Rigorous Derivation of Michaelis–Menten Kinetics in the Presence of Slow Diffusion

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5995-6024, October 2024.
Abstract. Reactions with enzymes are critical in biochemistry, where the enzymes act as a catalyst in the process. One of the most widely used mechanisms for modeling enzyme-catalyzed reactions is the Michaelis–Menten (MM) kinetics. On the ODE level, i.e., when concentrations are only time-dependent, this kinetics can be rigorously derived from mass action law using quasi-steady-state approximation. This issue in the PDE setting, for instance, when molecular diffusion is taken into account, is considerably more challenging, and only formal derivations have been established. In this paper, we prove this derivation rigorously and obtain MM kinetics in the presence of slow spatial diffusion. In particular, we show in this case that, in general, the reduced problem is a cross-diffusion-reaction system. Our proof is based on an improved duality method, heat regularization, and a suitable modified energy function. To the best of our knowledge, this work provides the first rigorous derivation of MM kinetics from mass action kinetics in the PDE setting.