Skip to main content
SpringerLink
Log in

On the Role of Physics in the Growth and Pattern Formation of Multi-Cellular Systems: What can we Learn from Individual-Cell Based Models?

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We demonstrate that many collective phenomena in multi-cellular systems can be explained by models in which cells, despite their complexity, are represented as simple particles which are parameterized mainly by their physical properties. We mainly focus on two examples that nevertheless span a wide range of biological sub-disciplines: Unstructured cell populations growing in cell culture and growing cell layers in early animal development. While cultured unstructured cell populations would apriori been classified as particularly suited for a biophysical approach since the degree to which they are committed to a genetic program is expected to be modest, early animal development would be expected to mark the other extreme—here the degree of determinism according to a genetic program would be expected to be very high. We consider a number of phenomena such as the growth kinetics and spatial structure formation of monolayers and multicellular spheroids, the effect of the presence of another cell type surrounding the growing cell population, the effect of mutations and the critical surface dynamics of monolayers. Different from unstructured cell populations, cells in early development and at tissue interfaces usually form highly organized structures. An example are tissue layers. Under certain circumstances such layers are observed to fold. We show that folding pattern again can largely be explained by physical mechanisms either by a buckling instability or active cell shape changes. The paper combines new and published material and aims at an overview of a wide range of physical aspects in unstructured populations and growing tissue layers.

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. S. Huang and D. Ingber, The structural and mechanical complexity of cell-growth control. Nature Cell Biol. 1:E131 (1999).

    Google Scholar 

  2. I. Salazar-Ciudad, J. Jernvall, and S. A. Newman, Mechanisms of pattern formation in development and evolution. Development 130:2027 (2003).

    Google Scholar 

  3. D. J. Tschumperlin, EGFR autocrine signaling in a compliant interstitial space: Mechanotransduction from the outside in. Cell Cycle 3:996 (2004).

    Google Scholar 

  4. A. Neagu, K. Jakab, R. Jamison, and G. Forgacs, Role of physical mechanisms in biological self-organization. Phys. Rev. Lett. 95:178104 (2005).

    ADS  Google Scholar 

  5. G. Forgacs and S. Newmann, Biological Physics of the Developing Embryo (Cambridge University Press, Cambridge, 2005).

    Google Scholar 

  6. G. Helmlinger, P. Netti, H. Lichtenfeld, R. Melder, and R. Jain, Solid stress inhibits the growth of multicellular tumor spheroids. Nature Biotechnol. 15:778 (1997).

    Google Scholar 

  7. C. Nelson, R. Jean, J. Tan, W. Liu, N. Sniadecki, A. Spector, and C. Chen, Mechanical control of tissue growth: Function follows form. Proc. Natl. Acad. Sci. (USA), 102:(2005).

  8. D. Ingber, Mechanical control of tissue morphogenesis during embryological development. Int. J. Dev. Biol. 50:255 (2006).

    Google Scholar 

  9. D. Ingber, Mechanical control of tissue growth: Function follows form. Proc. Natl. Acad. Sci. (USA) 102:11571 (2005).

    ADS  Google Scholar 

  10. L. D. Horb and J. M. Slack, Role of cell division in branching morphogenesis and differentiation of the embryonic pancreas. Int. J. Dev. Biol. 44:791 (2000).

    Google Scholar 

  11. B. Shraiman, Mechanical feedback as a possible regulator of tissue growth. Proc. Natl. Acad Sci. (USA) 102:3318 (2005).

    ADS  Google Scholar 

  12. H. Lodish, A. Berk, P. Matsudaria, C. Kaiser, M. Krieger, M. Scott, S. Zipursky, and J. Darnell, Molecular Cell Biology (Freeman, New York, 2004).

    Google Scholar 

  13. H. Byrne, J. King, D. McElwain, and L. Preziosi, A two-phase model of solid tumor growth, Appl. Math. Lett., pp. 1–15 (2001).

  14. C. Chen, H. Byrne and J. King, The influence of growth-induced stress from the surrounding medium on the development of multicell spheroids, J. Math. Biol. 43:191 (2001).

