Several land biogeochemical models used for studying carbon–climate
feedbacks have begun explicitly representing microbial dynamics. However, to
our knowledge, there has been no theoretical work on how to achieve a
consistent scaling of the complex biogeochemical reactions from microbial
individuals to populations, communities, and interactions with plants and
mineral soils. We focus here on developing a mathematical formulation of the
substrate–consumer relationships for consumer-mediated redox reactions of the
form

Near-surface soils hold more than twice the carbon in the current
atmosphere; therefore, a small change in land carbon dynamics can imply
significant feedbacks to the ongoing climate warming (Ciais et al., 2013).
This sensitivity has motivated research to better understand Earth's land
biogeochemical cycles, both for prediction and assessing the efficacy of
climate mitigation and adaptation strategies. To date, however, soil
biogeochemical models suffer from high uncertainty (e.g., Arora et al., 2013;
Bouskill et al., 2014; Friedlingstein et al., 2014; He et al., 2016). For
instance, eight CMIP5 Earth system models (ESMs) predicted that net land
carbon uptake varies from 22 to 456 Pg C for the 2006–2100 period under
the Representative Concentration Pathway 4.5 (RCP4.5; Shao et al., 2013).
Similarly, Todd-Brown et al. (2013) estimated that 16 CMIP5 ESMs predicted
contemporary global soil carbon stocks ranging from 510 to 3040 Pg C to
1 m depth, while the most recent empirical estimation is
1408

The predictive power of existing land biogeochemical models is diminished by
uncertainties from structural design, numerical implementation, model
parameterization, initial conditions, and forcing data (Tang and Zhuang,
2008; Tang et al., 2010; Luo et al., 2015; Wieder et al., 2015a; Blanke et
al., 2016; Tang and Riley, 2016). Among these, developing better model
structures and mathematical formulations have been identified as priorities.
One proposed structural improvement is to include explicit microbial dynamics
(Wieder et al., 2015b), which may enable better predictions of global soil
carbon stocks (Wieder et al., 2013), priming effects (Sulman et al., 2014),
vertical soil carbon profiles (Riley et al., 2014; Dwivedi et al., 2017), and
respiratory temperature sensitivity (Tang and Riley, 2015). A second proposal
is to explicitly resolve ecosystem nutrient cycles, following the hypothesis
that the potential for increasing land ecosystem carbon uptake from
atmospheric CO

A common feature that underlies these two proposed model structural improvements is substrate–consumer interactions, which affect microbial substrate decomposition (Grant et al., 1993; Tang and Riley, 2013a; Riley et al., 2014; Le Roux et al., 2016), mineral soil interactions with adsorptive substrates (Smith, 1979; Grant et al., 1993; Resat et al., 2012; Tang and Riley, 2015; Dwivedi et al., 2017), and plant–microbe competition for nutrients (Grant, 2013; Zhu et al., 2016a, b, 2017). In soil, because there are many consumers competing for many substrates in different places at different times, soil biogeochemical models must be able to scale consistently across space, time, and processes. Scaling across spatial and temporal dimensions is achieved through spatial and temporal discretization and integration, which has been intensively studied elsewhere (e.g., Kolditz et al., 1998; Mao et al., 2006). Here, we examine scaling along the less-studied third dimension (process), focusing on development of a consistent mathematical formulation of substrate–consumer interactions.

Previously, we studied a simple configuration of this consumer–substrate interaction, i.e., the network of single-substrate Monod-type reactions (discussed later), and developed a scaling method, the equilibrium chemistry approximation (ECA) kinetics (Tang and Riley, 2013a). ECA kinetics significantly improved the modeling of plant–microbial nutrient competition in the ACME land biogeochemical model (Zhu and Riley, 2015; Zhu et al., 2016a, b, 2017) and was recently cited as one of the most promising methods to improve representation of nutrient competition in ESMs (Achat et al., 2016; Niu et al., 2016). The ECA method also successfully explained why organomineral interactions can slow soil organic matter decomposition rates and how lignin–cellulose ratios (Melillo et al., 1989) can be stabilized during litter decomposition (Tang and Riley, 2013a, 2015).

