the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 12 Feb 2024
When and why microbial-explicit soil organic carbon models can be unstable
Abstract. Microbial-explicit soil organic carbon (SOC) cycling models are increasingly recognized for their advantages over linear models in describing SOC dynamics. These models are known to exhibit oscillations, but it is not clear when they yield stable vs. unstable equilibrium points (EPs) – i.e. EPs that exist analytically, but are not stable to small perturbations and cannot be reached by transient simulations. Occurrence of such unstable EPs can lead to unexpected model behaviour in transient simulations or unrealistic predictions of steady state soil organic carbon (SOC) stocks. Here we ask when and why unstable EPs can occur in an archetypal microbial-explicit model (representing SOC, dissolved OC [DOC], microbial biomass, and extracellular enzymes) and some simplified versions of it. Further, if a model formulation allows for physically meaningful but unstable EPs, can we find constraints in the model parameters (i.e. environmental conditions and microbial traits) that ensure stability of the EPs? We use analytical, numerical and descriptive tools to answer these questions. We found that instability can occur when the resupply of a growth substrate (DOC) is (via a positive feedback loop) dependent on its abundance. We identified a conservative, sufficient condition on model parameters to ensure stability of EPs. Interactive effects of environmental conditions and parameters describing microbial physiology point to the relevance of basic ecological principles for avoidance of unrealistic (i.e. unstable) simulation outcomes. These insights can help to improve applicability of microbial-explicit models, aid our understanding of the dynamics of these models and highlight the relation between mathematical requirements and (in silico) microbial ecology.
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RC1: 'Comment on egusphere-2024-348', Anonymous Referee #1, 19 Mar 2024
Schwarz and collaborators used analytical, numerical and descriptive tools to investigate when and why unstable equilibrium points occur in an archetypal four-pool microbial-explicit model and some simplified versions of it. This study has obtained sufficiently (and necessary) conditions for the stability of the equilibrium point (EPs) of different model version through rigorous mathematical derivations and numerical simulations, which is useful to understand the process or parameter interactions that cause unstable EPs to occur and to guide ecology-informed model developments.
The results of this study point to three strategies to avoid unstable EPs in microbial-explicit SOC models. Besides these mathematical implications, the authors should focus more on the microbial ecological rationale rather than simply forcing the theoretical mathematical model stable. For example, in the real world, whether there exists (1) a positive feedback coupling between microbial growth substrate available for uptake and its resupply? and (2) the dependency of uptake on microbial biomass? What if these two phenomena are indeed facts?
Regarding the third strategy, i.e., choosing parameter values to meet the sufficient and/or necessary conditions for stability, one may argue this could be feasible for a theoretical model with time-invariant carbon input rates and parameters. However, for the real field conditions, both the carbon input rates and parameters vary with time due to the biotic and abiotic effects. It could happen that the parameter values meet the so-called stability conditions sometimes but disobey the conditions at other times.
Therefore, I would like to see additional discussion on the limitation of applying theoretical analyses to real ecosystem, particularly regarding the stability analyses.
Minor comments:
Page 3, Line 36-39 (Fig. 1 caption): insert a comma between “dissolved organic carbon (DOC)” and “microbial biomass carbon (MBC)”
Page 5, Line 106: Table 4 or Table 2?
Page 8, Section 2.3.2 Specify the time step and period of numerical simulations
Page 11, Line 227: Clarify “If abiotic loss of SOC and are neglected”
Page 14, Line 291: Change “larger then” to “larger than”
Page 17, Caption of Figure 2: “four-pool SDB”, while “three-pool SDB” in the main text
Page 21, Table 7: briefly explain the meanings of marking the lower or upper threshold.
Citation: https://doi.org/10.5194/egusphere-2024-348-RC1 -
AC1: 'Reply on RC1', Erik Schwarz, 25 Apr 2024
We thank anonymous reviewer #1 for their very helpful and relevant comments.
