Understanding genetic regulatory networks

Abstract

Random Boolean networks (RBM) were introduced about 35 years ago as first crude models of genetic regulatory networks. RBNs are comprised of N on–off genes, connected by a randomly assigned regulatory wiring diagram where each gene has K inputs, and each gene is controlled by a randomly assigned Boolean function. This procedure samples at random from the ensemble of all possible NK Boolean networks. The central ideas are to study the typical, or generic properties of this ensemble, and see 1) whether characteristic differences appear as K and biases in Boolean functions are introducted, and 2) whether a subclass of this ensemble has properties matching real cells.

Such networks behave in an ordered or a chaotic regime, with a phase transition, ‘the edge of chaos’ between the two regimes. Networks with continuous variables exhibit the same two regimes. Substantial evidence suggests that real cells are in the ordered regime. A key concept is that of an attractor. This is a reentrant trajectory of states of the network, called a state cycle. The central biological interpretation is that cell types are attractors. A number of properties differentiate the ordered and chaotic regimes. These include the size and number of attractors, the existence in the ordered regime of a percolating ‘sea’ of genes frozen in the on or off state, with a remainder of isolated twinkling islands of genes, a power law distribution of avalanches of gene activity changes following perturbation to a single gene in the ordered regime versus a similar power law distribution plus a spike of enormous avalanches of gene changes in the chaotic regime, and the existence of branching pathway of ‘differentiation’ between attractors induced by perturbations in the ordered regime.

Noise is serious issue, since noise disrupts attractors. But numerical evidence suggests that attractors can be made very stable to noise, and meanwhile, metaplasias may be a biological manifestation of noise. As we learn more about the wiring diagram and constraints on rules controlling real genes, we can build refined ensembles reflecting these properties, study the generic properties of the refined ensembles, and hope to gain insight into the dynamics of real cells.