Model formulation of microbial CO2 production and efficiency can significantly influence short and long term soil C projections

Ballantyne IV, Ford; Billings, Sharon A.

doi:10.1038/s41396-018-0085-1

Perspective
Published: 22 March 2018

Model formulation of microbial CO₂ production and efficiency can significantly influence short and long term soil C projections

Ford Ballantyne IV¹ &
Sharon A. Billings²

The ISME Journal volume 12, pages 1395–1403 (2018)Cite this article

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Models of soil C dynamics continue to be rapidly developed in a effort to improve projections of terrestrial C fluxes under a range of climate scenarios. Recently developed models of soil organic C (SOC) dynamics that incorporate microbes as the agents of decomposition [1,2,3,4,5] capture a greater number of relevant processes than their predecessors, but it remains unclear whether the mathematical formulations and parameterization of such models are accurate [6,7,8,9]. Different model structures, parameterizations, and functions describing responses to changing environmental conditions exert great influence on projections of SOC stocks and CO₂ flux [1, 4, 9]. Discrepancies between projections of SOC stocks from microbial models and current Earth system model (ESM) soil C modules, which can span two orders of magnitude, are highly sensitive to assumptions regarding microbial respiration and efficiency [1, 4, 6].

Although physical structure, chemical composition, and edaphic conditions ultimately govern the accessibility of decomposable SOC [10], physiology dictates the fate of C once it is liberated and taken up from SOC by microbes, regardless of its original composition. A fundamental component of microbial physiology, heterotrophic respiration, which is the conduit for C flow from SOC to the atmosphere and one of the largest potential positive feedbacks to climate warming, is often incorporated in models of SOC dynamics heuristically in a term for efficiency. Alternatively, microbial respiration can be formulated more mechanistically to reflect the fact that heterotrophic respiration in soils is largely mass specific when microbes are actively decomposing SOC [11]. Because heterotrophic microbial respiration influences how much of the C liberated from SOC is retained in the soil and how much is lost to the atmosphere, it is important to understand both the short and long term dynamical consequences of formulating respiratory losses differently in models used to project future climate. Despite all the recent effort to develop better models, to date nobody has determined if the widespread, heuristic formulation of respiratory C losses gives rise to different dynamics than the more physiologically grounded formulations of microbial respiration, over a range of time scales. Here, we address this question for the first time.

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Acknowledgements

We dedicate this paper to Dr. Henry Gholz, who was an inspiration and a supporter of our work on soil C dynamics. We wish to thank Chao Song and Stefano Manzoni for conversations and comments that improved the manuscript. We also thank Bob Sinsabaugh, Josh Schimel, Will Wieder, and an anonymous reviewer for comments on previous versions of the manuscript. This work was supported by a grant from the US National Science Foundation (DEB-0950095).

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Odum School of Ecology, University of Georgia, 140 E. Green St., Athens, GA, 30602, USA
Ford Ballantyne IV
Department of Ecology and Evolutionary Biology, Kansas Biological Survey, University of Kansas, 2101 Constant Ave., Lawrence, KS, 66047, USA
Sharon A. Billings

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Ford Ballantyne IV
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Correspondence to Ford Ballantyne IV.

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Mathematical appendix

To simplify analysis, we non-dimensionalized Eq. 3 to arrive at

$$\begin{array}{l}\frac{{dS_C^\prime }}{{d\tau }} = I^\prime - B_C^\prime \frac{{v\prime S_C^\prime }}{{(1 + S_C^\prime )}} - S_C^\prime \lambda _S^\prime + B_C^\prime \varepsilon \\ \frac{{dB_C^\prime }}{{d\tau }} = \left\{ {\begin{array}{*{20}{l}} {B_C^\prime \left[ {CUE \times \frac{{v\prime S_C^\prime }}{{(1 + S_C^\prime )}} - 1} \right];} \hfill & {CUE\;{\mathrm{formulation}}} \hfill \\ {{\mathrm{or}}} \hfill & \, \hfill \\ {B_C^\prime \left[ {\frac{{v\prime S_C^\prime }}{{(1 + S_C^\prime )}} - MSR^\prime - 1} \right];} \hfill & {MSR\;{\mathrm{formulation}}} \hfill \end{array}} \right.\end{array}$$

