The ratio of relaxation time to cell duplication time controls the approach to equilibrium. (A) Time trace of the energy during a numerical simulation of relaxation toward the equilibrium phenotype. The growth of a cyst with no control on spindle orientation is simulated up to 16 cells. Afterward, cell divisions are blocked and purely relaxational dynamics are generated. Note the presence of multiple sequential energy barriers during the convergence to the equilibrium phenotype. Time is measured in cell division time units. Note that although the single lumen is rapidly attained, the convergence to the full equilibrium topology takes much longer. (B) Topological distribution for the simulated relaxation of a 12-cell cyst. (C) The time T needed to reach the equilibrium configuration in the same in silico relaxation experiment as in A, starting from cysts composed of different numbers of cells, grows exponentially with the number of cells. (insets) Distribution of cell–cell contacts at equilibrium for each value of ncells. (D) We simulated the growth of a normal cyst up to 16 cells (red trace and circles), and from that point on we let the cyst grow with different division rates. To compare different cysts with different division rates, we plot here the relative distance from equilibrium and nonequilibrium as a function of the number of cells. Slower cell divisions yield a topological shift toward equilibrium.