    MATH  MathSciNet  Google Scholar 

  15. U. Schwarz, N. Balaban, D. Riveline, A. Bershadsky, B. Geiger, and S. Safran, Calculation of forces at focal adhesions from elastic substrate data: The effect of localized force and the need for regularization, Biophys. J. 83:1380 (2002).

    ADS  Google Scholar 

  16. H. Byrne and L. Prezziosi, Modelling solid tumour growth using the theory of mixtures, Math. Med. Biol. 20:341 (2003).

    MATH  Google Scholar 

  17. I. Bischofs and U. Schwarz, Effect of poisson ratio on cellular structure formation. Phys. Rev. Lett. 95:068102 (2005).

    ADS  Google Scholar 

  18. I. Schiffer, S. Gebhard, C. Heimerdinger, A. Heling, J. Hast, U. Wollscheid, B. Seliger, B. Tanner, S. Gilbert, T. Beckers, S. Baasner, W. Brenner, C. Spangenberg, D. Prawitt, T. Trost, W. Schreiber, B. Zabel, M. Thelen, H. Lehr, F. Oesch, and J. Hengstler, Switching off her-2/ neu in a tetracyline-controlled mouse tumor model leads to apoptosis and tumorsize-dependent remission. Cancer Res. 63:7221 (2003).

    Google Scholar 

  19. M. Alison and C. Sarraf, Understanding Cancer (Cambridge University Press, Cambridge, 1998).

    Google Scholar 

  20. B. Sayan, G. Ince, A. Sayan, and M. Ozturk, Napo as a novel apoptosis marker. J. Cell Biol. 155:719 (2001).

    Google Scholar 

  21. J. Mombach and J. Glazier, Single cell motion in aggregates of embryonic cells. Phys. Rev. Lett. 76:3032 (1996).

    ADS  Google Scholar 

  22. J. Guck, R. Ananthakrishnan, H. Mahmood, T. Moon, C. Cunningham and J. Käs, The optical stretcher: A novel laser tool to micromanipulate cells. Biophys. J. 81:767 (2001).

    Google Scholar 

  23. J. Alcaraz, L. Buscemi, M. Grabulosa, X. Trepat, B. Fabry, R. Farre, and D. Navajas, Microrheology of human lung epithelial cells measured by atomic force microscopy. Biophys. J. 84:2071 (2003).

    Google Scholar 

  24. C. Laforsch, W. Ngwa, W. Grill, and R. Tollrian, An acoustic microscopy technique reveals hidden morphological defenses in daphnia. Proc. Natl. Acad. Sci. (USA) 101:15911 (2005).

    ADS  Google Scholar 

  25. S. Chesla, P. Selvaraj and C. Zhu, Measuring two-dimensional receptor-ligand binding kinetics by micropipette. Biophys. J. 75:1553 (1998).

    Google Scholar 

  26. X. Zhang, A. Chen, D. Leon, H. Li. E. Noiri, V. Moy, and M. Goligorsky, Atomic force microscopy measurement of leukocyte-endothelial interaction. Am. J. Physiol. Heart Circ. Physiol. 286:H359 (2004).

    Google Scholar 

  27. J. Galle, G. Aust, G. Schaller, T. Beyer, and D. Drasdo, Single-cell based mathematical models to the spatio-temporal pattern formation in multi-cellular systems, Cytometry A, in press (2006).

  28. U. Braumann, J. Kuska, J. Einenkel, L. Horn, M. Loeffler, and M. Hoeckel, Three-dimensional reconstruction and quantification of cervical carcinoma invasion fronts from histological serial sections. IEEE Trans. Med. Imaging 24:1286 (2005).

    Google Scholar 

  29. D. Helbing, Traffic and related self-driven many particle systems. Rev. Mod. Phys. 73:1067 (2001).

    ADS  Google Scholar 

  30. D. Drasdo, R. Kree and J. McCaskill, Monte-carlo approach to tissue-cell populations. Phys. Rev. E 52:6635 (1995).

    ADS  Google Scholar 

  31. J. Moreira and A. Deutsch, Cellular automata models of tumour development—a critical review. Adv. Complex Syst. 5: 247 (2002).

