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
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 the 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 SOMdegrading 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 the 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.
These findings increase our confidence in conclusions drawn from SESAM studies. Moreover, the new developments extend the range of conditions at which valid conclusions can be drawn. Further, based on these findings we formulated the constrained enzyme hypothesis. This hypothesis provides a complementary explanation why some substrates in soil are preserved over decades, although they are often decomposed within a few years in incubation experiments.
This study shows how optimality considerations lead to simplified models, new insights, and new hypotheses. This is another step in deriving a simple representation of an adaptive microbial community, which is required for coupled stoichiometric C–N–P dynamic models that are aimed to study decadal processes beyond the ecosystem scale.
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The Soil Enzyme Steady Allocation Model (SESAM) studies the effect of an adaptive soil microbial community on the coupling of element cycles in aerated soils at a decadal timescale. The coupling of the cycles of nitrogen (N), phosphorus (P), and carbon (C) is especially strong in soils because the stoichiometric requirements of soil organic matter (SOM) decomposers is much less flexible than the stoichiometric requirements of plants (Robert W. Sterner, 2002; Mooshammer et al., 2014b). The stoichiometric requirements, in turn, together with the stoichiometry of consumed substrates, determine the decomposer's carbon and nutrient use efficiencies, which are important controls on ecosystem dynamics. Carbon use efficiency (CUE) is key to controlling how much of the litter input is stored in soil or respired again to the atmosphere (Manzoni et al., 2017). Similarly, nitrogen use efficiency affects how much N in litter inputs is stored in organic matter or mineralized and made available for plant nutrition (Mooshammer et al., 2014a). These element use efficiencies are also affected by properties of the microbial community. Furthermore, the microbial community is hypothesized to adapt to a changing environment, such as increased litter inputs or litter stoichiometry or nitrogen deposition (Manzoni, 2017; Manzoni et al., 2021).
However, there is a gap between knowledge of microbial processes at smaller and effect at larger scales. On the one hand, knowledge of the complex microbial ecology and community adaptations accumulates at the soil pore scale. On the other hand, dynamic SOM models, which rely on nutrient efficiencies of the decomposers, focus on SOM changes at the ecosystem to global scale. Hence, we need to find ways to incorporate effects of soil microbial community adaptations on element use efficiencies (Kaiser et al., 2014) without the need to model all the microbial populations and microbial details.
Therefore, the SESAM model abstracts from microbial details by assuming that community composition develops towards maximizing growth of the entire microbial community. This assumption is in line with arguments from system ecology (Nielsen et al., 2020), which realized that open systems with positive internal feedback develop towards the best exploitation of a gradient of potential energy (Ulanowicz, 2002). This exploitation of the gradient is usually associated with maximizing the entropy production that can support the internal structure of the system (Kondepudi, 1998). For soil systems this mainly translates into efficiently degrading the chemical energy input provided by plant litter and rhizodeposition. In a first approximation this efficient degradation is achieved by maximum growth and respiration of soil microbes. This focus on the system perspective leads to complementary insights, compared to focusing on competition, and opens up new ways of studying living systems (Ulanowicz, 2009). One of the problems of this argument is the question at which scale to apply the maximum entropy production hypothesis. Application at different scales leads to different predictions of optimal system dynamics (Dewar, 2010). Hence, the assumption of optimal community growth is rational, but it is still an assumption to be challenged.
The heuristic approach of how community growth is optimized in SESAM requires a more rigorous treatment. The heuristics that is applied in SESAM v3.0 assumes the proportion of allocation for an enzyme, Z, to be proportional to its revenue, i.e. return per investment. Wutzler et al. (2017, Appendix B) provide a rationale for this approach, which argues that exploiting the full range of resources is beneficial. However, this attempt does not sufficiently well explain why this leads to optimal community growth. Hence, a better, i.e. more rigorous, rationale is required to increase confidence of assumptions made in SESAM.
Such a rigorous treatment of optimal enzyme allocation has now become possible because of recent model developments. The model developments of Wutzler et al. (2022) comprise a new formulation of decomposition based on the quasisteady state of enzymes and the new formulation of revenue with limitationweighted enzyme investments. They make it possible to express the revenue directly as a function of the enzyme allocation. This functional expression now allows us, in this study, to derive optimal community allocation by maximizing the total return from enzymatic processing. Further, it inspired another simpler heuristic optimality approach.
The aim of this study is to present and compare three approaches of computing enzyme allocation, i.e. the rigorous Optimal approach, the previously applied heuristic Relative approach, and the new heuristic Derivative approach. We compare approaches based on several scenarios of dynamic simulation and discuss the resulting insights and implications. One of those insights is the constrained enzyme hypothesis.
In this section, we first summarize the SESAM model and restate the equations that are most relevant for the optimality approaches (Sect. 2.1). Next, we present the three optimality approaches (Sect. 2.2). Finally, we describe the setup of the simulation experiments (Sect. 2.3).
^{a} Arbitrary high value so that plant uptake is constraint by ${u}_{{I}_{E},\mathrm{max}}\left(t\right)$.
^{b} Balancing nutrient input to the system.
2.1 The SESAM model
SESAM is described in the previous paper of this incremental model description paper series (Wutzler et al., 2022). A summary of the model is presented in Fig. 1, and state variables, model drivers, model parameters, and other symbols used in SESAM are listed in Tables 1, 2, and 3. The symbol d with subscripts denotes a form of decomposition or return flux, while the symbol d without subscripts denotes the derivative operator. The model version used in this study already anticipates ongoing unpublished model developments, which include phosphorus (P) cycling and microbial P limitation. While P is generally handled the same way as nitrogen (N), there is an additional class of Pbiomineralizing enzymes, E_{P}, that does not depolymerize substrates in stoichiometric quantities but cleaves phosphate groups. Moreover, E_{P}, is produced not only by microbes but also by plant roots. Those Prelated developments will be described in their own paper, but this paper presents formulas that can account for this new type of enzyme.
This study focuses on approximating optimal microbial community allocation, α. It modifies Eq. (3) in Wutzler et al. (2022) and compares several variants. SESAM substrate decomposition is controlled by the quantity of enzymes, which in turn, are controlled by the microbial community that adapts their allocation to different enzymes in order to maximize growth (Fig. 2). Allocation to different enzymes adapts to the return and revenue, i.e. return/investment, of those enzymes.
The details of how SESAM computes the return of an enzyme are restated in the following section, while revenue is described in Sect. 2.2.1.
2.1.1 Depolymerizing enzymes
The return of an enzyme, E_{Z}, which depolymerizes a substrate, S_{Z}, is the elementallimitationweighted average of the returns for the modelled elements required for microbial growth. The return equals the depolymerization flux that is taken up by microbes.
