On the relationships between the Michaelis–Menten kinetics, reverse Michaelis–Menten kinetics, equilibrium chemistry approximation kinetics, and quadratic kinetics
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 firstorder approximation to the quadratic kinetics that solves the equation of an enzyme–substrate complex exactly for a singleenzyme and singlesubstrate biogeochemical reaction with the law of mass action and the assumption of a quasisteady state for the enzyme–substrate complex and that the product genesis from enzyme–substrate complex is much slower than the equilibration between enzyme–substrate 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 tomore »
 Authors:

^{[1]}
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Department of Climate Sciences
 Publication Date:
 Grant/Contract Number:
 AC0205CH11231
 Type:
 Published Article
 Journal Name:
 Geoscientific Model Development (Online)
 Additional Journal Information:
 Journal Name: Geoscientific Model Development (Online); Journal Volume: 8; Journal Issue: 12; Journal ID: ISSN 19919603
 Publisher:
 European Geosciences Union
 Research Org:
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), Biological and Environmental Research (BER) (SC23)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 58 GEOSCIENCES; 97 MATHEMATICS AND COMPUTING; 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY
 OSTI Identifier:
 1227325
 Alternate Identifier(s):
 OSTI ID: 1378663
Tang, J. Y.. On the relationships between the Michaelis–Menten kinetics, reverse Michaelis–Menten kinetics, equilibrium chemistry approximation kinetics, and quadratic kinetics. United States: N. p.,
Web. doi:10.5194/gmd838232015.
Tang, J. Y.. On the relationships between the Michaelis–Menten kinetics, reverse Michaelis–Menten kinetics, equilibrium chemistry approximation kinetics, and quadratic kinetics. United States. doi:10.5194/gmd838232015.
Tang, J. Y.. 2015.
"On the relationships between the Michaelis–Menten kinetics, reverse Michaelis–Menten kinetics, equilibrium chemistry approximation kinetics, and quadratic kinetics". United States.
doi:10.5194/gmd838232015.
@article{osti_1227325,
title = {On the relationships between the Michaelis–Menten kinetics, reverse Michaelis–Menten kinetics, equilibrium chemistry approximation kinetics, and quadratic kinetics},
author = {Tang, J. Y.},
abstractNote = {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 firstorder approximation to the quadratic kinetics that solves the equation of an enzyme–substrate complex exactly for a singleenzyme and singlesubstrate biogeochemical reaction with the law of mass action and the assumption of a quasisteady state for the enzyme–substrate complex and that the product genesis from enzyme–substrate complex is much slower than the equilibration between enzyme–substrate 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 k2+ of the reaction velocity v with respect to the maximum product genesis rate k2+, persistently overpredict the normalized sensitivity ∂ ln v / ∂ ln k1+ of v with respect to the intrinsic substrate affinity k1+, persistently overpredict the normalized sensitivity ∂ ln v / ∂ ln [E]T of v with respect the total enzyme concentration [E]T, and persistently underpredict the normalized sensitivity ∂ ln v / ∂ ln [S]T of v with 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 chemistry approximation kinetics always gives consistent predictions of ∂ ln v / ∂ ln k2+, ∂ ln v / ∂ ln k1+, ∂ ln v / ∂ ln [E]T, and ∂ ln v / ∂ ln [S]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.},
doi = {10.5194/gmd838232015},
journal = {Geoscientific Model Development (Online)},
number = 12,
volume = 8,
place = {United States},
year = {2015},
month = {12}
}