    MATH  MathSciNet  Google Scholar 

  32. M. S. Alber, M. A. Kiskowski, J. A. Glazier, and Y. Jiang, On cellular automaton approaches to modeling biological cells. In Mathematical Systems Theory in Biology, Communication, and Finance, J. Rosenthal and D. S. Gilliam (eds.) (IMA 142, Springer-Verlag, New York, 2002), pp. 1–40.

    Google Scholar 

  33. D. Drasdo, On selected individual-based approaches to the dynamics of multicellular systems. In Multiscale modeling, J. L. W. Alt and M. Griebel (eds.) (Birkhäuser, 2003).

  34. T. Cickovski, C. Huang, R. Chaturvedi, T. Glimm, H. Hentschel, M. Alber, J. A. Glazier, S. A. Newman, and J. A. Izaguirre, A framework for three-dimensional simulation of morphogenesis. IEEE/ACM Trans. Comput. Biol. Bioinformatics 2:273 (2005).

    Google Scholar 

  35. R. Merks and J. Glazier, A cell-centered approach to developmental biology. Physica A 352:113 (2005).

    ADS  Google Scholar 

  36. A. Bru, J. Pastor, I. Fernaud, I. Bru, S. Melle, and C. Berenguer, Super-rough dynamics of tumor growth. Phys. Rev. Lett. 81:4008 (1998).

    ADS  Google Scholar 

  37. D. Balkovetz, Evidence that hepatocyte growth factor abbrogates contact inhibition of mitosis in madin-darby canine kidney cell monolayers. Life Sci. 64:1393 (1999).

    Google Scholar 

  38. L. Kunz-Schughart, Multicellular tumor spheroids: Intermediates between monolayer culture and in-vivo tumor. Cell Biol. Int. 23:157 (1999).

    Google Scholar 

  39. M. Locke, M. Heywood, S. Fawell, and I. Mackenzie, Retention of intrinsic stem cell hierarchies in carcinoma-derived cell lines. Cancer Res. 65:8944 (2005).

    Google Scholar 

  40. W. Mueller-Klieser, A review on cellular aggregates in cancer research. Cancer Res. Clin. Oncol. 113:101 (1987).

    Google Scholar 

  41. R. Sutherland, Cell and environment interactions in tumor microregions: The multicell spheroid model. Science 240:177 (1988).

    ADS  Google Scholar 

  42. M. Santini and G. Rainaldi, Three-dimensional spheroid model in tumor biology. Pathobiology 67:148 (1999).

    Google Scholar 

  43. S. Gilbert, Develoment (Sinauer Associates Inc., New York, 1997).

    Google Scholar 

  44. L. Wolpert, Principles of Development (Oxford Univ. Press, Oxford, 1998).

    Google Scholar 

  45. C. Booth and C. Potten, Gut instincts, thoughts on intestinal epithelial stem cells. Clin. Invest. 105:1493 (2000).

    Google Scholar 

  46. C. Potten and M. Loeffier, Stem cells: Attributes, cycles, spirals, pitfalls and uncertainties. Lessons for and from the crypt. Development 110:1001 (1990).

    Google Scholar 

  47. D. Drasdo and M. Löffler, Individual-based models on growth and folding in one-layered tissues: Intestinal crypts and blastulation. Nonl. Anal. 47:245 (2001).

    MATH  Google Scholar 

  48. C. Farrell, K. Rex, S. Kaufman, C. Dipalma, J. Chen, S. Scully and D. Layey, Effects of keratinocyte growth factor in the squamous epithelium of the upper aero-digestive tract of normal and irradiated mice. Int. J. Radiat. Biol. 75:609 (1999).

