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:

1. 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 f_I, 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).
2. 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

We thank anonymous reviewer #2 for their supportive comments and stimulating thoughts.

We have compiled our replies to all individual comments in the attached PDF to facilitate inclusion of suggested updates on figures and equations.