Closed Form Solution for Time-dependent Enzyme Kinetics

Based on the quasi-steady-state approximation, a closed form solution for the total time evolution of the reactant concentrations in the basic enzyme−substrate reaction is reported for the first time. Such a solution is given in terms of theomega function, which satisfies the transcendental equationW(x)exp(W(x))=x], and enables the generation of the corresponding time derivatives thus fully characterizing the system. The agreement with results obtained from both the inner (fast region) and outer (slow region) solutions of singular perturbation procedures is very good, but an advantage of the present formalism is the analytic representation of the transition where the perturbation methods are shown to be inaccurate.