# On Cellular Automaton Approaches to Modeling Biological Cells

## Abstract

We discuss two different types of Cellular Automata (CA): lattice-gas-based cellular automata (LGCA) and the cellular Potts model (CPM), and describe their applications in biological modeling.

LGCA were originally developed for modeling ideal gases and fluids. We describe several extensions of the classical LGCA model to self-driven biological cells. In particular, we review recent models for rippling in myxobacteria, cell aggregation, swarming, and limb bud formation. These LGCA-based models show the versatility of CA in modeling and their utility in addressing basic biological questions.

The CPM is a more sophisticated CA, which describes individual cells as extended objects of variable shape. We review various extensions to the original Potts model and describe their application to morphogenesis; the development of a complex spatial structure by a collection of cells. We focus on three phenomena: cell sorting in aggregates of embryonic chicken cells, morphological development of the slime mold *Dictyostelium discoideum* and avascular tumor growth. These models include intercellular and extracellular interactions, as well as cell growth and death.

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### References

- [1]J. Adam and N. Bellomo,
*A survey of models for tumor-immune system dynamics*, Birkhauser, Boston, 1997.MATHGoogle Scholar - [2]A. Adamatzky and O. Holland,
*Phenomenology of excitation in 2-D cellular automata and swarm systems*, Chaos Solitons Fractals,**9**(1998), pp. 1233–1265.MathSciNetMATHGoogle Scholar - [3]M. Alber and M. Kiskowski,
*On aggregation in CA models in biology*, J. Phys. A: Math. Gen.,**34**(2001), pp. 10707–10714.MathSciNetMATHGoogle Scholar - [4]M. Alber, M. Kiskowski, and Y. Jiang,
*A model of rippling and aggregation in Myxobacteria*, 2002 preprint.Google Scholar - [5]B. Alberts, M. Raff, J. Watson, K. Roberts, D. Bray, and J. Lewis,
*Molecular biology of the cell, 3rd edition*. Garland Publishing, NY, 1994.Google Scholar - [6]J. Ashkin and E. Teller,
*Statistics of two-dimensional lattices with four components*, Phys. Rev.,**64**(1943), pp. 178–184.Google Scholar - [7]E. Ben-Jacob, I. Cohen, A. Czirk, T. Vicsek, and D.L. Gutnick,
*Chemo-modulation of cellular movement, collective formation of vortices by swarming bacteria, and colonial development*, Physica A,**238**(1997), pp. 181–197.Google Scholar - [8]E. Ben-Jacob and H. Levine,
*The artistry of microorganisms*, Scientific American,**279**(1998), pp. 82–87.Google Scholar - [9]E. Ben-Jacob, I. Cohen, and H. Levine,
*Cooperative self-organization of microorganisms*, Advances in Physics,**49**(2000), pp. 395–554.Google Scholar - [10]L. Besseau and M. Giraud-Guille,
*Stabilization of ßuid cholesteric phases of collagen to ordered gelated matrices*, J. Mol. Bio.,**251**(1995), pp. 137–145.Google Scholar - [11]D. Beysens, G. Forgacs, and J.A. Glazier,
*Cell sorting is analogous to phase ordering in fluids*, Proc. Natl. Acad. Sci. USA**97**(2000) pp. 9467–9471.Google Scholar - [12]H. Bode, K. Flick, and G. Smith,
*Regulation of interstitial cell-differentiation in Hydra attenuata. I. Homeostatic control of interstitial cell-population size*, J. Cell Sci.,**20**(1976), pp. 29–46.Google Scholar - [13]E. Bonabeau, M. Dorigo, and G. Theraulaz,
*Swarm intelligence: From natural to artificial systems*, Oxford University Press, NY, 1999.MATHGoogle Scholar - [14]J. Boon., D. Dab, R. Kapral, and A. Lawniczak,
*Lattice gas automata for relative systems*, Physics Reports,**273**(1996), pp. 55–147.MathSciNetGoogle Scholar - [15]U. Börner, A. Deutsch, H. Reichenbach, and M. Bar,
*Rippling patterns in aggregates of myxobacteria arise from cell-cell collisions*, 2002 preprint.Google Scholar - [16]H. Bussemaker, A. Deutsch, and E. Geigant,
