the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Optimal enzyme allocation leads to the constrained enzyme hypothesis: The Soil Enzyme Steady Allocation Model (SESAM v3.1)
Christian Reimers
Bernhard Ahrens
Marion Schrumpf
Abstract. Describing the coupling of nitrogen (N), phosphorus (P), and carbon (C) cycles of land ecosystems requires understanding microbial element use efficiencies of soil organic matter (SOM) decomposition. These efficiencies are studied by the soil enzyme steady allocation model (SESAM) at decadal scale. The model assumes that the soil microbial communities and their element use efficiencies develop towards an optimum where the growth of the entire community is maximized. Specifically, SESAM approximated this growth optimization by allocating resources to several SOM degrading enzymes proportional to the revenue of these enzymes, called the Relative approach. However, a rigorous mathematical treatment of this approximation has been lacking so far.
Therefore, in this study we derive explicit formulas of enzyme allocation that maximize total return from enzymatic processing, called the Optimal approach. Further, we derive another heuristic approach that prescribes the change of allocation without the need of deriving a formulation for the optimal allocation, called the Derivative approach. When comparing predictions across these approaches, we found that the Relative approach was a special case of the Optimal approach valid at sufficiently high microbial biomass. However, at low microbial biomass, it overestimated allocation to the enzymes having lower revenues compared to the Optimal approach. The Derivativebased allocation closely tracked the Optimal allocation.
The model finding that the Relative approach was a special case of the more rigorous Optimal approach together with observing the same patterns across optimization approaches increases our confidence into conclusions drawn from SESAM studies. Moreover, the new developments extend the range of conditions at which valid conclusions can be drawn. The new model finding that a smaller set of enzyme types was expressed at low microbial biomass led us to formulate the constrained enzyme hypothesis, which provides a complementary explanation why some substrates in soil are preserved over decades although often being decomposed within a few years in incubation experiments. This study shows how optimality considerations lead to simplified models, new insights and new hypotheses. It is another step in deriving a simple representation of an adaptive microbial community, which is required for coupled stoichiometric CNP dynamic models that are aimed to study decadal processes beyond ecosystem scale.
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Thomas Wutzler et al.
Status: final response (author comments only)

RC1: 'Comment on egusphere20231492', Stefano Manzoni, 25 Aug 2023
The manuscript by Wutzler and coauthors presents a theoretical analysis of three approaches to define extracellular enzyme allocation by soil microbes. The topic is timely and important given the interest in developing microbial explicit models to predict carbon and nutrient cycling in soils. These three approaches build on concepts presented in earlier publications by Wutzler and coauthors, but are of sufficient novelty to warrant publication in a separate article. Specifically, the three approaches are based on the hypothesis that enzymes are optimally allocated to ensure maximum (instantaneous) microbial growth at the whole community level, but differ in the way optimization is implemented—from a rigorous maximization of total return to approximated relations that are easier to implement in models.
I do not have major concerns regarding the model setup and concept, but rather some comments to improve the presentation and clarify the model rationale and derivations.
Main comments
 Readers not familiar with the SESAM model will find it difficult to understand how the proposed developments fit into the broader model. I would add a model schematic in Section 2.1, pointing to the components in the model that affect (and are affected by) enzyme allocation. This could be a place to also define (graphically) the main fluxes used in the mathematical derivations.
 Maximization criterion: I agree that communitylevel maximization makes sense in practice, but I wonder about theoretical support for that (this is something I am also struggling with!). In a microbial community where interactions are dominated by facilitation, maximization of fitness of individual taxa might lead to maximization of community level growth, but what would happen in highly competitive environments? In section 4.3 there could be space for a short comment on the applicability of communitylevel maximization criteria.
 Presentation of main quantities: it would be easier to understand the theory if the main quantities were presented in mathematical form in the main text, whereas now they are spread between text and appendices. I would suggest providing equations and explanations for revenue, return, enzyme investment, microbial growth and other relevant fluxes in Section 2.2, using material now in the appendices. I would also suggest defining the element limitation weights, that are often mentioned but not explained nor listed in the symbol tables.
 Control simulations without adaptation: good idea to include a noadaptation scenario in some analyses, but why not showing it in all figures, to give an idea of the effect of dynamic allocation compared to a ‘control’ scenario?
 Shown variables: it would be helpful to show the same quantities in all (or most) figures, so readers can better appreciate how different numerical experiments affect the dynamics of the same quantities. For example, total respiration and microbial growth, as well as carbon in key model compartments could be shown almost in all figures (they are already shown in some). CUE is now only shown in a figure in appendix, but it is discussed in depth in the main text, so I would suggest showing it in the main text as well.
 Section 4.2: this is a very interesting discussion point and a key overall message. The flip side of the constrained enzyme hypothesis is that also low returns due to very diluted substrates would result in low production of enzymes and thus accumulation of those substrates (https://doi.org/10.1038/s4156102006123). Similar ideas had been proposed earlier in marine biogeochemistry and referred to as ‘dilution hypothesis’ (https://www.science.org/doi/10.1126/science.1258955). Perhaps this section could be supported a bit more using literature on aquatic systems, where return on enzyme investment has been studied for some time.