Following Tang and Riley (2013a), we start our analysis here by assuming a
certain homogeneous space–time–process unit in soil, within which there are
generally three types of substrate–consumer relationships:
(1) single-substrate Monod-type (also known as

Mathematically, the problem can be addressed with an explicit formulation of all kinetic processes using ordinary differential equations accounting for all substrates and consumers (Chellaboina et al., 2009). However, such a formulation would require too many parameters and would be numerically very difficult to solve because of its multi-temporal scale nature. By making the quasi-steady-state approximation (QSSA), i.e., assuming that product generation from the consumer–substrate complex is much slower than equilibration between consumers, substrates, and consumer–substrate complexes (Briggs and Haldane, 1925; Pedersen et al., 2008), the full kinetic problem is reduced to the simpler equilibrium chemistry (EC) form (e.g., Chellaboina et al., 2009). However, the EC formulation is also usually very difficult to solve numerically. Therefore, analytical approximations to the EC formulation are generally made.

Two classic analytical approximations for modeling redox reactions are dual
Monod (DM) kinetics (e.g., Yeh et al., 2001) and the synthesizing unit (SU)
approach (Kooijman, 1998; Brandt et al., 2003). Although both of these are
special cases of the EC formulation (Kooijman, 2010; Tang and Riley, 2013a),
they make different assumptions regarding the relative magnitudes of involved
kinetic parameters. For this, Kooijman (2010) has shown that DM kinetics
requires the consumer–substrate complex dissociation rate to be much larger
than the product generation rate from the complexes. In contrast,
single-substrate Monod kinetics (Monod, 1949) or Michaelis–Menten (MM)
kinetics (Michaelis and Menten, 1913, which is mathematically identical to
the empirical Monod kinetics) does not impose this requirement on its
parameters. Moreover, in applications to

We define a kinetic formulation to have consistent process scaling when the
formulated substrate–consumer relationship (1) can seamlessly transition
from a single substrate–consumer pair to a network of many substrate–consumer
pairs without changing its mathematical form (also known as the partition principle as
in Newton's second law of motion; Feynman et al., 1963) and (2) does not
predict any singularity over the range of substrate and consumer
concentrations (also known as the non-singular principle; e.g., Schnell and Maini,
2000; Tang, 2015). The full kinetics and EC formulations both satisfy these
two criteria, which can be explained using the following example network of
consumer–substrate relationships:

The full kinetic formulation for the network of Eq. (1) is

The first summation in Eqs. (2) and (4) satisfies the partition principle.
For instance, for Eq. (4), by defining an appropriate mean specific substrate
affinity

Meanwhile, that the full kinetic formulation satisfies the non-singular
principle can be verified by noting that, at any time,

Since the EC formulation is obtained by applying the QSSA to the full kinetic
formulation (i.e.,

Here,

For competitive Monod kinetics on the right-hand side of inequality (6), we
may define

In Tang (2015) (and also in Borghans et al., 1996 and Tang and Riley, 2013a),
it was shown that the linear dependence of

We therefore ask the question: how should we achieve a consistent scaling
from the simplest redox reaction

In the following, we address the above process-scaling question by first
presenting the step-by-step derivation of DM kinetics and SU kinetics from
the EC formulation of the redox reaction

We schematically represent the enzymatic redox reaction network as

By law of mass action and the total QSSA (tQSSA; see Borghans et al.,
1996; Tang and Riley, 2013a), we have the governing equations (Appendix A):

The derivation of substrate kinetics is therefore equivalent to solving for

To clarify, we note that obtaining the substrate kinetics only requires solving Eqs. (11)–(16); various production and destruction terms can be added to Eqs. (9) and (10) to form a full dynamic model (e.g., Maggi and Riley, 2009) without affecting our derivation below.