    Google Scholar 

  49. C. Klein, T. Blankenstein, O. Schmidt-Kittler, M. Petronio, B. Polzer, N. Stoecklein, and G. Riethmuller, Genetic heterogeneity of single disseminated tumour cells in minimal residual cancer. Lancet 360:683 (2002).

    Google Scholar 

  50. H. Eagle, Nutriention needs of mammalian cells in tissue culture. Science 122:43 (1955).

    Google Scholar 

  51. R. Ham, Clonal growth of mammalian cells in a chemically defined, synthetic medium. Proc. Natl. Acad. Sci. 53:288 (1965).

    ADS  Google Scholar 

  52. I. Hayashi and G. Sato, Replacement of serum by hormones permits growth of cells in defined medium. Nature 239:132 (1976).

    ADS  Google Scholar 

  53. G. Sato, A. Pardee and D. Sirbasku, Growth of Cells in Hormonally Defined Media (Cold Spring Harbour Laboratory, 1982).

  54. K. Burrige, Substrate adhesions in normal and transformed fibroblasts: Organization and regulation of cytoskeletal, membrane and extracellular matrix components at focal contacts. Cancer Review 4:18 (1986).

    Google Scholar 

  55. A. Bru, S. Albertos, J. Subiza, J. Garcia-Arsenio, and I. Bru, The universal dynamics of tumor growth. Biophys. J. 85: 2948 (2003).

    ADS  Google Scholar 

  56. L. Davidson, M. Koehl, R. Keller, and G. Oster, How do sea urchins invaginate? using bio-mechanics to distinguish between mechanisms of primary invagination. Development 121:2005 (1995).

    Google Scholar 

  57. M. Lekka, P. Laidler, D. Gil, J. Lekki, Z. Stachura, and A. Z. Hrynkiewicz, Elasticity of normal and cancerous human bladder cells studied by scanning force microscopy. European Biophys. J. 28:312 (1999).

    Google Scholar 

  58. J. Piper, R. Swerlick and C. Zhu, Determining force dependence of two-dimensional receptorligand binding affinity by centrifugation. Biophys. J. 74:492 (1998).

    Google Scholar 

  59. D. Beysens, G. Forgacs, and J. Glazier, Cell sorting is analogous to phase ordering in fluids. Proc. Natl. Acad. Sci. USA 97:9467 (2000).

    ADS  Google Scholar 

  60. M. Schienbein, K. Franke, and H. Gruler, Random walk and directed movement: Comparison between inert particles and self-organized molecular machines. Phys. Rev. E 49:5462 (1994).

    ADS  Google Scholar 

  61. R. A. Gatenby and P. K. Maini, Mathematical oncology: Cancer summed up. Nature 421:321 (2003).

    ADS  Google Scholar 

  62. J. Fidorra, T. Mielke, J. Booz, and L. Feinendegen, Cellular and nuclear volume of human cells during cell cycle. Radiat. Environ. Biophys. 19:205 (1981).

    Google Scholar 

  63. D. Landau, Theory of elasticity (Pergamon, 1975).

  64. R. Carpick, D. F. Ogletree, and M. Salmeron, A gerneral equation for fitting contact area and friction vs. load measurements. J. Colloid Interface Sci. 211:395 (1999).

    Google Scholar 

  65. Y.-S. C. et al.:Johnson-kendall-roberts theory applied to living cells. Phys. Rev. Lett. 280:312 (1999).

    Google Scholar 

  66. D. Drasdo and S. Hoehme, A single-cell based model to tumor growth in-vitro: Monolayers and spheroids. Phys. Biol. 2:133 (2005).

    ADS  Google Scholar 

  67. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, Equation of state calculations by fast computing machines. J. Chem. Phys. 21:1087 (1953).

    ADS  Google Scholar 

  68. D. Drasdo and G. Forgacs, Modelling the interplay of generic and genetic mechanisms in cleavage, blastulation and gastrulation. Dev. Dyn. 219:182 (2000).

    Google Scholar 

  69. D. Drasdo and S. Höhme, Individual-based approaches to birth and death in avascular tumors. Math. and Comp. Modelling 37:1163 (2003).