*Mean-field analysis of a dynamical phase transition in a cellular automaton model for collective motion*, Phys. Rev. Lett.,**78**(1997), pp. 5018–5027.Google Scholar - [17]M. Caterina and P. Devreotes,
*Molecular insights into eukaryotic Chemotaxis*, FASEB J.,**5**(1991), pp. 3078–3085.Google Scholar - [18]S. Chen, S.P. Dawson, G.D. Doolen, D.R. Janecky, and A. Lawniczak,
*Lattice methods and their applications to reacting systems*. Computers & Chemical Engineering,**19**(1995), pp. 617–646.Google Scholar - [19]B. Chopard and M. Droz,
*Cellular automata modeling of physical systems*, Cambridge University Press, NY, 1998.MATHGoogle Scholar - [20]L Cohen, LG. Ron, and E. Ben-Jacob,
*From branching to nebula patterning during colonial development of the Paenibacillus alvei bacteria*, Physica A,**286**(2000), pp. 321–336.Google Scholar - [21]J. Cook,
*Waves of alignment in populations of interacting, oriented individuals*. Forma,**10**(1995), pp. 171–203.MathSciNetMATHGoogle Scholar - [22]J. Cook, A. Deutsch, and A. Mogilner,
*Models for spatio-angular self-organization in cell biology*, in W. Alt, A. Deutsch and G. Dunn (Eds.)*Dynamics of cell and tissue motion*, Birkhuser, Basel, Switzerland, 1997, pp. 173–182.Google Scholar - [23]M. Cross and P. Hohenberg,
*Pattern-formation outside of equilibrium*, Rev. Mod. Phys.,**65**(1993), pp. 851–1112.Google Scholar - [24]A. Czirok, A. L. Barabasi, and T. Vicsek,
*Collective motion of organisms in three dimensions*, Phys. Rev. Lett.,**82**(1999), pp. 209–212.Google Scholar - [25]J. Dallon and J. Sherratt,
*A mathematical model for spatially varying extra cellular matrix alignment*, SIAM J. Appl. Math.,**61**(2000), pp. 506–527.MathSciNetMATHGoogle Scholar - [26]L.A. Davidson, M.A.R. Koehl, R. Keller, and G.F. Oster,
*How do sea-urchins invaginate — Using biomechanics to distinguish between mechanisms of primary invagination*, Development,**121**(1995), pp. 2005–2018.Google Scholar - [27]A.M. Delprato, A. Samadani, A. Kudrolli, and L.S. Tsimring,
*Swarming ring patterns in bacterial colonies exposed to ultraviolet radiation*, Phys. Rev. Lett.,**87**(2001), 158102.Google Scholar - [28]A. Deutsch,
*Towards analyzing complex swarming patterns in biological systems with the help of lattice-gas automaton model*, J. Biol. Syst.,**3**(1995), pp. 947–955.Google Scholar - [29]A. Deutsch,
*Orientation-induced pattern formation: Swarm dynamics in a lattice-gas automaton model*. Int. J. Bifurc. Chaos,**6**(1996), pp. 1735–1752.MATHGoogle Scholar - [30]A. Deutsch,
*Principles of morphogenetic motion: swarming and aggregation viewed as self-organization phenomena*, J. Biosc.,**24**(1999), pp. 115–120.Google Scholar - [31]A. Deutsch,
*Probabilistic lattice models of collective motion and aggregation: from individual to collective dynamics*, Mathematical Biosciences,**156**(1999), pp. 255–269.MathSciNetMATHGoogle Scholar - [32]A. Deutsch,
*A new mechanism of aggregation in a lattice-gas cellular automaton model*. Mathematical and Computer Modeling,**31**(2000), pp. 35–40.MathSciNetMATHGoogle Scholar - [33]A. Deutsch and S. Dormann,
*Cellular automata and biological pattern formation modeling*, 2002 preprint.Google Scholar - [34]S. Dormann,
*Pattern formation in cellular automation models, Dissertation, Angewandte Systemwissenschaft FB Mathematik/Informatik*, Universität Osnabrück, Austria, 2000.Google Scholar - [35]S. Dormann, A. Deutsch, and A. Lawniczak,
*Fourier analysis of Turing-like pattern formation in cellular automaton models*. Future Computer Generation Systems,**17**(2001), pp. 901–909.MATHGoogle Scholar - [36]S. Dormann and A. Deutsch,
*Modeling of self-organized avascular tumor growth with a hybrid cellular automaton*, Silico Biology,**2**(2002), 0035.Google Scholar - [37]D. Drasdo and G. Forgacs,
