the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 17 Dec 2025
Divergent carbon use efficiency-growth rate tradeoff in popular biological growth models
Abstract. Carbon use efficiency (CUE) is an important trait emerging from processes regulating biological growth. CUE can be computed either based on the growth of structural biomass or total biomass divided by substrate uptake rate. Nonequilibrium thermodynamics and observations suggest that, for an exponentially growing population, structural biomass CUE should first increase, then peak, and finally decrease with specific growth rate; meanwhile, total biomass CUE increases asymptotically with specific growth rate. We compared predictions from six popular models that are often used for plant and microbial growth in existing ecosystem models. We found that, for an exponentially growing population of biological cells, (1) the Pirt model and Compromise model predict that structural biomass CUE increase asymptotically with growth rate; (2) the modified Droop model predicts that structural biomass CUE decreases with growth rate; and (3) the variable internal storage model and two dynamic energy budget models predict that structural biomass CUE first increases, then peaks, and finally decreases with growth rate. Moreover, the modified Droop model predicts that total biomass CUE is constant with growth rate, while all other five models predict that total biomass CUE increases with growth rate asymptotically. For non-exponential biological growth, we show that there is no deterministic relationship between total biomass CUE or structural biomass CUE with respect to either growth rate or temperature. Therefore, we contend that biological growth models should explicitly represent interactions between substrate acquisition, substate transformation, and maintenance respiration to better capture observed CUE dynamics and thereby ecosystem biogeochemistry.
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Status: open (until 28 Jan 2026)
- RC1: 'Comment on egusphere-2025-5448', Anonymous Referee #1, 08 Jan 2026 reply
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RC2: 'Comment on egusphere-2025-5448', Stefano Manzoni, 18 Jan 2026
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Tang and co-authors investigate the relation between carbon-use efficiency (CUE) and growth rate, with specific examples on microbial growth in soil. This is an interesting topic in general and the examples on soils are particularly timely, given the recent efforts to measure, model, and map soil microbial growth and CUE, and their effects on soil organic matter. The topic itself fits well the scope of Biogeosciences. The authors selected six mathematical models describing growth processes and one more abstract describing energy flows in a generic system. The chosen models range from purely theoretical to phenomenological, to fairly detailed mechanistic, thus offering a comprehensive view of modelling approaches currently in use. The models are analyzed by plotting modeled CUE vs. growth rate under different scenarios. My main concerns are listed first, followed by line-by-line comments.
Note: I have not read the comments by Anonymous Referee #1 before writing my own comments, so it is possible that some overlap.
Main concerns
- Analytical solutions. Having analytical solutions for the exponential growth scenario is useful to see how CUE-growth relations emerge mathematically during a transient phase. I wonder if analytical solutions can also be found for a steady state scenario in which a single substrate is supplied at constant rate, leading to microbial growth with or without necromass recycling in the substrate pool. This alternative scenario would complement the transient analysis and represent an intermediate step before the current scenarios 2 and 3, though I am not sure analytical solutions can be found for all six models.
- Scenario setup. By reading the main text, it is not clear how the scenarios were setup. For the exponential growth case, what drives growth rate variations? In L154, it is stated that the rate of substrate supply is constant, but I suppose this means it is constant for each growth rate value, so to draw the CUE-growth curves in Figure 2, input rates are varied. It would be good to clarify that these curves are not result of temporal changes in growth rate as substrate is depleted during a transient simulation, but rather represent a range of experiments with different realized growth rates. What is the goal of scenarios 2 and 3? If I understand correctly, the aim is to show how variations in input rates (scenario 2) and environmental conditions (temperature in scenario 3) affect the CUE-growth relations. But imposing random fluctuations in addition to a smooth seasonal cycle does not help seeing these effects clearly. I would suggest forcing the system only with smooth seasonally varying input in scenario 2 and temperature in scenario 3 (sine functions should do the job). With this setup, the erratic behaviors shown in Figures 3-6 would disappear and more regular and interpretable trajectories would emerge. One can additionally play with the timing of the annual input and temperature peak to assess the consequences of synchronous vs. asynchronous drivers. I just want to emphasize that there is nothing wrong in the current scenarios, but I find them less informative and useful than they could be—and they leave many questions on the model behavior unanswered because of the large noise introduced by the driver variability.
