Traditional soil organic matter decomposition models are based on first-order kinetics, where decomposition scales to the pool size. The scaling factor represents recalcitrance of a specific pool and is modified by soil temperature, moisture, and other soil properties (e.g. van Veen et al., 1984; Parton et al., 1987; Molina et al., 1990; Li, 1996; Chertov and Komarov, 1997). Recent modelling efforts have specifically included catalysis of polymeric soil organic carbon to dissolved organic carbon (DOC) by extracellular enzymes. This depolymerisation step is thought to be a rate-limiting step in organic matter decomposition processes (Schimel and Weintraub, 2003; Fontaine and Barot, 2005).

In traditional models, microbes are only considered as a simple donor-controlled pool (i.e. microbial biomass has no impact on decomposition) or in an implicit manner (Gerber et al., 2010). In contrast, in microbial models, decomposition rates become a function of enzyme activity that is linked to microbial biomass (Allison et al., 2010; German et al., 2012). This leads to more complex dynamics because decomposers feed back into soil organic matter degradation via microbial enzyme production affecting depolymerisation. This positive feedback between microbial biomass and depolymerisation causes soil organic carbon stocks and microbial biomass to oscillate after a perturbation (Li et al., 2014; Wang et al., 2014). Nevertheless, microbial decomposition models have been shown to improve the prediction of soil carbon and perform well when compared to decomposition experiments (Lawrence et al., 2009; Wieder et al., 2013, 2014a, b, 2015b). Furthermore, when compared to traditional first-order models, microbial models also display an attenuated loss of soil organic matter to warming (Allison et al., 2010; Wieder et al., 2013).

Moreover, the response of soil organic matter to warming is very sensitive to microbial carbon use efficiency (CUE) because this parameter and its climate sensitivity define the fraction of carbon remaining in the soil as processed organic matter vs. carbon removed via respiratory CO2 (Allison et al., 2010; Frey et al., 2013; Kivlin et al., 2013; Tucker et al., 2013; Sinsabaugh et al., 2013; Wang et al., 2013; Li et al., 2014). Temperature dependence of CUE is typically not considered in traditional decomposition models (but see Frey et al., 2013), rather the ratios between respired CO2 and the transfer to different quality pools are mostly constant parameters or vary based on soil texture, soil recalcitrance, and organic or inorganic nutrient content (Parton et al., 1987; Gerber et al., 2010). Microbial respiration can be partitioned into a series of carbon expenditures that do not contribute to growth. These expenditures include growth respiration, maintenance respiration, respiratory cost for enzyme production, and overflow respiration (Manzoni et al., 2012; Moorhead et al., 2012). Each type of respiratory carbon expenditure may differ in its response to temperature.

Respiration may be parameterised based on different microbial properties. For example, maintenance respiration is assumed to scale with microbial biomass (Chapman and Gray, 1986; Fontaine and Barot, 2005) while growth respiration may scale to the amount of new tissues built. On the other hand, overflow respiration occurs during stoichiometric adjustment (Russell and Cook, 1995; Schimel and Weintraub, 2003; Frost et al., 2005; Franklin et al., 2003), whereas costs related to enzyme production may be governed by microbial demand and substrate availability and quality, resource diffusion, and microbial diversity (Allison, 2005). This differentiation can impact the dynamics of the microbial biomass: for example, maintenance respiration costs would be incurred even in the absence of carbon uptake, which can lead to a reduction in microbial biomass. In contrast, growth respiration is only due when substrate for growth is available. Because of the explicit and mechanistic link between microbial activity and soil organic matter degradation, inclusion of microbial models in Earth system models may have the potential to ultimately reduce uncertainty in climate–carbon feedback in the face of climate change because of the explicit link between microbial activity and soil organic matter degradation (Todd-Brown et al., 2012, 2013; Wieder et al., 2015a).

As microbial models are considered for broader application in Earth system models, it is essential to analyse and understand their structure and their dynamics. Here, we compare a series of microbial decomposition models with each other. Specifically, we analyse feedbacks between depolymerisation and microbial growth, consider constraints on depolymerisation and enzyme–substrate interactions, investigate the parameterisation of microbial enzyme productivity, and address the representation of microbial respiration and CUE.

Our main questions are as follows: a.

How do different model implementations of depolymerisation affect the feedback between microbial biomass and soil organic matter if subjected to warming?