*Modeling the interplay of generic and genetic mechanisms in cleavage, blastulation, and gastrulation*, Developmental Dynamics,**219**(2000), pp. 182–191.Google Scholar - [38]M. Dworkin and D. Kaiser,
*Myxobacteria II*, American Society for Microbiology, Washington, DC, 1993.Google Scholar - [39]M. Dworkin
*Recent advances in the social and developmental biology of the myxobacteria*, Microbiol. Rev.,**60**(1996), pp. 70–102.Google Scholar - [40]M. Eden, Vol. 4:
*Contributions to biology and problems of medicine*, in J. Neyman (Ed.),*Proceedings of the Fourth Berkeley Symposium in Mathematics, Statistics and Probability*, University of California Press, Berkeley, 1961, pp. 223–239.Google Scholar - [41]R. Engelhardt,
*Modeling pattern formation in reaction diffusion systems*. Master’s Thesis, Dept. of Chemistry, University of Copenhagen, Denmark, 1994.Google Scholar - [42]G. Ermentrout and L. Edelstein-Keshet,
*Cellular automata approach in biological modeling*, J. Theor. Biol.,**160**(1993), pp. 97–133.Google Scholar - [43]S.E. Esipov and J.A. Shapiro,
*Kinetic model of Proteus mirabilis swarm colony development*, J. Math. Biol.,**36**(1998), pp. 249–268.MathSciNetMATHGoogle Scholar - [44]M. Fontes and D. Kaiser,
*Myxococcus cells respond to elastic forces in their substrate*, Proc. Natl. Acad. Sci. USA,**96**(1999), pp. 8052–8057.Google Scholar - [45]G. Forgacs, R. Foty, Y. Shafrir, and M. Steinberg,
*Viscoelastic properties of living embryonic tissues: a quantitative study*, Biophys. J.,**74**(1998), pp. 2227–2234.Google Scholar - [46]R. Foty, G. Forgacs, C. Pfleger, and M. Steinberg,
*Liquid properties of embryonic tissues: measurements of interfacial tensions*, Phys. Rev. Lett.,**72**(1994), pp. 2298–2300.Google Scholar - [47]R. Foty, C. Pfleger, G. Forgacs, and M. Steinberg,
*Surface tensions of embryonic tissues predict their mutual envelopment behavior*, Development,**122**(1996), pp. 1611–1620.Google Scholar - [48]J. Freyer and R. Sutherland,
*Selective dissociation and characterization of cells from different regions of multicell spheroids during growth*. Cancer Research,**40**(1980), pp. 3956–3965.Google Scholar - [49]J. Freyer and R. Sutherland,
*Regulation of growth saturation and development of necrosis in EMT6/RO multicellular spheroids induced by the glucose and oxygen supply*. Cancer Research,**46**(1986), pp. 3504–3512.Google Scholar - [50]M. Gardner,
*The fantastic combinations of John Conway’s new solitaire game ’life’*. Scientific American,**223**(1970), pp. 120–123.Google Scholar - [51]F. Gianocotti,
*Integrin-signaling: specificity and control of cell survival and cell cycle progression*, Curr. Opin. Cell Biol,**9**(1997), pp. 691–700.Google Scholar - [52]J.A. Glazier,
*Dynamics of cellular patterns*, Ph.D. Thesis, The University of Chicago, USA, 1989.Google Scholar - [53]J.A. Glazier and F. Graner,
*Simulation of the differential adhesion driven rearrangement of biological cells*, Phys. Rev. E,**47**(1993), pp. 2128–2154.Google Scholar - [54]D. Godt and U. Tepass,
*Drosophila oocyte localization is mediated by differential cadherin-based adhesion*. Nature,**395**(1998), pp. 387–391.Google Scholar - [55]I. Golding, Y. Kozlovsky, I. Cohen, and E. Ben-Jacob,
*Studies of bacterial branching growth using reaction-diffusion models for colonial development*, Physica A,**260**(1998), pp. 510–554.Google Scholar - [56]A. Gonzalez-Reyes and D. St. Johnston,
*Patterning of the follicle cell epithe lium along the anterior-posterior axis during Drosophila oogenesis*. Development,**125**(1998), pp. 2837–2846.Google Scholar - [57]F. Graner and J.A. Glazier,
*Simulation of biological cell sorting using a two-dimensional Extended Potts Model*, Phys. Rev. Lett.,**69**(1992), pp. 2013–2016.Google Scholar - [58]J. Hardy, O. de Pazzis, and Y. Pomeau,
