Estimation of the number of extreme pathways for metabolic networks

Background: The set of extreme pathways (ExPa), {p_i}, defines the convex basis vectors used for the mathematical characterization of the null space of the stoichiometric matrix for biochemical reaction networks. ExPa analysis has been used for a number of studies to determine properties of metabolic networks as well as to obtain insight into their physiological and functional states in silico. However, the number of ExPas, p = {p_i} , grows with the size and complexity of the network being studied, and this poses a computational challenge. For this study, we investigated the relationship between the number of extreme pathways and simple network properties. Results: We established an estimating function for the number of ExPas using these easily obtainable network measurements. In particular, it was found that log [p] had an exponential relationship with log [∑_i=1^Rd₁d₊₁ c₁], where R = R_eff is the number of active reactions in a network, d_-i and d_+i the incoming and outgoing degrees of the reactions r_i ∈ R_eff, and c_i the clustering coefficient for each active reaction. Conclusion: This relationship typically gave an estimate of the number of extreme pathways to within a factor of 10 of the true number. Such a function providing an estimate for the total number of ExPas for a given system will enable researchers to decide whether ExPas analysis is an appropriate investigative tool.