch intercellular signaling. J. Theor. Biol. 183, 429–446.PubMedCrossRefGoogle Scholar
  14. 14.
    Honda, H. and Mochizuki, A. (2002) Formation and maintenance of distinctive cell patterns by coexpression of membrane-bound ligands and their receptors. Dev. Dynam. 223, 180–192.CrossRefGoogle Scholar
  15. 15.
    Moreira, J. and Deutsch, A. (2005) Pigment pattern formation in zebrafish during late larval stages: A model based on local interactions. Dev. Dynam. 232, 33–42.CrossRefGoogle Scholar
  16. 16.
    Wearing, H. J., Owen, M. R., and Sherratt, J. A. (2000) Mathematical modelling of juxtacrine patterning. Bull. Math. Biol. 62, 293–320.PubMedCrossRefGoogle Scholar
  17. 17.
    Zhdanov, V. P. and Kasemo, B. (2004) Simulation of the growth of neurospheres. Europhys. Lett. 68, 134–140.CrossRefGoogle Scholar
  18. 18.
    Ambrosi, D., Gamba, A., and Serini, G. (2005) Cell directional persistence and chemotaxis in vascular morphogenesis. Bull. Math. Biol. 67, 195–195.CrossRefGoogle Scholar
  19. 19.
    Gamba, A., Ambrosi, D., Coniglio, A., de Candia, A., di Talia, S., Giraudo, E., Serini, G., Preziosi, L., and Bussolino, F. (2003) Percolation, morphogenesis, and Burgers dynamics in blood vessels formation. Phys. Rev. Lett. 90, 118101.PubMedCrossRefGoogle Scholar
  20. 20.
    Novak, B., Toth, A., Csikasz-Nagy, A., Gyorffy, B., Tyson, J. A., and Nasmyth, K. (1999) Finishing the cell cycle. J. Theor. Biol. 199, 223–233.PubMedCrossRefGoogle Scholar
  21. 21.
    Peirce, S. M., van Gieson, E. J., and Skalak, T. C. (2004) Multicellular simulation predicts microvascular patterning and in silico tissue assembly. FASEB J. 18, 731–733.PubMedGoogle Scholar
  22. 22.
    Merks, R. M. H., Brodsky, S. V., Goligorksy, M. S., Newman, S. A., and Glazier, J. A. (2006) Cell elongation is key to in silico replication of in vitro vasculogenesis and subsequent remodeling. Dev. Biol. 289, 44–54.PubMedCrossRefGoogle Scholar
  23. 23.
    Merks, R. M. H. and Glazier, J. A. (2005) Contact-inhibited chemotactic motility can drive both vasculogenesis and sprouting angiogenesis. q-bio/0505033. Google Scholar
  24. 24.
    Kesmir, C. and de Boer, R. J. (2003) A spatial model of germinal center reactions: Cellular adhesion based sorting of B cells results in efficient affinity maturation. J. Theor. Biol. 222, 9–22.PubMedCrossRefGoogle Scholar
  25. 25.
    Meyer-Hermann, M., Deutsch, A., and Or-Guil, M. (2001) Recycling probability and dynamical properties of germinal center reactions. J. Theor. Biol. 210, 265–285.PubMedCrossRefGoogle Scholar
  26. 26.
    Nguyen, B., Upadhyaya, A., van Oudenaarden, A., and Brenner, M. P. (2004) Elastic instability in growing yeast colonies. Biophys. J. 86, 2740–2747.PubMedCrossRefGoogle Scholar
  27. 27.
    Walther, T., Reinsch, H., Grosse, A., Ostermann, K., Deutsch, A., and Bley, T. (2004) Mathematical modeling of regulatory mechanisms in yeast colony development. J. Theor. Biol. 229, 327–338.PubMedCrossRefGoogle Scholar
  28. 28.
    Borner, U., Deutsch, A., Reichenbach, H., and Bar, M. (2002) Rippling patterns in aggregates of myxobacteria arise from cell–cell collisions. Phys. Rev. Lett. 89, 078101.PubMedCrossRefGoogle Scholar
  29. 29.
    Bussemaker, H. J., Deutsch, A., and Geigant, E. (1997) Mean-field analysis of a dynamical phase transition in a cellular automaton model for collective motion. Phys. Rev. Lett. 78, 5018–5021.CrossRefGoogle Scholar
  30. 30.
    Dormann, S., Deutsch, A., and Lawniczak, A. T. (2001) Fourier analysis of Turing-like pattern formation in cellular automaton models. Future Gener. Comput. Syst. 17, 901–909.CrossRefGoogle Scholar
  31. 31.