C depolymerization flux, ${{d}_{Z}}_{\mathrm{C}}$, is described by reverse Michaelis–Menten kinetics (Schimel and Weintraub, 2003), which is first order to the source pool, k_{Z}S_{Z}, and saturating with the amount of the respective enzyme. By assuming enzymes to be near quasisteady state at the timescale of larger than a month, their amount is proportional to microbial enzyme production flux (α_{Z}a_{E}B). This enzyme production flux then is used in the Michaelis–Menten kinetics together with a lumped affinity parameter, ${{k}_{mN}}_{Z}$.
Nutrient, $E\in \mathit{\{}\mathrm{N},\mathrm{P}\mathit{\}}$, depolymerization fluxes are derived by dividing the C flux by the C:E ratio, ${{\mathit{\beta}}_{\mathrm{E}}}_{Z}$, of the source pool. These depolymerization fluxes are then converted to C units by C:E ratio of microbial biomass, ${{\mathit{\beta}}_{E}}_{B}$, so that a weighted return, ${{d}_{Z}}_{\mathrm{w}}$, can be computed. Further the depolymerization fluxes are multiplied by a dynamically computed proportion, ν_{TE}, that describes what part of the flux currently reaches microbes rather than plants, leaching, or sequestration at minerals (Wutzler et al., 2022, Appendix B).
The limitationweighted return can be expressed as a potential return, d_{Z}, multiplied a factor that reduces return due to low enzyme levels. Hence, the potential return denotes the return potentially achieved at infinitely high enzyme levels. It is the potential Csubstrate depolymerization flux, multiplied by the combined elemental weighting factor, ω_{Z}.
Similarly to defining an elementalweighted limitation factor for enzyme returns, such an elementalweighted factor is defined for the enzyme synthesis flux (Eq. 3).
The elemental weights, w_{E}, are the same in both ω_{Z} and ω_{Enz}. Therefore, they do not need to be normalized within ratios of these two quantities, e.g. the revenue calculation in Sect. 2.2.1.
How strongly microbial biomass is limited by either of the elements $E\in \mathit{\{}C,N,P\mathit{\}}$ is described by the elementallimitation weights (Eq. 4) (Wutzler et al., 2022, A15).
It exponentially decreases with the difference between flux potentially available for microbial biomass synthesis by this element, C_{synBE}, and the actual synthesis flux, syn_{B}, which is constrained also by other elements. To derive a unitless quantity, it is scaled by the microbial turnover flux, tvr_{B}. Parameter δ controls how steep is the transition near colimitation by several elements. For an more detailed presentation of the elemental limitation we refer the reader to Wutzler et al. (2022).
2.1.2 Biomineralizing enzymes
The phosphatases only cleave phosphate groups from soil organic matter. Hence, they only make P available for uptake, without making C and N available. They attack both labile and residue organic matter. Although the P cycle in SESAM will be described in its own paper, we state the return and revenue here.
The potential return of action of Pdegrading enzymes, d_{P}, includes the Plimitation weights, w_{P}, only, contrary to the depolymerizing enzymes (Eq. 2). Moreover, it does not divide by the C:P ratio of the substrate, as the mineralization flux is already expressed in P units:
In addition, a limitation factor, ${l}_{{{\mathit{\beta}}_{\mathrm{P}}}_{S}}\in (\mathrm{0},\mathrm{1})$, decreases the potential rate of a biomineralizing enzyme with an increasing C:P ratio, ${{\mathit{\beta}}_{\mathrm{P}}}_{S}$, of a substrate, S. Parameter ${{\mathit{\beta}}_{\mathrm{P}}}_{m}$ is the C:P ratio at which the limitation factor decreased to $\mathrm{1}/\mathrm{2}$.
Moreover, these phosphatases are also produced by plant roots at a rate, e_{P}. Hence, one needs to calculate the return of microbeproduced enzymes, ${{d}_{\mathrm{P}}}_{m}$, by subtracting the flux due to plantproduced enzymes from total biomineralization flux (Table B1).
2.2 Allocation optimization approaches
The derivative of the total return, u_{T}, with respect to each enzyme allocation share, α_{Z}, called “the derivative” for short, is the central quantity to inspect. The differences across those derivatives across enzymes determine the direction of changes in enzyme allocation, i.e. changes in the microbial community. Allocation is changed towards the enzyme, Z, that has the highest derivative, i.e. highest increase in return per additional allocation, at the expense of decreasing allocation to enzymes with the lowest derivative. Hence, derivatives are equal at the optimum (Appendix B1). The derivatives decrease with increasing allocation because the return saturates at high enzyme levels. Therefore, it is often beneficial for the community to distribute investment in enzymes across different enzymes rather than investing solely in the enzyme with the highest potential return (Fig. 3).
The revenue of allocation to an enzyme, Z, another important quantity, is the return from enzymatic processing (Sect. 2.1.1 and 2.1.2) divided by the investment in enzyme production: ${u}_{Z}=\frac{{{d}_{Z}}_{\mathrm{w}}\left({\mathit{\alpha}}_{Z}\right)}{{\mathit{\alpha}}_{Z}{\mathit{\omega}}_{\mathrm{Enz}}{a}_{\mathrm{E}}B}$. The investment is the share, α_{Z}, invested in the production of an enzyme, Z, multiplied by total elementallimitationweighted carbon flux allocated to enzyme production, ω_{Enz}a_{E}B.
Hence the total return and revenue depend on the potential decomposition flux, i.e. the amount and the decomposition rate of the substrate, as well as its stoichiometry via weighting by the current elemental limitation of the microbes. In addition, they depend on enzyme levels, i.e. the size of the microbial biomass producing the enzymes, and on the current enzyme allocation, i.e. the shares of the total enzyme production in the alternative enzymes.
Three approaches of estimating the time development of enzyme allocation, α, are presented in this study. The Optimal approach is the mathematically exact formulation of the hypothesis of the maximum return of enzyme investment, but it is only practical for simple cases. Therefore, two heuristic approximations are added. First, the Relative approach assumes that the optimal allocation can be estimated by setting the allocation proportional to the revenue. Second, the Derivative approach describes the direction of change in allocation without explicitly computing the optimal allocation. The optimum, to which microbial community in SESAM develops towards, is characterized by maximum growth of the entire microbial community, which in turn is achieved when the return from extracellular enzymatic processing is maximized.
2.2.1 Total return of enzyme action
We seek the enzyme allocation, α, that maximizes the total limitationweighted return, i.e. the action of enzymes, depolymerization, and biomineralization. We exclude the trivial case of investing only into a single enzyme (α_{Z}=1) and exclude enzymes that are not allocated to (α_{Z}=0).