*Molecular dynamics of a classical lattice gas: Transport properties and time correlation functions*, Phys. Rev. A,**13**(1976), pp. 1949–1961.Google Scholar - [59]P. Hogeweg,
*Evolving mechanisms of morphogenesis: On the interplay between differential adhesion and cell differentiation*, J. Theor. Biol.,**203**(2000), pp. 317–333.Google Scholar - [60]P. Hogeweg,
*Shapes in the shadow: Evolutionary dynamics of morphogenesis*, Artificial Life,**6**(2000), pp. 611–648.Google Scholar - [61]E. Holm, J.A. Glazier, D. Srolovitz, and G. Crest,
*Effects of lattice anisotropy and temperature on domain growth in the 2-dimensional Potts model*, Phys. Rev. A,**43**(1991), pp. 2262–2268.Google Scholar - [62]A. Howe, A. Aplin, S. Alahari, and R. Juliano,
*Integrin signaling and cell growth control*, Curr. Opin. Cell Biol.,**10**(1998), pp. 220–231.Google Scholar - [63]O. Igoshin, A. Mogilner, D. Kaiser, and G. Oster,
*Pattern formation and traveling waves in myxobacteria: Theory and modeling*, Proc. Natl. Acad. Sci. USA,**98**(2001), pp. 14913–14918.Google Scholar - [64]L. Jelsbak and L. Sogaard-Andersen,
*The cell surface-associated intercellular C-signal induces behavioral changes in individual Myxococcus xanthus cells during fruiting body morphogenesis*, Devel. Bio,**96**(1998), pp. 5031–5036.Google Scholar - [65]L. Jelsbak and L. Sogaard-Andersen,
*Pattern formation: Fruiting body morphogenesis in Myxococcus xanthus*. Current Opinion in Microbiology,**3**(2000), pp. 637–642.Google Scholar - [66]Y. Jiang and J.A. Glazier,
*Extended large-Q Potts model simulation of foam drainage*, Philos. Mag. Lett.,**74**(1996), pp. 119–128.Google Scholar - [67]Y. Jiang,
*Cellular pattern formation*, Ph.D. Thesis, University of Notre Dame, USA, 1998.Google Scholar - [68]Y. Jiang, H. Levine, and J.A. Glazier,
*Possible cooperation of differential adhesion and Chemotaxis**in mound formation of Dictyostelium*, Biophys. J.,**75**(1998), pp. 2615–2625.Google Scholar - [69]B. Julien, D. Kaiser, and A. Garza,
*Spatial control of cell differentiation in Myxococcus xanthus*, Proc. Natl. Acad. Sci. USA,**97**(2000), pp. 9098–9103.Google Scholar - [70]L.P. Kadanoff, G.R. McNamara, and G. Zanetti,
*From automata to fluid-flow-Comparisons óf**simulation and theory*, Phys. Rev. A,**40**(1989), pp. 4527–4541.Google Scholar - [71]D. Kaiser,
*How and why myxobacteria talk to each other*, Current Opinion in Microbiology,**1**(1998), pp. 663–668.MathSciNetGoogle Scholar - [72]D. Kaiser,
*Intercellular signaling for multicellular morphogenesis*, Society for General Microbiology Symposium 57, Cambridge University Press, Society for General Microbiology Ltd., UK, 1999.Google Scholar - [73]A. Kansal, S. Torquato, E. Chiocca, and T. Deisboeck,
*Emergence of a sub-population in a computational model of tumor growth*, J. Theor. Biol.,**207**(2000), pp. 431–441.Google Scholar - [74]N. Kataoka, K. Saito, and Y. Sawada,
*NMR microimaging of the cell sorting process*, Phys. Rev. Lett.,**82**(1999), pp. 1075–1078.Google Scholar - [75]E.F. Keller and L.A. Segal,
*Initiation of slime mold aggregation viewed as an instability*, J. Theor. Bio.,**26**(1970), pp. 399–415.Google Scholar - [76]P. Kiberstis and J. Marx,
*Frontiers in cancer research*, Science,**278**(1977), pp. 1035–1035.Google Scholar - [77]S. Kim and D. Kaiser,
*Cell alignment in differentiation of Myxococcus xanthus*, Science,**249**(1990), pp. 926–928.Google Scholar - [78]S. Kim and D. Kaiser,
*C-factor has distinct aggregation and sporulation thresholds during Myxococcus development*, J. Bacteriol.,**173**(1991), pp. 1722–1728.Google Scholar - [79]M. Kiskowski, M. Alber, G. Thomas, J. Glazier, N. Bronstein, and S. Newman,
*Interaction between reaction-diffusion process and cell-matrix adhesion in a cellular automata model for chondrogenic pattern formation: a prototype study for developmental modeling*, 2002, in preparation.Google Scholar - [80]J. Kuner and D. Kaiser,
