Combinatorial explosion in model gene networks

The explosive growth in knowledge of the genome of humans and other organisms leaves open the question of how the functioning of genes in interacting networks is coordinated for orderly activity. One approach to this problem is to study mathematical properties of abstract network models that capture the logical structures of gene networks. The principal issue is to understand how particular patterns of activity can result from particular network structures, and what types of behavior are possible. We study idealized models in which the logical structure of the network is explicitly represented by Boolean functions that can be represented by directed graphs on n-cubes, but which are continuous in time and described by differential equations, rather than being updated synchronously via a discrete clock. The equations are piecewise linear, which allows significant analysis and facilitates rapid integration along trajectories. We first give a combinatorial solution to the question of how many distinct logical structures exist for n-dimensional networks, showing that the number increases very rapidly with $\mathrm{n.}$ We then outline analytic methods that can be used to establish the existence, stability and periods of periodic orbits corresponding to particular cycles on the n-cube. We use these methods to confirm the existence of limit cycles discovered in a sample of a million randomly generated structures of networks of 4 genes. Even with only 4 genes, at least several hundred different patterns of stable periodic behavior are possible, many of them surprisingly complex. We discuss ways of further classifying these periodic behaviors, showing that small mutations (reversal of one or a few edges on the n-cube) need not destroy the stability of a limit cycle. Although these networks are very simple as models of gene networks, their mathematical transparency reveals relationships between structure and behavior, they suggest that the possibilities for orderly dynamics in such networks are extremely rich and they offer novel ways to think about how mutations can alter dynamics.