    Börner, U., Deutsch, A., Reichenbach, H., and Bär, M. (2002) Rippling patterns in aggregates of myxobacteria arise from cell–cell collisions. Phys. Rev. Lett. 89, 078101.PubMedCrossRefGoogle Scholar
  32. 32.
    Zhdanov, V. P. and Kasemo, B. (2004) Simulation of the growth and differentiation of stem cells on a heterogeneous scaffold. Phys. Chem. Chem. Phys. 6, 4347–4350.CrossRefGoogle Scholar
  33. 33.
    Knewitz, M. A. and Mombach, J. C. (2006) Computer simulation of the influence of cellular adhesion on the morphology of the interface between tissues of proliferating and quiescent cells. Comput. Biol. Med. 36, 59–69.PubMedCrossRefGoogle Scholar
  34. 34.
    Marée, A. F. M. and Hogeweg, P. (2001) How amoeboids self-organize into a fruiting body: Multicellular coordination in Dictyostelium discoideum. Proc. Natl Acad. Sci. USA 98, 3879–3883.PubMedCrossRefGoogle Scholar
  35. 35.
    Marée, A. F. M. and Hogeweg, P. (2002) Modelling Dictyostelium discoideum morphogenesis: the culmination. Bull. Math. Biol. 64, 327–353.PubMedCrossRefGoogle Scholar
  36. 36.
    Marée, A. F. M., Panfilov, A. V., and Hogeweg, P. (1999) Migration and thermotaxis of Dictyostelium discoideum slugs, a model study. J. Theor. Biol. 199, 297–309.CrossRefGoogle Scholar
  37. 37.
    Savill, N. J. and Hogeweg, P. (1997) Modelling morphogenesis: From single cells to crawling slugs. J. Theor. Biol. 184, 229–235.CrossRefGoogle Scholar
  38. 38.
    Hogeweg, P. (2000) Evolving mechanisms of morphogenesis: On the interplay between differential adhesion and cell differentiation. J. Theor. Biol. 203, 317–333.PubMedCrossRefGoogle Scholar
  39. 39.
    Johnston, D. A. (1998) Thin animals. J. Phys. A 31, 9405–9417.CrossRefGoogle Scholar
  40. 40.
    Groenenboom, M. A. and Hogeweg, P. (2002) Space and the persistence of male-killing endosymbionts in insect populations. Proc. Biol. Sci. 269, 2509–2518.PubMedCrossRefGoogle Scholar
  41. 41.
    Groenenboom, M. A., Maree, A. F., and Hogeweg, P. (2005) The RNA silencing pathway: the bits and pieces that matter. PLoS Comp. Biol. 1, 155–165.CrossRefGoogle Scholar
  42. 42.
    Kesmir, C., van Noort, V., de Boer, R. J., and Hogeweg, P. (2003) Bioinformatic analysis of functional differences between the immunoproteasome and the constitutive proteasome. Immunogenetics 55, 437–449.PubMedCrossRefGoogle Scholar
  43. 43.
    Pagie, L. and Hogeweg, P. (2000) Individual- and population-based diversity in restriction-modification systems. Bull. Math. Biol. 62, 759–774.PubMedCrossRefGoogle Scholar
  44. 44.
    Silva, H. S. and Martins, M. L. (2003) A cellular automata model for cell differentiation. Phys. A 322, 555–566.CrossRefGoogle Scholar
  45. 45.
    Zajac, M., Jones, G. L., and Glazier, J. A. (2000) Model of convergent extension in animal morphogenesis. Phys. Rev. Lett. 85, 2022–2025.PubMedCrossRefGoogle Scholar
  46. 46.
    Zajac, M., Jones, G. L., and Glazier, J. A. (2003) Simulating convergent extension by way of anisotropic differential adhesion. J. Theor. Biol. 222, 247–259.PubMedCrossRefGoogle Scholar
  47. 47.
    Savill, N. J. and Sherratt, J. A. (2003) Control of epidermal stem cell clusters by Notch-mediated lateral induction. Dev. Biol. 258, 141–153.PubMedCrossRefGoogle Scholar
  48. 48.
    Mombach, J. C. M., de Almeida, R. M. C., Thomas, G. L., Upadhyaya, A., and Glazier, J. A. (2001) Bursts and cavity formation in Hydra cells aggregates: Experiments and simulations. Phys. A 297, 495–508.CrossRefGoogle Scholar
  49. 49.