Recently developed microbial decomposition models (Schimel and Weintraub, 2003; Allison et al., 2010; German et al., 2012) highlight the importance of microbial processes and microbial physiology during decomposition. Their application specifically highlights the role of extracellular enzymes during decomposition and how these constraints will further affect the release of soil organic matter as a consequence of warming. While microbial decomposition models are able to improve prediction of organic carbon stock globally and can successfully recreate litter decomposition dynamics, the long-term trajectory of a warming response needs further evaluation (Wang et al., 2014; Hararuk et al., 2015). In particular, a positive feedback between depolymerisation and microbes can only be curbed via the longer-term adjustment of soil organic matter and therefore lead to oscillation in both microbial biomass and soil organic matter (Wang et al., 2014). The oscillation is the consequence of a positive feedback between depolymerisation and microbial growth caused by a knife's edge or unstable equilibrium in the short term (unstable QSS for microbes; Schimel and Weintraub, 2003). A break in this feedback and stabilisation only occurs via the slow changing soil organic matter pool. We note that some attenuation of the oscillation may occur via direct input into a DOC pool that does not require depolymerisation (Allison et al., 2010), a feature not considered here.

The display of oscillation in the FWD model has been a point of critique as it has not been observed in laboratory and field incubation studies (Wang et al., 2014). Here, we introduce mechanisms that curb the positive feedback between substrate and microbial biomass.We portray two scenarios, where each increment in microbial biomass or enzyme concentration yields a smaller increase in depolymerisation than the previous increment (i.e. diminishing return). The scenarios we worked out are (1) microbial biomass feeds on active extracellular enzymes and (2) limited sites for substrate–enzyme reactions (see Appendix B). We derived the forms of depolymerisation from the original Michaelis–Menten kinetics and the resulting formulations presented in the “Materials and methods” section are simplified and more illustrative versions of more complex functions. The simplified formulation of depolymerisation and microbial consumption we obtained has been dubbed a reverse Michaelis–Menten formulation (Schimel and Weintraub, 2003) because microbial biomass (or enzyme concentration) instead of the substrate concentration now occurs in the denominator of the depolymerisation term, invoking the diminishing return. Wang and Post (2013) arrived at a reverse Michaelis–Menten depolymerisation function if enzymes only adsorbed to a fraction of binding sites because of complex substrates. Transitions between FWD and REV model behaviour have also been detailed in the more complex Equilibrium Chemistry Approximation model that also included sorption of enzymes and substrates to mineral surfaces (Tang and Riley, 2015; Tang, 2015). Our analysis shows that the positive feedback between decomposition and microbial growth is removed, as our REV model now has a stable short-term QSS.

Limited sites may play a role if the substrate has a high volume-to-surface ratio, or if the substrate is associated with minerals (Davidson and Janssens, 2006; Gillabel et al., 2010; Conant et al., 2011; Davidson et al., 2012, 2014; Cotrufo et al., 2013; Wagai et al., 2013; Benbi et al., 2014; Wieder et al., 2014a; Tang and Riley, 2015). Our implementation of limited substrate causes a surplus of free enzymes that compete for binding to substrates, similar to the Langmuir adsorption isotherm theory (Vetter et al., 1998; Schimel and Weintraub, 2003; Wang and Post, 2013; and see Appendix B, “Model with limited available substrate”), leading to diminishing depolymerisation returns and a REV model formulation. Effects of microbial scavenging for enzymes cause a diminishing return because more microbial biomass will lead to an increased probability of enzymes being consumed before they interact with soil organic matter. Other mechanisms of diminishing return as enzymes increase may be the stabilisation of enzymes into an organic-matter–humate complex (Allison, 2006) or sorption to minerals, soil organic matter, or microbes (Tang and Riley, 2015). Diminishing returns also occur with rate–yield trade-offs (Allison, 2014).

Many microbial decomposition models work under the assumption that enzyme production is proportional to microbial biomass; however, it is also conceivable that microbes are adjusting production to maximise return or growth (Cooney, 2009; Merchant and Helmann, 2012; Tang and Riley, 2015). In our OPT model, we relax the proportionality of microbial enzyme production and microbial biomass and instead allow a best possible return given the cost of enzyme synthesis. While the exact cost of enzyme production is not known, we fixed parameters (the product of Kp and c) that relate to the fractional expense of carbon depolymerised upon initialisation (i.e. at steady state and reference temperature; Eqs. 8 and 16). Importantly, enzyme production optimisation is not possible for some of the models presented here. Higher enzyme production would always lead to further microbial growth in the FWD model, and the highest yield would occur with infinite enzyme production. Similarly, in the case of microbial scavenging for enzymes, additional investments into enzymes always increase depolymerisation.