    MATH  Google Scholar 

  70. M. Allen and D. Tildersley, Computer Simulation of Liquids (Oxford Science Publ., Oxford, 1987).

    MATH  Google Scholar 

  71. D. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, 2000).

  72. M. Eden, A two-dimensional growth process. In Proceedings of the 4th. Berkeley Symposium on Mathematics and Probability, vol. IV, J. Neyman (ed.) (University of California Press, 1961), pp. 223–239.

  73. R. Weinberg, The biology of cancer (Garland Science, New York and Oxford, 2007).

    Google Scholar 

  74. J. Xin, Front propagation in heterogeneous media. SIAM Rev. 42:161 (2000).

    MathSciNet  Google Scholar 

  75. K. Swanson, E. Alvord, and J. Murray, quantitativ model for differential motility of gliomas in gey and white matter. Cell Prolif. 33:317 (2000).

    Google Scholar 

  76. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, The cell (Garland Science Publ., New York, 2002).

    Google Scholar 

  77. J. Galle, M. Loeffler, and D. Drasdo, Modelling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in-vitro. Biophys. J. 88: 62 (2005).

    Google Scholar 

  78. G. Schaller and M. Meyer-Hermann, Multicellular tumor spheroid in an off-lattice voronoi-delaunay cell model. Phys. Rev. E. 71:051910 (2005).

    ADS  MathSciNet  Google Scholar 

  79. J. Piper, R. Swerlick, and C. Zhu, Determining force dependence of two-dimensional receptorligand binding affinity by centrifugation. Biophys. J. 74:492 (1998).

    Article  Google Scholar 

  80. J. Dhont, An introduction to dynamics of colloids (Elsevier, Amsterdam, 1996).

    Google Scholar 

  81. L. Li, J. Backer, A. Wong, E. Schwanke, B. Stewart, and M. Pasdar, Bcl-2 expression decreases cadherin-mediated cell-cell adhesion. J. Cell Sci. 116:3687 (2003).

    Google Scholar 

  82. M. Warchol, Cell density and n-cadherin interaction regulates cell proliferation in the sensory epithelia of the inner ear. J. Neurosci. 22:2607 (2002).

    Google Scholar 

  83. P. Klekotka, S. Santoro, A. Ho, S. Dowdy, and M. Zutter, Mammary epithelial cell-cycle progression via the αβ-integrin. Am. J. Path. 159:983 (2001).

    Google Scholar 

  84. L. Junqueira and J. Carneiro, Basic histology (McGraw Hill, 2005).

  85. D. Stupack and D. Cheresh, Get a ligand, get a life: Integrins, signaling and cell survival. J. Cell Sci. 115:3729 (2002).

    Google Scholar 

  86. K. Orford. C. Orford, and S. W. Byers, Exogenous expression of β-catenin regulates contact inhibition, anchorage-independent growth, anoikis and radiation-induced cell cycle arrest. J. Cell Biol. 146:855 (1999).

    Google Scholar 

  87. Z. Yan, M. Chen, M. Perucho, and E. Friedman, Oncogenic ki-ras but not oncogenic ha-ras blocks integrin? 1-chain maturation in colon epithelial cells. J. Biol. Chem. 272:2607 (1997).

    Google Scholar 

  88. P. Lu, Q. Lu, A. Rughetti, and J. Taylor-Papadimitriou, bcl-2 overexpression inhibits cell death and promotes the morphogenesis, but not tumorigenesis of human mammary epithelial cells. J. Cell Biol. 129:1363 (1995).

    Google Scholar 

  89. R. Bates, N. Edwards, and J. Yates, Spheroids and cell survival. Crit. Rev. Oncol./Hematol. 36:61 (2000).

    Google Scholar 

  90. M. Santini, G. Rainaldi, and P. Indovina, Apoptosis, cell adhesion and the extracellular matrix in the three-dimensional growth of multicellular tumor spheroids. Crit. Rev. Oncology/Hematology 36:75 (2000).

