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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. 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.

Summary: All proteins are translated in the cytoplasm, yet many, including transcription factors, play vital roles in the nucleus. While previous research has concentrated on molecular motors for the transport of these proteins to the nucleus, recent observations reveal perinuclear accumulation even in the absence of an energy source, hinting at alternative mechanisms. Here, we propose that structural properties of the cellular environment, specifically the endoplasmic reticulum (ER), can promote molecular transport to the perinucleus without requiring additional energy expenditure. Specifically, physical interaction between proteins and the ER impedes their diffusion and leads to their accumulation near the nucleus. This result explains why larger proteins, more frequently interacting with the ER membrane, tend to accumulate at the perinucleus. Interestingly, such diffusion in a heterogeneous environment follows Chapman’s law rather than the popular Fick’s law. Our findings suggest a novel protein transport mechanism arising solely from characteristics of the intracellular environment.