Generalized Michaelis-Menten rate law with time-varying molecular concentrations

Abstract

The Michaelis-Menten (MM) rate law has been the dominant paradigm of modeling biochemical rate processes for over a century with applications in biochemistry, biophysics, cell biology, systems biology, and chemical engineering. The MM rate law and its remedied form stand on the assumption that the concentration of the complex of interacting molecules, at each moment, approaches an equilibrium (quasi-steady state) much faster than the molecular concentrations change. Yet, this assumption is not always justified. Here, we relax this quasi-steady state requirement and propose the generalized MM rate law for the interactions of molecules with active concentration changes over time. Our approach for time-varying molecular concentrations, termed the effective time-delay scheme (ETS), is based on rigorously estimated time-delay effects in molecular complex formation. With particularly marked improvements in protein-protein and protein-DNA interaction modeling, the ETS provides an analytical framework to interpret and predict rich transient or rhythmic dynamics (such as autogenously-regulated cellular adaptation and circadian protein turnover), which goes beyond the quasi-steady state assumption. The Michaelis-Menten (MM) rate law has enjoyed for over a century the status of the de facto standard of modeling enzymatic reactions. Despite its simple and intuitive interpretation for a wide range of applications in biochemistry, biophysics, cell biology, systems biology, and chemical engineering, the MM rate law and its modified form stand on the quasi-steady state assumption, which is not necessarily justified under active molecular concentration changes over time. Here, we relax this assumption and propose the generalized MM rate law where the effective time delay in molecular complex formation comes into pivotal play. This scheme allows the analytical interpretation and prediction of various biochemical processes with transient or rhythmic dynamics, opening a new avenue of applications beyond the previous approaches.