    Google Scholar 

  91. C. Chen, M. Mrksich, S. Huang, G. Whitesides, and D. Ingber, Geometric control of cell life and death. Science 276: 1425 (1997).

    Google Scholar 

  92. N. Gloushankova, N. Alieva, M. Krendel, E. Bonder, H. Feder, J. Vasiliev, and I. Gelfand, Cell-cell contact changes the dynamics of lamellar activity in nontransformed epitheliocytes but not in their ras-transformed descendants. Proc. Natl. Acad. Sci. USA 94:879 (1997).

    ADS  Google Scholar 

  93. A. Barabasi and H. Stanley, Fractal concepts in surface growth (Cambridge University Press, Cambridge, 1995).

    MATH  Google Scholar 

  94. E. Moro, Internal fluctuations effects on fisher waves. Phys. Rev. Lett. 87:238303 (2001).

    ADS  Google Scholar 

  95. T. Halpin-Healy and Y. Zhang, Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Phys. Rep. 254:215 (1995).

    ADS  Google Scholar 

  96. M. Block, E. Schoell, and D. Drasdo, Classifying the expansion kinetics and critical surface dynamics of growing cell populations, Cond. mat. physics/0610146 (2006).

  97. D. Drasdo, Coarse graining in simulated cell populations. Adv. Complex Syst. 8:319 (2005).

    MATH  MathSciNet  Google Scholar 

  98. J. Ramasco, J. Lopez, and M. Rodriguez, Generic dynamic scaling in kinetic roughening. Phys. Rev. Lett. 84:2199 (2000).

    ADS  Google Scholar 

  99. F. Family and T. Vicsek, Scaling of the active zone in the eden process on percolation networks and the ballistic deposition model. J. Phys. A: Math. Gen. 18:L75 (1985).

    ADS  Google Scholar 

  100. J. Buceta and J. Galeano, Comments on the article—the universal dynamics of tumor growth. Biophys. J. 88:3734 (2005).

    Google Scholar 

  101. A. Wong and B. Gumbiner, Adhesion-independent mechanism for suppression of tumor cell invasion by e-cadherin. J. Cell Biol. 161:1191 (2003).

    Google Scholar 

  102. P. Friedl, Prespecification and plasticity: Shifting mechanisms of cell migration. Curr. Opin. Cell. Biol. 16:14 (2004).

    Google Scholar 

  103. J. Freyer and R. Sutherland, A reduction in the in situ rates of oxygen and glucose consumption of cells in emt6/ro spheroids during growthregulation of growth. J. Cell. Physiol. 124:516 (1985).

    Google Scholar 

  104. J. Freyer and R. Sutherland, Regulation of growth saturation and development of necrosis in emt6/ro multicellular spheroids by the glucose and oxygene supply. Cancer Res. 46:3504 (1986).

    Google Scholar 

  105. J. Casciari, S. Sotirchos, and R. Sutherland, Glucose diffusivity in multicellular tumor spheroids. Cancer Res. 48:3905 (1988).

    Google Scholar 

  106. J. Casciari, S. Sotirchos, and R. Sutherland, Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration and extracellular ph. J. Cell. Physiol. 151:386 (1992).

    Google Scholar 

  107. Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell, and J. Freyer, A multiscale model for avascular tumor growth. Biophys. J. 89:3884 (2005).

    Google Scholar 

  108. E. Stott, N. Britton, J. Glazier, and M. Zajac, Stochastic simulation of benign avascular tumor growth using the potts model. Math. Comput. Modelling 30:183 (1999).

    Google Scholar 

  109. S. Dormann and A. Deutsch, Modeling of self-organized avascular tumor growth with a hybrid cellular automaton. Silico Biol. 2:0035 (2002).

    Google Scholar 

  110. J. Folkman and M. Hochberg, Self-regulation of growth in three dimensions. J. Exp. Med. 138:745 (1973).

    Google Scholar 

  111. J. Glazier and F. Graner, Simulation of the differential adhesion driven rearrangement of biological cells. Phys. Rev. E 47:2128 (1993).