*Fruiting body morphogenesis in submerged cultures of Myxococcus xanthus*, J. Bacteriol.,**151**(1982), pp. 458–46LGoogle Scholar - [81]S. Kyriacou, C. Davatzikos, S. Zinreich, and R. Bryan,
*Nonlinear elastic registration of brain images with tumor pathology using a biomechanical model*, IEEE Transactions On Medical Imaging,**18**(1999), pp. 580–592.Google Scholar - [82]J. Landry, J. Freyer, and R. Sutherland,
*A model for the growth of multicellular spheroids*, Cell Tiss. Kinet.,**15**(1982), pp. 585–594.Google Scholar - [83]C. Leonard, H. Fuld, D. Frenz, S. Downie, Massagué, and S. Newman,
*Role of transforming growth factor-β**in chondrogenic pattern formation in the embryonic limb: Stimulation of mesenchymal condensation and fibronectin**gene expression by exogenous TGF-β-like**activity*, Devel. Bio.,**145**(1991), pp. 99–109.Google Scholar - [84]H. Levine, I. Aranson, L. Tsimring, and T. Truong,
*Positive genetic feedback governs CAMP spiral wave formation in Dictyostelium*, Proc. Natl. Acad. Sci. USA, 93 (1996), pp. 6382–6386.Google Scholar - [85]A. Nicol, W.J. Rappel, H. Levine, and W.F. Loomis,
*Cell-sorting in aggregates of Dictyostelium discoideum*, J. Cell. Sci.,**112**(1999), pp. 3923–3929.Google Scholar - [86]H. Levine, W-J. Rappel, and I. Cohen,
*Self-organization in systems of self-propelled particles*, Phys. Rev. E,**63**(2001), 017101.Google Scholar - [87]S. Li, B. Lee and L. Shimkets,
*csgA expression entrains Myxococcus Xanthus development*. Genes Development,**6**(1992), pp. 401–410.Google Scholar - [88]W. Loomis,
*Lateral inhibition and pattern formation in Dictyostelium*, Curr. Top. Dev. Biol.,**28**(1995), pp. 1–46.Google Scholar - [89]F. Lutscher,
*Modeling alignment and movement of animals and cells*, J. Math. Biol.,**DOI**: 10.1007/s002850200146, 2002.Google Scholar - [90]F. Lutscher and A. Stevens,
*Emerging patterns in a hyperbolic model for locally interacting cell systems*. Journal of Nonlinear Sciences, 2002 preprint.Google Scholar - [91]P. Maini,
*Mathematical models in morphogenesis*, pp. 151–189. In V. Capasso and O. Dieckmann (Eds.),*Mathematics Inspired Biology*, Springer, Berlin, 1999.Google Scholar - [92]A. Maree, A. Panfilov, and P. Hogeweg,
*Migration and thermotaxis of Dictyostelium discoideum slugs, a model study*, J. Theor. Biol.,**199**(1999), pp. 297–309.Google Scholar - [93]A. Maree,
*From pattern formation to morphogenesis: Multicellular coordination in Dictyostelium discoideum*, Ph.D. Thesis., Utrecht University, the Netherlands, 2000.Google Scholar - [94]A. Maree and P. Hogeweg,
*How amoeboids self-organize into a fruiting body: Multicellular coordination in Dictyostelium discoideum*, Proc. Natl. Acad. Sci. USA,**98**(2001), pp. 3879–3883.Google Scholar - [95]M. Marusic, Z. Bajzer, J. Freyer, and S. Vuk-Pavlovic,
*Modeling autostimulation of growth in multicellular tumor spheroids*. Int. J. Biomed. Comput.,**29**(1991), pp. 149–158.Google Scholar - [96]M. Marusic, Z. Bajzer, J. Freyer, and S. Vuk-Pavlovic,
*Analysis of growth of multicellular tumor spheroids by mathematical models*. Cell Prolif.,**27**(1994), pp. 73–94.Google Scholar - [97]J. Marrs and W. Nelson,
*Cadherin cell adhesion molecules in differentiation and embryogenesis*. Int. Rev. Cytol.,**165**(1996), pp. 159–205.Google Scholar - [98]N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller,
*Combinatorial minimization*, J. Chem. Phys.,**21**(1953), pp. 1087–1092.Google Scholar - [99]A. Mogilner and L. Edelstein-Keshet,
*Spatio-angular order in populations of self-aligning objects: formation of oriented patches*, Physica D,**89**(1996), pp. 346–367.MathSciNetMATHGoogle Scholar - [100]A. Mogilner, L. Edelstein-Keshet, and G. Ermentrout,
*Selecting a common direction*.*II. Peak-like solutions representing total alignment of cell clusters*, J. Math. Biol.,**34**(1996), pp. 811–842.MathSciNetMATHGoogle Scholar - [101]A. Mogilner and L. Edelstein-Keshet,
