Quasi-steady-state kinetics at enzyme and substrate concentrations in excess of the Michaelis–Menten constant

In vitro enzyme reactions are traditionally conducted under conditions of pronounced substrate excess since this guarantees that the bound enzyme is at quasi-steady-state (QSS) with respect to the free substrate, thereby justifying the Briggs–Haldane approximation (BHA). In contrast, intracellular reactions, amplification assays, allergen digestion assays and industrial applications span a range of enzyme-to-substrate ratios for which the BHA is invalid, including the extreme of enzyme excess. The quasi-equilibrium approximation (QEA) is valid for a subset of enzyme excess states. Previously, we showed that the total QSSA (tQSSA) overlaps and extends the validity of the BHA and the QEA, and that it is at least roughly valid for any total substrate and enzyme concentrations. The analysis of the tQSSA is hampered by square root nonlinearity. Previous simplifications of the tQSSA rate law are valid in a parameter domain that overlaps the validity domains of the BHA and the QEA and only slightly extends them. We now integrate the tQSSA rate equation in closed form, without resorting to further approximations. Moreover, we introduce a complimentary simplification of the tQSSA rate law that is valid in states of enzyme excess when the absolute difference between total enzyme and substrate concentrations greatly exceeds the Michaelis–Menten constant. This includes a wide range of enzyme and substrate concentrations where both the BHA and the QEA are invalid and allows us to define precisely the conditions for zero-order and first-order product formation. Remarkably, analytical approximations provided by the tQSSA closely match the expected stochastic kinetics for as few as 15 reactant molecules, suggesting that the conditions for the validity of the tQSSA and for its various simplifications are also of relevance at low molecule numbers.