This report is the result of the use of the Python 3.4 package Sympy (for symbolic mathematics), as means to translate published models to a common language. It was created by Holger Metzler (Orcid ID: 0000-0002-8239-1601) on 11/03/2016, and was last modified on lm.
The model depicted in this document considers soil organic matter decomposition. It was originally described by Schimel & Weintraub (2003).
Traditional models of soil organic matter (SOM) decomposition are all based on first order kinetics in which the decomposition rate of a particular C pool is proportional to the size of the pool and a simple decomposition constant (\(dC/dt = kC\)). In fact, SOM decomposition is catalyzed by extracellular enzymes that are produced by microorganisms. We built a simple theoretical model to explore the behavior of the decomposition – microbial growth system when the fundamental kinetic assumption is changed from first order kinetics to exoenzymes catalyzed decomposition (\(dC/dt = KC\times Enzymes\)). An analysis of the enzyme kinetics showed that there must be some mechanism to produce a non-linear response of decomposition rates to enzyme concentration—the most likely is competition for enzyme binding on solid substrates as predicted by Langmuir adsorption isotherm theory. This non-linearity also induces C limitation, regardless of the potential supply of C. The linked C and N version of the model showed that actual polymer breakdown and microbial use of the released monomers can be disconnected, and that it requires relatively little N to maintain the maximal rate of decomposition, regardless of the microbial biomass’ ability to use the breakdown products. In this model, adding a pulse of C to an N limited system increases respiration, while adding N actually decreases respiration (as C is redirected from waste respiration to microbial growth). For many years, researchers have argued that the lack of a respiratory response by soil microbes to added N indicates that they are not N limited. This model suggests that conclusion may be wrong. While total C flow may be limited by the functioning of the exoenzyme system, actual microbial growth may be N limited.
differential equations, nonlinear, time invariant
heterogeneity of speed of decay, internal transformations of organic matter, substrate interactions
| Abbreviation | Description |
|---|---|
| Set 1 | original values from linked model (no nitrogen cycle considered in this model here) |
The following table contains the available information regarding this section:
| Name | Description | Units |
|---|---|---|
| \(S\) | soil organic carbon | \(g\) |
| \(D\) | dissolved organic carbon | \(g\) |
| \(M\) | microbial biomass | \(g\) |
| \(E\) | exoenzymes | \(g\) |
The following table contains the available information regarding this section:
| Name | Description | Type | Units | Values Set 1 |
|---|---|---|---|---|
| \(k_{d}\) | collapsed decomposition constant | parameter | \(d^{-1}\) | \(1\) |
| \(k_{m}\) | microbial maintenance rate | parameter | \(d^{-1}\) | \(0.01\) |
| \(k_{t}\) | rate of microbial biomass that dies in each time interval | parameter | \(d^{-1}\) | \(0.012\) |
| \(k_{l}\) | decay constant for exoenzymes | parameter | \(d^{-1}\) | \(0.05\) |
The following table contains the available information regarding this section:
| Name | Description | Type | Values Set 1 |
|---|---|---|---|
| \(SUE\) | substrate use efficiency | parameter | \(0.5\) |
| \(K_{e}\) | fraction of of dissolved organic carbon that is allocated to exoenzymes | parameter | \(0.05\) |
| \(K_{r}\) | proportion of dead microbial biomass that is available for microbial use | parameter | \(0.85\) |
The following table contains the available information regarding this section:
| Name | Description | Expressions | Values Set 1 |
|---|---|---|---|
| \(R_{e}\) | respiration to support enzyme synthesis | \(R_{e}=\frac{K_{e}\,D\,\left(1-SUE\right)}{SUE}\) | - |
| \(R_{m}\) | respiration to support maintenance energy needs | \(R_{m}=k_{m}\,M\) | - |
| \(R_{g}\) | respiration to support growth of the microbial biomass (not allowed to be negative) | \(R_{g}=\operatorname{Max}\left(\left(D-K_{e}\,D-R_{e}-R_{m}\right)\,\left(1-SUE\right), 0\right)\) | - |
The following table contains the available information regarding this section:
| Name | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}S\\D\\M\\E\end{matrix}\right]\) |
| \(T\) | transition operator | \(T=\left[\begin{matrix}-1 & 0 & 0 & 0\\1 & -1 &\frac{K_{r}\cdot k_{t}}{SUE\cdot k_{m} + k_{t}} & 0\\0 & - K_{e} + SUE & -1 & 0\\0 & K_{e} & 0 & -1\end{matrix}\right]\) |
| \(N\) | decomposition operator | \(N=\left[\begin{matrix}\frac{E}{S}\cdot k_{d} & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & SUE\cdot k_{m} + k_{t} & 0\\0 & 0 & 0 & k_{l}\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=T\,N\,C\) |
| Flux description | |
|---|---|
 Figure 1: Pool model representation |
Output fluxes\(D: D\cdot\left(- SUE + 1\right)\) Internal fluxes\(S > D: E\cdot k_{d}\)\(D > M: D\cdot\left(- K_{e} + SUE\right)\) \(D > E: D\cdot K_{e}\) \(M > D: K_{r}\cdot M\cdot k_{t}\) |
\(\left[\begin{matrix}- E\cdot k_{d}\\- D + E\cdot k_{d} + K_{r}\cdot M\cdot k_{t}\\D\cdot\left(- K_{e} + SUE\right) + M\cdot\left(- SUE\cdot k_{m} - k_{t}\right)\\D\cdot K_{e} - E\cdot k_{l}\end{matrix}\right]\)
\(\left[\begin{matrix}0 & 0 & 0 & - k_{d}\\0 & -1 & K_{r}\cdot k_{t} & k_{d}\\0 & - K_{e} + SUE & - SUE\cdot k_{m} - k_{t} & 0\\0 & K_{e} & 0 & - k_{l}\end{matrix}\right]\)
\(S = S\)
\(D = 0\)
\(M = 0\)
\(E = 0\)
\(S: S\), \(D: 0.0\), \(M: 0.0\), \(E: 0.0\)
\(\lambda_{1}: 0.000\)
\(\lambda_{2}: 0.010-0.000j\)
\(\rho_{2}: -1.000000\)
\(\lambda_{3}: -0.022+0.000j\)
\(\rho_{3}: 1.000000\)
\(\lambda_{4}: -1.054-0.000j\)
\(\rho_{4}: 1.000000\)
Schimel, J. P., & Weintraub, M. N. (2003). The implications of exoenzyme activity on microbial carbon and nitrogen limitation in soil: A theoretical model. Soil Biology and Biochemistry, 35(4), 549–563.