The total return that is optimized is the sum of each revenue multiplied by enzyme investment.
u_{T} fulfils the conditions of Lemma 1 (Appendix B1). Therefore, potential optima are located at the borders or at condition $\frac{\mathrm{d}\left({\mathit{\alpha}}_{Z}{\mathrm{rev}}_{Z}\right)}{\mathrm{d}{\mathit{\alpha}}_{Z}}={C}_{\mathrm{3}}$. This implies that the derivatives of the total return, $\frac{\mathrm{d}{u}_{\mathrm{T}}}{\mathrm{d}{\mathit{\alpha}}_{Z}}={\mathit{\omega}}_{\mathrm{Enz}}{a}_{\mathrm{E}}B\frac{\mathrm{d}\left({\mathit{\alpha}}_{Z}{\mathrm{rev}}_{Z}\right)}{\mathrm{d}{\mathit{\alpha}}_{Z}}$, are equal at the optimum.
The revenue for a depolymerizing enzyme and its derivative are
Revenue, rev_{P}, and its derivative of a biomineralizing enzyme are slightly more complex due to plant enzyme production, but they are similar to the ones of the depolymerizing enzymes. They are presented in Table B1.
2.2.2 Optimal approach
The Optimal approach computes the target allocation that maximizes the total return by computing where the derivatives of the total return across the set of allocated enzymes are equal (Appendix B1). Such a derivative of the return with respect to enzyme allocation, α_{Z}, is proportional to the derivative of the allocation times the revenue, $\frac{\mathrm{d}{u}_{\mathrm{T}}}{\mathrm{d}{\mathit{\alpha}}_{Z}}\propto \frac{\mathrm{d}\left({\mathit{\alpha}}_{Z}{\mathrm{rev}}_{Z}\right)}{\mathrm{d}{\mathit{\alpha}}_{Z}}$ (Sect. 2.2.1). While the maximum change of return is realized at an arbitrarily small allocation, ${{\mathit{\rho}}_{Z}}_{\mathrm{max}}=\frac{\mathrm{d}\left({\mathit{\alpha}}_{Z}{\mathrm{rev}}_{Z}\right)}{\mathrm{d}{\mathit{\alpha}}_{Z}}{}_{{\mathit{\alpha}}_{Z}\to \mathrm{0}}$, the optimal allocation, α^{*}, often involves several enzymes (Fig. 3). However, if the maximum change of return for an enzyme, Z_{j}, is lower than the return of allocating only to other enzymes, the optimal allocation to this enzyme is zero; i.e. it is excluded from the set of allocated enzymes. The set of allocated enzymes, i.e. enzymes among which to distribute resources, can be found by the following algorithm.

Order the enzymes according to their maximum change in return, ${{\mathit{\rho}}_{Z}}_{\mathrm{max}}$; index them by i; set i=1; and start with a mix that includes only the most efficient enzyme, {Z_{1}}.

Solve for the optimal allocation strategy, α_{i}, equalizing derivatives
$$\frac{\mathrm{d}{u}_{\mathrm{T}}}{\mathrm{d}{\mathit{\alpha}}_{Z}}}\propto {\displaystyle \frac{\mathrm{d}\left({\mathit{\alpha}}_{Z}{\mathrm{rev}}_{Z}\right)}{\mathrm{d}{\mathit{\alpha}}_{Z}}}={\mathit{\rho}}_{i}\phantom{\rule{0.125em}{0ex}}\text{for all}Z\in \mathit{\{}{Z}_{\mathrm{1}},\mathrm{\dots},{Z}_{i}\mathit{\}},$$and allocate nothing to enzymes that are not part of the current mix.

For ρ_{i} computed in step 2, if ${\mathit{\rho}}_{i}>{{\mathit{\rho}}_{{Z}_{i+\mathrm{1}}}}_{\mathrm{max}}$, stop and report the found optimum, ${\mathit{\alpha}}^{*}={\mathit{\alpha}}_{i}$. Otherwise increase i; i.e. include the enzyme, Z_{i+1}, in the mix and go to step 2.
Step 2 needs explicit solutions for different numbers and types of enzymes in the mix. Appendix B3 provides such explicit solutions for up to three enzymes across depolymerizing and biomineralizing enzymes.
2.2.3 Relative approach
The Relative approach, which was used up to SESAM v3.0 (Wutzler et al., 2022), estimates optimal allocation to be proportional to revenue based on current allocation (Eq. 8).
where rev_{Z} is the revenue for an enzyme, Z.
Appendix C shows that it is a special case of the Optimal approach given several assumptions. It approximates optimal allocation well for the case of sufficiently high microbial biomass levels.
2.2.4 Derivative approach
The Derivative approach computes the rate change in α_{Z} over time. It assumes that enzymes allocation changes faster the larger the corresponding derivative is away from the average, i.e. the optimal state where all derivatives are equal. More precisely, it assumes the change rate of allocation over time to be $\frac{\mathrm{d}{\mathit{\alpha}}_{Z}}{\mathrm{d}t}\propto \frac{\mathrm{d}{u}_{\mathrm{T}}}{\mathrm{d}{\mathit{\alpha}}_{Z}}{\mathrm{mean}}_{\mathrm{i}}\left(\frac{\mathrm{d}{u}_{\mathrm{T}}}{\mathrm{d}{\mathit{\alpha}}_{i}}\right)$ across the enzymes in the current mix (Appendix D). It does not rely on an optimal solution, α^{*}. This is beneficial because formulas in the Optimal approach for a higher number of enzymes or more types of enzymes quickly grow and involve higherorder polynomials of α_{Z} with multiple roots and additional mathematically possible solutions outside the reasonable bound ${\mathit{\alpha}}_{Z}\in [\mathrm{0},\mathrm{1}]$.
The Derivative approach assumes that a higher increase in the total return leads to faster shifts of allocation towards this enzyme. It takes care, similar to the Optimal approach, to compute the average only across enzymes that are part of the current mix (Appendix D1).
2.3 Simulation experiments
In order to study the effects of using different allocation optimization approaches on model behaviour, we set up different simulation experiments and compared differences in predictions among the approaches.
2.3.1 Immediate response: Prescribed potential returns
The “Prescribed potential returns” simulation experiment fixed the direct inputs to the function computing allocation changes. It neglected all other model feedback and focused and compared computation of optimum allocation for prescribed conditions.
