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Abstract. Decomposition of soil organic matter (SOM) is limited by both the available substrate and the active decomposer community. The understanding of this colimitation strongly affects the understanding of feedbacks of soil carbon to global warming and its consequences. This study compares different formulations of soil organic matter (SOM) decomposition. We compiled formulations from literature into groups according to the representation of decomposer biomass on the SOM decomposition rate a) non-explicit (substrate only), b) linear, and c) non-linear. By varying the SOM decomposition equation in a basic simplified decomposition model, we analyzed the following questions. Is the priming effect represented? Under which conditions is SOM accumulation limited? And, how does steady state SOM stocks scale with amount of fresh organic matter (FOM) litter inputs? While formulations (a) did not represent the priming effect, with formulations (b) steady state SOM stocks were independent of amount of litter input. Further, with several formulations (c) there was an offset of SOM that was not decomposed when no fresh OM was supplied. The finding that a part of the SOM is not decomposed on exhaust of FOM supply supports the hypothesis of carbon stabilization in deep soil by the absence of energy-rich fresh organic matter. Different representations of colimitation of decomposition by substrate and decomposers in SOM decomposition models resulted in qualitatively different long-term behaviour. A collaborative effort by modellers and experimentalists is required to identify formulations that are more or less suitable to represent the most important drivers of long term carbon storage.

position formulations have been applied in SOM decomposition models, what are their underlying assumptions, and how can they be classified? What are their long-term implication for soil carbon storage? This is approached first, by reviewing the assumptions of several formulations of decomposition and second, by comparing the steady states of a basic minimal model, in which the decomposition equation was modified. 10 The colimitation of decomposition is represented in various ways in models that describe decomposition at daily resolution at plot-scale. However, the assumption of different decomposer communities that mutually independent decompose different kinds of substrates has led to a widely used representation of decomposition at decadal to millennial time scales that is focused on substrate only . 15 However, the observation of the priming effect (e.g. Kuzyakov et al., 2000) challenges this assumption of independent decomposition. In the modelling context we define priming as the effect that decomposition of the one soil carbon pool is influenced by the dynamics of another soil carbon pool. Based on observations of priming Fontaine and Barot (2005) suggest a formulation of SOM decomposition that results in SOM 20 accumulation that is only limited by nitrogen availability for the SOM decomposers. This implies a completely different long-term dynamics compared to most commonly used models, where the cabon pools at steady state are constant and depend linearly on the fresh organic matter input. This contradiction warrants a closer review of different representations of various used decomposition equations and their underlying 25 assumptions.
There are several good reviews and comparisons of SOM decomposition models. Van Veen and Frissel (1981a) group models that particularly take account of the role of microorganisms in mineralization versus simplified models that are more generically EGU applicable. Paustian (1994) contrast organism oriented versus process oriented models. McGill (1996) compares 10 process-based models against long term field data and propose a classification scheme. The scheme distinguishes amongst others for kinetic versus biochemical or functional litter and SOM compartmentalization, which relates to representation of decomposer biomass in the models. Molina and Smith (1998) give 5 a good general introduction into the historical evolution and various concepts of SOM models. Smith et al. (1998) focus on the purpose of the various models and summarize the reviews of Paustian, McGill, and Molina and Smith. Paustian et al. (1997) compare short-term decomposition and equilibrium states of several conceptual model formulations and three full models. Chertov et al. (2007) compare three conceptually different models against data from incubation studies. There are also more current reviews emphasizing on soil models for cropping systems (Shibu et al., 2006;Manlay et al., 2007), the stabilization by micro aggregates (Six et al., 2004), and the general role of soils (Yadav and Malanson, 2007). However, all the above reviews are difficult to interpret in respect to different rep- 15 resentations of colimitation of decomposition by substrate and decomposers because they compare full soil carbon models that differ in many aspects. Hence, this study reviews modelling literature with the explicit focus on this colimitation and compares various formulations by substituting them into a common basic model and by calculating steady states as a representation of the essentials of long- 20 term behaviour.
The study shows that that the long-term consequences of formulations of decomposition qualitatively differ by the representation of the active decomposer in the description of SOM decomposition.