- Numerical simulations. The trajectories in some of the figures show sharp edges, which I guess are due to changes in temperature affecting the rates, but I wonder to what degree this is an artifact of the way the numerical simulations are conducted. How are temperature variations imposed in the numerical solution? In L157 it is stated that temperature is varied within a day, but in L252 variations are defined as “daily”. In addition to clarifying what is the scale of temperature variations (within a day or across days?), how are such variations implemented in the numerical solution of the mass balance equations? Are the equations solved at finer temporal resolution for numerical stability, so that temperature is piecewise constant? This comment is not relevant if the authors decide to simplify the setup of scenarios 2 and 3 as suggested in my previous comment.
- I would suggest a slight restructuring of the Introduction paragraphs 2 and 3. Now paragraph 2 starts with a description of soil systems, then presents general concepts, and is followed by paragraph 3 on plants. I would present general concepts first and then examples on soil, plants, or other organisms.
- Discussion and conclusions. Section 4.2 develops arguments to support the claim that dynamic energy budget models (in particular mDEB) are “superior” compared to the other models. My impression is that this manuscript is not about comparing model performance, but rather about comparing model emergent behaviors in terms of CUE-growth relations. There is no comparison with data, so statements regarding how “good” a model is are not well supported (e.g., L310, L313). For example, one could argue that CUE-growth data tend to only populate the growing branch of the CUE-growth curve (Figure 3 in Hu et al. 2025, https://doi.org/10.1111/gcb.70036), suggesting that Pirt and Compromise models might be able to capture the essential dynamics without the added complexity of DEB models. Similarly, the conclusion that environmental factors should not be included as multiplier functions is not supported by evidence shown in this work. To sum up, I would try to limit discussion and conclusions to the patterns presented in the analyses.
Other comments
L12: I would provide a concise definition of “structural biomass” (as opposed to “total biomass”)
L44: Tao et al. assumed Monod kinetics for substrate assimilation and Michaelis-Menten kinetics for enzymatic decomposition, so their model was not linear
L105: I find this thermodynamic toy model very useful to understand the origin of CUE-growth curves, but I would suggest to explain in simple terms what negative efficiencies mean, and how the forces are related to substrate and biomass thermodynamic properties
L156: please check the units of the input rate—here it’s a rate (mass/time), while in Supplementary Section E it is expressed a flux (mass/area/time). It would be good to have consistent units in the main text and in the supplementary materials
Bottom of P4: the terminology introduced here is not always clear; “food capture” in my mind is the process of ingesting food, while “assimilation” is the process of absorbing resources from the guts, but here it seems J2 is a rate supplying the reserve pool if I interpreted correctly, so not “food capture”
Section 4.1, L325: the trajectories shown in Figures 3-6 might look complicated because the drivers were not smooth functions, but they are (likely) still deterministic. My suggestions to test the effect of smooth driver functions could also help assess if trajectories are indeed deterministic or if they exhibit chaotic behavior. In the current version, I don’t think it can be concluded that the relationships are not deterministic (though I agree with the authors that they are not unique)
L273: it is not clear what “averaging method” refers to
L287-289: are the details on the dynamics of this particular carbon storage pool important here?
Supplementary Section A: I am a bit confused by the definition of cellular quota—I have often seen it defined as content of a given element per unit total biomass (or per cell), while here it is defined as the inverse of the common definition
Supplementary Section B: in contrast to Section A where B_X represented the total biomass, here the same symbol B_X represents storage biomass (B_X – B_V in Section A). Please use the same definition for a given symbol throughout the manuscript, otherwise it becomes very difficult to follow the derivations
Supplementary Section E: “one microbe” meaning a homogeneous microbial population (as opposed to one microbial cell)?