*A non-local model for a swarm*, J. Math. Biol.,**38**(1999), pp. 534–570.MathSciNetMATHGoogle Scholar - [102]J. Mombach, J.A. Glazier, R. Raphael, and M. Zajac,
*Quantitative comparison between differential adhesion models and cell sorting in the presence and absence of ßuctuations*, Phys. Rev. Lett.,**75**(1995), pp. 2244–2247.Google Scholar - [103]J. Mombach and J.A. Glazier,
*Single cell motion in aggregates of embryonic cells*, Phys. Rev. Lett.,**76**(1996), pp. 3032–3035.Google Scholar - [104]F. Monier-Gavelle and J. Duband,
*Cross talk between adhesion molecules: Control of N-cadherin activity by intracellular signals elicited by beta 1 and beta 3 integrins in migrating neural crest cells*, J. Cell. Biol.,**137**(1997), pp. 1663–1681.Google Scholar - [105]
- [106]V. Nanjundiah,
*Chemotaxis*,*signal relaying and aggregation morphology*, J. Theor. Bio.,**42**(1973), pp. 63–105.Google Scholar - [107]S. Newman and H. Frisch,
*Dynamics of skeletal pattern formation in developing chick limb*. Science,**205**(1979), pp. 662–668.Google Scholar - [108]S. Newman,
*Sticky fingers: Hox genes and cell adhesion in vertebrate development*, Bioessays,**18**(1996), pp. 171–174.Google Scholar - [109]K. O’Connor and D. Zusman,
*Patterns of cellular interactions during fruiting-body formation in Myxococcus xanthus*, J. Bacteriol.,**171**(1989), pp. 6013–6024.Google Scholar - [110]G.M. Odell and J.T. Bonner,
*How the Dictyostelium discoideum grex crawls*, Philos. Trans. Roy. Soc. London, B.,**312**(1985), pp. 487–525.Google Scholar - [111]C. Ofria, C. Adami, T.C. Collier, and G.K. Hsu, E
*volution of differentiated expression patterns in digital organisms*; Lect. Notes Artif. Intell.,**1674**(1999), pp. 129–138.Google Scholar - [112]H.G. Othmer, S. Dunbar, and W. Alt,
*Models of dispersal in biological systems*, J. Math. Biol.,**26**(1988), pp. 263–298.MathSciNetMATHGoogle Scholar - [113]H.G. Othmer and T. Hillen,
*The diffusion limit of transport equations II: Chemotaxis equations*, SIAM J. Appl. Math.,**62**(2002), pp. 1222–1250.MathSciNetMATHGoogle Scholar - [114]J.K. Parrish and W. Hamner, (Eds.),
*Animal groups in three dimensions*, Cambridge University Press, Cambridge, 1997.Google Scholar - [115]J.K. Parrish and L. Edelstein-Keshet,
*From individuals to aggregations: Complexity, epiphenomena, and evolutionary trade-offs of animal aggregation*, Science,**284**(1999), pp. 99–101.Google Scholar - [116]A. Pelizzola,
*Low-temperature phase of the three-state antiferromagnetic Potts model on the simple-cubic lattice*, Phys. Rev. E,**54**(1996), pp. R5885-R5888.Google Scholar - [117]J. Pjesivac and Y. Jiang,
*A cellular model for avascular tumor growth*, unpublished (2002).Google Scholar - [118]T. Pollard and J. Cooper,
*Actin**and act in-binding proteins. A critical evaluation of mechanisms and function*, Ann. Rev. Biochem.,**55**(1986), pp. 987–1035.Google Scholar - [119]R. Potts,
*Some generalized order-disorder transformations*, Proc. Cambridge Phil. Soc.,**48**(1952), pp. 106–109.MathSciNetMATHGoogle Scholar - [120]I. Prigogine and R. Herman,
*Kinetic theory of vehicular traffic*, American Elsevier, New York, 1971.MATHGoogle Scholar - [121]S. Rahman, E. Rush, and R. Swendsen,
*Intermediate-temperature ordering in a three-state antiferromagnetic Potts model*, Phys. Rev. B.**58**(1998). pp. 9125–9130.Google Scholar - [122]W.J. Rappel, A. Nicol, A. Sarkissian, H. Levine, and W.F. Loomis
*Self-organized vortex state in two-dimensional Dictyostelium dynamics*, Phys. Rev. Lett.,**83**(1999), pp. 1247–1250.Google Scholar - [123]H. Reichenbach,
*Myxobacteria: A most peculiar group of social prokaryotes*, in*Myxobacteria development and cell interactions*, E. Rosenburg (Ed.) Springer-Verlag, NY, 1984, pp. 1–50.Google Scholar - [124]C.W. Reynolds,
