Cell phenotypes as macrostates of the GRN dynamics

The two most fundamental processes describing change in biology, development, and evolution, occur over drastically different timescales. Development involves temporal sequences of cell states controlled by hierarchies of regulatory structures. It occurs over the lifetime of a single individual and is associated with gene expression level change of a given genotype. Evolution, by contrast, entails genotypic change through mutation, the acquisition/loss of genes and changes in the network topology of interactions among genes. It involves the emergence of new, environmentally selected phenotypes over the lifetimes of many individuals. We start by reviewing the most limiting aspects of the theoretical modeling of gene regulatory networks (GRNs) which prevent the study of both timescales in a common, mathematical language. We then consider the simple framework of Boolean network models of GRNs and point out its inadequacy to include evolutionary processes. As opposed to one-to-one maps to specific attractors, we adopt a many-to-one map which makes each phenotype correspond to multiple attractors. This definition no longer requires a fixed size for the genotype and opens the possibility for modeling the phenotypic change of a genotype, which is itself changing over evolutionary timescales. At the same time, we show that this generalized framework does not interfere with established numerical techniques for the identification of the kernel of controlling nodes responsible for cell differentiation through external signals.