Revised Michaelis–Menten Rate Law with Time-Varying Molecular Concentrations

Despite over a century’s use as a dominant paradigm in the description of biochemical rate processes, the Michaelis–Menten (MM) rate law stands on the restrictive assumption that the concentration of the complex of interacting molecules, at each moment, approaches an equilibrium much faster than the molecular concentration changes. The increasingly-appreciated, remedied form of the MM rate law is also based on this quasi-steady state assumption. Although this assumption may be valid for a range of biochemical systems, the exact extent of such systems is not clear. In this study, we relax the quasi-steady state requirement and propose the revised MM rate law for the interactions of molecules with active concentration changes over time. Our revised rate law, characterized by rigorously-derived time delay effects in molecular complex formation, improves the accuracy of models especially for protein–protein and protein–DNA interactions. Our simulation and empirical data analysis show that the improvement is not limited to the quantitatively better characterization of the dynamics, but also allows the prediction for qualitatively new patterns in the systems of interest. The latter include the oscillation condition and period patterns of the mammalian circadian clock and the spontaneous rhythmicity in the degradation rates of circadian proteins, both not properly captured by the previous approaches. Moreover, our revised rate law is capable of more accurate parameter estimation. This work offers an analytical framework for understanding rich dynamics of biomolecular systems, which goes beyond the quasi-steady state assumption.