Co-evolving cellular automata for morphogenesis

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