We address each individual comment below. The original reviewer’s comments are stated in regular typeface and our responses in italic.Schwarz and collaborators used analytical, numerical and descriptive tools to investigate when and why unstable equilibrium points occur in an archetypal four-pool microbial-explicit model and some simplified versions of it. This study has obtained sufficiently (and necessary) conditions for the stability of the equilibrium point (EPs) of different model version through rigorous mathematical derivations and numerical simulations, which is useful to understand the process or parameter interactions that cause unstable EPs to occur and to guide ecology-informed model developments.
The results of this study point to three strategies to avoid unstable EPs in microbial-explicit SOC models. Besides these mathematical implications, the authors should focus more on the microbial ecological rationale rather than simply forcing the theoretical mathematical model stable. For example, in the real world, whether there exists (1) a positive feedback coupling between microbial growth substrate available for uptake and its resupply? and (2) the dependency of uptake on microbial biomass? What if these two phenomena are indeed facts?
Our study was motivated by the need for more diversified descriptions of soil organic carbon dynamics in Earth-system models (ESMs), and the ensuing proposition of using microbial-explicit models in ESMs. To meet this end, it is relevant to understand the mathematical properties of these models and potential problems. We thus focused our mathematical analysis on a suite of archetypal microbial-explicit model formulations that are being tested for implementation in ESMs. We found that for these microbial-explicit SOC models whether or not the temporal dynamics of a labile substrate (dissolved organic carbon, DOC) pool are explicitly represented has important consequences for a models’ mathematical properties – because of the positive feedback they create. Whether these models are best suited to realistically represent SOC dynamics was not the aim of our study. A debate on suitable structures of microbial-explicit models is currently ongoing (He et al. (2024) and replies – as well as a recent opinion piece by Lennon et al. (2024)). Our study adds another aspect to this important discussion. It might also be important to recall that a practical reason why we need to understand instabilities is that analytically determined steady state values can be used for model initialization. If such equilibrium points (EPs) are unstable, the model will exhibit erratic (and probably unrealistic) behavior. Thus, our work is important not only for understanding of ecological contexts that might lead to collapse of soil functions (probably at the micro-scale, as discussed below), but also for avoiding instabilities in models applying nonlinear equations at large scales.
Within this context, we acknowledge the importance of the raised questions. A short answer would be that both (1) the feedback mechanism and (2) the microbial biomass dependence of uptake are rooted in ecological theory (or at least are based on our ecological understanding). We explain in more details as follows:
- Generally, for conceptualizing substrate-microbe interaction in soil, the relevance of depolymerization of polymeric substrates through extracellular enzymes is well established in soil ecology. If we consider that these extracellular enzymes are produced predominantly by microbes and they make previously inaccessible substrate available for microbial consumption, this generates the basis for the positive feedback loop represented by the analyzed models. Therefore, this feedback loop is a direct consequence of our conceptual understanding of microbial resource acquisition. The priming effect might be regarded as an empirical example of this positive feedback: the temporary provision of labile substrate allows microbes to produce enzymes and break down substrate that was previously not degraded because of energetic limitations (Kuzyakov, 2010; Kuzyakov et al., 2000). However, several factors can influence the strength of such a positive feedback. Importantly, not all inputs of labile carbon are due to microbial depolymerization, such as root exudation, desorption or leaching from litter. The relative importance of these inputs can vary locally (e.g. between the rhizosphere and the bulk soil). If these inputs represent a considerable and constant carbon flux, the positive feedback coupling is partly broken – i.e. the positive feedback exists on a continuum from strong feedback when enzymatic reactions contribute the most to DOC formation to weak feedback when inputs independent of microbial activity are dominant. In the analyzed models this continuum is described through the factor fI, which prescribes how the organic carbon input is partitioned between SOC and DOC. As expected, in the extreme case of all input going to the labile DOC pool, we did also not observe instability any longer (p. 24, line 459-461, and Supplementary Information (SI) Section 2.3 and Fig. S5).