    ADS  Google Scholar 

  112. T. Newman, Modeling multi-cellular systems using sub-cellular elements. Math. Biosciences Eng. 2:613 (2005).

    MATH  Google Scholar 

  113. N. Wright and M. Alison, The Biology of Epithelial Cell Population (Clarendon Press, Oxford, 1984).

    Google Scholar 

  114. D. Drasdo, Buckling instabilities in one-layered growing tissues. Phys. Rev. Lett. 84:4244 (2000).

    ADS  Google Scholar 

  115. S. Hörstadius, The mechanics of sea urchin development, studied by operative methods. Biol. Rev. 14:132 (1939).

    Google Scholar 

  116. K. Dan, Cytoembryology of echinoderms and amphibia. Int. Rev. Cytol 9:321 (1960).

    Article  Google Scholar 

  117. L. Wolpert and E. Mercer, An electron microscope study of the development of the blastula of the sea urchin embryo and its radial polarity. Exp. Cell Res. 30:280 (1963).

    Google Scholar 

  118. M. Leptin and B. Grunewald, Cell shape changes during gastrulation in drosophila. Development 110:73 (1990).

    Google Scholar 

  119. J. Gere and S. Timoshenko, Mechanics of Materials, 4th edn. (PWS-Publishing Co., Boston, 1997).

    Google Scholar 

  120. J. Dallon and H. Othmer, How cellular movement determines the collective force generated by the dictyostelium discoideum slug. J. theor. Biol. 231:203 (2004).

    MathSciNet  Google Scholar 

  121. E. Palsson and H. Othmer, A model for individual and collective cell movement in dictyostelium discoideum. Proc. Natl. Acad. Sci. USA 12:10448 (2000).

    Google Scholar 

  122. Z. Kam, J. Minden, D. Agard, J. Sedat, and M. Leptin, Drosophila gastrulation: Analysis of cell shape changes in living embryos by three-dimensional fluorescence mircroscopy. Development 112:365 (1991).

    Google Scholar 

  123. A. Cairnie and B. Millen, Fission of crypts in the small intestine of the irradiated mouse. Cell Tissue Kinet. 8:89 (1975).

    Google Scholar 

  124. K. Araki, T. Ogata, M. Kobayashi, and R. Yatani, A morphological study on the histogenesis of human colorectal hyperplastic crypts. Gastroenterology 109:1468 (1995)

    Google Scholar 

  125. A. Bru, S. Albertos, J. L. Garcia-Asenjo, and I. Bru, Pinning of tumoral growth by enhancement of the immune response. Phys. Rev. Lett. 92:238101 (2004).

    ADS  Google Scholar 

  126. Y. Boucher, L. Baxter, and R. Jain, Interstitial pressure gradients in tissue-isolated and subcutaneous tumors: Implications for therapy. Cancer Res. 50:4478 (1990).

    Google Scholar 

  127. Y. Boucher, J. Salehi, B. Witwer, and R. Jain, Interstitial fluid pressure in intracranial tumors in patients and in rodents. Br. J. Cancer 75:829 (1997).

    Google Scholar 

  128. E. Filipski, F. Delaunay, V. King, B. C. MW Wu, A. Grechez-Cassiau, C. Guettier, M. Hastings, and F. Levi, Effects of chronic jet lag on tumor progression in mice. Cancer Res. 64:7879 (2004).

    Google Scholar 

  129. A. Goriely and M. B. Amar, Differential growth and instability in elastic shells. Phys. Rev. Lett. 94:198103 (2005).

    ADS  Google Scholar 

  130. J. Dunphy, Wound healing (MedCom-Press, New York, 1978).

    Google Scholar 

  131. P. Hogeweg, Evolving mechanisms of morphogenesis: On the interplay between differential adhesion and cell differentiation. J. Theor. Biol. 203:317 (2000).

    Google Scholar 

  132. D. Drasdo and M. Kruspe, Emergence of cell migration and aggregation strategies in a simulated evolutionary process. Adv. Complex Syst. 8 (2005).

