Co-response Coefficients, Monovalent Units, and Combinatorial Rules: Unification of Concepts in Metabolic Control Analysis

The JacobianHof a linear metabolic pathway without feedback loops is tridiagonal. Its inverse,H−1, which is needed for calculating control coefficients or elasticities, can be decomposed into two regions of mutually dependent rows and columns. For each of these regions ofH−1, all sub-determinants of order two are zero. The existence of the two regions is shown to cause certain invariance properties of the ratios of control coefficients that can be expressed as co-response coefficients. Also the concept of monovalent functional units seems to be related to the existence of the regions. Moreover, the combinatorial rules for selecting the right modulations originate from the fact that certain sub-determinants ofH−1are zero when located within the regions. The regions of zero sub-determinants of order two ofH−1are thus a central element for the analysis of the linear pathway. The unifying potential ofH−1is not restricted to the linear chain but is also expected to be valid for complex metabolic systems.