Estimation of the number of extreme pathways for metabolic networks

ABSTRACT: BACKGROUND: The set of extreme pathways (ExPa), {pi}, 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 = |{pi}|, 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[ summation operatori=1Rd-id+ici] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacyGGSbaBcqGGVbWBcqGGNbWzdaWadaqaamaaqadabaGaemizaq2aaSbaaSqaaiabgkHiTmaaBaaameaacqWGPbqAaeqaaaWcbeaakiabdsgaKnaaBaaaleaacqGHRaWkdaWgaaadbaGaemyAaKgabeaaaSqabaGccqWGJbWydaWgaaWcbaGaemyAaKgabeaaaeaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGsbGua0GaeyyeIuoaaOGaay5waiaaw2faaaaa@4414@, where R = |Reff| is the number of active reactions in a network, d-i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGKbazdaWgaaWcbaGaeyOeI0YaaSbaaWqaaiabdMgaPbqabaaaleqaaaaa@30A9@ and d+i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGKbazdaWgaaWcbaGaey4kaSYaaSbaaWqaaiabdMgaPbqabaaaleqaaaaa@309E@ the incoming and outgoing degrees of the reactions ri in Reff, and ci 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.