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Estimation of the number of extreme pathways for metabolic networks.

Yeung M, Thiele I, Palsson BO - BMC Bioinformatics (2007)

Bottom Line: In particular, it was found that log [p] had an exponential relationship with log[ summation operatori=1Rd-id+ici] where R = /Reff/ is the number of active reactions in a network, d-i and d+i 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.

View Article: PubMed Central - HTML - PubMed

Affiliation: Dept, of Bioengineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0412, USA. palsson@ucsd.edu.

ABSTRACT
ABSTRACT: BACKGROUND: The set of extreme pathways (ExPa), {pi}, defines the convex basis vectors used for the mathematical characterization of the 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] where R = /Reff/ is the number of active reactions in a network, d-i and d+i 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.

No MeSH data available.


Relationship between Reactions with Non-zero Clustering-coefficient and Alternative Routes. Diagram showing relationship between non-zero clustering coefficients and alternative pathways. (i) shows three possible routes for a simple system; (ii) is the non-directed representation of this system using the above projection. The system has non-zero clustering coefficients, emphasizing alternative routes are possible; (iii) is the projection conforming to that shown in Figure 5, where non-zero clustering coefficient is found for 5 of the reactions that are involved in the branching points of alterative routes.
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Figure 6: Relationship between Reactions with Non-zero Clustering-coefficient and Alternative Routes. Diagram showing relationship between non-zero clustering coefficients and alternative pathways. (i) shows three possible routes for a simple system; (ii) is the non-directed representation of this system using the above projection. The system has non-zero clustering coefficients, emphasizing alternative routes are possible; (iii) is the projection conforming to that shown in Figure 5, where non-zero clustering coefficient is found for 5 of the reactions that are involved in the branching points of alterative routes.

Mentions: where ki is the number of reactions that ri is connected to, i.e., ki is the number of non-zero elements of the vector of the matrix . The set {ep, q} denotes the set of edges going from rp to rq with both rp and rq being connected to ri, i.e., rp and rq are connected to ri and are themselves adjacent. The denominator is the number of all such possible edges for a given ri. Note that since we are dealing with a directed graph, ep, q is not the same as eq, p. Alternative pathways from one set of substrates to one set of products almost always exist in biological networks, especially in metabolic networks. Figure 6 shows that non-zero clustering coefficients are related to alternative routes, and could then be an important factor for determining the number of extreme pathways calculated.


Estimation of the number of extreme pathways for metabolic networks.

Yeung M, Thiele I, Palsson BO - BMC Bioinformatics (2007)

Relationship between Reactions with Non-zero Clustering-coefficient and Alternative Routes. Diagram showing relationship between non-zero clustering coefficients and alternative pathways. (i) shows three possible routes for a simple system; (ii) is the non-directed representation of this system using the above projection. The system has non-zero clustering coefficients, emphasizing alternative routes are possible; (iii) is the projection conforming to that shown in Figure 5, where non-zero clustering coefficient is found for 5 of the reactions that are involved in the branching points of alterative routes.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2089122&req=5

Figure 6: Relationship between Reactions with Non-zero Clustering-coefficient and Alternative Routes. Diagram showing relationship between non-zero clustering coefficients and alternative pathways. (i) shows three possible routes for a simple system; (ii) is the non-directed representation of this system using the above projection. The system has non-zero clustering coefficients, emphasizing alternative routes are possible; (iii) is the projection conforming to that shown in Figure 5, where non-zero clustering coefficient is found for 5 of the reactions that are involved in the branching points of alterative routes.
Mentions: where ki is the number of reactions that ri is connected to, i.e., ki is the number of non-zero elements of the vector of the matrix . The set {ep, q} denotes the set of edges going from rp to rq with both rp and rq being connected to ri, i.e., rp and rq are connected to ri and are themselves adjacent. The denominator is the number of all such possible edges for a given ri. Note that since we are dealing with a directed graph, ep, q is not the same as eq, p. Alternative pathways from one set of substrates to one set of products almost always exist in biological networks, especially in metabolic networks. Figure 6 shows that non-zero clustering coefficients are related to alternative routes, and could then be an important factor for determining the number of extreme pathways calculated.

Bottom Line: In particular, it was found that log [p] had an exponential relationship with log[ summation operatori=1Rd-id+ici] where R = /Reff/ is the number of active reactions in a network, d-i and d+i 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.

View Article: PubMed Central - HTML - PubMed

Affiliation: Dept, of Bioengineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0412, USA. palsson@ucsd.edu.

ABSTRACT
ABSTRACT: BACKGROUND: The set of extreme pathways (ExPa), {pi}, defines the convex basis vectors used for the mathematical characterization of the 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] where R = /Reff/ is the number of active reactions in a network, d-i and d+i 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.

No MeSH data available.