Specifically, the experiment prescribed elementallimitationweighted potential return fluxes, d_{Z} (Sect. 2.1.1), which otherwise had been dynamically computed in the model from pools and parameters. It assigned values for enzymes decomposing residue litter and biomineralizing phosphorus of ${d}_{\mathrm{R}}=\mathrm{0.7}\phantom{\rule{0.125em}{0ex}}\mathrm{g}\phantom{\rule{0.125em}{0ex}}\mathrm{C}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{\mathrm{1}}$ and ${d}_{\mathrm{P}}=\mathrm{0.5}\phantom{\rule{0.125em}{0ex}}\mathrm{g}\phantom{\rule{0.125em}{0ex}}\mathrm{C}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{\mathrm{1}}$ and varied the flux for enzymes decomposing labile substrates, d_{L}∈{0…1}, in units $\mathrm{g}\phantom{\rule{0.125em}{0ex}}\mathrm{C}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{\mathrm{1}}$. It simulated the allocation state until it converged to its estimated optimum for each d_{L}. For complete reference we list the other relevant parameters without further explanation here: ${a}_{\mathrm{E}}=\mathrm{0.1}{\mathrm{yr}}^{\mathrm{1}}$, $B=\mathrm{1}\mathrm{gC}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}$, ${e}_{\mathrm{P}}=\mathrm{0}\mathrm{gC}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{\mathrm{1}}$, $\mathit{\tau}=\mathrm{365}/\mathrm{30}{\mathrm{yr}}^{\mathrm{1}}$, ${k}_{mN}={a}_{\mathrm{E}}B/\mathrm{2}$, and ω_{Enz}=1. The experiment included further runs with a 5fold increase in microbial biomass levels, B.
2.3.2 Decadal term: FACE
The Free Air Carbon Enrichment (FACE) simulation experiment simulated the decadalterm response of the system to increased labile substrate inputs. It started with model pools in steady state with litter inputs. Next it prescribed a jump of labile substrate inputs by 20 % simulated for 50 years and prescribed another jump of labile substrate inputs to initial values. It simulated Nlimited conditions and excluded P limitation by prescribing an arbitrary high value of potential P immobilization and very low P leaching (Table 2). The experiment included two additional scenarios where parameters with the Relative approach had been adjusted to match the initial steadystate conditions of the Optimal approach. These additional scenarios allowed testing if the differences in predictions could be compensated by other model parameters.
2.3.3 Subannual: Incubation
The Incubation simulation experiment added a portion of labile substrate to a previously labilesubstratedepleted soil. Next, it tracked the carbon use efficiency (CUE) of the microbial community over time and across different C:N ratios of the added labile substrate. Specifically, it first simulated model pools in steady state with continuous annual inputs, then simulated no inputs for 1 year in order to deplete the labile substrate pool, and next it simulated a step increase of the labile substrate C and N pools. In addition to the three scenarios that differed by optimality approach, it simulated a scenario where microbial community allocation was fixed to α_{R}=0.5. This scenario allowed comparing results to a model where microbial community is not adaptive.
We do not expect simulating a correct timedynamics with SESAM at this short timescale, because SESAM explicitly omits detailed microbial processes that are relevant at this scale such as storage, resting stages, or dynamics of the enzyme pools. However, the experiment allows inspecting general dynamics with smooth annual changes and differences between model variants as the labile substrate pool gets depleted.
In this section, we present the results of the simulation experiments in turn.
3.1 Immediate response: Prescribed potential returns
The Derivative approach yielded the same allocation as the Optimal approach with the “Prescribed potential returns” simulation experiment. The Relative approach yielded similar results as the Optimal approach for high microbial biomass levels, i.e. levels that resulted in an enzyme synthesis flux of 10 times the halfsaturation constant of enzyme action k_{mN}, which in SESAM is a flux, specifically the product of a halfsaturation enzyme concentration and enzyme turnover rate. For moderate microbial biomass levels it overestimated allocation to the enzymes with low revenue (Fig. 4). With the Optimal and Derivative approaches there was no investment in enzymes with very low revenue at moderate biomass levels.
Since all state variables are held constant in this experiment, there is no change in respiration, microbial growth, CUE, as well as C in key model compartments. These subsequent changes are explored in the following experiments.
3.2 Decadal term: FACE
The Derivative approach yielded the same allocation as the Optimal approach with the FACE simulation experiment. The Relative approach differed by overestimating the allocation to the enzyme with lowest revenue, α_{R}. Hence, it not only predicted smaller initial steadystate stocks but also predicted relatively less mining of residue OM during the period of increased carbon inputs (Fig. 5). By adjusting parameters related to organic matter decomposition in the simulation with the Relative approach, the same steadystate stocks were simulated, but still the decrease in residue OM was smaller (Fig. A1).
3.3 Subannual: Incubation
The difference between optimization approaches were small compared to the differences to the variant without adaptation (Fig. 6). All three optimization approaches showed decreased fluctuations in CUE, both in time, as well as across C:N ratios of added labile substrate compared to a nonoptimized fixed allocation. The Derivative approach's predictions matched the Optimal approach's predictions, while the Relative approach initially slightly underestimated allocation to the residuedegrading enzyme (α_{R}), resulting in decreased biomass synthesis (Fig. 7).
The Relative approach's predictions differed from the Optimal approach after 1 year of incubation when microbial biomass and enzyme levels declined (Fig. 8). It still allocated to the enzymes degrading the labile substrate (α_{R}<1), while with the Optimal approach, the microbial community did not invest in degrading the small labile substrate pool anymore. Hence, some of the labile substrate pool was not decomposed, i.e. was apparently persistent with the Optimal approach.
The purpose of this work was to more rigorously define and implement the optimal growth hypothesis for SESAM and study the consequences of two simplifications. We found that the previously used Revenue approach could be derived from the more rigorous Optimal approach for a set of conditions. Therefore, we are more confident in conclusions drawn from previous SESAM studies. Further, we found no or only marginal differences between the Derivative and Optimal approaches. Therefore, we will use the Derivative approach to further develop SESAM. The following section discusses the optimization approaches in more detail.
4.1 Optimization approaches
The Optimal approach constitutes the mathematical formalization of the hypothesis of community enzyme allocation, optimizing microbial biomass growth for SESAM. The Relative approach was used in previous SESAM versions. It was shown in this study to be a special case of the mathematically formalized Optimal approach. It is valid for enzyme allocation fluxes larger than the halfsaturation constant in the decomposition equation, which is usually the case at not too low a microbial biomass. The Derivative approach is another heuristic of optimal enzyme allocation that relies on derivatives of the enzyme returns but does not require explicit formulas for the optimal allocation.
The three approaches predicted the same patterns in longterm as well as subannualscale simulation experiments. Hence, the conclusions drawn with SESAM so far were corroborated. Specifically, the following patterns emerge as a consequence of microbial community adaptation of enzyme allocation: the priming effect (Kuzyakov et al., 2000) and the Nbanking mechanism (Perveen et al., 2014) (Fig. 5) and the dampening of CUE fluctuations with an adaptive microbial biomass (Kaiser et al., 2014) (Fig. 6).
While the Optimal approach is exact, it is tedious to implement and to update with further developments of SESAM. It requires the developers to derive explicit formulas for the optimal allocation for enzymes that are in the subset of enzyme types that microbes allocate resources to. With including more enzymes or more types of enzymes, the formulas grow complex and an increasing number of potential optima need to be checked and compared. Therefore, we also consider the simpler Relative and Derivative approaches and discuss their effect on model predictions.
