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 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

The recent recognition that the typical turnover-pool-based soil carbon models cannot model the priming effect has revived the interest in developing microbe explicit soil biogeochemistry models. This has been manifested in a long list of microbial models that were published in the last few years (e.g., Schimel and Weintrub, 2003; Moorhead and Sinsabaugh, 2006; Allison et al., 2010; German et al., 2012; Wang et al., 2013; Wieder et al., 2013; Li et al., 2014; He et al., 2014; Riley et al., 2014; Xenakis and Williams, 2014; Tang and Riley, 2015; Sulman et al., 2014; Wieder et al., 2015). To build a microbial model, the substrate kinetics is fundamental as it describes the rate at which microbes would take up a substrate and represents the first step towards describing how microbes would decompose the soil organic matter. Under the assumption that a single “master reaction” limits the growth of microbes (Johnson and Lewin, 1946), the substrate kinetics even completely determines the microbial dynamics as done in many models (e.g., the Monod model). Among the many mathematical formulations of substrate kinetics (see Tang and Riley, 2013 for a review), the Michaelis–Menten (MM) kinetics is often used because it has succeeded in many applications ever since its creation in the early 20th century (Michaelis and Menten, 1913). However, Schimel and Weintraub (2003) proposed in their study that the decomposition rate should vary more like an asymptotic function of enzyme abundance such that the reverse Michaelis–Menten (RMM) kinetics would better model the soil carbon decomposition dynamics. The proposal of RMM kinetics was motivated by the empirical observation that, as enzyme concentration increases, microbial growth cannot increase continuously without a limit; therefore, some dynamic feedbacks between the different components must stabilize the system. In contrast, the MM kinetics predicts that substrate degradation is proportional to enzyme concentration and, therefore, like the linear kinetics, as used in Schimel and Weintraub (2003), it will predict unstable decomposition dynamics. The success by Schimel and Weintraub has led to a number of studies using the RMM kinetics as the backbone of their microbial models, including the Moorhead and Sinsabaugh (2006) model of litter decomposition, the Drake et al. (2013) model for root priming, the Waring et al. (2013) model for change in microbial community structure in soil carbon and nitrogen cycling, and the Averill (2014) model for change in microbial allocation in soil carbon decomposition.

Wang and Post (2013) pointed out that both the MM kinetics and RMM kinetics (although the latter is empirical) are special approximations to the quadratic kinetics that exactly solves for the enzyme–substrate complex under the quasi-steady-state approximation (QSSA), which states that the enzyme–substrate complexes are in instantaneous equilibrium with enzyme and substrate concentrations (Borghans et al., 1996). They further concluded that the MM kinetics is applicable when the substrate concentration far exceeds the enzyme concentration, and the RMM kinetics is applicable when either the enzyme concentration far exceeds the substrate concentration or vice versa. The condition for the MM kinetics to be applicable as provided by Wang and Post (2013) was however much narrower than that was proposed in some earlier studies. For instance, Borghans et al. (1996) showed that the MM kinetics is a good approximation to the quadratic kinetics when enzyme concentration is far smaller than the sum of the substrate concentration and the Michaelis–Menten constant (Palsson, 1987; Segel, 1988; Segel and Slemrod, 1989). To handle enzyme–substrate interactions under high enzyme concentrations, Borghans et al. (1996) proposed the total quasi-steady-state approximation (tQSSA) and obtained a substrate kinetics that was a special case of the later proposed equilibrium chemistry approximation (ECA) kinetics by Tang and Riley (2013). Tang and Riley (2013) applied the law of mass action with tQSSA and derived the ECA kinetics to describe the formation of enzyme–substrate complexes in a network of an arbitrary number of enzymes and substrates.

The consistent application of mathematical formulations to describe a
dynamic system is critical for the model to successfully resolve the
empirical measurements that observe the dynamic system. This consistency
requirement has been raised in several studies using microbe explicit
models. For instance, Maggi and Riley (2009) have found the MM kinetics has
to be revised to resolve the evolution of

