Derivation of stationary distributions of biochemical reaction networks via structure transformation

摘要

a The non-weakly reversible and deficiency (δ) one network is translated to the weakly reversible deficiency zero network by merging two reactions, which have the same stoichiometric vectors (green dotted lines). λ~k denotes the propensities of the translated network. b Factorize λ~k with constants κk and functions θ(n) and ω(n) as λ~k(n)=κkθ(n)ω(n−νk)1{n≥νk} on Γ = {n∣n ≥ b} at which λ~k(n)>0 if n ≥ νk + b, and λ~k(n)=0 otherwise. νk is the source complex vector of the kth reaction. c Compute a complex balanced equilibrium (CBE) of the deterministic mass action model for the translated network with rate constants {κk}. d Using the θ(n) and the CBE, the stationary distribution can be derived analytically. Here, M is a normalizing constant.