Use of implicit methods from general sensitivity theory to develop a systematic approach to metabolic control. II. Complex systems

In the accompanying paper (Cascante et al., this issue) we have used general sensitivity theory to develop a matrix algebra that, in the case of sequential reactions, directly relates global and local properties of a given system. In complex biochemical systems this direct relationship is not possible due to the existence of linear dependencies among fluxes and among metabolite concentrations (conserved aggregate concentrations in BST or moiety-conserved concentrations in MCT). In this paper our matrix algebra is applied to conserved cycles and branched pathways, and it is shown that with minor modifications it again relates global properties to the local properties of the enzymes in the system. In the case of conserved cycles, elasticities become modified due to the existence of linear dependencies among the concentration variables in the cycle. In branched pathways, new matrix elements involving ratios of fluxes appear. With these modifications, one can show that the so-called theorems of metabolic control theory specific to these types of pathways are special cases of more general relationships. Rules for the construction of matrices relating global and local properties are given that apply to an arbitrary system of cycles and branches. The implicit approach developed in these papers, which is a generalization of that used in MCT, allows one to make more direct comparisons with the general explicit approach originally developed in BST.