    Rieu, J. P., Upadhyaya, A., Glazier, J. A., Ouchi, N. B., and Sawada, Y. (2000) Diffusion and deformations of single hydra cells in cellular aggregates. Biophys. J. 79, 1903–1914.PubMedCrossRefGoogle Scholar
  50. 50.
    Mochizuki, A. (2002) Pattern formation of the cone mosaic in the zebrafish retina: A cell rearrangement model. J. Theor. Biol. 215, 345–361.PubMedCrossRefGoogle Scholar
  51. 51.
    Takesue, A., Mochizuki, A., and Iwasa, Y. (1998) Cell-differentiation rules that generate regular mosaic patterns: Modelling motivated by cone mosaic formation in fish retina. J. Theor. Biol. 194, 575–586.PubMedCrossRefGoogle Scholar
  52. 52.
    Dallon, J., Sherratt, J., Maini, P. K., and Ferguson, M. (2000) Biological implications of a discrete mathematical model for collagen deposition and alignment in dermal wound repair. IMA J. Math. Appl. Med. Biol. 17, 379–393.PubMedCrossRefGoogle Scholar
  53. 53.
    Maini, P. K., Olsen, L., and Sherratt, J. A. (2002) Mathematical models for cell–matrix interactions during dermal wound healing. Int. J. Bifurcat. Chaos 12, 2021–2029.CrossRefGoogle Scholar
  54. 54.
    Kreft, J. U., Picioreanu, C., Wimpenny, J. W. T., and van Loosdrecht, M. C. M. (2001) Individual-based modelling of biofilms. Microbiology 147, 2897–2912.PubMedGoogle Scholar
  55. 55.
    Picioreanu, C., van Loosdrecht, M. C. M., and Heijnen, J. J. (2001) Two-dimensional model of biofilm detachment caused by internal stress from liquid flow. Biotechnol. Bioeng. 72, 205–218.PubMedCrossRefGoogle Scholar
  56. 56.
    van Loosdrecht, M. C. M., Heijnen, J. J., Eberl, H., Kreft, J., and Picioreanu, C. (2002) Mathematical modelling of biofilm structures. Antonie Van Leeuwenhoek Int. J. General Mol. Microbiol. 81, 245–256.CrossRefGoogle Scholar
  57. 57.
    Pop awski, N. J., Shirinifard, A., Swat, M., and Glazier, J. A. (2008) Simulations of single-species bacterial-biofilm growth using the Glazier–Graner–Hogeweg model and the CompuCell3D modeling environment. Math. Biosci. Eng. 5, 355–388.Google Scholar
  58. 58.
    Chaturvedi, R., Huang, C., Izaguirre, J. A., Newman, S. A., Glazier, J. A., and Alber, M. S. (2004) A hybrid discrete-continuum model for 3-D skeletogenesis of the vertebrate limb. Lect. Notes Comput. Sci. 3305, 543–552.CrossRefGoogle Scholar
  59. 59.
    Pop awski, N. J., Swat, M., Gens, J. S., and Glazier, J. A. (2007) Adhesion between cells, diffusion of growth factors, and elasticity of the AER produce the paddle shape of the chick limb. Phys. A 373, 521–532.CrossRefGoogle Scholar
  60. 60.
    Glazier, J. A. and Weaire, D. (1992) The kinetics of cellular patterns. J. Phys.: Condens. Matter 4, 1867–1896.CrossRefGoogle Scholar
  61. 61.
    Glazier, J. A. (1993) Grain growth in three dimensions depends on grain topology. Phys. Rev. Lett. 70, 2170–2173.PubMedCrossRefGoogle Scholar
  62. 62.
    Glazier, J. A., Grest, G. S., and Anderson, M. P. (1990) Ideal two-dimensional grain growth, in Simulation and Theory of Evolving Microstructures (Anderson, M. P. and Rollett, A. D., eds.), The Minerals, Metals and Materials Society, Warrendale, PA, pp. 41–54. Google Scholar
  63. 63.
    Glazier, J. A., Anderson, M. P., and Grest, G. S. (1990) Coarsening in the two-dimensional soap froth and the large-Q Potts model: a detailed comparison. Philos. Mag. B 62, 615–637.CrossRefGoogle Scholar
  64. 64.
    Grest, G. S., Glazier, J. A., Anderson, M. P., Holm, E. A., and Srolovitz, D. J. (1992) Coarsening in two-dimensional soap froths and the large-Q Potts model. Mater. Res. Soc. Symp. 237, 101–112.Google Scholar
  65. 65.