The call for a substrate kinetics that can consistently work across a wide range of substrate and enzyme (or more broadly competitor) concentrations becomes more imperative when land biogeochemical models are required to resolve plant–microbe interactions. In plant–microbe interactions, both substrates and competitors evolve constantly and their concentration ratios could vary by orders of magnitude. For instance, when a soil is seriously nitrogen limited, the aqueous nitrogen concentration is much lower than the volumetric density of competitors and substrate uptake may follow more linearly with respect to the substrate concentration and be of an asymptotic function of competitors as described by the RMM kinetics. However, when a large dose of fertilizer is added, the soil quickly becomes nitrogen saturated, such that the uptake dynamics would follow more linearly with respect to the variation of competitors (or enzymes) as represented in the MM kinetics. To consistently model the soil nitrogen dynamics that fluctuates between status of nitrogen limitation and nitrogen saturation, one therefore has to constantly choose between the MM kinetics and RMM kinetics, making a consistent mathematical formulation theoretically impossible. Therefore, an approach that includes the advantages from both the MM kinetics and RMM kinetics will significantly advance our capability in modeling soil biogeochemical processes. Fortunately, such kinetics (a.k.a. the ECA kinetics) was already derived in Tang and Riley (2013), but my coauthor and I did not give a theoretical analysis for the relationships between MM kinetics, RMM kinetics, and the ECA kinetics, nor did we explain how the parametric sensitivity would vary depending on the choice of substrate kinetics and whether the ECA kinetics is superior across the whole range of feasible kinetic parameters. Because all model calibration methods either explicitly or implicitly rely on the parametric sensitivity to tune model predictions with respect to observations (e.g., Tang and Zhuang, 2009; Zhu and Zhuang, 2014), correct parametric sensitivity of the model formulation is a requisite for delivering a robust model. An analysis of the differences in their predicted parametric sensitivities will also help to reveal the pitfalls that may exist in biogeochemical models that rely on the MM kinetics (Allison et al., 2010) or RMM kinetics (e.g., Averill, 2014) or a combination of the two (e.g., Sihi et al., 2015) when the model is otherwise benchmarked against its equilibrium-chemistry-based formulation that solves the biogeochemical system exactly under the tQSSA (readers please refer to Tang and Riley (2013) for a thorough discussion on why the equilibrium chemistry formulation should be the benchmark for models based on the MM kinetics, RMM kinetics, and ECA kinetics).

In this study, I first review how the ECA kinetics could be derived from the quadratic kinetics and how the MM kinetics and RMM kinetics could be derived from the ECA kinetics or directly from the equilibrium chemistry formulation of the enzyme–substrate interaction. Then I analyze how accurate the MM kinetics, the RMM kinetics, and the ECA kinetics could approximate the parametric sensitivity, as one would derive from the quadratic kinetics that is exact for the one-enzyme and one-substrate biogeochemical reaction. Based on these analyses, I finally give recommendations on how to obtain more robust microbial models for soil biogeochemical modeling. Note that, although the following analysis is for a single-enzyme and single-substrate system in an aqueous solution, the results are applicable to a wide range of problems, including predator–prey, microbial growth, Langmuir adsorption, and any process that can be appropriately formulated as an equilibrium binding problem (Tang and Riley, 2013).

Below I first review how one could obtain the quadratic kinetics under the QSSA for a biogeochemical reaction that involves one enzyme and one substrate. Then I show how one could derive the ECA kinetics, the MM kinetics, and the RMM kinetics.

The biogeochemical reaction of the system is

By the law of mass action, the governing equations for biogeochemical
Reaction (1) are

Here and below, I use square brackets to designate the concentration (mol m

Under the QSSA, Eq. (4) is approximated as

For a small temporal window when the amount of new product is negligible, it
holds that

By solving

Therefore, if one applies the quadratic formula to Eq. (9) and takes the physically meaningful solution,

To obtain the ECA formulation of the enzyme–substrate complex, one assumes

Then by substitution of the first-order approximation

The application of Eq. (12) implies

The MM kinetics can be derived in two different approaches. In the first
approach, by assuming

In the second approach, one solves

Note

When

The sufficient condition

There are also two approaches to derive the RMM kinetics. In the first
approach, one assumes

In the second approach, one solves

Note

Here the condition

In the following I analyze the sensitivities of the reaction velocity with respect to
the four parameters as predicted by the four kinetics. The four parameters
are (1) maximum product genesis rate

In evaluating the parametric sensitivity, I made the conventional assumption
that

In addition, to simplify the presentation, I define

In all the analyses below, I represent the parametric sensitivity using the
normalized form

The normalized sensitivities of the reaction velocity vs.

From the equations above, it is observed that both the MM kinetics and RMM kinetics predict a less variable and lower parametric sensitivity than the ECA kinetics, because the ECA kinetics predicts a more variable and larger denominator in the second term (in Eq. 24) as compared to that by the MM kinetics (Eq. 25) and RMM kinetics (Eq. 26). Large deviations between predicted sensitivities by the MM kinetics and the ECA kinetics are expected at high enzyme concentrations, whereas large deviations between predictions by the RMM kinetics and ECA kinetics are expected at high substrate concentrations. Predicted sensitivities by the MM kinetics and RMM kinetics are also smaller than those by the quadratic kinetics (green and black dots in Fig. 1d). In contrast, the ECA kinetics consistently captures the variability of the normalized sensitivity, with some overestimation (but the relative difference is no greater than 5 %) under moderate enzyme and substrate concentrations (Fig. 1c), where the normalized sensitivity is, however, small or moderate (Fig. 1a).

