The basis of easy controllability in Boolean networks.

Effective control of biological systems can often be achieved through the control of a surprisingly small number of distinct variables. We bring clarity to such results using the formalism of Boolean dynamical networks, analyzing the effectiveness of external control in selecting a desired final state when that state is among the original attractors of the dynamics. Analyzing 49 existing biological network models, we find strong numerical evidence that the average number of nodes that must be forced scales logarithmically with the number of original attractors. This suggests that biological networks may be typically easy to control even when the number of interacting components is large. We provide a theoretical explanation of the scaling by separating controlling nodes into three types: those that act as inputs, those that distinguish among attractors, and any remaining nodes. We further identify characteristics of dynamics that can invalidate this scaling, and speculate about how this relates more broadly to non-biological systems.

Evolution of default genetic control mechanisms.

We present a model of the evolution of control systems in a genome under environmental constraints. The model conceptually follows the Jacob and Monod model of gene control. Genes contain control elements which respond to the internal state of the cell as well as the environment to control expression of a coding region. Control and coding regions evolve to maximize a fitness function between expressed coding sequences and the environment. The model was run 118 times to an average of 1.4∙106 'generations' each with a range of starting parameters probed the conditions under which genomes evolved a 'default style' of control. Unexpectedly, the control logic that evolved was not significantly correlated to the complexity of the environment. Genetic logic was strongly correlated with genome complexity and with the fraction of genes active in the cell at any one time. More complex genomes correlated with the evolution of genetic controls in which genes were active ('default on'), and a low fraction of genes being expressed correlated with a genetic logic in which genes were biased to being inactive unless positively activated ('default off' logic). We discuss how this might relate to the evolution of the complex eukaryotic genome, which operates in a 'default off' mode.

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.