Elasticity analysis and design for large metabolic responses produced by changes in enzyme activities

Metabolic control analysis has been extensively used to describe how the sensitivity properties of the component enzymes in a metabolic pathway (represented by the elasticity coefficients) determine the way in which metabolic variables respond (described by the control coefficients). Similarly, metabolic control design addresses the inverse problem of obtaining the sensitivity properties of the component enzymes that are required for the system to show a pre-established pattern of responses. These formalisms, including what is called elasticity analysis and design, were developed for small, strictly speaking infinitesimal, changes. Here we extend them to large metabolic responses. The new approach can be applied to simple two-step pathways or to any arbitrary metabolic system divided into two groups linked by one intermediate. General expressions that relate control and elasticity coefficients for large changes are derived. Concentration and flux connectivity relationships are obtained. The relationships for large changes indicate that the pattern of responses is not necessarily the same as the one obtained with the traditional infinitesimal approach, in some cases the patterns being qualitatively different. The general analysis is used to study the control of ketogenesis in rat liver mitochondria, starting from data available in the literature. The control profile of the pathway subject to large changes shows both quantitative and qualitative differences from the one obtained from an analysis that is performed with infinitesimal coefficients. This exemplifies the type of errors that may be introduced when drawing conclusions about large metabolic responses from results obtained with an infinitesimal treatment.