    Jiang, Y. and Glazier, J. A. (1996) Extended large-Q Potts model simulation of foam drainage. Philos. Mag. Lett. 74, 119–128.CrossRefGoogle Scholar
  66. 66.
    Jiang, Y., Levine, H., and Glazier, J. A. (1998) Possible cooperation of differential adhesion and chemotaxis in mound formation of Dictyostelium. Biophys. J. 75, 2615–2625.PubMedCrossRefGoogle Scholar
  67. 67.
    Jiang, Y., Mombach, J. C. M., and Glazier, J. A. (1995) Grain growth from homogeneous initial conditions: Anomalous grain growth and special scaling states. Phys. Rev. E 52, 3333–3336.CrossRefGoogle Scholar
  68. 68.
    Jiang, Y., Swart, P. J., Saxena, A., Asipauskas, M., and Glazier, J. A. (1999) Hysteresis and avalanches in two-dimensional foam rheology simulations. Phys. Rev. E 59, 5819–5832.CrossRefGoogle Scholar
  69. 69.
    Ling, S., Anderson, M. P., Grest, G. S., and Glazier, J. A. (1992) Comparison of soap froth and simulation of large-Q Potts model. Mater. Sci. Forum 94–96, 39–47.CrossRefGoogle Scholar
  70. 70.
    Mombach, J. C. M. (2000) Universality of the threshold in the dynamics of biological cell sorting. Phys. A 276, 391–400.CrossRefGoogle Scholar
  71. 71.
    Weaire, D. and Glazier, J. A. (1992) Modelling grain growth and soap froth coarsening: Past, present and future. Mater. Sci. Forum 94–96, 27–39.CrossRefGoogle Scholar
  72. 72.
    Weaire, D., Bolton, F., Molho, P., and Glazier, J. A. (1991) Investigation of an elementary model for magnetic froth. J. Phys.: Condens. Matter 3, 2101–2113.CrossRefGoogle Scholar
  73. 73.
    Glazer, J. A., Balter, A., and Pop awski, N. (2007) Magnetization to morphogenesis: A brief history of the Glazier—Graner—Hogeweg model, in Single-Cell-Based Models in Biology and Medicine (Anderson, A. R. A., Chaplain, M. A. J., and Rejniak, K. A., eds.), Birkhauser Verlag, Basel, pp. 79–106.Google Scholar
  74. 74.
    Walther, T., Reinsch, H., Ostermann, K., Deutsch, A., and Bley, T. (2005) Coordinated growth of yeast colonies: Experimental and mathematical analysis of possible regulatory mechanisms. Eng. Life Sci. 5, 115–133.CrossRefGoogle Scholar
  75. 75.
    Keller, E. F. and Segel, L. A. (1971) Model for chemotaxis. J. Theor. Biol. 30, 225–234.PubMedCrossRefGoogle Scholar
  76. 76.
    Glazier, J. A. and Upadhyaya, A. (1998) First steps towards a comprehensive model of tissues, or: A physicist looks at development, in Dynamical Networks in Physics and Biology: At the Frontier of Physics and Biology (Beysens, D. and Forgacs, G., eds.), EDP Sciences, Berlin, pp. 149–160.Google Scholar
  77. 77.
    Glazier, J. A. and Graner, F. (1993) Simulation of the differential adhesion driven rearrangement of biological cells. Phys. Rev. E 47, 2128–2154.CrossRefGoogle Scholar
  78. 78.
    Glazier, J. A. (1993) Cellular patterns. Bussei Kenkyu 58, 608–612.Google Scholar
  79. 79.
    Glazier, J. A. (1996) Thermodynamics of cell sorting. Bussei Kenkyu 65, 691–700.Google Scholar
  80. 80.
    Glazier, J. A., Raphael, R. C., Graner, F., and Sawada, Y. (1995) The energetics of cell sorting in three dimensions, in Interplay of Genetic and Physical Processes in the Development of Biological Form (Beysens, D., Forgacs, G., and Gaill, F., eds.), World Scientific, Singapore, pp. 54–66.Google Scholar
  81. 81.
    Graner, F. and Glazier, J. A. (1992) Simulation of biological cell sorting using a 2-dimensional extended Potts model. Phys. Rev. Lett. 69, 2013–2016.PubMedCrossRefGoogle Scholar
  82. 82.
    Mombach, J. C. M. and Glazier, J. A. (1996) Single cell motion in aggregates of embryonic cells. Phys. Rev. Lett. 76, 3032–3035.PubMedCrossRefGoogle Scholar
  83. 83.