in which $S_C^\prime = \frac{{S_C}}{K}$, $B_C^\prime = \frac{{B_C}}{K}$, $\tau = \lambda _Bt$, $I^\prime = \frac{I}{{K\lambda _B}}$, $v\prime = \frac{v}{{\lambda _B}}$, $\lambda _S^\prime = \frac{{\lambda _S}}{{\lambda _B}}$, and $MSR^\prime = \frac{{MSR}}{{\lambda _B}}$.

The equilibria are given by

$$\begin{array}{l}\hat S_C = \left\{ {\begin{array}{*{20}{l}} {\frac{1}{{\left( {v\prime CUE - 1} \right)}};} \hfill & {CUE\,{\mathrm{formulation}}} \hfill \\ {{\mathrm{or}}} \hfill & \, \hfill \\ {\frac{1}{{\left( {\frac{{v\prime }}{{MSR^\prime }} - 1} \right)}};} \hfill & {MSR^\prime \,{\mathrm{formulation}}} \hfill \end{array}} \right.\\ \hat B_C = \left\{ {\begin{array}{*{20}{l}} {\frac{{I^\prime - \left( {\frac{{\lambda _S^\prime }}{{v\prime CUE - 1}}} \right)}}{{\left( {\frac{1}{{CUE}} - \varepsilon } \right)}};} \hfill & {CUE\;{\mathrm{formulation}}} \hfill \\ {{\mathrm{or}}} \hfill & \, \hfill \\ {\frac{{I^\prime - \frac{{\lambda _S^\prime }}{{\left( {\frac{{v\prime }}{{MSR^\prime + 1}} - 1} \right)}}}}{{MSR^\prime + 1 - \varepsilon }};} \hfill & {MSR\;{\mathrm{formulation}}} \hfill \end{array}} \right.\end{array}$$

(6)

In the absence of life,

$$\hat S_C^\prime = \frac{{I^\prime }}{{\lambda _S^\prime }} = \frac{I}{{\lambda _S}},$$

(7)

and for biota to persist, v′ > CUE and $\lambda _S^\prime$ < I′(v′CUE−1) for the CUE formulation, and v′ > MSR′ + 1 and $\lambda _S^\prime < I^\prime \left( {\frac{{v\prime }}{{MSR^\prime + 1}} - 1} \right)$ for the MSR formulation.

The ratio of equilibrium SOC for the MSR formulation to equilibrium SOC for the CUE formulation is given by

$$\frac{{\hat S_C^{\prime MSR}}}{{\hat S_C^{\prime CUE}}} = \frac{{\left( {v\prime CUE - 1} \right)}}{{\left( {\frac{{v\prime }}{{MSR^\prime + 1}} - 1} \right)}}.$$

(8)

The ratio will equal 1 if $CUE = \frac{1}{{MSR^\prime + 1}}$, denoted by the bold line in the figure panels.

To address the discrepancy in the response to perturbation, we analyze the characteristic equation resulting from the determinants of the Jacobians for the two formulations. The Jacobian evaluated at the equilibrium for both systems assumes the general form

$$\hat J_X = \left[ {\begin{array}{*{20}{c}} {\frac{{ - v\prime \hat B_C^\prime }}{{\left( {1 + \hat S^\prime_C } \right)^2}} - \lambda^{\prime} _{S}} & {\frac{{ - v\prime \hat S^\prime_C }}{{1 + \hat S^\prime_C }} + \epsilon } \\ {\frac{{v\prime \hat B_C^\prime X}}{{\left( {1 + \hat S^\prime_C } \right)^2}}} & 0 \end{array}} \right]$$

