12/01/2015

# Improving the Simulation Treatment of Microbe-Substrate Kinetics

## Summary

The Michaelis–Menten (MM) kinetics and reverse Michaelis–Menten (RMM) 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. A recent Department of Energy-supported study shows 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. The popular MM kinetics and RMM kinetics are thus inconsistent approximations to their foundation–law of mass action, in that the MM kinetics fails to consider the mass balance constraint from substrate abundance, and the RMM kinetics fails to consider the mass balance constraint from organism abundance. In contrast, when benchmarked with the quadratic kinetics, which is the exact solution to the substrate-uptake problem formulated with the total quasi-steady-state approximation for a single-substrate-single-enzyme system, the ECA appropriately incorporates the mass balance constraints from both substrates and organisms, and predicts consistent parametric sensitivity across a wide range of substrate and organism abundances. This finding resolves the ambiguity in choosing which substrate kinetics for a consistent biogeochemical modeling. The ECA kinetics is expected to motivate a new generation of more robust biogeochemical models for earth system models.

## References

Tang, J. Y. 2015. “On the Relationships Between the Michaelis–Menten Kinetics, Reverse Michaelis–Menten Kinetics, Equilibrium Chemistry Approximation Kinetics, and Quadratic Kinetics,” *Geoscientific Model Development* **8**, 3823–35. DOI: 10.5194/gmd-8-3823-2015.