  133. T. Alarcon. H. Byrne, and P. Maini, A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells. J. Theor. Biol. 229:395 (2004).

    MathSciNet  Google Scholar 

  134. K. Frame and W. Hu, A model for density-dependent growth of anchorage-dependent mammalian cells. Biotechnol. Bioengineering 32:1062 (1988).

    Article  Google Scholar 

  135. K. Hawboldt, N. Kalogerakis, and L. Behie, A cellular automaton model for micro-carrier cultures. Biotechnol. Bioengineering 43:90 (1993).

    Google Scholar 

  136. L. Arakelyan, Y. Merbl, and Z. Agur, Vessel maturation effects on tumour growth: Validation of a computer model in implanted human ovarian carcinoma spheroids. Eur. J. Cancer 41:159 (2005).

    Google Scholar 

  137. C. Basdevant, J. Clairambault, and F. Levi, Optimisation of time-scheduled regimen for anti-cancer drug infusion. Math. Modelling Numerical Anal. 39:1069 (2005).

    MATH  MathSciNet  Google Scholar 

  138. B. Ribba, K. Marron, Z. Agur, T. A. T, and P. Maini, A mathematical model of doxorubicin treatment efficacy for non-hodgkin's lymphoma: Investigation of the current protocol through theoretical modelling results. Bull. Math. Biol. 67:79 (2005).

    MathSciNet  Google Scholar 

  139. B. Ribba, T. Colin, and S. Schnell, A multiscale mathematical model of cancer and its use in analyzing irradiation therapies. Theor. Biol. Med. Model 3:7 (2006).

    Google Scholar 

  140. N. Grabe and K. Neuber, A multicellular systems biology model predicts epidermal morphology, kinetics and ca+-flow. Bioinformatics 21:3541 (2005).

    Google Scholar 

  141. S. Hoehme, J. Hengstler, M. Brulport, M. Schaefer, A. Bauer, R. Gebhardt, and D. Drasdo, Mathematical modelling of liver regeneration after intoxification with ccl4, Chemico-Biological Interactions, in revision (2007).

  142. G. Michalopoulos and M. DeFrances, Liver regeneration. Science 276:60 (1997).

    Google Scholar 

  143. R. Goldstein and S. Langer, Nonlinear dynamics of stiff polymers. Phys. Rev. Lett. 75:1094 (1995).

    ADS  Google Scholar 

  144. M. Doi and S. F. Edwards, The theory of polymer dynamics (Oxford University Press, 1986).

  145. U. Seifert, Adhesion of vesicles in two dimensions. Phys. Rev. A 43:6803 (1991).

    ADS  MathSciNet  Google Scholar 

  146. D. Kessler, J. Koplik, and H. Levine, Pattern selection in fingered growth, phenomena. Adv. Phys. 37:255 (1988).

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute and Center for Systems Biology, University of Warwick, Coventry, CV4 7AL, UK

    Dirk Drasdo

  2. French National Institute for Reasearch in Computer Science and Control (INRIA), Rocquencourt, B.P. 105, 78153, Le Chesnay Cedex, France

    Dirk Drasdo

  3. Interdisciplinary Centre for Bioinformatics, University of Leipzig, Härtelstr. 14–16, D-04107, Leipzig, Germany

    Dirk Drasdo & Stefan Hoehme

  4. Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22–26, D-0 4103, Leipzig, Germany

    Dirk Drasdo

  5. Institute for Theoretical Physics, Technical University of Berlin, D-10623, Berlin, Germany

    Michael Block

Authors
  1. Dirk Drasdo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Hoehme
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Block
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dirk Drasdo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Drasdo, D., Hoehme, S. & Block, M. On the Role of Physics in the Growth and Pattern Formation of Multi-Cellular Systems: What can we Learn from Individual-Cell Based Models?. J Stat Phys 128, 287–345 (2007). https://doi.org/10.1007/s10955-007-9289-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-007-9289-x

Keywords

Search

Navigation