The response to temperature in our OPT model closely resembles the traditional first-order decay model (FOD). In the limit of enzyme production cost approaching zero, depolymerisation occurs at the maximum rate (Vmax⋅S), essentially turning the OPT model into a first-order model (Fig. 2). In the OPT model, reductions in depolymerisation via Kp are alleviated when enzyme synthesis is inexpensive, where the reduction in the maximum depolymerisation rate becomes a function of the product of Kp⋅c (Eq. 7 and Table 2). The results of the OPT model also show the effects on assumptions regarding microbial enzyme production rates. In many microbial models, enzyme production is scaled to microbial biomass. Lifting the tight coupling between microbial biomass and enzyme production leads to a more dynamic enzyme concentration and ultimately affects the temperature sensitivity of decomposition. Thus, the cost and trade-offs associated with microbial enzyme production are potential important areas to better quantify the long-term response of soil carbon storage to climate change.

The response of decomposition to warming can be viewed as a response occurring on multiple timescales. For example, while enzyme activity likely produces an immediate response, microbial respiration responses may also be triggered quickly, although longer-term acclimation may occur (Frey et al., 2013). It may take longer for microbial biomass to respond to temperature changes (weeks to months). Finally, because the rate of decomposition is slow compared to the overall abundance of soil organic matter, discernible changes in this pool occur on timescales of months to years. Based on the distinct rates of adjustments, timescales can – in principle – be separated by assuming a quasi-steady state of pools that turn over fast.

The assumption that both enzyme concentrations and DOC (i.e. the depolymerisation products) are at quasi-steady state cuts across all models presented here (FWD, REV, and OPT; see Appendix A). When we extend our assumption of steady state to the microbial timescale (quasi-steady state of microbial biomass), we find that for both the REV and the OPT model, the short-term response of microbial biomass and respiration is influenced by the adjustment of microbial dynamics to the warmer temperature (Fig. 3). Because microbial biomass jumps immediately to a higher level after the temperature increase in our QSS assumption, depolymerisation, and thus respiration, are affected. However, the QSS assumption affects the trajectory of the soil carbon pool only minimally. On timescales that allow microbes to turn over a couple of times (several months), the quasi-steady state poses a suitable approximation to represent respiration and microbial biomass, even after a sharp perturbation in the form of a step change. In the QSS assumption, depolymerisation becomes independent of the microbial biomass (but is still dependent on a combination of microbial parameters; see Table 2).

The introduction of QSS microbial biomass allows addressing and comparing the long-term responses of the different models to warming. In particular, the comparison of the QSS-derived depolymerisation of the FOD with the REV and the OPT directly show the effect of how enzyme–substrate affinity and enzyme production costs dampen the rate of depolymerisation and its response to temperature. In other words, the long-term response of the FOD is equivalent to the long-term response of our OPT or REV model, when (1) Ke is low (high enzyme production, high enzyme–substrate affinity, and low enzyme turnover) and/or (2) costs of enzyme production are low, and (3) and CUE (the fraction of the organic matter pool that is depolymerised but not respired and instead cycled back into soil) is also temperature dependent in the FOD, a feature typically not included in traditional decomposition models.

CUE ultimately is the result of different microbial respiration terms. Here, we consider three processes that may affect microbial respiration under a warming scenario. We first consider a partitioning into growth and maintenance respiration across our three models. Growth respiration is simply assumed to be a proportion of carbon allocated to microbial growth. In contrast, maintenance respiration scales to microbial biomass in our models, where the proportionality factor increases with temperature. This partitioning is motivated by formulations of plant respiration in terrestrial biosphere models. We find that this separation affects the short-term responses of respiration because microbial biomass lags behind the increase in depolymerisation. The temperature response of CUE is thus delayed. The partitioning of the respiration terms also has a particular impact on the transient dynamics of the FWD model, in that the lag in maintenance respiration amplifies the oscillation (Fig. 3). However, in the REV and the OPT model, effects of separation are only discernible on the microbial timescale, before microbial biomass is approaching quasi-steady-state values.

