A model for the growth of a gene-regulatory network by gene duplication: each new…
PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems
Many Parts, One Behaviour, Ch Growing networks: preferential attachment and scale-free degree distributions
Problem
A model for the growth of a gene-regulatory network by gene duplication: each new gene arises by duplicating an existing 'parent' gene, chosen uniformly at random among the t genes already present, and inherits each of the parent's m regulatory links independently. For each inherited link, the new gene copies the parent's link (probability 1−ρ) or is instead given a link to a uniformly random gene (probability ρ), the same ρ for every link. This mechanism gives a gene i with current in-degree (number of regulators pointing to it) qi the attachment probability Πi=tρ+(1−ρ)mtqi.
((a)) Rewrite Πi in the form Πi=mt1−ρ(qi+a) for some constant a, and identify a in terms of ρ and m. By matching this to the Price-model attachment probability Πq=(q+a)/[t(c+a)] for a network with c links added per new node, show that this copying model has effective offset a=ρm/(1−ρ) and effective out-degree c=m.
((b)) Use the Price-model exponent γ=2+a/c to show that the copying model gives γ=1−ρ2−ρ.
((c)) A measured regulatory network has a fitted in-degree exponent γ^=2.6 (maximum-likelihood estimate). Find the value of ρ in the copying model consistent with this exponent, and state the corresponding copying probability 1−ρ.