One method to linearize Eqs. (11)–(16) is to assume that concentrations of
consumer–substrate complexes are so small that free substrate concentrations
are effectively equal to bulk concentrations (e.g., for substrate

We now derive DM kinetics. Adopting the equilibrium approximation that
forward binding between consumer and substrate is in rapid equilibrium with
backward dissociation of consumer–substrate complex (e.g., Michaelis and
Menten, 1913; Pyun, 1971), we have the following:

By solving for

As one substrate, e.g.,

We note that the half-saturation coefficient

In deriving SU kinetics for the redox reaction network represented in
Eq. (8), consumer

From Eqs. (23)–(25), we obtain (see Appendix C)

The two-substrate SU kinetics (Eq. 26) can also be viewed as a special case
of the general SU kinetics for any number of complementary substrates, which
was first derived by Kooijman (1998) based on queue theory (e.g., Gross et
al., 2011). Kooijman (1998) assumed that consumers act like synthesizing
units, which process substrates in two steps: binding and production. He then
assumed that all flux rates (including production rates

As one substrate, e.g.,

In Tang and Riley (2013a) and Tang (2015), it was shown that the derivation of MM kinetics ignores the substrate mass balance constraint, resulting in MM kinetics predicting inaccurate parametric sensitivity over the wide range of substrate to consumer ratios (e.g., Fig. 1 in Tang, 2015). This problem is particularly acute when consumer abundances are high with respect to their substrates, a situation that may occur in in vivo conditions (Sols and Marco, 1970; Schnell and Maini, 2000) or when consumers interact with mineral surfaces, such as microbial decomposition of soil organic matter or plant–microbial competition for soil nutrients (Schimel and Bennett, 2004; Vitousek et al., 2010; Resat et al., 2012; Tang and Riley, 2015; Zhu et al., 2016a). In the above, we also note that the substrates' mass balance constraints (Eqs. 14 and 15) are not used in deriving DM and SU kinetics, suggesting that both DM and SU kinetics may suffer from the same deficiency as MM kinetics. Further, since DM kinetics fails to consistently scale from one to two substrates, we focus below on combining SU and ECA kinetics into SUPECA kinetics to achieve a scalable and non-singular formulation of redox reactions.

As implied in Eqs. (9)–(16), the derivation of substrate kinetics requires
solving for

Graphical representation of ECA kinetics as derived in Tang and
Riley (2013a). The equation below the table shows the uptake of substrate

In actual biogeochemical systems, it is more common for many substrates to be
processed by many consumers concurrently (and such an assumption is
implicitly assumed in the space–time–process unit of any biogeochemical
model). To consistently handle such situations, Tang and Riley (2013a)
derived ECA kinetics (see Fig. 1 for a graphic demonstration) for calculating
the consumption of a substrate

By defining the normalized substrate flux (with subscript “c” designating
that the summation is over a column of the graph in Fig. 1),

Equation (30) can then be rewritten as

The normalized substrate flux as defined in Eq. (31) and its conjugate in
Eq. (32) implies that the consumption of substrate

We note that Eqs. (31) and (32) define some very interesting scaling
relationships. For instance, from Eq. (31), we can define the effective
substrate affinity for the bulk substrates (

Similarly, we can define the effective affinity for substrate

Then, by substituting Eqs. (34) and (35) into Eq. (33), we obtain

By applying the above two scaling relationships and the three consistent
scaling criteria (as we proposed in the introduction section) to SUPECA
kinetics in Eq. (28), we obtain (in Appendix E) the network form of SUPECA
kinetics:

For Eq. (37), one can verify that if

Graphic representation for the relationships between substrates, consumers, and normalized fluxes and their conjugates for a block unit of a large substrate–consumer network.

