Mathematics of the genome (Q1683737)

The authors consider a directed network, where each node represents a gene and there is a connection from node \(i\) to node \(j\) if gene \(i\) has some control over gene \(j\). The state of each node is described by a non-negative scalar representing the expression level of the associated gene. Interactions between nodes are assumed to be governed by Hill-type equations of the form \(dy/dt=H(x)-by\), where \(x\) and \(y\) are expression levels, \(b\) is a positive constant and \(H\) is a Hill function. From such a network one derives a general equation \(dx/dt=F(x)\), where \(F:(\mathbb{R}^+)^n\rightarrow \mathbb{R}^n\). The main result consists in showing that if the system is at a stable equilibrium and a parameter is varied (and ``the equilibrium does not leave its basin''), the generic ``first'' bifurcation to occur will be either a supercritical pitchfork or a supercritical Hopf. The authors then give examples in 2- and 3-gene networks.