*Flocks, herds, and schools: A distributed behavioral model, ACM Computer Graphics*, SIGGRAPH ’87,**21**(1987), pp. 25–34.Google Scholar - [125]D. Richardson
*Random growth in a tessellation*, Proc. Camb. Phil. Soc.,**74**(1973), pp. 563–573.Google Scholar - [126]J. Rieu, A. Upadhyaya, J.A. Glazier, N. Ouchi, and Y. Sawada,
*Diffusion and deformations of single hydra cells in cellular aggregates*, Biophys. J,**79**(2000), pp. 1903–1914.Google Scholar - [127]J. Rubin and A. Robertson,
*The tip of the Dictyostelium pseudoplasmodium as an organizer*, J. Embryol. Exp. Morphol.,**33**(1975), pp. 227–241.Google Scholar - [128]B. Sager and D. Kaiser,
*Two cell-density domains within the Myxococcus xanthus fruiting body*, Proc. Natl. Acad. Sci.,**90**(1993), pp. 3690–3694.Google Scholar - [129]B. Sager and D. Kaiser,
*Intercellular C-signaling and the traveling waves of Myxococcus xanthus*. Genes Development,**8**(1994), pp. 2793–2804.Google Scholar - [130]P. Sahni, G. Grest, M. Anderson, and D. Srolovitz,
*Kinetics of the Q-state Potts model in 2 dimensions*, Phys. Rev. Lett., 50 (1983), pp. 263–266.Google Scholar - D. Srolovitz, M. Anderson, G. Grest, and P. Sahni,
*Grain-growth in 2 dimensions*, Scripta Met., 17 (1983), pp. 241–246.Google Scholar - D. Srolovitz, M. Anderson, G. Grest, and P. Sahni,
*Computer-simulation of grain-growth. 2. Grain-size distribution, topology, and local dynamics*. Acta Met.,**32**(1984), pp. 793–802.Google Scholar - D. Srolovitz, M. Anderson, G. Grest, and P. Sahni,
*Computer-simulation of grain-growth. 3. Influence**of a particle dispersion*. Acta Met.,**32**(1984), pp. 1429–1438.Google Scholar - G. Grest, D. Srolovitz, and M. Anderson,
*Kinetics of domain growth: universality of kinetic exponents*, Phys. Rev. Letts,.**52**(1984), pp. 1321–1329.Google Scholar - D. Srolovitz, G. Grest, and M. Anderson,
*Computer-simulation of grain growth. 5. Abnormal grain-growth*, Acta Met.,**33**(1985), pp. 2233–2247.Google Scholar - [131]N. Savill and p. Hogeweg,
*Modelling morphogenesis: From single cells to crawling slugs*, J. Theor. Bio.,**184**(1997), pp. 229–235.Google Scholar - [132]M. scalerandi, B. Sansone, and C. Condat,
*Diffusion with evolving sources and competing sinks: Development of angiogenesis*, Phys. Rev. E,**65**(2002), 011902.Google Scholar - [133]J.A. Shapiro,
*Bacteria as multicellular organisms*. Scientific American,**258**(1988), pp. 82–89.Google Scholar - [134]J. A. Shapiro,
*The significances of bacterial colony patterns*, Bioessays,**17**(1995), pp. 597–607.Google Scholar - [135]J.A. Shapiro,
*Thinking about bacterial populations as multicellular organisms*, Annual Review of Microbiology,**52**(1998), pp. 81–104.Google Scholar - [136]N. Shimoyama, K. Sugawara, T. Mizuguchi, Y. Hayakawa, and M. Sano,
*Collective motion in a system of motile elements*, Phys. Rev. Lett.,**76**(1996), pp. 3870–3873.Google Scholar - [137]E. Siggia,
*Late stages of spinodal decomposition in binary mixtures*, Phys. Rev. A,**20**(1979), pp. 595–605.Google Scholar - [138]S. Simpson, A. McCaffery, and B. Hagele,
*A behavioural analysis of phase change in the desert locust*. Bio. Rev. of the Cambridge Philosophical Society,**74**(1999), pp. 461–480.Google Scholar - [139]D. Soll,
*Computer-assisted three-dimensional reconstruction and motion analysis of living, crawling cells*, Computerized Medical Imaging and Graphics,**23**(1999), pp. 3–14.Google Scholar - [140]D. Soll, E. Voss, O. Johnson, and D. Wessels,
*Three-dimensional reconstruction and motion analysis of living, crawling cells*, Scanning,**22**(2000), pp. 249–257.Google Scholar - [141]J. Stavans,
*The evolution of cellular structures*. Rep. Prog. Phys.,**56**(1993), pp. 733–789.Google Scholar - [142]M. Steinberg,
