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

Abstract 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 (kf), reverse (kr), and catalytic (kcat). Under suitable assumptions, the full system reduces to the classical Michaelis-Menten model, depending on two experimentally measurable parameters: the half saturation constant (Km), and the limiting rate Vmax. This paper addresses the inverse problem of recovering kf and kr from Km and Vmax (or equivalently kcat, retrievable from Vmax). 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. Competing Interest Statement The authors have declared no competing interest. Abbreviations - C - enzyme-substrate complex - c - complex concentration - c0 - complex initial concentration - CRN - Chemical Reaction Network - E - enzyme - e - enzyme concentration - e0 - enzyme initial concentration - kcat - microscopic catalitic constant - kf - microscopic forward constant - Km - Michaelis–Menten constant - kr - microscopic reverse constant - MM - Michaelis–Menten - ODE - Ordinary Differential Equation - P - product - p - product concentration - S - substrate - s - substrate concentration - e0 - substrate initial concentration - Vmax - catalytic maximum velocity