25
In a first step we compiled formulations of SOM decomposition from literature and summarized their underlying assumptions. The original formulations were simplified in Introduction EGU a way so that only the factors and terms relating to substrate and decomposer biomass were included. Other drivers such as temperature, moisture, soil texture or nutrient availability were assumed to be constant and lumped into constants. We grouped the equations in the three groups a) "non-explicit", b) "linear", and c) "nonlinear" according to representation of decomposer biomass in the SOM decompo-5 sition equations. In most cases this corresponded to similar assumptions and consequences for long term carbon storage.
In a second step we compared long-term consequences of the formulations. Following the conclusion of Jans-Hammermeister and McGill (1997) we compared only one contrasting component of system models, in our case the decomposition equations. 10 We accomplished this by setting up a simplified minimal model system (Sect. 2.1) and substituted different versions of decomposition equations into this common model. Next, we calculated carbon pool sizes and fluxes as a function of model parameters at system steady state. Steady state represented the essential characteristics of the long-term behaviour and long-term consequences of the formulation of SOM decom-15 position. The following questions were addressed.
-Is the priming effect simulated? In order to compare the different decomposition formulations, we inserted them into the same minimal SOM decomposition model. A flowchart of the system is given in Fig. 1. The minimal system considered only one pool of SOM (S) of a single quality. The SOM was decomposed according to the equations d s that we compared. During 5 decomposition a part ǫ of the decomposed SOM was assimilated by the active decomposers A and the other part was repired as growth respiration. The carbon in active decomposers was respired as maintenance respiration r or entered the SOM as a flux s of microbial metabolites or dead microbial biomass. Booth fluxed were described by a first order kinetics. Pool sizes were expressed in weight per volume (kg/m 3 ) and the 10 time was expressed in years.
We assumed an additional source of carbon i F that is available to the active SOM decomposers. Active SOM decomposers can either directly feed on FOM or they indirectly feed on metabolites of a fast cycling FOM decomposer community.
We were interested in the qualitative behaviour in steady state for a given input 15 of FOM and given, i.e. constant, environmental conditions. Therefore, the minimal model abstracted from the effects of environmental conditions such as temperature, soil moisture or texture, on the various model parameters. Further the minimal model did not account for interactions with other elements such as nitrogen or phosphorus. The system was described by the following equations.
The formulation assumes that substrate of each quality, i.e. the ease of mineralisation , has it's own decomposer community associated with, and that this decomposer community is in equilibrium with the available substrate most of the 10 time and therefore decomposition is only limited by substrate (McGill and Myers, 1987). Decomposition therefore scales linear with available substrate. Microbiology studies of substrate decomposition, however, show that decomposition often follows standard enzyme kinetics (Paul and Clark, 1989), where the rate of decomposition saturates at a maximum rate with increasing substrate availability (Eq. 4). 15 Hence, serveral models use Michaelis-Menten type equations (e.g., van Dam and van Breemen, 1995).
Where, S is the quantity of carbon in recalcitrant SOM, k is the maximum decomposition rate and k m is the quantity of S where decomposition rate is half of it's maximum.