Eq. F2 and F3: to avoid confusion with variable and parameter symbols, I would use normal font for mathematical functions exp and ln
Citation: https://doi.org/10.5194/egusphere-2025-5448-RC2
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Modeling code Jinyun Tang https://github.com/jinyun1tang/cue_paper
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- 1
This manuscript presents a theoretical and numerical analysis of carbon use efficiency (CUE) dynamics across six commonly used biological growth formulations. By grounding the analysis in nonequilibrium thermodynamics, the authors argue that under exponential growth conditions, structural biomass CUE should exhibit a non-monotonic tradeoff with growth rate, analogous to power–efficiency tradeoffs in thermal engines. They further demonstrate that only models explicitly representing internal storage and sink-driven growth (VIS, sDEB, mDEB) reproduce this behavior, whereas the Pirt and Compromise models do not.
Overall, the manuscript is well written and addresses an important and timely question regarding the mechanistic consistency of microbial growth representations in ecosystem models. The comparison across multiple models is systematic and informative, and the conclusions are potentially useful for future land surface and Earth system model development.
I have several suggestions that I believe would strengthen the interpretation and applicability of the results, as outlined below.
The manuscript clearly distinguishes between structural biomass CUE and total biomass CUE, and convincingly demonstrates that these two metrics exhibit markedly different behaviors across models. I suggest that the authors further strengthen the manuscript by explicitly connecting this conceptual distinction to empirical CUE measurements, particularly in the Introduction and Discussion.
For example, in the DEB framework, the denominator of structural biomass CUE includes all energetic sinks, including transient accumulation in reserve pools. In contrast, some empirical approaches (e.g., the ¹⁸O–H₂O labeling method) estimate CUE based on the ratio of growth to respiration, without explicitly accounting for reserve dynamics. Consequently, the structural biomass CUE analyzed here is not strictly equivalent to commonly used empirical CUE estimates, except under special conditions where changes in reserve pools are negligible.
Explicitly discussing this distinction would help readers better interpret the comparability (and limitations thereof) between the model-derived CUE metrics and observational datasets.
The manuscript repeatedly refers to plant growth CUE alongside microbial CUE. While this comparison is potentially valuable, the purpose of comparing plant and microbial CUE is not always entirely clear.
For instance, the paragraph on plant CUE in the Introduction (lines 55–62) appears rather suddenly and could benefit from clearer framing. In the Discussion (lines 291–292), the authors note that representations of plant growth processes in Earth system models are generally more advanced than those of soil microbes. This is an important point that many readers would appreciate seeing further developed.
I suggest that the authors more clearly explain why plant CUE is introduced, how it conceptually relates to microbial CUE in the context of this study, and whether plant growth modeling can serve as a useful reference for improving microbial growth representations. Strengthening this linkage—potentially by using plant CUE as a motivating example in the Introduction—would improve narrative coherence.
A key conceptual assumption (or analogy) in the manuscript is that biological growth can be treated as a finite-time engine operating under nonequilibrium conditions, and therefore subject to a universal power–efficiency tradeoff. I find this analogy appealing and conceptually stimulating.
At the same time, I would appreciate seeing the authors more explicitly acknowledge and discuss the potential limitations or conditions of applicability of this analogy. In real soil microbial systems, processes such as dormancy, physiological acclimation, and adaptation to fluctuating environmental conditions are common. It would be helpful for readers if the authors briefly discussed whether, and to what extent, such processes might challenge or constrain the strict applicability of the thermal engine analogy.
Another important concept of this manuscript is the distinction between source-driven and sink-driven growth, with the results suggesting that sink-driven formulations more robustly capture emergent CUE dynamics. I encourage the authors to expand the discussion on how this conceptual framework applies to existing soil carbon models.
If my understanding is correct, widely used models such as MIMICS, Millennial, and even traditional non-microbial models like CENTURY would fall largely into the source-driven category. From the perspective in this manuscript, this would imply that most microbial growth representations currently used in Earth system models are structurally limited.
I suggest that the authors:
This expanded discussion would greatly enhance the practical relevance of the manuscript for ongoing soil carbon and land surface model development.