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Co-evolving cellular automata for morphogenesis

This work is partially supported by the Simons Foundation, award # 359510

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  • Morphogenesis, the shaping of an organism, is a complex biological process accomplished through an well organized interplay between growth, differentiation and cell movement.It is still today one of the major outstanding problems in the biological sciences. Pattern formation has been well-addressed in the literature with the development of many mathematical models including the famous reaction-diffusion ones. We here take a different approach, introducing a controlled cellular automaton in order to model the signal molecules, known as growth factors, that convey information from one cell to another during an organism's development and help maintain the viability of the adult. This control represents extracellular structures that have been associated with the regulation of stem cell proliferation and are called fractones. In this paper we introduce two co-evolving automata, one describing the perturbed diffusion of growth factors and one accounting for the rules of basic cellular functions (proliferation, differentiation, migration and apoptosis). Fractones are introduced as an external input to control the shaping of multi-cellular organisms; we analyze their influence on the emerging shape. We illustrate our theory with 2 and 3 dimensional simulations. This work presents the foundation upon which to develop cellular automata as a tool to simulate the morphodynamics in embryonic development.

    Mathematics Subject Classification: Primary: 92Bxx; Secondary: 93Cxx.


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  • Figure 1.  Fractones are visible as the small puncta arranged around the aqueduct. The larger elongated shapes surrounding aqueduct are blood vessels. Image of the cerebral aqueduct in an adult mouse taken by 20x PlanApo objective lens. The aqueduct connects the third and fourth ventricles together. The image is from [1], see [18] for information about the experimental technique. It was obtained in Dr. Mercier's laboratory at the University of Hawaii

    Figure 2.  Illustration of the superposition of the grids on which the two co-evolving automata act. (a) Grid for the diffusion automaton. (b) Grid for the cell-functions automaton. The yellow displays the space occupied by a biological cell and green units represent fractones

    Figure 3.  Our definition of neighborhood is based on the Manhattan distance. Blue units are at a distance 1 from the central unit $ u $, yellow at a distance 2 and red at a distance 3. $ N_c(u) $ with $ r = 3 $ is the union of all colored units in this picture beside unit $ u $

    Figure 4.  A representation of the 3D neighborhood. The colors are as in Figure 3. Picture (1) displays the sphere of radius 3, and a view with the front units made transparents to show the inside is provided in (2). In (3) we can see a cross section of the sphere that passes through the middle

    Figure 5.  Starfish-like shape. Starting from the top left, going right and down, the snapshots are at timesteps 1, 35,100,130,160 and 198. Each cell in the simulation is marked with its age

    Figure 6.  3D simulation of a uniform mass starting from a unique cell and using a unique fractone throughout the simulation. The cell color indicates age – a red cell is the product of a recent mitosis, a yellow cell is old enough to undergo mitosis. The color of the diffusion units indicate concentration with blue representing low concentration and red high concentration. Note that for 3D simulations, we have hidden the diffusion concentrations except close to each of the cells in order to make the shape of the cell mass visible

    Figure 7.  3D simulation of an elongated shape. (a) represents the original cell and fractone as well as the heterogenous concentration of growth factor around the cell. (b) corresponds to time step # 77. At this point in the simulation, there are 13 cells. The visible fractone on the image is becomes active at this time step #78 represented in (c) which explains the new cell daughter cell on the bottom. Finally (d) is a side view (left) and a top view (right) of the final mass of cells, it corresponds to time step #240 and there are 41 cells

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