Linear representation of decomposer biomass in SOM decomposition
The assumption that decomposition is limited by substrate only has been questioned (Fontaine and Barot, 2005 EGU is limited by the quantity of enzymes and not by the quantity of substrate. With the assumption that the quantity of enzymes is proportional to quantity of carbon in the decomposer pool they propose Eq. (5), which was also already used by van Wensem et al. (1997). The first order kinetics (Eq. 3) and Fontaine's equation (Eq. 5) can be seen as two 5 extremes of a colimitation of decomposition by substrate and decomposers. There are several equations that take into account both quantities. The probably simplest assumption is that decomposition is proportional to both quantities (Eq. 6) (Manzoni and Porporato, 2007;Fang et al., 2005;Knapp et al., 1983). According to Liebigs law of minimum Moorhead and Sinsabaugh (2006) use Eq. (7), which is the minimum of of Eq.
(3) and Eq. (5), to describe decomposition. Also the formulation of a mass-action law to describe the fraction of the substrate that is decomposed by Neill and Gignoux (2006) essentially leads to a decomposition that is smaller or equal to this minimum. A classic formulation (Monod, 1949) is based on standard enzyme kinetics (Eq. 8) 15 with variable amount of enzymes, which are assumed to be proportional to the quantity of decomposers. It also has been frequently employed (Parnas, 1975;Smith, 1979;Van Veen and Frissel, 1981b;Ladd et al., 1995;Blagodatsky and Richter, 1998;Kersebaum and Richter, 1994). 2008 Comparison of SOM decomposition formulations There are further formulations of colimitation that we distinguished from the previous ones because they are nonlinear in respect to the the decomposer quantity. Besides standard enzyme kinetics, microbes may inhibit each other (Suzuki et al., 1989). This kinetics can be described by an increase of the k m constant with increasing 5 microbial pool in the Monod-formulation (Eq. 8). Hence, Grant et al. (2001) applied Eq. (9) to SOM decomposition.
In a theoretical modelling study Schimel and Weintraub discussed several decomposition formulations and eventually used Eq. (10) (Schimel and Weintraub, 2003). The same equation has been used by other studies as well (Garnier et al., 2003;Raynaud et al., 2006). The formulation is structurally opposite to the formulation of Monod (Eq. 8), and assumes that the decomposition rate saturates with increasing enzyme availability instead of increasing substrate availability.
Using a simple simulation experiment of spatial accessibility of microbial communities to a small soil volume Wutzler (2008) 1 inferred an exponential equation of the 15 accessible proportion given the size of the decomposer pool. With the simplifying assumption that substrate is randomly distributed within a small soil volume, decomposition then can be described by Eq. (11).
There are also more complex formulations of Eq. (12) in the ITE model (Arah, 1996) and Eq. (13) in the SOMKO model (Gignoux et al., 2001), for which we did not calculate 20 equilibrium states.
BGD 5,2008 Comparison of SOM decomposition formulations 3.1.4 Formulations of SOM decomposition with additional states Agren and Bosatta (1996) propose a conceptual view of the decomposition process, 5 that involves a continuous spectrum of quality of organic matter. Microbial access to the organic matter, decomposition rate, and microbial efficiency depend on the quality q of a litter cohort (Eq. 14) that changes during decomposition. Blagodatsky and Richter (1998) propose a view on decomposition that depends on the proportion of active to dormant microbial biomass (Eq. 15). This proportion is ex-10 pressed as an activity state r which in turn is expressed as an additional state variable. r approaches a value that is a function of the substrate φ(S) (Eq. 16).
These two cases did not fit with our simple basic model and we could not calculate steady states. However, we refer to steady states in the original models in the discussion section.

Steady states
The steady state of the decomposer biomass A * is given by Eq. (18) for almost all the formulations. The only exception was Formulation (5), where steady state of decomposer biomass follows Eq. (19).
The steady state for SOM (S * ) for the various formulations is given in Table 1. Figure 2 displays the effect of the assimilation of FOM i F on the steady state for SOM (S * ).

15
The Formulations (10), and (11), which are non-linear in respect to decomposer biomass A * , exhibit a monotonous increase of steady state SOM S * with carbon inputs i F (Fig. 2). When carbon inputs approach zero, also the decomposer biomass A goes to zero. However, S * does not decrease to zero but stabilizes at a low level. Hence, in the absence of FOM assimilation there exists a fraction of SOM that is not decomposed.

20
A similar behaviour is exhibited by Formulation (9) for k>s, i.e. when decomposition is greater than the turnover of microbial biomass. For s→k, S * approaches infinity. And for k < s there is an infinite accumulation of SOM.
The continuous quality concept (Ågren and Bosatta, 1996) was not studied with our minimalistic model, which assumed only a single pool of SOM with given quality. Both, 25 limited and unlimited accumulation can be simulated with the continuous quality concept. The decay depends on the functions of microbial efficiency e(q), specific growth rate u(q). However, currently the priming effect is not simulated. All cohorts, i.e. carbon that entered the soil within the same time frame, decompose independently.

EGU
The steady state r in the model of Blagodatsky and Richter (1998) is given by φ(S * ) Eq. (15) and Eq. (16). In the original model also the turnover of the microbes is modified by r. With assuming a constant microbial turnover Formulation (15) yields qualitatively same results for steady state as the Monod-kinetics (Eq. 8).

5
Our study provides the first review and comparison of soil organic matter decomposition models that explicitly focuses on the colimiation of decomposition by substrate and decomposers.
By abstracting from other factors such as fluctuations in environmental conditions or nutrient availability and by using a basic minimalistic model for all the equations, 10 we could show that long term consequences of formulations do differ qualitatively. These differences could be grouped according to the assumptions about decomposer biomass and to the resulting representation of decomposer biomass in the decomposition equation. 15 The priming effect, i.e. the decomposition of SOM is influenced by the assimilation of FOM, was simulated with all formulations of SOM decomposition that accounted for active microbial biomass in an explicit manner (Table 1). The non-explicit formulations, used in many models (Sect. 3.1.1) were based on the assumption that decomposition of SOM can be considered in equilibrium with the available SOM at timescales larger 20 than a few month . If, however, the active SOM decomposers can feed on an additional carbon source related to FOM, this assumption does not longer hold. Contrary, the active decomposer biomass is near an equilibrium with the assimilation of FOM i F . Hence, in order to simplify models at larger time scales, we suggest to replace active decomposer biomass in model decomposition equations with EGU the assimilation flux i F and then simplify the system equations.

