Using cellular automata (CA), we performed simulations of: i) the growth of vessels in blood capillaries and in leaves (see e.g. the Gingko in the figure below: a) simulations; b) real leave); ii) the pigmentation of animal skins; iii) the pigmentation on the shells of molluscs.

The advantages of cellular automata, as compared to partial differential equations, are: i) faster calculations; ii) biologically more realistic discretization (these systems are, in fact, composed of discrete cells); iii) holistic approach that allows clear recognition of generic features. The latter advantage is particularly evident in the simulations of mollusc pigmentations. In fact, all investigated shells could be simulated by only one set of simple rules, independently of the complexity of the structure and including even "class 4" behaviour (erratic alternation of periodic and chaotic patterns).

We also used CA to simulate waves in excitable media, such as those ocurring during the aggregation of the slime mold. These waves are formally analogous to those in heart muscle, in neural tissue, and in the Belousov-Zhabotinsky reaction.

Furthermore, we simulated (using molecular dynamics) the self-organization of a colony of unicellular green algae.

Simulation of growth if Gingko leaf vessels:

Gingko leaf:



M. Markus, D. Boehm and M. Schmick, "Simulations of vessel morphogenesis using cellular automata", Mathematical Biosciences 156, 191-206 (1999)

I. Kusch and M. Markus, "Mollusc shell pigmentation: cellular automaton simulations and evidence for undecidability", Journal of theoretical Biology 178, 333-340 (1996)

M. Markus and I. Kusch, "Cellular automata for modelling the shell pigmentation of molluscs", Journal of Biological Systems 3, 999-1011 (1995)

H. Schepers and M. Markus, "Two types of performance of aniIsotropic cellular automaton: stationary (Turing) patterns and spiral waves", Physica A 188 (1992), pp. 337-343

M. Grewe and M. Markus, "Aggregation of the green alga Pediastrum: experiments and simulations", J. biol. Systems 8, 373-398 (2000)

M. Markus, "Modelling morphogenetic processes in excitable tissues using novel cellular automata". Biomed. Biochim. Acta 49, 681-696 (1990)

M. Markus, S.C. Mueller and G. Nicolis (eds.), "From chemical to biological organization", Springer-Verlag, Heidelberg (1988)

M. Markus, K.Koetter, M. Schmick, M. Grewe and E. Goles, "Self-organization of interacting, round particles into symmetric or asymmetric aggregates", in: 'Symmetry 2000' (I. Hargittai & T. Laurent, eds.), Portland Press, London, 377-398 (2002)

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