(9)

in which X = CUE or 1 and $\frac{{ - v\prime \hat S^\prime }}{{1 + \hat S^\prime }} = \frac{1}{{CUE}}$ or MSR′ + 1 for the CUE and MSR formulations, respectively. The eigenvalues for the associated characteristic equation are

$$\frac{{ - (A + \lambda _S^\prime ) \pm \sqrt {(A + \lambda _S^\prime )^2 + 4AX( \epsilon - B)} }}{2}$$

(10)

in which $A = \frac{{v\prime \hat B_C^\prime }}{{\left( {1 + \hat S^\prime_C } \right)^2}}$ for both formulations and $B = \frac{1}{{CUE}}$ or MSR′ + 1 for the CUE and MSR formulations, respectively. In the absence of life, we simply recover $\lambda _S^\prime$ as the only eigenvalue. Thus, the addition of living microbes to the soil system increases the absolute magnitude of the dominant eigenvalue, thereby decreasing return time to equilibrium and increasing system stability. The real parts of all eigenvalues for both formulations are negative, reflecting the stability of the steady-state. To most directly compare stability between formulations, we assume that the equilibria are the same for both, i.e $CUE = \frac{1}{{MSR^\prime + 1}}$. The discrepancy in stability between the two formulations is most easily seen for the case of no recycling, ε = 0, because the only difference in the eigenvalues arises in the AXB term. For the CUE formulation, AXB simply reduces to A because X = CUE and $B = \frac{1}{{CUE}}$, which also means that the response to perturbation is independent of CUE in the absence of recycling. For the MSR formulation, AXB = A(MSR′ + 1) because X = 1 and B = MSR′ + 1. A is the same by assumption, which means that in the case of no recycling the return to identical equilibrium will be faster for the CUE formulation. In general, increased rates of recycling, ε > 0, promote a faster return to equilibrium and shorter duration transient dynamics. MSR′ > 1, which is substantiated in other work [18], is a sufficient condition to guarantee the same damped response to perturbation and longer transient dynamics of the MSR formulation as for the case of no recycling. The influence of CUE on stability is more severely constrained because both ε and CUE are bounded by 0 and 1, whereas MSR′ can, and will often be significantly greater than 1.

Equations 3 were simulated with I′ = 1, v′ = 16, CUE = [0.2,0.5], MSR′ = [4,10], and $\lambda _S^\prime$ = 0.1 to generate temporal trajectories of S_C and B_C. To simulate the response to temperature variation, we made the standard assumption that v exhibits an Arrhenius temperature dependence [1], and incorporated variation in the following manner,

$$v^{\prime \prime } = v\prime \alpha e^{\frac{{ - E_a}}{{R\left( {288 + 5sin\frac{{\pi \tau }}{2}} \right)}}},$$

(11)

with α = 10⁹, E_a = 50,000 and R = 8.134.

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Ballantyne IV, F., Billings, S.A. Model formulation of microbial CO₂ production and efficiency can significantly influence short and long term soil C projections. ISME J 12, 1395–1403 (2018). https://doi.org/10.1038/s41396-018-0085-1

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Received: 21 August 2017
Revised: 22 December 2017
Accepted: 31 December 2017
Published: 22 March 2018
Version of record: 22 March 2018
Issue date: June 2018
DOI: https://doi.org/10.1038/s41396-018-0085-1

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Model formulation of microbial CO₂ production and efficiency can significantly influence short and long term soil C projections

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Ecological stoichiometry as a foundation for omics-enabled biogeochemical models of soil organic matter decomposition

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Acknowledgements

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Ecological stoichiometry as a foundation for omics-enabled biogeochemical models of soil organic matter decomposition

Influence of CO2 on Water Chemistry and Bacterial Community Structure and Diversity: An Experimental Study in the Laboratory

Evaluating soil microbial carbon use efficiency explicitly as a function of cellular processes: implications for measurements and models

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