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

Abstract

Abstract. The Michaelis-Menten kinetics and the reverse Michaelis-Menten kinetics are two popular mathematical formulations used in many land biogeochemical models to describe how microbes and plants would respond to changes in substrate abundance. However, the criteria of when to use either of the two are often ambiguous. Here I show that these two kinetics are special approximations to the equilibrium chemistry approximation (ECA) kinetics, which is the first-order approximation to the quadratic kinetics that solves the equation of an enzyme-substrate complex exactly for a single-enzyme and single-substrate biogeochemical reaction with the law of mass action and the assumption of a quasi-steady state for the enzyme-substrate complex and that the product genesis from enzyme-substrate complex is much slower than the equilibration between enzymesubstrate complexes, substrates, and enzymes. In particular, I show that the derivation of the Michaelis-Menten kinetics does not consider the mass balance constraint of the substrate, and the reverse Michaelis-Menten kinetics does not consider the mass balance constraint of the enzyme, whereas both of these constraints are taken into account in deriving the equilibrium chemistry approximation kinetics. By benchmarking against predictions from the quadratic kinetics for a wide range of substrate and enzyme concentrations, the Michaelis-Menten kinetics was found to persistently underpredict the normalized sensitivity ∂ ln v/∂ ln k + 2 of the reaction velocity v with respect to the maximum product genesis rate k T , indicating that ECAbased models will be more calibratable if the modeled processes do obey the law of mass action. Since the equilibrium chemistry approximation kinetics includes advantages from both the Michaelis-Menten kinetics and the reverse Michaelis-Menten kinetics and it is applicable for almost the whole range of substrate and enzyme abundances, land biogeochemical modelers therefore no longer need to choose when to use the Michaelis-Menten kinetics or the reverse Michaelis-Menten kinetics. I expect that removing this choice ambiguity will make it easier to formulate more robust and consistent land biogeochemical models.

Merging a mechanistic enzymatic model of soil heterotrophic respiration into an ecosystem model in two AmeriFlux sites of northeastern USA

“…Consistent with the "big microsite" approach articulated for the DAMM model structure (Davidson et al, 2014), we assume that the majority of extracellular enzymes can be represented by similar kinetic parameters, such as Ea and Km. Obviously, this is not universally true (Tang, 2015;Tang and Riley, 2013), just as all leaves simulated in "big leaf" models do not behave identically (Baldocchi and Meyers, 1998), but representing soil microbial metabolism at a "big microsite" permits model parsimony, and we test here its effectiveness.…”

“…Incorporation of similar MCNiP algorithms into DAMM-FöBAAR or other ecosystem models and ESMs may provide further improvements, although it adds more parameters and reduces parsimony. Because it simulates both variation in enzyme production and diffusional constraints of substrate supply, DAMM-MCNiP was also able to apply equilibrium chemistry approximation (ECA) kinetics, which may be superior to using forward or reverse M-M approaches when enzyme and substrate concentrations can be simulated or measured independently (Tang, 2015;Tang and Riley, 2013). Similarly, incorporating microbial CUE could be valuable, as it largely dictates the uncertainty in longterm soil C stocks (Sihi et al, 2017), but also requires additional parameterizations that remain challenging due to variation in its fundamental definitions and measurement techniques.…”

“…For Howland Forest we ran the evergreen version of the model (Carbone et al, 2016) and for Harvard Forest we ran the deciduous version (Keenan et al, 2012) of the model. We optimized the model using the period when we had Rh constraints for the DAMM module from the trenching experiments along with the total soil R measurements (from control plots) using automated chambers and eddy-covariance measurements from tower -2015for Howland Forest and 2013-2014. Model validations were conducted for decadal-scale soil R data as far back in as we have soil moisture measurements within the tower footprint.…”

Comparing models of microbial–substrate interactions and their response to warming

“…arrived at a reverse Michaelis-Menten depolymerisation function if enzymes only adsorbed to a fraction of binding sites because of complex substrates. Transitions between FWD and REV model behaviour have also been detailed in the more complex Equilibrium Chemistry Approximation model that also included sorption of enzymes and substrates to mineral surfaces (Tang and Riley, 2015;Tang, 2015). Our analysis shows that the positive feedback between decomposition and microbial growth is removed, as our REV model now has a stable short-term QSS.…”

“…This simplification can be justified with fast and efficient scavenging of microbes and, thus, fast turnover of the DOC pool. Further sensitivity analysis may shed light on the dynamics across the full parameter space, while using the simplified linear terms (Appendices B and C; Tang, 2015), particularly also because many of the parameters are difficult to estimate. Furthermore, we did not include nutrient requirements of microbes where considering the stoichiometric requirements can change the allocation of resources to optimise enzyme synthesis.…”