The Derivative approach yielded predictions that were so close to the predictions of the Optimal approach that they can hardly be spotted in the plots (Figs. 5, 7). However, it works similarly to gradientbased numerical optimization schemes and also shares its risks. First, it might result in limit cycles, where residue organic matter and microbial biomass oscillate instead of converge to a stable optimal allocation. We argue that this actually may really happen in soil, although perturbations with fluctuating litter input and decomposition fluxes changing with environmental conditions may quickly shift the decomposer system into states away from the basin of such a limit cycle (Strogatz, 1994). If the Derivative approach yields predictions with a decadalscale limit cycle, perturbations of model drivers are expected to drive the simulation away from the limit cycle. Second, the Derivative approach might get stuck in local optima and saddle points where the derivative of the return approaches 0. Gradientbased optimization schemes implement some notion of momentum to get past such points. There is also some momentum in the soil system because enzyme levels need some time to develop towards its quasisteady state and microbes use storage compounds to support developments in an hourly to weekly timescale where returns from enzymatic processing do not support further growth. Because SESAM explicitly tries to abstract from such microbial details that are important for reacting on shortterm fluctuations, the Derivative approach is prone to the risk of getting stuck at saddle points. We have not yet encountered such conditions in our simulations. However, in case such issues pose a problem, we need to think of ways how to implement simple notions of momentum in SESAM.
The Relative approach yielded predictions that differed from the predictions of the Optimal approach, specifically for low microbial biomass levels and for enzymes with low revenue. This was expected with the derivation of the conditions where the Relative approach is valid (Appendix C). Although small differences in enzyme allocation yield also only small differences in relative steadystate stocks, a small relative difference in the stock of the residue pool can result in considerable differences in total soil organic matter stocks. Such behaviour is observed in the FACE simulation experiments (Fig. 5). With this experiment, the Relative approach predicted an initial share of enzyme allocation towards residuedegrading enzymes of 30 % compared to about 26 % with the Optimal approach. This led to a decrease in residue steadystate stocks from about 3600 to about 3400 g C m^{−2} (Fig. A1), which is an absolute difference that was larger than the entire labile substrate pool. This, in turn, resulted in a predicted relative change in residue stocks with the FACE simulation experiment that significantly differed from the predictions with the Optimal approach (Fig. 5).
Based on these findings, we will continue developing SESAM, focusing on the Derivative approach.
4.2 The constrained enzyme hypothesis
The Optimal approach's predictions differed most from the previously used Relative approach's predictions at low microbial biomass levels. The Optimal approach excluded enzymes with low revenue from the set of enzymes to allocate to. For example, the allocation to the enzyme depolymerizing the labile substrate pool was zero for a potential return of this enzyme below 0.2 $\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{\mathrm{1}}$ in the “Prescribed potential returns” simulation (Fig. 4). Optimal enzyme allocation is determined primarily by availability of carbon and nutrients from organic and inorganic uptake. However, with the Optimal approach, optimal enzyme allocation in addition depends on the size of the microbial biomass because they control the relative size of the enzyme pools compared to saturating levels. The lower the microbial biomass, the farther away enzyme production is from levels where organic matter decomposition saturates. Hence at low microbial biomass it is not beneficial to distribute enzyme allocation across several enzymes including enzymes with low potential revenue. Similarly, the Optimal approach predicted in the Incubation simulation experiment that a small fraction of added organic matter, L, is not decomposed (Fig. 8). This insight into optimal allocation with SESAM generates an additional hypothesis as to why we observe high ages of some organic matter in soil and an additional insight into priming mechanisms (Fig. 9): the microbial community expresses a smaller set of enzyme types at low biomass levels. This hypothesis predicts that some organic matter is not decomposed in the presence of microbes that potentially can decompose it if biomass levels are low and if there are alternative substrates decomposable with higher revenue.
The constrained enzyme hypothesis is able to account for observed rhizosphere priming (Cheng et al., 2014) or increased SOM loss after disturbance. When microbial biomass grows, e.g. by making enough labile substrate available, the focus solely on the enzymes with the highest revenue is not beneficial anymore and the optimal microbial community also invests in decomposition of the organic matter with lower revenue. Apparently persistent organic matter becomes decomposed.
This noninvestment in enzymes of relatively low revenue is a complementary hypothesis to the existing hypotheses of SOM preservation (von Lützow et al., 2006, 2008). Since the chemical recalcitrance hypothesis has largely been dismissed (Schmidt et al., 2011; Dungait et al., 2012; KögelKnabner, 2017), most hypotheses focus on restricted accessibility of SOM to soil microbial decomposition. One set of hypotheses emphasizes protection by association with minerals (Schrumpf et al., 2013; Ahrens et al., 2015; Mathieu et al., 2015; Woolf and Lehmann, 2019), another set emphasizes protection inside soil aggregates (Six et al., 2000; Lehmann, 2007; Schrumpf et al., 2013), and yet another set emphasizes soil heterogeneity and spatial separation (Ekschmitt et al., 2005, 2008; Salome et al., 2009) or environmental conditions (Or et al., 2007; Keiluweit et al., 2017). They are related to the hypothesis of microbial energy limitation and are modified by inputs of fresh organic matter, i.e. the priming effect (Keiluweit et al., 2015; Henneron et al., 2022). Recently, the diversity hypothesis has gained attention for SOM preservation (Lehmann et al., 2020; Weverka et al., 2023), which has formerly been discussed in aquatic literature (Jannasch, 1967; Jiao et al., 2010; Arrieta et al., 2015). It emphasizes the low return on investment for very heterogeneous substrates and the associated required investment in a broad set of enzymes. The constrained enzyme hypothesis goes beyond the diversity hypothesis. While in the diversity hypothesis the preservation is controlled by the heterogeneity of available substrates, the constrained enzyme hypothesis predicts that preservation additionally depends on the amount or density of microbial biomass and on the availability of alternative substrates.
Very similar conclusions have been drawn by a modelling study that was published shortly after the discussion paper of this study. Weverka et al. (2023) modelled the revenue of intracellular enzymes or metabolic pathways that need to be expressed to assimilate a diverse set of substrates. Similar to Wutzler et al. (2017), they compared different strategies of microbes investing in different enzymes including a strategy of investing only in the enzyme with the highest revenue and a strategy that the corresponds to the Relative approach. They also assumed microbes to maximize growth. Their model structures, formulas for allocation, and insights are comparable to this study. They differ by focusing on assimilation and intracellular enzymes, rather than decomposition and extracellular enzymes, and they focus on the number of different substrates rather than stoichiometry of substrates. Instead of computing optimal allocation they assumed that microbes would not invest in enzymes where change in return was less than investments (Harvey et al., 2016).
