  Gene regulatory network (GRN)-based morphogenetic models have recently gained
an increasing attention. However, the relationship between microscopic
properties of intracellular GRNs and macroscopic properties of morphogenetic
systems has not been fully understood yet. Here we propose a theoretical
morphogenetic model representing an aggregation of cells, and reveal the
relationship between criticality of GRNs and morphogenetic pattern formation.
In our model, the positions of the cells are determined by spring-mass-damper
kinetics. Each cell has an identical Kauffman's $NK$ random Boolean network
(RBN) as its GRN. We varied the properties of GRNs from ordered, through
critical, to chaotic by adjusting node in-degree $K$. We randomly assigned four
cell fates to the attractors of RBNs for cellular behaviors. By comparing
diverse morphologies generated in our morphogenetic systems, we investigated
what the role of the criticality of GRNs is in forming morphologies. We found
that nontrivial spatial patterns were generated most frequently when GRNs were
at criticality. Our finding indicates that the criticality of GRNs facilitates
the formation of nontrivial morphologies in GRN-based morphogenetic systems.
