Mathematical modelling of dynamics and control in metabolic networks. IV. Local stability analysis of single biochemical control loops

The objective of the study presented herein is to describe the dynamic behavior of a single biochemical control loop, a simple system but an important element of metabolic networks. This loop is a self-regulated sequence of reactions that converts an initial substrate (S) into a final product (P). It consists of three basic elements: (1) a regulated reaction, where the concentration of P controls the flux (I) into the system. This element serves as the control element in the feedback circuit. (2) a sequence of unregulated reactions that leads to the formation of P. This process is to be regulated so that the production rate of P meets a desired target. (3) a process (R) that removes P from the loop to another part of the metabolic network. A mathematical description is formulated that consists of two differential equations and two unspecified functions that represent the reaction rates of I and R. This description is scaled to clarify functional dependence and to attempt a separation of genetic and process determined parameters. The global dynamic behavior of the model is assessed qualitatively by examining the occurrence of static and dynamic bifurcations, multiple steady states or sustained oscillations respectively, via local stability analysis. General criteria for both types of bifurcations are developed without specifying the functional form of I and R, but explicitly accounting for the kinetic properties of the reaction chain. A particularly simple criterion is found for static bifurcations which can appear only for loops with positive feedback, i.e. when the regulated reaction is activated by P. This criterion only contains the properties of I and R. The criteria for dynamic bifurcations, which occur when the feedback interaction is inhibitory, are more complex. These depend strongly on the properties of the reaction chain, and oscillations are favored if the dynamic operator describing the reaction sequence is of high order or if it contains time delays.