Priming effect and steady states
There was a finite steady state of SOM S * with all formulations except the Formulations (5) and (7). With the latter equations and also with Formulations (4), (8) and (9) an unlimited accumulation of SOM was possible. In these cases other factors must limit SOM accumulation in order to not to lock away all nutrients in SOM. Fontaine and 5 Barot (2005) showed that competition for nitrogen eventually limits carbon assimilation. Especially in older ecosystems also other nutrients such as phosphorus might be important. With these formulations the long-term balance is not determined by the quantity of litter input and decomposition rates only. Rather, parameters of the nitrogen cycle and nitrogen deposition become important. However, the understanding of nitro-10 gen cycle is not equivocal in literature. For example there are competing hypothesis about direct or indirect nitrogen uptake (Manzoni and Porporato, 2007), damping or amplification of the priming effect by nitrogen fertilization (Fontaine et al., 2004;Conde et al., 2005), and the role of plants in competition for organic nitrogen (Schimel and Bennett, 2004). Further the nitrogen cycle may be strongly influenced by micro sites 15 (Li et al., 2000).
The equilibrium state of SOM increased monotonically with input of fresh organic matter (FOM) in the non-explicit group of SOM decomposition formulations and the nonlinear group of formulations (Table 1). Contrary, with all formulations within the linear group the steady state was independent of FOM. This independence seems to 20 contradict observations of environmental gradients of litter inputs, which are assumed to correlate with primary production, where carbon stocks are increasing with input of carbon (Jobbagy and Jackson, 2000;Paul et al., 1997). One argument was, that this positive correlation between primary production and SOM stocks is not due to litter production but due to other confounding factors. We think, that this arguement is 25 unlikely, because the most important other factor temperature ususally also increases with primary productivity, leading to increasing decomposition rates and lower and not higher SOM stocks (Table 1, Eq. 7).
The steady state for the case when FOM assimilation approached zero differed be- 5,2008 Comparison of SOM decomposition formulations EGU tween the groups of formulations. Within the non-explicit group of formulations SOM steady state S * was zero, i.e., all SOM is eventually decomposed (Fig. 2). With all the other formulations, there was an offset for SOM steady state S * for reasonable model parameterization and initial conditions. For the Formulations (5) and (7), which did not lead to a general steady state, the amount of the SOM pool did not change and 5 stayed at the amount before FOM assimilation decreased to zero (Table 1). For the formulations in the non-linear group, the SOM pool decreased but approached a positive amount. Hence, there was a part of the SOM that is not decomposed at all in the absence of available fresh organic matter. This finding corresponds to observations of Fontaine et al. (2007) of millenia-years old carbon (Rumpel et al., 2002;Jobbagy and Jackson, 2000) in deeper soil layers where FOM supply is very low (von Lützow et al., 2006). It also corresponds to observations of litter bag studies, which can be best modelled by inferring a limit of decomposition where there is a part of the initial mass that is not decomposed in finite time (e.g. Berg et al., 1996;Bottner et al., 2000).

BGD
The strength of the approach of using a common basic model to compare different 15 formulations of SOM decomposition is also its biggest limitation. We could not compare the behaviour of the continuous quality model (Ågren and Bosatta, 1996) and the activity state model (Blagodatsky and Richter, 1998). The abstraction from other factors such as temperature, moisture, and nutrients discards aspects that are important in the original context of the equations. However, inclusion of other aspects would yield 20 in more complex and quantitatively larger differences in the steady state behaviour of the different models.

