Morphogenesis modelization of a fractone-based model

Figure 1.  Fractones in the fourth ventricle of a rat brain. The dot-like structures arrayed around the boundary of the 4th ventricle are the fractones.

Figure 2.  An individual fractone. Note the branching structure that lends it its name. Image taken by transmission electron microscope and used with permission from Dr. Frederic Mercier.

Figure 3.  Fractones in the third ventricle of a rat brain. The dot-like structures arrayed in the funnel-shaped ventricle (indicated by the arrows) are the fractones.

Figure 4.  Examples of an (a) admissible set of cell bodies and (b-c) two inadmissible sets of cell bodies. The dark spheres are the cell bodies, and the lighter surrounding ellipsoids to form the associated $\hat{B}$. The set in (b) is not admissible since $\hat{B}$ is not connected. The set in (c) is not admissible since two of the cell bodies intersect.

Figure 5.  Examples of two overlapping cell spaces, one in black and one in white. Each cell has a radius of 4.5 $\mu$m. The geometric distance between the cell spaces are (a) 4.5 $\mu$m and (b) 9 $\mu$m. Note that the existence of an outlier cell in (b) causes the much larger distance even though many of the other overlapping cells are close to one another.

Figure 6.  Example of two cells with the same radius and with centers 24 $\mu$m apart. The cell-birth parameters are $90$ minutes and $120$ minutes. The age distance $d_a$ (with $kappa=4.5$ $\mu$m/min) is therefore $4.5|120-90|+24=159$ $\mu$m.

Figure 7.  (a) A 2D figure of a biological structure and diffusion space and (b) a 3D figure of a biological structure with diffusion space. Cells are in dark grey, fractones in black, meninges in white, and diffusion space in light grey (although the diffusion space also includes the fractones).

Figure 8.  An example initial condition with a single cell (in black) and distribution of a single growth factor in the diffusion space surrounding it. The growth factor concentration is represented by the color intensity of the grey; close to the cell, the concentration is $0$ concentration units, and at outer boundary of the diffusion space, it is $20$ concentration units.

Figure 9.  The interaction of the domain, guard conditions, and edges of our model. Left to right, we begin in the domain of biological structure $q_1$. The guard condition for edge $(q_1,q_2)$ is then met when the positive fractone captures $100$ concentration units of growth factor. This triggers growth and we instantaneously switch discrete states to biological structure $q_2$.

Figure 10.  An example of the growth algorithm. (a) A mother cell (grey) about to undergo mitosis. The position of the new cell is determined by the direction of the tangent vector where the fractone (black circle) touches the cell. (b) The two daughter cells. Note that one of the new cells is in the same position as the mother cell, and the other is created a distance $d_m$ away along the tangent axis. Both cells are assigned a cell-birth parameter $\lambda$ equal to the time at which the growth occurs. In addition, the fractone is relocated to be on the displaced new cell.

Figure 11.  An example biological structure $q$ with cells, $c_1$, $c_2$, $c_3$, in dark gray; positive fractones, $f^+_1$, $f^+_2$, in black; negative fractone, $f^-_3$, is shaded; and diffusion space in light gray. Corresponding cell-birth parameters $\lambda_{c_1}$, $\lambda_{c_2}$, $\lambda_{c_3}$ for $c_1$, $c_2$, $c_3$, respectively, are shown and given in minutes.

Figure 12.  (a) Two-dimensional example of an initial cell set (black) and a final target set (white). (b) The adjusted target set (grey) formed by adding cells in the proposition algorithm. Each grey cell is 4.5 $\mu$m away from a neighboring cell.

Figure 13.  Top view of the neurulation simulation. In black, the cells. In grey, the fractones. We start with a U-shape configuration and wish to end with a closed ring of cells.

Figure 14.  Progression of the growth of the tube. Shown is the mass of cells at selected times. Cells (black), fractones (light grey), and immature cells (dark grey). Meninges is not shown in the image for clarity -it is present surrounding the mass, however. From $t=0$ to $t=210$, we simply lengthen the sides of the U-shape. Until $t=560$, we continue to extend upwards, but with staggered growth to create the curved edge. Around $t=1000$ we have closed the ring, and at $t=1150$ we have completed the top of the ring.

Figure 15.  Left: Final target cell configuration. Center: Initial cell configuration ($9 \times 9 \times 9$ cube of cells). Right: Transparency of target cell configuration showing relative location of initial cell configuration in black. Positive axes are as shown.

Figure 16.  Progression of growth of cells (black) compared to target space (light grey) over time. Images shown correspond to times at which the control exerted a change on the position of fractones (dark grey).