Similar to this study, they observed in their model that substrates at a low concentration persist because it is not beneficial for microbes to produce respective enzymes. Moreover, they explained cometabolization of a substrate of low revenue by the assumptions that assimilation/degradation saturates at high enzyme levels and that it is therefore beneficial for microbes to distribute their investments also in enzymes of a lower potential return (Fig. 3). The current study differs from Weverka et al. (2023) by actually computing optimal enzyme allocation and consequently predicts different allocation and conditions at which specific enzymes are produced.
4.3 Optimality assumptions
The conclusions of this paper depend on several assumptions. First, they depend on the formulation of depolymerization (Eq. 1) and biomineralization (Eq. 5). Specifically, they depend on the assumption that the decomposition fluxes saturate at high enzyme levels (Schimel and Weintraub, 2003; Tang and Riley, 2019). With alternative formulations (Wutzler and Reichstein, 2008) that assume a linear dependence of decomposition on enzyme levels (or alternatively microbial biomass) it would be optimal to allocate to the single enzyme that yields the highest decomposition flux of the currently limiting element.
Moreover, we assumed that the instantaneous growth rate of the microbial community is optimized. Alternatively to instantaneous growth, the cumulative growth over a microbial characteristic time span could be optimized, e.g. the time for decomposing a single portion of carbon (Manzoni et al., 2023). The instantaneous strategy is suboptimal for dynamical strategies if legacy effects are present that are internal to the optimized system. At the same time the two strategies yield similar performance when legacy effects are external to the optimized system (Feng et al., 2022) because competition alters the tradeoff between current and future gains. Hence, optimizing at a different system boundary, which is usually associated with a different timescale, results in different optimal strategies (Dewar, 2010). The focus of SESAM on the entire microbial community calls for a dynamic strategy because it renders many factors internal, compared to a focus on competing microbial populations that renders soil organic matter an external factor. However, SESAM is intended to model decadalterm changes and to be driven with annually averaged drivers. The two strategies will presumably converge at such conditions. This is because enzyme pools and decomposition develop towards a quasisteady state where current and future gains are similar within a subannual timescale of microbial growth.
SESAM focused on the partitioning of allocation of the total enzyme investment towards different enzymes. In addition, the total allocation in enzyme production can be a trait that adapts to optimize microbial growth (Calabrese et al., 2022). Future SESAM developments will explore if a joint optimization of total allocation and allocation partitioning can be derived and whether such a joint optimization alters the consequences for the longterm dynamics of SOM stocks.
4.4 Observational evidence
The constrained enzyme hypothesis is a consequence of several model assumptions. It was derived without reference to observed patterns. However, there is already some observational evidence supporting the hypothesis of lower diversity of expressed enzymes at low microbial activity.
Metatranscriptomics (Carvalhais et al., 2012) directly studies functional diversity of expressed enzymes in soils. Evidence for the constrained enzyme hypothesis resulting from such studies is mixed. Straw amendments increased microbial activity diversity of an agricultural soil and let microbes upregulate several enzyme families (Kozjek et al., 2023). This result is in line with the constrained enzyme hypothesis. Contrary, microbes downregulated enzyme families with straw amendment to a soil of an already diverse grassland soil in the same study.
A novel approach combines isotopically labelled measurements of microbial growth with quantitative stable isotope probing (Hungate et al., 2015). It can assess microbial diversity of the active part of the microbial community. It revealed a reduction in diversity of actively growing microorganisms with lower microbial activity under drought (Richter, 2023), which is in line with an expected reduction in diversity of expressed enzymes with lower microbial biomass as predicted with the constrained enzyme hypothesis. However, low diversity of actively growing microorganisms under drought could also be due to stressinduced shifts toward nonactive conditions rather than due to optimal allocation with lower active microbial biomass.
Analysis of potential activities of specific enzymes (Marx et al., 2001) and the spatially resolved zymography version (Spohn et al., 2013) do not directly explain the diversity of enzyme expression because only specific enzymes are analysed. However, in line with the constrained hypothesis, zymography of a temperate forest soil revealed that common enzymes are hardly expressed outside hotspots and before fostering microbial growth by amendments (Heitkötter and Marschner, 2018).
In summary, studies that specifically look at enzyme diversity in relation to microbial biomass levels are still lacking. However, we can find observations from other studies that are in line with the constrained enzyme hypothesis.
The Optimal approach is the mathematical formulation of the hypothesis that microbial community enzyme allocation develops in a way that optimizes growth in SESAM. The finding of similar predictions by the heuristic approaches compared to the Optimal approach increases our confidence in conclusions drawn with SESAM. The heuristic Relative approach is shown to be a special case of the Optimal approach valid at sufficiently high microbial biomass levels. The Derivative approach, another heuristic of the Optimal approach, is valid also for low microbial biomass levels. Given that the Derivative approach is a good heuristic of the Optimal approach that is better scalable to more enzyme types than the Optimal approach, we will continue the SESAM developments with the Derivative approach.
The Optimal and Derivative approaches yield predictions at low microbial biomass that differ from the predictions of the Relative approach. Specifically, they predict that enzymes with low revenue are not expressed at low microbial biomass. This finding generated the constrained enzyme hypothesis for the preservation of organic matter in soils.
This section provides figures that detail some of the results and provide consistent presentation of main quantities across the experiments. The consistent presentation of all the quantities can not avoid some overplotting.
First, predictions of the FACE simulations experiment for the nonnormalized residue pool, R, and for additional scenarios with adjusted decomposition parameters are shown in Fig. A1.
Next, Figs. A2, A3, and A4 present common quantities across experiments. They also include a “Fixed” scenario, where enzyme allocation is not adaptive but constant, where specifying the initial value corresponds to specifying another model parameter.
This section derives explicit formulas of optimal enzyme allocation by finding the allocation that maximizes the total return. It starts with a lemma that states conditions for which the optimum is attained when derivatives are equal. The lemma is then used in subsequent derivations of optimal allocation.
B1 Optima at equality of derivatives
Lemma 1. Let ${u}_{\mathrm{T}}\left(\mathit{\alpha}\right)={C}_{\mathrm{2}}{\sum}_{Z}{\mathit{\alpha}}_{Z}{\mathrm{rev}}_{Z}$ be a function that is a weighted sum of components, rev_{Z} up to some constant C_{2}≠0, where weights ${\mathit{\alpha}}_{Z}\in (\mathrm{0},\mathrm{1})$ add up to 1, ${\sum}_{Z}^{n}{\mathit{\alpha}}_{Z}=\mathrm{1}$, and a component, rev_{Z}, may depend on a weight, α_{Z}, but not on the other weights. Further, let rev_{Z} be differentiable to α_{Z} and let potential optima $\in (\mathrm{0},\mathrm{1})$. Then at the optima of u_{T}(α) all derivatives, $\frac{\mathrm{d}\left({\mathit{\alpha}}_{Z}{\mathrm{rev}}_{Z}\right)}{\mathrm{d}{\mathit{\alpha}}_{Z}}$, are equal.
