2013/10/16 by Shirshendu Chatterjee, Chatterjee, Shirshendu
Biochemistry, Genetics and Molecular Biology · Engineering · Physics and Astronomy · #Artificial Immune Systems Applications #Complex Network Analysis Techniques #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Gene Regulatory Network Analysis #Mathematical Physics (math-ph) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1310.4474
openalex publication_date 2013/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a model for heterogeneous 'gene regulatory networks' that is a generalization of the model proposed by Chatterjee and Durrett (2011) as an "annealed approximation" of Kauffmann's (1969) random Boolean networks. In this model, genes are represented by the nodes of a random directed graph on n vertices with specified in-degree distribution (resp. out-degree distribution or joint distribution of in-degree and out-degree), and the expression bias (the expected fraction of 1's in the Boolean functions) p is same for all nodes. Following a standard practice in the physics literature, we use a discrete-time threshold contact process with parameter q=2p(1-p) (in which a vertex with at least one 'occupied' input at time t will be occupied at time t+1 with probability q, and 'vacant' otherwise) on the above random graph to approximate the dynamics of the Boolean network. We show that there is a parameter r (which can be written explicitly in terms of first few moments of the degree distribution) such that, with probability tending to 1 as n goes to infinity, if 2p(1-p)r>1, then starting from all occupied sites the threshold contact process maintains a positive (\it quasi-stationary) density of occupied sites for time which is exponential in n, whereas if 2p(1-p)r<1, then the persistence time of the threshold contact process is at most logarithmic in n. These two phases correspond to the 'chaotic' and 'ordered' behavior of the gene networks.