Following Tang and Riley (2013a), we assume that the EC formulation is a good
approximation to the law of mass action and use it to evaluate the numerical
accuracy of SUPECA kinetics. Because of numerical complexity, we restricted
the comparison to the

We evaluated the numerical accuracy of SUPECA (Eq. 37) and SU (Eq. 26) over a
wide range of parameter values. We fixed both substrates at a nominal value
of 40 mol m

Benchmark of the SU (left column) and SUPECA (right column)
predictions against those by the full EC formulation. We note that the

Model predicted consumer–substrate complexes as a function of the
relative abundance of consumers with respect to substrates. Corresponding to
Fig. 3, panels

Our comparison (Figs. 3 and 4) indicates that SUPECA kinetics is superior to
SU kinetics in computing the microbe–substrate complex in the presence of
substrate binding competition between microbes and sorbent. SUPECA
predictions are more accurate than SU predictions in terms of goodness of
linear fitting and RMSE (for which the linear regressions are shown as black
solid lines in Fig. 3). In magnitude, the RMSEs of SUPECA predictions are
less than 10 % of that of SU predictions (and also note that the

As a proof-of-concept example, we applied SUPECA kinetics to predict the
moisture stress on aerobic soil respiration. We note that we are not
suggesting that SUPECA kinetics should replace existing soil biogeochemical (BGC) models, but rather
that mechanistic analysis using a SUPECA-based model can inform process
understanding and thereby improve such models. Following the CENTURY-like
models' approach in modeling topsoil soil carbon dynamics (Coleman and
Jenkinson, 1999; Parton and Rasmussen, 1994) and the set up of Franzluebbers'
(1999) soil incubation experiments (from which the data were used for our
model evaluation), this example (Appendix G) considers a homogenous 10 cm
thick topsoil with 2.0 mol C m

Our approach assumes that the inter-microsite (or aggregate) transport
dominates intra-aggregate transport, consistent with pore-scale simulations
(Yang et al., 2014). The model is solved to steady state by assuming that the
microbes, atmospheric oxygen, and DOC are in balance under the influence of
Langmuir-type DOC sorption by soil minerals. Calculations are conducted for
three levels of soil minerals (with adsorption capacities at 0, 90, and
180 mol C m

Comparison of predicted normalized soil moisture response functions
to that derived from incubation data from Franzluebbers (1999). All response
functions are normalized with their respective peak respiration. The

Simulated moisture response functions using elevated affinity
parameter for O

The calculation with elevated

When the respiration curves are normalized, we found that all curves have the
same pattern where soil respiration first increases from dry soil with
increasing moisture and then levels off after reaching a peak value (where
the respiration is co-limited by oxygen and DOC bioavailability). The curve
with the highest mineral soil carbon adsorption capacity
(180 mol C m

When the oxygen affinity parameter is reduced to its default value (while
keeping the adsorption capacity at 180 mol C m

When the moisture response function is evaluated, we found a higher

Higher adsorption capacity resulted in significantly lower soil respiration (Fig. 6), consistent with results for temperature sensitivity described in Tang and Riley (2015). Combining results from Figs. 5 and 6, we conclude that because the soil moisture response function emerges from interactions between biotic and abiotic factors that co-regulate soil organic carbon decomposition (Manzoni et al., 2016), its functional shape is not deterministic. This result contradicts the popular approach used in many soil BGC models (including our own, e.g., Koven et al., 2013; Tang et al., 2013; and others, e.g., Sierra et al., 2015), where a deterministic soil moisture response function is applied to the moisture-unstressed decomposition rate. We also note that there are many different functional forms for the soil moisture response function used in soil BGC models (Sierra et al., 2015).

At the default oxygen affinity value (

Besides the example application above, SUPECA kinetics could be a powerful
tool for trait-based modeling in various biogeochemical systems (e.g.,
Follows et al., 2007; Bouskill et al., 2012; Litchman and Klausmeier, 2008;
Merico et al., 2009). As we show above and below, SUPECA kinetics will enable
more robust predictions with better numerical consistency and smaller
parametric sensitivities than the popular family of Monod kinetics, and
SUPECA will be applicable for any biogeochemical system that involves
substrate–consumer binding and binding competition (of the

The assertion of smaller parametric sensitivity as predicted by SUPECA (than
by Monod kinetics) can be verified using the single-substrate reaction
network as an example. In this case, SUPECA is reduced to ECA kinetics, and
for some substrate

