General Overview

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 21/03/2016, and was last modified on lm.

About the model

The model depicted in this document considers soil organic matter decomposition. It was originally described by Wang, Post, & Mayes (2013).

Further references

Li, Wang, Allison, Mayes, & Luo (2014):

Abstract

We developed a microbial-enzyme-mediated decomposition (MEND) model, based on the Michaelis-Menten kinetics, that describes the dynamics of physically defined pools of soil organic matter (SOC). These include particulate, mineral-associated, dissolved organic matter (POC, MOC, and DOC, respectively), microbial biomass, and associated exoenzymes. The ranges and/or distributions of parameters were determined by both analytical steady-state and dynamic analyses with SOC data from the literature. We used an improved multi-objective parameter sensitivity analysis (MOPSA) to identify the most important parameters for the full model: maintenance of microbial biomass, turnover and synthesis of enzymes, and carbon use efficiency (CUE). The model predicted that an increase of 2°C (baseline temperature 12°C) caused the pools of POC-cellulose, MOC, and total SOC to increase with dynamic CUE and decrease with constant CUE, as indicated by the 50% confidence intervals. Regardless of dynamic or constant CUE, the changes in pool size of POC, MOC, and total SOC varied from −8% to 8% under +2°C. The scenario analysis using a single parameter set indicates that higher temperature with dynamic CUE might result in greater net increases in both POC-cellulose and MOC pools. Different dynamics of various SOC pools reflected the catalytic functions of specific enzymes targeting specific substrates and the interactions between microbes, enzymes, and SOC. With the feasible parameter values estimated in this study, models incorporating fundamental principles of microbial-enzyme dynamics can lead to simulation results qualitatively different from traditional models with fast/slow/passive pools.

Keywords

differential equations, nonlinear, time invariant, Michaelis-Menten

Principles

mass balance, substrate dependence of decomposition, heterogeneity of speed of decay, internal transformations of organic matter, substrate interactions

State Variables

The following table contains the available information regarding this section:

Parameters

The following table contains the available information regarding this section:

Additional Variables

The following table contains the available information regarding this section:

Input Components

The following table contains the available information regarding this section:

Components

The following table contains the available information regarding this section:

Pool model representation

Flux description

Figure 1: Pool model representation

Input fluxes \(P: I_{P}\) \(D: I_{D}\) Output fluxes \(B: -\frac{B\cdot D}{E_{C}\cdot\left(D + K_{D}\right)}\cdot\left(E_{C}\cdot V_{D} + E_{C}\cdot m_{R} - V_{D} - m_{R}\right)\) Internal fluxes \(P > M: -\frac{EP\cdot P\cdot V_{P}}{K_{P} + P}\cdot\left(f_{D} - 1\right)\) \(P > D: \frac{EP\cdot P\cdot V_{P}}{K_{P} + P}\cdot f_{D}\) \(M > D: \frac{EM\cdot M\cdot V_{M}}{K_{M} + M}\) \(Q > D: \frac{K_{des}}{Q_{max}}\cdot Q\) \(B > P: B\cdot m_{R}\cdot\left(g_{D} - 1\right)\cdot\left(p_{EM} + p_{EP} - 1\right)\) \(B > D: B\cdot g_{D}\cdot m_{R}\cdot\left(- p_{EM} - p_{EP} + 1\right)\) \(B > EP: B\cdot m_{R}\cdot p_{EP}\) \(B > EM: B\cdot m_{R}\cdot p_{EM}\) \(D > Q: -\frac{D}{Q_{max}}\cdot K_{ads}\cdot\left(Q - Q_{max}\right)\) \(D > B: \frac{B\cdot D\cdot\left(V_{D} + m_{R}\right)}{E_{C}\cdot\left(D + K_{D}\right)}\) \(EP > D: EP\cdot r_{EP}\) \(EM > D: EM\cdot r_{EM}\)

The right hand side of the ODE

\(\left[\begin{matrix}B\cdot m_{R}\cdot\left(- g_{D} + 1\right)\cdot\left(- p_{EM} - p_{EP} + 1\right) -\frac{EP\cdot P\cdot V_{P}}{K_{P} + P} + I_{P}\\-\frac{EM\cdot M\cdot V_{M}}{K_{M} + M} +\frac{EP\cdot P\cdot V_{P}}{K_{P} + P}\cdot\left(- f_{D} + 1\right)\\D\cdot K_{ads}\cdot\left(-\frac{Q}{Q_{max}} + 1\right) -\frac{K_{des}}{Q_{max}}\cdot Q\\\frac{B\cdot D\cdot\left(V_{D} + m_{R}\right)}{E_{C}\cdot\left(D + K_{D}\right)} + B\cdot\left(-\frac{D}{D + K_{D}}\cdot\left(-1 +\frac{1}{E_{C}}\right)\cdot\left(V_{D} + m_{R}\right) - m_{R}\right)\\B\cdot g_{D}\cdot m_{R}\cdot\left(- p_{EM} - p_{EP} + 1\right) + D\cdot\left(-\frac{B\cdot\left(V_{D} + m_{R}\right)}{E_{C}\cdot\left(D + K_{D}\right)} - K_{ads}\cdot\left(-\frac{Q}{Q_{max}} + 1\right)\right) +\frac{EM\cdot M\cdot V_{M}}{K_{M} + M} + EM\cdot r_{EM} +\frac{EP\cdot P\cdot V_{P}}{K_{P} + P}\cdot f_{D} + EP\cdot r_{EP} + I_{D} +\frac{K_{des}}{Q_{max}}\cdot Q\\B\cdot m_{R}\cdot p_{EP} - EP\cdot r_{EP}\\B\cdot m_{R}\cdot p_{EM} - EM\cdot r_{EM}\end{matrix}\right]\)