The normalized sensitivities of the reaction velocity vs.

From Eqs. (28)–(30), it is
inferred that both the MM kinetics and RMM kinetics predict a less variable
and higher normalized sensitivity with respect to

Similar to Fig. 1, but the sensitivity is evaluated against the
intrinsic substrate affinity

The normalized sensitivities of the reaction velocity vs.

From the equations above, it is observed that the MM kinetics predicts a constant
normalized sensitivity of the reaction velocity with respect to the total
enzyme concentration

The normalized sensitivities of the reaction velocity vs.

Similar to Fig. 1, but the sensitivity is evaluated against the
total enzyme concentration

Because

Similar to Fig. 1, but the sensitivity is evaluated against the
total substrate concentration

From the above analyses, I showed that the ECA kinetics is a better
approximation to the quadratic kinetics, which, obtained from the law of
mass action and the quasi-steady-state approximation, is the exact solution
to the governing equations of substrate-enzyme interaction (as indicated by
Eqs. 6–8). In
contrast, the Michaelis–Menten kinetics and the reverse Michaelis–Menten
kinetics are inferior in approximating the quadratic kinetics over the wide
range of enzyme and substrate concentrations. The worse performance of the
MM kinetics than the ECA kinetics in approximating the quadratic kinetics
stems from the negligence of mass balance constraint of the substrate during
the derivation of the MM kinetics; while the worse performance of the RMM
kinetics in approximating the quadratic kinetics is caused by the negligence
of mass balance constraint of the enzyme during the derivation of the RMM
kinetics. The failure to consider the mass balance constraints for both
enzyme and substrate during their derivations caused the MM kinetics and the
RMM kinetics to predict significantly biased normalized sensitivity of the
reaction velocity with respect to the two kinetic parameters

In modeling complex soil biogeochemical dynamics, the consistency between the kinetics used and the equilibrium chemistry formulation of the relationships between enzymes, substrates, and enzyme–substrate complexes might be critical (Tang and Riley, 2013), but it has been unfortunately underappreciated in many previous studies. In Tang and Riley (2013), it was shown that, for a system involving three microbes competitively decomposing three carbon substrates, the MM kinetics failed wildly even with industrious calibration (see their Fig. 12). In an earlier study, Moorhead and Sinsabaugh (2006) had to prescribe the relative decomposition between lignin and cellulose in order to resolve the lignocellulose index dynamics. The ECA kinetics was able to consistently resolve the lignin–cellulose dynamics during the litter decomposition because it explicitly considers the mass balance constraints for each of the substrates and enzymes (or, effectively, abundance of competitors; Tang and Riley, 2013). Both the success of the ECA kinetics and the failure of the MM kinetics in abovementioned studies can be traced back to their capability in approximating the actual parametric sensitivities of the specific dynamic system. Because all model calibration techniques rely on a model's parametric sensitivity to obtain improved agreement between model predictions and measurements, wrong parametric sensitivity as formulated in the adopted substrate kinetics would result in a non-calibratable or poorly calibratable model, which could be manifested as systematic model biases or completely unreasonable model predictions. This explained well why the MM-kinetics-based model in Tang and Riley (2013) failed wildly even with intensive Bayesian model calibration.

Therefore, if the ecological dynamics involved in substrate processing by
microbes does approximately obey the law of mass action and the
total quasi-steady-state approximation (as it is already implied in any
microbe explicit model that uses the MM kinetics or the RMM kinetics), then
the analytically tractable ECA kinetics is a much more powerful and
mathematically more consistent tool than the popular MM kinetics and RMM
kinetics that are currently used in many microbial models. Indeed, a recent
application (Zhu and Riley, 2015) indicated that by representing
plant–microbe competition of soil mineral nitrogen using the ECA kinetics,
the predicted global nitrogendynamics became much more consistent with that
inferred from the

Using the definitions

Then, from Eq. (A1), one has

By substitution of Eqs. (A3), (A7), and (A11) into (A13), and using the
definition of

Similarly, from Eq. (A1), one has

Taking the partial derivative with respect to

Note, because switching the order of

Using the definitions of

Similarly, from Eq. (B1), one has

For

Then, by dividing both sides of Eq. (B7) with

By using the symmetry between

J. Y. Tang developed the theory, conducted the analyses, and wrote the paper.

This research is supported by the Director, Office of Science, Office of Biological and Environmental Research of the US Department of Energy under contract no. DE-AC02-05CH11231 as part of the Next-Generation Ecosystem Experiments (NGEE-Arctic) and the Accelerated Climate Model for Energy project in the Earth System Modeling program. I sincerely thank Thomas Wutzler, Joshua Schimel, an anonymous reviewer, and the handling editor for their constructive comments, which significantly improved the paper.Edited by: C. Sierra