From Michaelis–Menten parameters to microscopic rate constants: an inversion approach for enzyme kinetics

A basic model of enzyme kinetics is the enzyme-catalyzed reaction, in which a single enzyme E binds a substrate S to form a complex C , which subsequently releases a product P and regenerates the enzyme. According to the mass-action law, this process can be described by a system of four ordinary differential equations for the four unknown concentrations, governed by the microscopic rate constants: forward ( k _f ), reverse ( k _r ), and catalytic ( k _cat ). Under suitable assumptions, the full system reduces to the classical Michaelis-Menten model, depending on two experimentally measurable parameters: the half saturation constant ( K _m ), and the limiting rate V _max . This paper addresses the inverse problem of recovering k _f and k _r from K _m and V _max (or equivalently k _cat , retrievable from V _max ). The proposed method exploits a new identity satisfied by any substrate concentration s ( t ) in the full mass-action system, which provides the starting point for the inversion procedure. Here, a detailed presentation of an algorithm for reaching this goal is provided, along with an assessment of reconstruction accuracy and an outline of simulations and applications. Beyond enzyme kinetics, the approach supports the construction of novel chemical reaction networks, with potential applications to modeling complex biochemical pathways involved in diseases such as cancer.