Reciprocal enzyme regulation as a source of bistability in covalent modification cycles

Covalent modification cycles (CMCs) are the building blocks of many regulatory networks in biological systems. Under proper kinetic conditions such mono-cyclic enzyme systems can show a higher sensitivity to effectors than enzymes subject to direct allosteric regulation. Using methods from reaction network theory it has been argued that CMCs can potentially exhibit multiple steady states if the converter enzymes are regulated in a reciprocal manner, but the underlying mechanism as well as the kinetic requirements for the emergence of such a behavior remained unclear. Here, we reinvestigate CMCs with reciprocal regulation of the converter enzymes for two common regulatory mechanisms: allosteric regulation and covalent modification. To analyze the steady state behavior of the corresponding mass-action equations, we derive reduced models by means of a quasi-steady state approximation (QSSA). We also derive reduced models using the total QSSA which often better reproduces the transient dynamics of enzyme-catalyzed reaction systems. Through a steady state analysis of the reduced models we show that the occurrence of bistability can be associated with the presence of a double negative feedback loop. We also derive constraints for the model parameters which might help to evaluate the potential significance of the mechanisms described here for the generation of bistability in natural systems. In particular, our results support the view of a possible bistable response in the metabolic PFK1/F1,6BPase cycle as observed experimentally in rat liver extracts, and it suggests an alternative view on the origin of bistability in the Cdk1-Wee1-Cdc25 system.