There is some evidence that enzyme synthesis and compound decomposition is
concentration dependent. If this is the case for most soil organic matter, then a
forward kinetics approach may be required to describe microbial allocation, rather
than the reverse kinetics implemented in EnzMax. To address this I re-solved the
model using forward kinetics for decomposition fluxes. I also included carbon
uptake functions, required to stabilize model behavior in forward kinetic soil
organic matter models that are enzyme and microbial explicit (Allison 2006, Allison
et al. 2010, Wieder et al. 2013).
Rather than maximizing decomposition, as is done in the original EnzMax model,
uptake of carbon is maximized in the forward kinetics model. This is because in the
reverse kinetic formulation of the model all decomposition fluxes are taken up by
the microbial biomass, while in forward kinetic models there are explicit microbial
uptake functions that regulate the maximum uptake of any particular carbon
compound.
(1) UC  UC1 UC 2

UC1 and UC2 can be replace by their respective uptake functions (sensu Allison et al.
2010).
(2) UC 

Vmax up * DOC1 Vmax up * DOC 2

K m up  DOC1 K m up  DOC 2
The DOC terms in each uptake function can be replaced by their forward kinetic
equations:
k d *  * E * S1
k * (1   ) * E * S2
Vmax up * d
K m  S1
K m  S2
(3) UC 

k *  * E * S1
k * (1   ) * E * S2
K m up  d
K m up  d
K m  S1
K m  S2
Vmax up *

By making the simplifying assumption that both S1 and S2 pools are >> Km, then this
equation can be reduced to:
(3a) UC 

Vmax up * kd *  * E Vmax up * kd * (1   ) * E

K m up  kd *  * E K m up  kd * (1   ) * E
Then, by taking the derivative of equation 3a, setting it to 0, and solving for alpha,
the allocation behavior that maximizes carbon uptake can be determined. The
derivative of equation 3a is:
(4)

Vmax up * E 2 * K d 2 * K m up * (2 *  1) * (E * K d  2 * K m up )
(E * K d *   K m up ) 2 * (E * (K d  K d *  )  K up ) 2
Equation 4 equals 0 when alpha=0.5, or enzyme allocation is 1:1. Therefore, both
forward and reverse kinetic assumption of decomposition dynamics converge on
the same solution under carbon limitation, so long as a simplifying assumption is
made that substrate concentrations are >> Km values.
Taking the derivative of Equation 3 and solving for alpha returns an expression that
depends on both substrate concentrations. This allocation pattern should dominate
when substrate concentrations are not >> Km values.
The assumption that substrate concentrations are >> Km values will not hold for all
compounds. When this assumption is violated concentration dependence of enzyme
allocation and decomposition should be observed, and this has been verified with
starch compounds by German et al. (2011). However, starch compounds
represented <1% of soil organic matter in this German et al. (2011) study. For
compounds that were more common in organic matter, such as cellulose and
hemicellulose, which represent >25% of organic matter in the study soils, no
concentration dependence of decomposition was observed, suggesting that the
concentrations of these substrates are indeed much greater than their Km values.
Given that simple substrates dominate the composition of soil organic matter in
many soils (Amelung et al. 2008, Kleber et al. 2010, Schmidt et al. 2011), and that
soil organisms produce enzymes that target multiple classes of substrates
(Valaskova and Baldrian 2006, Baldrian and Lopez-Mondejar 2014), it seems likely
that the dominant classes of soil compounds in the soil organic matter profile will
exceed their Km values by several fold. Furthermore, there is evidence that soil
organisms produce different classes of enzymes to catalyze the same reaction in
order to minimize Km effects (Nannipieri et al. 1982).
A simple calculation based on soil enzyme assay optimization protocols also supports the
idea that soil substrate concentrations are likely higher than their Km values. 500uM is a
common substrate concentration that achieves Vmax for a soil enzyme assay. At this point
the substrate concentration id large enough to ignore Km effects, necessary for
determining Vmax. This substrate added as a 50ul aliquot to 200ul of soil slurry. Soil
slurries are prepared by spinning ~2g of field moist soil in 100mL of buffer. If we assume
a soil moisture of 50%, a soil C concentration of 5% C / g soil, and that substrates contain
100 g C / mol substrate, then these reactions reach Vmax when a target substrate represents
just 2.5% of the soil organic C pool.
Finally, if relative substrate concentrations were driving microbial allocation
patterns in this analysis, that at very low C:N ratios we should observe very low C:N
enzyme ratios, as this would maximize both C and N uptake for microbial
decomposers. The ratio of C:N enzymes should increase with substrate C:N up to the
microbial threshold element ratio. This is not predicted by neither the EnzMax,
EnzOpt, or the original EEZY model (Moorhead 2012). Analysis of the empirical data
however, do not support this (Figure 4). Therefore, it is unlikely concentration
dependence is driving enzyme allocation patterns in this data set.
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