Proof. Because of the sumto1 constraint, we express one of the weights as a function of the other weights and have only n−1 free weights.
We are interested in the optima of u_{T} away from the borders, i.e. ${\mathit{\alpha}}_{\mathrm{Opt}}\in (\mathrm{0},\mathrm{1})$. In the derivative to α_{Z} all terms vanish except the term involving rev_{Z} and the term involving rev_{n} because there α_{n} is a function of α_{Z}. By the chain rule we have
Hence, for C_{2}≠0 each $\frac{\mathrm{d}\left({\mathit{\alpha}}_{Z}{\mathrm{rev}}_{Z}\right)}{\mathrm{d}{\mathit{\alpha}}_{Z}}$ has to be equal to $\frac{\mathrm{d}\left({\mathit{\alpha}}_{n}{\mathrm{rev}}_{n}\right)}{\mathrm{d}{\mathit{\alpha}}_{n}}$; i.e. all these derivatives have to be equal.
B2 Return, revenue, and derivative for a biomineralizing enzyme
The return, revenue, and its derivative of a biomineralizing enzyme are slightly more complex than the corresponding quantities of depolymerizing enzymes (Sect. 2.1.2 and 2.2.1). They are presented in Table B1 because of the onecolumn constraint of normal text in this journal.
B3 Explicit optimum formulas
We seek the community composition, here represented by enzyme allocation, α, that maximizes the total return. This maximizer is located either at the borders of the domain or at a location where all derivatives of the total return are 0. We only look at cases where we know which enzymes take part in the mix with positive allocation, i.e. having ${\mathit{\alpha}}_{Z}\in (\mathrm{0},\mathrm{1})$, and therefore do not need to look at the borders.
The strategy is first to find the small set of allocations where all the derivatives are 0, which includes maxima, minima, and saddle points. Second, we constrain the set to conditions ${\mathit{\alpha}}_{Z}\in (\mathrm{0},\mathrm{1})$ and select that element that results in the highest return.
In order to simplify formulas, we make the assumption that all halfsaturation parameters are equal: ${{k}_{mN}}_{Z}={k}_{mN}$.
B3.1 Two depolymerizing enzymes
Utilizing Lemma 1 (Appendix B1) we have
where ${\mathit{\alpha}}_{\mathrm{R}}=\mathrm{1}{\mathit{\alpha}}_{\mathrm{L}}$.
This provides a quadratic equation of α_{L}, which one can solve. We used the SymPy symbolic math tool. One of the two roots where ${\mathit{\alpha}}_{\mathrm{L}}\in (\mathrm{0},\mathrm{1})$ and yielding a higher u_{T}(α_{L}) provides the optimal α_{L}.
B3.2 Depolymerizing and biomineralizing enzyme
B3.3 Two depolymerizing and one biomineralizing enzyme
We set ${\mathit{\alpha}}_{\mathrm{R}}=\mathrm{1}{\mathit{\alpha}}_{\mathrm{L}}{\mathit{\alpha}}_{\mathrm{P}}$ and have equations of Table B2.
One of the four roots where ${\mathit{\alpha}}_{\mathrm{P}}\in (\mathrm{0},\mathrm{1})$ and derived ${\mathit{\alpha}}_{\mathrm{L}}\left({\mathit{\alpha}}_{\mathrm{P}}\right)\in (\mathrm{0},\mathrm{1})$ and yielding the highest u_{T}(α) provides the optimal α.
B4 Excursion: replacing revenue by relative profit
Revenue, here, is defined as return per investment, ${\mathrm{rev}}_{Z}={{d}_{Z}}_{\mathrm{w}}/{{\mathrm{inv}}_{Z}}_{\mathrm{w}}$. One could argue that one should rather maximize the profit, i.e. return – investment and corresponding profit revenue, ${{\mathrm{rev}}_{p}}_{Z}$, i.e. profit $/$ investment by optimizing enzyme allocation. Here we show that optimizing the profit yields the same optimal allocation as optimizing the return.
The total profit is the sum of profit revenues multiplied by total enzyme investment, inv_{w}.
This equation fulfils the conditions of Lemma 1 (Appendix B1), and at the optima all derivatives are equal.
The last line corresponds to the same condition as when optimizing returns. Hence, they lead to the same optima.
The Relative approach approximates optimal allocation by setting optimal allocation proportional to revenue (Sect. 2.2.3). Hence, we seek the conditions for which the following relationship holds:
At the solution of the Optimal approach all the derivatives of (revenue times α_{Z}) for all enzymes in the mix are equal (Appendix 2.2.1). By using $\frac{\mathrm{d}\left({\mathit{\alpha}}_{Z}{\mathrm{rev}}_{Z}\right)}{\mathrm{d}{\mathit{\alpha}}_{Z}}\approx {\mathrm{rev}}_{Z}\frac{{e}_{Z}+{{k}_{mN}}_{Z}}{{\mathit{\alpha}}_{Z}{a}_{\mathrm{E}}B}$, as shown below, for any two enzymes, i and j, we have
The last approximation holds only for similar halfsaturation parameters across enzymes ${{k}_{mN}}_{Z}\approx {k}_{mN}$, with plant enzyme production being low compared to the following halfsaturation: e_{Z}≪k_{mN}.
The first approximation in the second line is only valid for an enzyme production flux that is not larger than the halfsaturation, ${{k}_{mN}}_{Z}$ (see below). This is violated at low microbial biomass or very low α_{Z}.
Hence, the optimal allocation is approximately proportional to the revenue for the combination of the following conditions:

all enzymes having a nonnegligible share

microbial biomass being sufficiently high

plant biomineralizingenzyme production being low.
The derivation above used the following relationship that still needs to be shown: $\frac{\mathrm{d}\left({\mathit{\alpha}}_{Z}{\mathrm{rev}}_{Z}\right)}{\mathrm{d}{\mathit{\alpha}}_{Z}}\approx {\mathrm{rev}}_{Z}\frac{{e}_{Z}+{{k}_{mN}}_{Z}}{{\mathit{\alpha}}_{Z}{a}_{\mathrm{E}}B}$.
For depolymerizing enzymes we use the following approximations. For ${\mathit{\alpha}}_{Z}{a}_{\mathrm{E}}B\gg {{k}_{mN}}_{Z}$, i.e. $\mathrm{2}{{k}_{mN}}_{Z}+{\mathit{\alpha}}_{Z}{a}_{\mathrm{E}}B\approx {\mathit{\alpha}}_{Z}{a}_{\mathrm{E}}B$, the halfsaturation ${{k}_{mN}}_{Z}$ can be neglected in the denominator of the revenue. Note that ${\mathit{\alpha}}_{Z}{a}_{\mathrm{E}}B\gg {{k}_{mN}}_{Z}$ implies $({\mathit{\alpha}}_{Z}{a}_{\mathrm{E}}B{)}^{\mathrm{2}}\gg {{k}_{mN}^{\mathrm{2}}}_{Z}$.
where the first two relationships have been derived in Appendix 2.1.1. For depolymerizing enzymes we have e_{Z}=0 because they are not produced by plant roots.