In the OPT model, we introduce an additional respiration term, namely the cost of enzyme production. In this model, we allow microbes to adjust enzyme production in order to optimise growth. It is interesting that increasing costs lead to a smaller immediate response in respiration and more resilient soil organic matter pool in the long term, when subject to warming (Fig. 4). The early respiration response in the OPT model is a product both of higher rates of depolymerisation (increased Vmax) and of a higher rate of enzyme production. However, the enhancement relative to the rates at the reference temperature becomes smaller with higher enzyme production cost. In the long term, the decrease in soil organic matter is reduced when enzyme production costs are considered. This reduction is accompanied by a smaller reduction in CUE under higher enzyme production, even though there is a subsequent CUE reduction occurring as S declines. The changing yield trade-off overall acts to buffer respiration increases that could be expected from physiological responses alone (Vmax), although the effects are smaller and may be well within the uncertainty of the temperature response of any parameters considered here. We note that enzyme expenditure relative to depolymerisation is a function of the product of Kp and c.

We acknowledge that we used a simplified set-up of our model suite. For example, we assumed that depolymerised carbon in soil solution (DOC) is always in steady state with the microbial biomass (see also German et al., 2012, and Moorhead et al., 2012). This simplification can be justified with fast and efficient scavenging of microbes and, thus, fast turnover of the DOC pool. Further sensitivity analysis may shed light on the dynamics across the full parameter space, while using the simplified linear terms (Appendices B and C; Tang, 2015), particularly also because many of the parameters are difficult to estimate. Furthermore, we did not include nutrient requirements of microbes where considering the stoichiometric requirements can change the allocation of resources to optimise enzyme synthesis. Finally, our model does not include interaction that may occur with adsorption to mineral surfaces, which may occur with the substrate, the enzymes and microbial biomass, and which has important short- and long-term consequences for to temperature fluctuations and changes (Wieder et al., 2014a; Tang and Riley, 2015). Nevertheless, our suite of models shows the importance of formulating the depolymerisation step in mathematical models when evaluating the response of decomposition under warming.

Microbial models are considered to be more realistic because of the mechanistic representation of the decomposition steps, yet the oscillatory behaviour has been viewed as an unrealistic response to perturbation (Wang et al., 2014). Perhaps on a more fundamental level, first-order decomposition models inherently assume substrate limitation while the FWD model incorporates enzyme availability (and enzyme production) as the limiting step during decomposition. Here, we show that first-order models can be viewed as a special case of a microbial model that considers a limitation other than enzyme availability (i.e. diminishing returns) and low values of the half-saturation constant (REV model), or alternatively, a decoupling of microbial enzyme production from microbial biomass (OPT model). While moving from the FWD to the REV model (diminishing return) introduced a form of substrate limitation, optimising enzyme production can be viewed as a further alleviation (or removal under marginal production cost) of enzyme limitation. Since the response to warming is vastly different across our suite of models, our results suggest that the degree of enzyme limitation and the microbial response to enzyme limitation are potential areas that could help constrain the quantification of the long-term response of soil organic matter to warming.

Our findings suggest that different formulations of microbial substrate acquisition will have a significant impact on the short- vs. long-term consequences of warming. Here, we present simple, yet feasible, mechanisms of microbial dynamics. We show that substrate limitation in the form of decreasing marginal return can create a break in the positive feedback between microbial biomass and depolymerisation, turning a forward Michaelis–Menten model into a reverse model. We further separate out three types of respiration that have possible consequences for the temporal trend of CUE in response to warming. Although such separation is more mechanistic, it remains an open question whether the addition of extra parameters is justified at this point, given the uncertainty in models, and because much of the effects of this separation diminishes on timescales longer than the microbial lifespan. Finally, among our suite of models, our OPT model most closely resembles the traditional first-order decomposition model. In our modelling framework, a first-order model is a special case of a microbial decomposition model where (1) mechanisms of diminishing returns break the feedback between substrate and microbes, (2) the proportionality of enzyme production and microbial biomass is relaxed and adjusted to yield optimum return of enzyme investments, (3) costs associated with enzyme synthesis are small (and/or enzyme–substrate affinity is high), and (4) microbes turn over relatively fast compared to soil organic matter. Our results thus suggest that a better grasp of the limiting steps of decomposition and mechanisms of microbial enzyme production will help to constrain the long-term response to warming.