- Mechanistically we could argue that without microbial biomass, there would be no uptake (we would arrive at the “abiotic” EP), and further that the larger the (active) microbial community/biomass, the larger the uptake flux. However, whether this dependency of uptake on microbial biomass is first-order, as assumed by multiplicative and forward Michaelis-Menten kinetics (fMM) can be debated – e.g. density effects could limit the per-biomass uptake rate as the microbial community grows; and importantly only the active fraction of the microbial community contributes to metabolic processes. These phenomena might (implicitly) be better captured by reverse Michaelis-Menten (rMM) or the equilibrium chemistry approximation (ECA) kinetics. However, Tang & Riley (2019) argued that fMM might be a valid approximation of microbial uptake kinetics – though this might not hold in carbon-rich organic soils where microbial biomass can be larger than in carbon-poor mineral soils.
Based on our findings, removing the positive feedback coupling between abundance and resupply of the growth substrate (by removing an explicit representation of DOC) or removing the dependence of microbial uptake on microbial biomass can help to avoid the occurrence of unstable equilibrium points in microbial-explicit models. However, with this we do not mean that these model formulations are to be preferred above others that for instance have an explicit representation of DOC. In fact, we state e.g. that models neglecting an explicit representation of DOC “might have shortcomings in cases where DOC dynamics become important e.g. if drying-rewetting dynamics or leaching are relevant” (p. 24, lines 470-471), and that the choice of how unstable EP’s are avoided depends on the research question at hand and highlight the potential of the third approach—i.e., adapting parameter values in an ecologically consistent way (p 25-26, lines 523-528).
Following the reviewers’ suggestion to “focus more on the microbial ecological rationale rather than simply forcing the theoretical mathematical model stable” we will extend the discussion based on the above reply and by highlighting that while the first two approaches described in Section 4.2 are based on simplifications of the modeled system that might require further justification, the third approach in fact aims to add realism to microbial-explicit models, as it proposes to better acknowledge microbial ecology.
Regarding the third strategy, i.e., choosing parameter values to meet the sufficient and/or necessary conditions for stability, one may argue this could be feasible for a theoretical model with time-invariant carbon input rates and parameters. However, for the real field conditions, both the carbon input rates and parameters vary with time due to the biotic and abiotic effects. It could happen that the parameter values meet the so-called stability conditions sometimes but disobey the conditions at other times.
Spatial and temporal variability are undoubtedly important controls on ecosystem dynamics and functioning. Locally, on small spatial and temporal scales it is not difficult to imagine that environmental fluctuations can lead to diverse behaviors including also the collapse of local microbial populations. However, at large spatial scales this is not observed. E.g. the litter removal data compiled by Georgiou et al. (2017) did not show an extinction of microbes even after decades without inputs. That being said, microbes in model simulations might not directly go extinct if parameter values would temporarily lead to unstable EP’s but could recover if parameter values change back fast enough (though once the “abiotic” state is reached, recovery is impossible). Yet, determining stochastic steady states and their stability is beyond the scope of our study.
Adding to the complexity, not only do carbon input rates and rate parameters fluctuate but so does the microbial community and its functions. Microbial community adaptation to changing environmental condition happens at different timescales ranging from hours (stress response), and weeks (changes in community composition), to years and decades (evolutionary changes) (Abs et al., 2024). These considerations are still largely missing in microbial-explicit models trailed for application in ESM. The third strategy we lined out proposes to capture some of these processes implicitly by introducing appropriate relationships between parameters encoding microbial traits with ecological relevance and environmental conditions. We argue that allowing for adaptation of the modeled microbial community to different environmental conditions by varying microbial trait parameters as a function of environmental conditions could be an effective tool to avoid instability in these models while at the same time making them more consistent with ecological theory.
Therefore, I would like to see additional discussion on the limitation of applying theoretical analyses to real ecosystem, particularly regarding the stability analyses.
Thank you for these helpful questions and reflections. We suggest to expand the discussion with the above outlined arguments.
Minor comments:
Page 3, Line 36-39 (Fig. 1 caption): insert a comma between “dissolved organic carbon (DOC)” and “microbial biomass carbon (MBC)”
Done.
Page 5, Line 106: Table 4 or Table 2?
We meant to point to the summary of analyzed scenarios but this might not be appropriate at this point. We moved the reference to Table 4 to the end of the previous sentence.
Page 8, Section 2.3.2 Specify the time step and period of numerical simulations
We numerically computed steady state values and eigenvalues of the Jacobian matrix by substituting numerical values in the analytical solutions. We did not run transient simulations requiring ODE solvers.