Relation with temperature sensitivity
The formulations of decomposition based on substrate only have lead to the models with the smallest number of state variables and parameters, properties that are 25 favourable in modelling. However, the assumptions that decomposer biomass is in equilibrium with the SOM pool neglects the priming effect and results in long-term behaviour that can explain very old carbon only by assuming a very low intrinsic decom-176 Introduction EGU posability. With these equations the decomposition of old carbon is primarily controlled by the temperature sensitivity of the old carbon (e.g. Reichstein et al., 2005). Contrary, temperature sensitivity may be overruled by other factors when explicitly modelling the priming effect by a second food source to the SOM decomposers and when explicitly accounting for decomposers in the SOM decomposition. The accumu-5 lation or decomposition of the old SOM depends either on other limiting factors such as nitrogen (Fontaine and Barot, 2005) (linear group of formulation). Or it depends on the availability of energy-rich fresh organic matter, belowground litter, or root exudates (Godbold et al., 2006;Göttlicher et al., 2006) which vary with soil depth (Bruun et al., 2007;Rasse et al., 2006;Gill and Burke, 2002;Frey et al., 2003;Elzein and Balesdent, 10 1995) (nonlinear group of formulation). The latter dependency is sensitive to land use changes, management practices and soil perturbations.
The importance of temperature sensitivity of SOM decomposition strongly affects our understanding of the feedback of SOM to global warming and we conclude that it is necessary to study which assumptions are appropriate and to discriminate between 15 the assumptions underlying the various studied formulations of SOM decomposition.

Challenging models with experiments to discriminate between formulations
In the following section we discuss approaches of discriminating between the different formulations. Often experiments are designed to calibrate a given model, or a model is designed to explain the observed data. Most of the cited models have been repeat-20 edly compared to observations that were collected to validate the model. However, science usually works the opposite direction where inappropriate hypotheses are falsified or ranked down by comparison against observation data (Popper, 1934;Kuhn, 1962;Lakatos, 1977). Therefore, we argue to design experiments in a way that models can be falsified in the best way (Hunter and Reiner, 1965;Atkinson and Donev, 1992;25 Reynolds and Ford, 1999). When sorting out inappropriate models, we also challenge the assumptions that underlie the models and the formulations of SOM decomposition.
The first idea of discriminating between the models is to challenge the long-term 177 Introduction EGU behaviour of the models (Fig. 2) by observations of carbon stocks for soil that are assumed to be in steady state and to compare the scaling of the soil carbon stocks with the mean litter input. The finding of increasing SOM stocks with increasing primary productivity and litter input (Jobbagy and Jackson, 2000;Paul et al., 1997) renders the formulations in the linear group unlikely. However, we already discussed the pos-Hence, we suggest to study the transient behaviour of soil under laboratory conditions, where the confounding factors and the input of fresh organic matter are controlled. We propose to challenge models by observations of patterns of several variables, which is used in multiple constraint model identification (Raupach et al., 2005;Reichstein et al., 2003) or pattern oriented modelling (Wiegand et al., 2003). When 15 using respiration data alone, it is hard to distinguish between the models. This was demonstrated by an artificial model calibration experiment (Fig. 5a). However, if the FOM was labelled, the time course of isotopic ratio of the produced CO 2 would differ between the models (Fig. 5b). Hence, explicitly modelling the isotopic ratio and calibrating the models to both outputs, resulted already in a better discrimination of the 20 models. The model of first order kinetics slightly, but consistently underestimated the respiration during days 5 to 40 (Fig. 5c) and overestimated the isotopic ratio during these days (Fig. 5d). In this artificial model calibrating experiment we assumed no discrimination of the carbon and no loss by dissolved organic carbon. A closer collaboration between soil scientists, microbiologists, modellers and experimentalists is 25 required in order to set up sound models and experiments in order to solve the model identification task.
If experiments can show that the priming effect is important for the dynamics of SOM, the formulations of the non-explict group are not appropriate to describe long- EGU term SOM dynamics. The discussion on the positive correlation between litter input and steady state SOM stocks rendered also the formulations of the linear group unlikely. Hence, we argue that the formulations of SOM decomposition where the active decomposers are represented in a nonlinear manner are most appropriate to describe long-term SOM dynamics.

Conclusions
This study reviews and compares different assumptions and formulations of colimitation of SOM decomposition by substrate and decomposers. The substitution of several formulations into a common basic model and the calculation of steady states enabled to compare the long term consequences of the formula-10 tions and their underlying assumptions.
We showed that the consequences of various formulations can be grouped according to the representation of active decomposer biomass in the decomposition of SOM.
-The assumption that decomposition kinetics of various OM pools is independent of each other together with the assumptions that decomposers are quickly in 15 steady state with substrate supply leads to formulation of decomposition that use substrate only. The priming effect is not simulated and SOM pools eventually decrease to zero on exhaust of FOM supply.
-The assumption that SOM decomposition is linearly related to decomposer biomass leads to steady states of soil organic matter that is independent of as-20 similation of FOM. Other factors such as nutrient limitation must be invoked to limit carbon sequestration.