*Mechanism of tissue reconstruction by dissociated cells, II. Time-course of events*. Science,**137**(1962), pp. 762–763.Google Scholar - [143]
- [144]A. Stevens,
*A stochastic cellular automaton modeling gliding and aggregation of Myxobacteria*, SIAM J. Appl. Math.,**61**(2000), pp. 172–182.MathSciNetMATHGoogle Scholar - [145]E. Stott, N. Britton, J. A. Glazier, and M. Zajac,
*Stochastic simulation of benign avascular tumour growth using the Potts model*, Mathematical and Computer Modelling,**30**(1999), pp. 183–198.Google Scholar - [146]U. Technau and T. Holstein,
*Cell sorting during the regeneration of hydra from reaggregated cells*, Devel. Bio,**151**(1992), pp. 117–127.Google Scholar - [147]
- [148]A. Turing,
*The chemical basis of morphogenesis*, Phil. Trans. R. Soc. London,**237**(1952), pp. 37–72.Google Scholar - [149]A. Upadhyaya,
*Thermodynamics and fluid properties of cells, tissues and membranes*, Ph.D. Thesis., The University of Notre Dame, USA, 2001.Google Scholar - [150]A. Upadhyaya, J. Rieu, J. A. Glazier and Y. Sawada,
*Anomalous diffusion and non-Gaussian velocity distribution of Hydra cells in cellular aggregates*, Physica A,**293**(2001), pp. 49–558.Google Scholar - [151]P. Van Haaster,
*Sensory adaptation of Dictyostelium discoideum cells to chemotactic signals*, J. Cell Biol.,**96**(1983), pp. 1559–1565.Google Scholar - [152]B. Vasiev, F. Siegert and C.J. Weijer,
*A hydrodynamic model approach for Dictyostelium mound formation*, J. Theor. Biol.,**184**(1997), pp. 441–450.Google Scholar - [153]T. Vicsek, A. Czirok, E. Ben-Jacob, I Cohen, O. Shochet, and A. Tenenbaum,
*Novel type of phase transition in a system of self-driven particles*, Phys. Rev. Lett.,**75**(1995), pp. 1226–1229.Google Scholar - [154]J. von Neumann,
*Theory of self-reproducing automata*, (edited and completed by A. W. Burks), University of Illinois Press, Urbana, 1966.Google Scholar - [155]J. Wartiovaara, M. Karkinen-Jääskelänen, E. Lehtonen, S. Nordling, and L. Saxen,
*Morphogenetic cell interactions in kidney development*, in N. Müller-Bér) (Ed.),*Progress in differentiation research*. North-Holland Publishing Company, Amsterdam, 1976, 245–252.Google Scholar - [156]D. Weaire and N. Rivier,
*Soap, cells and statistics: random patterns in 2 dimensions*, Contemp. Phys.**25**(1984) pp. 59–99.Google Scholar - [157]H. Williams, S. Desjardins, and F. Billings,
*Two-dimensional growth models*, Phys. Lett. A,**250**(1998), pp. 105–110.Google Scholar - [158]J. Williams,
*Regulation of cellular differentiation during Dictyostelium morphogenesis*, Curr. Opin. Genet. Dev.,**1**(1991), pp. 338–362.Google Scholar - [159]J. Wejchert, D. Weaire, and J. Kermode,
*Monte-Carlo simulation of the evolution of a two-dimensional soap froth*, Phil. Mag. B,**53**(1986), pp. 15–24.Google Scholar - [160]R. Welch and D. Kaiser,
*Cell behavior in traveling wave patterns of myxobacteria*, Proc. Natl. Acad. Sci. USA,**98**(2001), pp. 14907–14912.Google Scholar - [161]T. Witten and L. Sander,
*Diffusion-limited aggregation*, Phys. Rev. B,**27**(1983), pp. 5686–5697.MathSciNetGoogle Scholar - [162]D. Wolf-Gladrow,
*Lattice-gas cellular automata and lattice Boltzmann models — An introduction*. Springer-Ver lag, Berlin, Lecture Notes in Mathematics**1725**(2000).MATHGoogle Scholar - [163]S. Wolfram,
*Statistical mechanics of cellular automata*. Rev. Mod. Phys.,**55**(1983), pp. 601–604.MathSciNetMATHGoogle Scholar - [164]
- [165]
- [166]C. wolgemuth and E. Hoiczyk,
*How Myxohactevia glide*. Current Biology,**12**(2002), pp. 369–377.Google Scholar - [167]
- [168]M. Zajac, G. Jones, and J.A. Glazier,
*Model of convergent extension in animal morphogenesis*, Phys. Rev. Lett.,**85**(2000), pp. 2022–2025.Google Scholar - [169]M. Zajac,
*Modeling convergent extension by way of anisotropic differential adhesion*. Ph.D. thesis. The University of Notre Dame, USA, 2002.Google Scholar