Page 11, Line 227: Clarify “If abiotic loss of SOC and are neglected”
Changed to “If abiotic loss of SOC is neglected”.
Page 14, Line 291: Change “larger then” to “larger than”
Done.
Page 17, Caption of Figure 2: “four-pool SDB”, while “three-pool SDB” in the main text
We changed the first sentence in the caption to read “Simplified causal loop diagrams of the SBE (a) and SDB and SDBE models (b).” to avoid confusion.
Page 21, Table 7: briefly explain the meanings of marking the lower or upper threshold.
We suggest to add the below schematics to Table 7 to clarify the meaning of the thresholds, and further explain it in the caption.
References used in replies
Abs, E., Chase, A. B., Manzoni, S., Ciais, P., & Allison, S. D. (2024). Microbial evolution—An under‐appreciated driver of soil carbon cycling. Global Change Biology, 30(4), e17268. https://doi.org/10.1111/gcb.17268
Georgiou, K., Abramoff, R. Z., Harte, J., Riley, W. J., & Torn, M. S. (2017). Microbial community-level regulation explains soil carbon responses to long-term litter manipulations. Nature Communications, 8(1), 1223. https://doi.org/10.1038/s41467-017-01116-z
He, X., Abramoff, R. Z., Abs, E., Georgiou, K., Zhang, H., & Goll, D. S. (2024). Model uncertainty obscures major driver of soil carbon. Nature, 627(8002), E1–E3. https://doi.org/10.1038/s41586-023-06999-1
Kuzyakov, Y. (2010). Priming effects: Interactions between living and dead organic matter. Soil Biology and Biochemistry, 42(9), 1363–1371. https://doi.org/10.1016/j.soilbio.2010.04.003
Kuzyakov, Y., Friedel, J. K., & Stahr, K. (2000). Review of mechanisms and quantification of priming effects. Soil Biology and Biochemistry, 32(11–12), 1485–1498. https://doi.org/10.1016/S0038-0717(00)00084-5
Lennon, J. T., Abramoff, R. Z., Allison, S. D., Burckhardt, R. M., DeAngelis, K. M., Dunne, J. P., Frey, S. D., Friedlingstein, P., Hawkes, C. V., Hungate, B. A., Khurana, S., Kivlin, S. N., Levine, N. M., Manzoni, S., Martiny, A. C., Martiny, J. B. H., Nguyen, N. K., Rawat, M., Talmy, D., … Zakem, E. J. (2024). Priorities, opportunities, and challenges for integrating microorganisms into Earth system models for climate change prediction. mBio, e00455-24. https://doi.org/10.1128/mbio.00455-24
Tang, J., & Riley, W. J. (2019). Competitor and substrate sizes and diffusion together define enzymatic depolymerization and microbial substrate uptake rates. Soil Biology and Biochemistry, 139, 107624. https://doi.org/10.1016/j.soilbio.2019.107624
Citation: https://doi.org/10.5194/egusphere-2024-348-AC1
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AC1: 'Reply on RC1', Erik Schwarz, 25 Apr 2024
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RC2: 'Comment on egusphere-2024-348', Anonymous Referee #2, 11 Apr 2024
General Comments
Schwartz et al. present a thorough and robust stability analysis of microbial-explicit biogeochemical models, showing how model structure, kinetics, and parameter space can create unstable equilibria. The phenomenon of instability in microbial models has been mentioned in passing in existing literature (e.g. Schimel and Weintraub 2003, Georgiou et al., 2017) but the analysis presented in this manuscript represents the most thorough investigation of this phenomenon to date. This makes the manuscript novel and interesting to the community of people interested in developing microbial-explicit soil carbon models. I have two main areas of feedback to improve the manuscript.
