A graph-theoretic analysis of metabolic regulation in linear pathways with multiple feedback loops and branched pathways

Abstract

With the aid of signal flow graphs, we have analyzed the flux-control distribution in linear metabolic pathways with multiple feedback loops and branched pathways. It is shown that the flux control coefficients of the enzymes in a linear pathway with multiple feedback loops can be evaluated by modifying the signal flow graph of the unregulated pathway in a step-by-step fashion. On the basis of the results obtained with the signal flow graphs, a principle of superposition is suggested for calculating the flux control coefficients of a linear pathway with a general pattern of feedback inhibition. Using this superposition principle, it is possible to determine the flux control coefficients directly from an examination of the topology of the feedback loops in the metabolic pathway, without drawing a signal flow graph. In a branched pathway the control coefficients of the enzymes depend on the fluxes through the various branches in addition to the enzyme elasticities. We show how these fluxes can be incorporated into a signal flow graph from which the flux control coefficients are found. We also describe a systematic procedure for converting a signal flow graph to a simpler form which may significantly reduce the effort necessary for calculating the flux control coefficients. Modifications of a signal flow graph for assessing the relative importance of the enzymes in flux control are also discussed. Based on our findings from the signal flow graphs, we have presented a heuristic method for determining the flux control coefficients directly from the reaction sequence of the pathway, without drawing a signal flow graph. The present analysis applies to metabolic pathways in a steady state.