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The Michaelis–Menten kinetics and the reverse Michaelis–Menten kinetics aretwo popular mathematical formulations used in many land biogeochemical modelsto describe how microbes and plants would respond to changes in substrateabundance. However, the criteria of when to use either of the two are oftenambiguous. Here I show that these two kinetics are special approximations tothe equilibrium chemistry approximation (ECA) kinetics, which is the first-orderapproximation to the quadratic kinetics that solves the equation ofan enzyme–substrate complex exactly for a single-enzyme and single-substratebiogeochemical reaction with the law of mass action and the assumption ofa quasi-steady state for the enzyme–substrate complex and that the productgenesis from enzyme–substrate complex is much slower than the equilibrationbetween enzyme–substrate complexes, substrates, and enzymes. In particular, Ishow that the derivation of the Michaelis–Menten kinetics does not considerthe mass balance constraint of the substrate, and the reverseMichaelis–Menten kinetics does not consider the mass balance constraint ofthe enzyme, whereas both of these constraints are taken into account inderiving the equilibrium chemistry approximation kinetics. By benchmarkingagainst predictions from the quadratic kinetics for a wide range of substrateand enzyme concentrations, the Michaelis–Menten kinetics was found topersistently underpredict the normalized sensitivity ∂ ln v /∂ ln k2+ of the reaction velocity v with respect to themaximum product genesis rate k2+, persistently overpredict thenormalized sensitivity ∂ ln v / ∂ ln k1+ of vwith respect to the intrinsic substrate affinity k1+, persistentlyoverpredict the normalized sensitivity ∂ ln v / ∂ ln[E]T of v with respect the total enzyme concentration[E]T, and persistently underpredict the normalizedsensitivity ∂ ln v / ∂ ln [S]T of vwith respect to the total substrate concentration [S]T.Meanwhile, the reverse Michaelis–Menten kinetics persistently underpredicts∂ ln v / ∂ ln k2+ and ∂ ln v /∂ ln [E]T, and persistently overpredicts∂ ln v / ∂ ln k1+ and ∂ ln v /∂ ln [S]T. In contrast, the equilibrium chemistryapproximation kinetics always gives consistent predictions of ∂ lnv / ∂ ln k2+, ∂ ln v / ∂ ln k1+, ∂ ln v / ∂ ln [E]T, and ∂ln v / ∂ ln [S]T, indicating that ECA-basedmodels will be more calibratable if the modeled processes do obey the law ofmass action. Since the equilibrium chemistry approximation kinetics includesadvantages from both the Michaelis–Menten kinetics and the reverseMichaelis–Menten kinetics and it is applicable for almost the whole range ofsubstrate and enzyme abundances, land biogeochemical modelers therefore nolonger need to choose when to use the Michaelis–Menten kinetics or thereverse Michaelis–Menten kinetics. I expect that removing this choice ambiguitywill make it easier to formulate more robust and consistent landbiogeochemical models

geoscientific model development

On the relationships between the Michaelis–Menten kinetics, reverse Michaelis–Menten kinetics, equilibrium chemistry approximation kinetics, and quadratic kinetics