First, I would like the authors to make sure that this manuscript is accessible to the community of researchers who would benefit from understanding its main conclusions. To this end, I count 46 numbered equations in the manuscript in addition to those contained in tables, which is really quite a lot. I appreciate that these equations build support for the main findings of the paper, but I suspect that very few people will read this paper and work through all these equations. There is a significantly broader audience that will benefit from understanding the outcome of this analysis without working through all the math. With this in mind, I suggest the authors take steps to ensure that the main findings are easy to locate and understand for this audience. Although the supplement is quite large already, moving some of the less crucial equations to the supplement could improve readability. In the figures, finding ways to represent data in less abstract ways and providing conceptual interpretations will make the conclusions more actionable for readers of the manuscript. Specific comments below.
My second main comment is that the paper could benefit from some discussion of stability as a realistic ecological/biogeochemical phenomenon. We should expect microbial-explicit models to produce realistic predictions at the scale of the mechanisms that they represent. Generally the mechanisms in these models can be described over very small spatial scales where conditions can be assumed to be homogenous. Directly upscaling non-linear uptake and depolymerization kinetics in heterogenous environments does not preserve model behavior (Chakrawal et al., 2020). It is probably worth some discussion then whether equilibrium is a realistic way to represent microbial dynamics in soil at all. Microbial populations in reality may very well oscillate and exhibit instability at the scale of mechanisms represented in microbial models, while still producing stable emergent behavior at larger scales.
Specific Comments
218-222: This is one place where grouping parameters helps express the equilibrium solutions in a concise way, but it becomes very difficult for me to interpret any of these equations with the added layers of abstraction. When the authors then mention ω > 0 or , the significance of this fact is not clear because the expression is hidden behind a layer of abstraction. Is there any conceptual interpretation that can be added to help readers understand? Similar issue lines 285-290.8
Table 5 and 6: What do the vertical bars in the B and E columns signify?
339-340: This feedback is interesting because it makes mathematical sense but I’m not sure if it makes sense in an eco-evolutionary view. Doesn’t this imply that microbes could decrease constitutive or inducible enzyme production rates without losing access to SOC? If depolymerization is substrate-limited at equilibrium, then isn’t producing extracellular enzymes a losing strategy from an individual fitness perspective? Discussion of this potential paradox could strengthen the paper and help identify future lines of research.
Figure 3: There are a few ways that labeling and captioning on this figure could be clarified to help readers interpret the figure
- For figure 3a, I think it may be useful to label the axes with a conceptual description of the mathematical expression. If I understand correctly, I think we are seeing the sensitivity of depolymerization to changes in soil carbon on the x axis and sensitivity of depolymerization to changes in enzymes on the Y. Points below the line fit the conservative condition for stability, but points with damping values > 1 are stable (if oscillatory). Clearly indicating this in the figure labels, axes, and caption will help readers interpret this figure independently, even if they haven’t worked through all equations. It may also be helpful to specify in the caption that the proposed condition isn’t arbitrary and rather an extension of the criteria found in the simpler model.
- It is confusing that 3b and 3c have essentially the same axis but are shown on two graphs. Figure should include a color legend for stability instead of explaining in the caption. Alternatively, can the same color gradient be used in 3b and 3c that is in 3a?
Figure 4: Color scheme for 4b-d would benefit from a legend on the graph rather than in-text description. It is also confusing that the color scheme is the same as 4a, but corresponds to a different variable. I would also suggest labeling axes with variable names instead of single letters (i.e. Biomass decay rate) so that readers don’t have to refer back to the parameter table to interpret the figure.
Figure 5: Please clarify – are grey points stable but not plausible?
465: This is good and helpful for readers. It would be helpful to add 1-2 sentences in the abstract that summarize these approaches to avoiding instability.
515: This may be a good place to include some discussion about whether microbial equilibrium is , either as a scale issue, or an eco-evolutionary issue, as discussed in comments above.
Technical Corrections
382: analytical analysis is redundant
Citation
Chakrawal, A., Herrmann, A. M., Koestel, J., Jarsjö, J., Nunan, N., Kätterer, T., & Manzoni, S. (2020). Dynamic upscaling of decomposition kinetics for carbon cycling models. Geoscientific Model Development, 13(3), 1399-1429.
Citation: https://doi.org/10.5194/egusphere-2024-348-RC2 - AC2: 'Reply on RC2', Erik Schwarz, 26 Apr 2024
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