    Mombach, J. C. M., Glazier, J. A., Raphael, R. C., and Zajac, M. (1995) Quantitative comparison between differential adhesion models and cell sorting in the presence and absence of fluctuations. Phys. Rev. Lett. 75, 2244–2247.PubMedCrossRefGoogle Scholar
  84. 84.
    Cipra, B. A. (1987) An introduction to the Ising-model. Am. Math. Monthly 94, 937–959.CrossRefGoogle Scholar
  85. 85.
    Metropolis, N., Rosenbluth, A., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953) Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092.CrossRefGoogle Scholar
  86. 86.
    Forgacs, G. and Newman, S. A. (2005). Biological Physics of the Developing Embryo. Cambridge University Press, Cambridge.Google Scholar
  87. 87.
    Alber, M. S., Kiskowski, M. A., Glazier, J. A., and Jiang, Y. (2002) On cellular automation approaches to modeling biological cells, in Mathematical Systems Theory in Biology, Communication and Finance (Rosenthal, J. and Gilliam, D. S., eds.), Springer, New York, NY, pp. 1–40. Google Scholar
  88. 88.
    Alber, M. S., Jiang, Y., and Kiskowski, M. A. (2004) Lattice gas cellular automation model for rippling and aggregation in myxobacteria. Phys. D 191, 343–358.CrossRefGoogle Scholar
  89. 89.
    Upadhyaya, A., Rieu, J. P., Glazier, J. A., and Sawada, Y. (2001) Anomalous diffusion in two-dimensional Hydra cell aggregates. Phys. A 293, 549–558.CrossRefGoogle Scholar
  90. 90.
    Cickovski, T., Aras, K., Alber, M. S., Izaguirre, J. A., Swat, M., Glazier, J. A., Merks, R. M. H., Glimm, T., Hentschel, H. G. E., and Newman, S. A. (2007) From genes to organisms via the cell: A problem-solving environment for multicellular development. Comput. Sci. Eng. 9, 50–60.PubMedCrossRefGoogle Scholar
  91. 91.
    Izaguirre, J. A., Chaturvedi, R., Huang, C., Cickovski, T., Coffland, J., Thomas, G., Forgacs, G., Alber, M., Hentschel, G., Newman, S. A., and Glazier, J. A. (2004) CompuCell, a multi-model framework for simulation of morphogenesis. Bioinformatics 20, 1129–1137.PubMedCrossRefGoogle Scholar
  92. 92.
    Armstrong, P. B. and Armstrong, M. T. (1984) A role for fibronectin in cell sorting out. J. Cell Sci. 69, 179–197.PubMedGoogle Scholar
  93. 93.
    Armstrong, P. B. and Parenti, D. (1972) Cell sorting in the presence of cytochalasin B. J. Cell Sci. 55, 542–553.Google Scholar
  94. 94.
    Glazier, J. A. and Graner, F. (1993) Simulation of the differential adhesion driven rearrangement of biological cells. Phys. Rev. E 47, 2128–2154.CrossRefGoogle Scholar
  95. 95.
    Glazier, J. A. and Graner, F. (1992) Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys. Rev. Lett. 69, 2013–2016.PubMedCrossRefGoogle Scholar
  96. 96.
    Ward, P. A., Lepow, I. H., and Newman, L. J. (1968) Bacterial factors chemotactic for polymorphonuclear leukocytes. Am. J. Pathol. 52, 725–736.PubMedGoogle Scholar
  97. 97.
    Lutz, M. (1999) Learning Python. O'Reilly & Associates, Sebastopol, CA.Google Scholar
  98. 98.
    Balter, A. I., Glazier, J. A., and Perry, R. (2008) Probing soap-film friction with two-phase foam flow. Philos. Mag. Lett. 88, 679–691.CrossRefGoogle Scholar
  99. 99.
    Dvorak, P., Dvorakova, D., and Hampl, A. (2006) Fibroblast growth factor signaling in embryonic and cancer stem cells. FEBS Lett. 580, 2869–2287.PubMedCrossRefGoogle Scholar

Copyright information

© Humana Press 2009

Authors and Affiliations

  • Maciej H. Swat
  • Susan D. Hester
  • Ariel I. Balter
  • Randy W. Heiland
  • Benjamin L. Zaitlen
  • James A. Glazier
    • 1
  1. 1.Biocomplexity Institute and Department of PhysicsIndiana UniversityBloomingtonUSA

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