Mathematical simulation of gonad compaction.

<p>(A) A scheme of the Cellular Potts model (CPM). In the CPM, each cell is described over a region of multiple sites on a square lattice, and the shape of individual cells is determined by simulating the minimization process of a given potential energy for the system (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0052649#s4" target="_blank">Materials and Methods</a> for details). Briefly, in each Monte Carlo step (MCS), a flip of a randomly chosen pixel at the cell boundary (see the <i>ij</i>-th pixel) is adopted with the probability determined by the difference in potential energy between the two configurations. (B) Left: The average values of gonad compaction (circularity) between 12,000 and 20,000 MCS were calculated, and are shown in a color map against different values of <i>E_SGP-SGP</i> and <i>E_SGP-out</i>. The circularity increases for smaller contracting forces (larger adhesive forces) among SGPs, and for larger contracting force along the surfaces between SGPs and the outer space. Right: The cell configurations obtained from the CPM at 19,660 MCS. Values of <i>E_SGP-SGP</i> and <i>E_SGP-out</i> are indicated with the corresponding number in the left panel. Other parameter values are described in Methods. (C–D) Temporal changes in gonad morphology in the CPM. Selected frames from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0052649#pone.0052649.s006" target="_blank">Movies S4</a> (C) and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0052649#pone.0052649.s007" target="_blank">S5</a> (D) are shown. (C) and (D) correspond to 1 and 2 in the phase diagram in (B). (E) A mechanical view of gonad compaction. For clarity, only SGPs are shown. Blue lines indicate the SGP-SGP contact surface. Redistribution of DE-cad to the SGP-SGP surface generates a large adhesive force between the SGPs (i.e., small <i>E_SGP-SGP</i>), which increases the SGP-SGP contact surface. The adhesive force, therefore, functions as a glue to move the SGPs inward (left).