The reversible Hill equation: how to incorporate cooperative enzymes into metabolic models

Realistic simulation of the kinetic properties of metabolic pathways requires rate equations to be expressed in reversible form, because substrate and product elasticities are drastically different in reversible and irreversible reactions. This presents no special problem for reactions that follow reversible Michaelis-Menten kinetics, but for enzymes showing cooperative kinetics the full reversible rate equations are extremely complicated, and anyway in virtually all cases the full equations are unknown because sufficiently complete kinetic studies have not been carried out. There is a need, therefore, for approximate reversible equations that allow convenient simulation without violating thermodynamic constraints.

We show how the irreversible Hill equation can be generalized to a reversible form, including effects of modifiers. The proposed equation leads to behaviour virtually indistinguishable from that predicted by a kinetic form of the Adair equation, despite the fact that the latter is a far more complicated equation. By contrast, a reversible form of the Monod-Wyman-Changeux equation that has sometimes been used leads to predictions for the effects of modifiers at high substrate concentration that differ qualitatively from those given by the Adair equation.