Cellular Automaton Modeling of Biological Pattern Formation by Andreas Deutsch & Sabine Dormann

Author:Andreas Deutsch & Sabine Dormann
Language: eng
Format: epub
Publisher: Birkhäuser Boston, Boston, MA

Figure 10.3:Sketch of horizontal and vertical pigment pattern formation in two salamander species (Triturus alpestris (left) and Ambystoma mexicanum); cp. explanations in the text.

10.2 LGCA Model with Adhesiveand Orientational Interaction 2

A three-species LGCA is defined (ς = 3, cp. sec. 4.3, p. 76). In the LGCA, pigment cells and the extracellular matrix (ECM) are represented by oriented cells (particles) of type σ, where :

melanophore,

:

xanthophore and

:

ECM.

We assume that all cells interact but only “pigment cells” (of types σ = 1, 2) move on a two-dimensional hexagonal lattice (b = 6). We distinguish top and bottom regions of the lattice representing the neural crest and the ventral regions of the embryo, respectively. Thereby, we introduce a reference frame on the lattice. Various boundary conditions to be specified in the corresponding context will be considered.

The cells occupy lattice channels corresponding to nearest neighbor vectors c i, associated with each node r. An exclusion principle prevents two (or more) cells of the same type to inhabit the same channel. Accordingly, a state is specified by . s i σ  ∈ { 0, 1} = 0 or 1 denotes the absence resp. presence of a cell (particle) of type σ in channel c i. We have .

Cell-Cell Interactions. States may change according to local interactions. States of cell type σ = 3 (ECM) are assumed to maintain their initial value and not to change in time, thereby introducing an orientational anisotropy influencing the dynamics of the moving pigment cells.

Let the number of cells of type σ corresponding to state be denoted by

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