To quantitatively evaluate our assertion that SUPECA kinetics predicts lower
parametric sensitivity, we applied Eq. (46) to 100 competing substrate fluxes
of equal magnitude. We then have

The assertion of wide applicability with SUPECA kinetics has been demonstrated by a number of successful applications that we have published with ECA kinetics. In a series of studies (Zhu and Riley, 2015; Zhu et al., 2016a, b, 2017), we showed that ECA kinetics significantly improved the modeling of nutrient competition between plants, microbes, and mineral soils. In Tang and Riley (2013a), where ECA kinetics was first proposed, lignin decomposition dynamics were accurately captured without a priori imposing a target lignocellulose index. In Tang and Riley (2013a, 2015) and this study, ECA kinetics was able to seamlessly incorporate Langmuir-type substrate adsorption without invoking an ad hoc numerical order that is prerequisite to MM (or Monod) kinetics for modeling mineral, microbe, and substrate interactions.

Finally, we expect SUPECA kinetics will provide a robust approach to resolve
the redox ladder in soil biogeochemistry. Existing approaches have imposed
the redox ladder following some specific order, e.g.,

In this study, we showed that the popular Monod family kinetics and
SU kinetics are not scaling consistently for a reaction
network involving mixed

The source code and data used in this paper are available upon request to the corresponding author.

The law of mass action formulation of the redox reaction (Eq. 8) is

We now apply the total quasi-steady-state approximation (e.g., Borghans et
al., 1996) to obtain the equilibrium chemistry formulation of the system.
Specifically, we obtain Eqs. (11)–(13) by, respectively, setting the time
derivatives of Eqs. (A1)–(A3) to zero. Equation (9) is obtained by adding
together Eqs. (A1), (A3), and (A5), and using the definition of

Replacing

By solving

Now we solve

We thence obtain dual Monod kinetics by entering Eq. (B3) into Eq. (9).

Since SU kinetics assumes that substrates are not limiting the biogeochemical
reaction, we then, from Eqs. (23) and (24), obtain

By entering Eqs. (C1) and (C2) into Eq. (13), and solving for

We first derive the set of linear equations using the first-order closure
approach (i.e., the perturbation method truncated to first-order accuracy;
Shankar, 1994; Tang et al., 2007). By entering Eqs. (14)–(16) into Eq. (23),
we have

Now, if we expand Eq. (D1) and keep only the zeroth- and first-order terms of

Using the definitions of

Because substrates

Now, by substituting Eqs. (14)–(16), (23), and (24) into Eq. (25) and assuming

Once again, by dropping the second- and higher-order terms of the
consumer–substrate complexes, Eq. (D6) can be reduced to

Now, we solve for

Equations (D9) and (D10) together will lead to

which, when entered into Eq. (9), leads to Eq. (28).

In the second equation of Eq. (33), we show that the consumption of a certain
substrate as represented in ECA kinetics is determined by the consumer
reaction potential

Following Eqs. (23)–(25), the EC problem used to
benchmark synthesizing unit (SU) and SUPECA predictions is defined as

To relate these equations to a dynamic system,

For benchmarking,

The aerobic respiration problem is formulated as

where

In this aerobic respiration problem, microbes are assumed to form microsites
sitting uniformly inside pores of the bulk soil. O

The bulk aqueous diffusivity in Eq. (G3) is

Now, if we assume the steady state (also known as

Now assume that the ball-like microbe is covered with

With a similar procedure for DOC, we have the following:

Below, we provide some estimates for the parameters to support the above
model of moisture dependence of microbial decomposition. The microbial cell
radius

Since, at 25

For model applications, microbes are often in units of mol C m

J-YT designed the theory and conducted the analysis. J-YT and WJR discussed the results and wrote the paper.

The authors declare that they have no conflict of interest.

This research was 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 Accelerated Climate Model for Energy in the Earth system modeling program and the Next Generation Ecosystem Experiment-Arctic project. Financial support does not constitute an endorsement by the Department of Energy of the views expressed in this study.Edited by: Jason Williams Reviewed by: four anonymous referees