Emergence of diversity in homogeneous coupled Boolean networks

The origin of multicellularity in metazoa is one of the fundamental questions of evolutionary biology. We have modeled the generic behaviors of gene regulatory networks in isogenic cells as stochastic nonlinear dynamical systems—coupled Boolean networks with perturbation. Model simulations under a variety of dynamical regimes suggest that the central characteristic of multicellularity, permanent spatial differentiation (diversification), indeed can arise. Additionally, we observe that diversification is more likely to occur near the critical regime of Lyapunov stability.

Figure 1

Convergence of maximum chaotic, maximum ordered, and critical regimes for $10\times 10$ tissues.

Figure 2

Diversity vs Lyapunov exponent in a $10\times 10$ tissue, all experiments. Panels (a) and (b) are simulation results for $\theta =1$; panels (c) and (d) are simulation results for $\theta =2$; and panels (e) and (f) are simulation results for $\theta =3$.

Figure 3

Steady-state distributions of nine upper-left hand corner cells of a $10\times 10$ “outlier” tissue on logarithmic scale. For each cell, the $x$ axis is the binary-encoded decimal representation of 1024 Boolean states, and the $y$ axis is the relative frequency of the states.

Figure 4

(a) Randomly initialized states. (b) Formation of checkerboard pattern of cellular states at $t=2986$. The individual squares are binary-encoded color representations of 1024 Boolean states of the $10\times 10$ cells. See Supplemental Material video S2 for the full temporal behaviors of the states [44].