The Jacobian (derivative of the ODE w.r.t. state variables)

\(\left[\begin{matrix}\frac{EP\cdot P\cdot V_{P}}{\left(K_{P} + P\right)^{2}} -\frac{EP\cdot V_{P}}{K_{P} + P} & 0 & 0 & m_{R}\cdot\left(- g_{D} + 1\right)\cdot\left(- p_{EM} - p_{EP} + 1\right) & 0 & -\frac{P\cdot V_{P}}{K_{P} + P} & 0\\-\frac{EP\cdot P\cdot V_{P}}{\left(K_{P} + P\right)^{2}}\cdot\left(- f_{D} + 1\right) +\frac{EP\cdot V_{P}}{K_{P} + P}\cdot\left(- f_{D} + 1\right) &\frac{EM\cdot M\cdot V_{M}}{\left(K_{M} + M\right)^{2}} -\frac{EM\cdot V_{M}}{K_{M} + M} & 0 & 0 & 0 &\frac{P\cdot V_{P}}{K_{P} + P}\cdot\left(- f_{D} + 1\right) & -\frac{M\cdot V_{M}}{K_{M} + M}\\0 & 0 & -\frac{D}{Q_{max}}\cdot K_{ads} -\frac{K_{des}}{Q_{max}} & 0 & K_{ads}\cdot\left(-\frac{Q}{Q_{max}} + 1\right) & 0 & 0\\0 & 0 & 0 & -\frac{D}{D + K_{D}}\cdot\left(-1 +\frac{1}{E_{C}}\right)\cdot\left(V_{D} + m_{R}\right) +\frac{D\cdot\left(V_{D} + m_{R}\right)}{E_{C}\cdot\left(D + K_{D}\right)} - m_{R} & -\frac{B\cdot D\cdot\left(V_{D} + m_{R}\right)}{E_{C}\cdot\left(D + K_{D}\right)^{2}} + B\cdot\left(\frac{D}{\left(D + K_{D}\right)^{2}}\cdot\left(-1 +\frac{1}{E_{C}}\right)\cdot\left(V_{D} + m_{R}\right) -\frac{1}{D + K_{D}}\cdot\left(-1 +\frac{1}{E_{C}}\right)\cdot\left(V_{D} + m_{R}\right)\right) +\frac{B\cdot\left(V_{D} + m_{R}\right)}{E_{C}\cdot\left(D + K_{D}\right)} & 0 & 0\\-\frac{EP\cdot P\cdot V_{P}}{\left(K_{P} + P\right)^{2}}\cdot f_{D} +\frac{EP\cdot V_{P}\cdot f_{D}}{K_{P} + P} & -\frac{EM\cdot M\cdot V_{M}}{\left(K_{M} + M\right)^{2}} +\frac{EM\cdot V_{M}}{K_{M} + M} &\frac{D}{Q_{max}}\cdot K_{ads} +\frac{K_{des}}{Q_{max}} & -\frac{D\cdot\left(V_{D} + m_{R}\right)}{E_{C}\cdot\left(D + K_{D}\right)} + g_{D}\cdot m_{R}\cdot\left(- p_{EM} - p_{EP} + 1\right) &\frac{B\cdot D\cdot\left(V_{D} + m_{R}\right)}{E_{C}\cdot\left(D + K_{D}\right)^{2}} -\frac{B\cdot\left(V_{D} + m_{R}\right)}{E_{C}\cdot\left(D + K_{D}\right)} - K_{ads}\cdot\left(-\frac{Q}{Q_{max}} + 1\right) &\frac{P\cdot V_{P}\cdot f_{D}}{K_{P} + P} + r_{EP} &\frac{M\cdot V_{M}}{K_{M} + M} + r_{EM}\\0 & 0 & 0 & m_{R}\cdot p_{EP} & 0 & - r_{EP} & 0\\0 & 0 & 0 & m_{R}\cdot p_{EM} & 0 & 0 & - r_{EM}\end{matrix}\right]\)

References