Beyond homogeneity: Assessing the validity of the Michaelis-Menten rate law in spatially heterogeneous environments

Abstract

The Michaelis-Menten (MM) rate law has been a fundamental tool in describing enzyme-catalyzed reactions for over a century. When substrates and enzymes are homogeneously distributed, the validity of the MM rate law can be easily assessed based on relative concentrations: the substrate is in large excess over the enzyme-substrate complex. However, the applicability of this conventional criterion remains unclear when species exhibit spatial heterogeneity, a prevailing scenario in biological systems. Here, we explore the MM rate law's applicability under spatial heterogeneity by using partial differential equations. In this study, molecules diffuse very slowly, allowing them to locally reach quasi-steady states at each time point. We find that the conventional criterion for the validity of the MM rate law cannot be readily extended to heterogeneous environments solely through spatial averages of molecular concentrations. That is, even when the conventional criterion for the spatial averages is satisfied, the MM rate law fails to capture the enzyme catalytic rate under spatial heterogeneity. In contrast, a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA), is accurate. Specifically, the tQSSA-based modified form, but not the original MM rate law, accurately predicts the drug clearance via cytochrome P450 enzymes and the ultrasensitive phosphorylation in heterogeneous environments. Our findings shed light on how to simplify spatiotemporal models for enzyme-catalyzed reactions in the right context, ensuring accurate conclusions and avoiding misinterpretations in in silico simulations. For over a century, scientists have relied on the simple Michaelis-Menten (MM) rate law to explain how enzymes function. The conventional criterion for using the MM rate law has been derived in homogeneous environments where species are evenly dispersed (e.g., in vitro experiments). Here, we found that the conventional criterion does not work in intracellular environments where species are heterogeneously distributed and diffuse slowly. However, we find that a slightly modified formula of the MM rate law, based on total quasi-steady state approximation (tQSSA), is accurate even in heterogeneous environments. In particular, this modified formula, unlike the MM rate law, accurately predicts the rate of drug metabolism and ultrasensitive phosphorylation when species are not evenly distributed. Our results provide insight into how to use simplified models for describing enzyme functions in such non-evenly distributed environments.