Minor comments
L62 and elsewhere: I would use the term ‘mortality’ instead of ‘turnover’ if the process being modelled is indeed just mortality
L8082: I would reverse the order of these sentences, first saying what ‘revenue’ is and then explaining why it is an important quantity
Figure 1 is difficult to understand. How do we read that \alpha_L increases by looking at the shown curves? The sentence starting “Starting at levels…” is also quite convoluted. Maybe it would help including additional panels showing trends in return and microbial growth rate as a function of \alpha_L?
L8990: convoluted sentence, hard to understand
L99: point to corresponding equations in the appendices
Section 2.3.1: please add units next to numerical values
Section 2.3.3: worth adding a short motivation for modelling seasonal processes with a model designed for decadal scale dynamics
L151: suggested edit “In the Incubation experiment, a labile Cdepleted soil was amended…”
Figure 2: check units on xaxis (should be “…yr^{1}”?) and “… labile C…” (missing “C”)
L162: “Derivative”
Figure 3, caption: units of L should not contain “yr^{1}”
L168: “By adjusting”
L188190: convoluted sentence, hard to understand
L209: “should help” to achieve what? Not quite clear
End of P10: I see the point of discussing why certain implementation approaches might be numerically problematic, but it is not easy to follow the arguments without visual support; also, typical time scales for enzyme deactivation and microbial mortality are in the order of weeks to few months, so much shorter than the scale at which SESAM is meant to be run
L284288: I do not fully agree with this interpretation, as during drought physiological mechanisms lead to slower growth at the community level (some taxa go dormant, others spend energy to maintain turgor via osmolytes…). My impression is that in dry conditions the production of enzymes per se is not the first priority of the microbes
L326: in other words, enzymes do no interact with each other so rev_Z depends only on \alpha_Z—is this a reasonable interpretation?
P15: I like how this derivation is presented—very elegant!
Supplementary tables: please add missing symbols, such as \omega_C, \omega_N, \omega_P, \nu_{TC}, \nu_{TN}, \nu_{TP} (unless the \nu_{TE} are the same as \nu_{T}?)
L346: “this condition implies…”
Figure B2: it would help to show also \alpha values in this figure
P18, equation for d_{ZC} and following equations: is \alpha the same as \alpha_C? Also, the weighing factors \omega are not defined
L393: is \alpha_{P1,2} the same as \alpha_{1,2} in Section C3.1?
Section C4: just a semantic comment, profit is maximized by optimizing enzyme allocation (to be consistent with the rest of the manuscript)
L498, L405: in both lines, “profit/investment”, not just “profit” (I think)
L411: “By using…”
L433434: convoluted sentence, hard to understand
L437: “it is possible…”
Citation: https://doi.org/10.5194/egusphere20231492RC1 
RC2: 'Comment on egusphere20231492', Sergey Blagodatsky, 25 Sep 2023
The paper by Wutzler et al., "Optimal enzyme allocation leads to the constrained enzyme hypothesis: The Soil Enzyme Steady Allocation Model (SESAM v3.1)" describes the current development of the SESAM model, where three approaches for the calculation of enzyme activity allocation by soil microorganisms were compared. The main conclusion is that the derivative approach proposed in this MS differs from the relative approach and gives more realistic values in model outputs in the case of low microbial biomass levels. The authors argue that this approach is promising and plan to use it in further model development. Based on model predictions (namely from three simulation experiments), the authors put forward the constrained enzyme hypothesis, which they believe explains the longterm persistence of SOM. In the discussion, the authors mention that this hypothesis is complementary to other current explanations of this phenomenon, but I remain sceptical about the relative importance of the constrained enzyme hypotheses in explaining SOM persistence. This view of the authors is not supported by experimental data, and the results of the presented simulation experiments are strongly dependent on the prescribed model parameterisation and the chosen classification of enzyme groups. From section 4.4, which discusses the observational evidence for the proposed hypothesis, it becomes clear that there is no data yet to confirm the constrained enzyme hypothesis. Figure 6 illustrates the spatial separation of substrates and enzymes at low biomass levels in the soil (right part) rather than the production of fewer enzyme types. However, these remarks do not imply that the work is of low quality or incomplete. The idea of allocating resources to the production of different enzyme groups as a response of the microbial community to specific limitation by one or two key nutrients is very valuable and timely. The current MS provides an excellent basis for further solutions in modelling soil microbial biomass and SOM dynamics. The next step would be to test the new model version against experimental data, where the current division into enzyme groups could be confirmed or rejected with suggestions for further model structure modifications.
Specific comments and technical notes
Section 2.1: I would suggest providing a model scheme for readers who are facing the SESAM model for the first time. In order not to exactly repeat the scheme of Wutzler et al. 2022, it would be desirable to show the parts (or application points of allocation optimisation approaches) newly developed in this study. It is also not clear from which pool P is obtained. In the 2022 paper there are two enzyme groups for P and in your current publication there seems to be 1, or?
L150: There is a misprint, please correct. Seasonal does not seem to be the right term to describe this simulation experiment. Does the temperature and the moisture vary in this case? If not, perhaps 6month incubation would be a more accurate name for this part.
L151: Please edit the sentence, as it is not fully clear.
Fig.2. The legend on the figure can be expanded to make it easier to understand. The meaning of R, L and P can be deduced from the method section, where the numerical experiment is described, but a direct description, i.e. residues, labile and phosphorustargeted enzymes, would improve the figure.Citation: https://doi.org/10.5194/egusphere20231492RC2
Thomas Wutzler et al.
Thomas Wutzler et al.
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