Similarly, for biomineralizing enzymes we require ${\mathit{\alpha}}_{Z}{a}_{\mathrm{E}}B\gg {{k}_{mN}}_{Z}+{e}_{Z}$, where e_{Z} is the production of an enzyme, Z, by plants.
SESAM assumes that microbial community develops in a way to maximize growth of the entire community. Growth increases with uptake and hence increases with decomposition flux for given enzyme allocation. The revenue of allocation to an enzyme, Z, is ${\mathrm{rev}}_{Z}=\frac{{{d}_{Z}}_{\mathrm{w}}\left({\mathit{\alpha}}_{Z}\right)}{{\mathit{\alpha}}_{Z}{\mathit{\omega}}_{\mathrm{Enz}}{a}_{\mathrm{E}}B}$. The return, ${{d}_{Z}}_{\mathrm{w}}$, is a limitationweighted mineralization flux or uptake flux of nutrients and carbon (Sect. 2.1.1 and 2.1.2). The investment is the share, α_{Z}, invested in the production of an enzyme, Z, multiplied by total limitationweighted flux, ω_{Enz}a_{E}B, allocated to enzyme production.
Although it is possible to derive explicit formulas for the allocation that optimize the total return for simple cases, the formulas quickly grow and involve higherorder polynomials of α_{Z} with several solutions outside the reasonable bound ${\mathit{\alpha}}_{Z}\in [\mathrm{0},\mathrm{1}]$.
Here we follow an alternative local approach where we assume the rate change in α_{Z} over time to be proportional to the deviation of the derivative of change of the total return with respect to α_{Z} to the average across the derivatives for different enzymes. The higher the increase in the total return for shifting allocation towards a specific enzyme, the faster the community changes in this direction.
The total return is a weighted sum of enzyme revenues, and derivatives of $\frac{\mathrm{d}\left({\mathit{\alpha}}_{Z}{\mathrm{rev}}_{Z}\right)}{\mathrm{d}{\mathit{\alpha}}_{Z}}$ have been derived for depolymerizing and biomineralizing enzymes (Sect. 2.2.1).
We assume that the larger the change in return with increasing allocation, i.e. the derivative to allocation coefficient, α_{Z}, the larger the change in allocation is. In addition to the assumption of proportionality to the derivative, we assume that the community changes at a rate of the same magnitude as synthesis and turnover of microbial biomass.
where m_{du} is the average across derivatives of return across enzymes. If all changes are the same, i.e. equal to the mean, the allocation is optimal because it does not increase in any direction.
We want the change to be proportional to the change in return compared to the average return. Subtracting this mean ensures that the sum of all the changes in α sums to 0 so that the sum across α_{Z} is preserved. The proportionality factor normalizes the change in return and multiplies this relative change by the rate of microbial turnover, composed of biomass synthesis and biomass turnover.
D1 Exclude enzymes whose negative relative change is larger than its share
The community may not allocate to all enzymes. Hence, m_{due} (an updated version of m_{du}) averages only across a subset of enzymes. The derivative optimization strategy assumes that nothing is allocated to an enzyme if its normalized change towards 0 is larger than its current share, i.e. is more negative than −α_{Z}.
where $\left\mathit{\right\{}Z\mathit{\}}\setminus {Z}_{\mathrm{0}}$ denotes the number of enzymes allocated, i.e. the cardinality of the set of all enzymes without those in Z_{0}. The relative change of those excluded enzymes is set to −α_{Z}, resulting in negative changes going to 0 as α_{Z} approaches 0. Hence, the relative change is lowerbounded by −α_{Z}.
m_{du} has to be adjusted to m_{due} so that ${\sum}_{i}\frac{\mathrm{d}{\mathit{\alpha}}_{i}}{\mathrm{d}t}=\mathrm{0}$ holds, i.e. ${\sum}_{\mathit{\zeta}\notin {Z}_{\mathrm{0}}}\frac{\frac{\mathrm{d}{u}_{\mathrm{T}}}{\mathrm{d}{\mathit{\alpha}}_{Z}}{m}_{\mathrm{due}}}{{m}_{\mathrm{due}}}+{\sum}_{\mathit{\zeta}\in {Z}_{\mathrm{0}}}{\mathit{\alpha}}_{\mathit{\zeta}}=\mathrm{0}$.
This definition is recursive because m_{due} is computed across a set that is defined using m_{due}. In order to determine Z_{0} one can start with the empty set and add all enzymes that fulfil the condition. If enzymes were added, then the mean across remaining derivatives increases and the condition has to be checked again. Hence, adding enzymes to Z_{0} is repeated until no more enzymes fulfil the condition and the mean does not change any more.
SESAM (v3.1) is available coded in R at https://github.com/bgctw/sesam (last access: 19 February 2024) (https://doi.org/10.5281/zenodo.8026318, Wutzler, 2023a) and coded in Julia at https://github.com/bgctw/Sesam.jl (last access: 19 February 2024) (https://doi.org/10.5281/zenodo.8026366, Wutzler, 2023b). R source code is released using the GPL v2 (GNU General Public License) because it uses other GPL libraries. Julia code is released using the more permissive MIT License.
The simulation experiments are part of the R repository. They use the derivSesam3P model variant. The “Prescribed potential returns” code is provided in the “Allocation” section of file develop/23_optimAlloc/sesamess/sesam_LRP_deriv.Rmd. The decadalterm FACE code is provided with file develop/23_optimAlloc/Face1_3P.Rmd. The subannual Incubation code is provided with file SimBareSoilPulse_opt.Rmd. This paper presents prototypical artificial simulations based on synthetic data. All the necessary information is listed in the tables as well as in the referenced source code.
TW and CR developed the mathematical concepts. TW implemented the concepts in the model and led the writing of the manuscript. TW, CR, BA, and MS contributed to the discussion of results and writing of the manuscript.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.
We thank Lin Yu for fruitful discussion. We thank the Max Planck Society for funding.
The article processing charges for this openaccess publication were covered by the Max Planck Society.
This paper was edited by Tomomichi Kato and reviewed by Sergey Blagodatsky and Stefano Manzoni.
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 Abstract
 Introduction
 Methods
 Results
 Discussion
 Conclusions
 Appendix A: Additional figures
 Appendix B: Optimal enzyme allocation
 Appendix C: Derivation of the relative approach
 Appendix D: Derivativebased change of community allocation
 Code and data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Conclusions
 Appendix A: Additional figures
 Appendix B: Optimal enzyme allocation
 Appendix C: Derivation of the relative approach
 Appendix D: Derivativebased change of community allocation
 Code and data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References