1209301129soil fertility decomposition

Soil Biology & Biochemistry 38 (2006) 803–811 www.elsevier.com/locate/soilbio Soil organic matter decomposition driven ...

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Soil Biology & Biochemistry 38 (2006) 803–811 www.elsevier.com/locate/soilbio

Soil organic matter decomposition driven by microbial growth: A simple model for a complex network of interactions Cathy Neill*, Jacques Gignoux Ecole Normale Supee´rieure, Laboratoire d’e´cologie, CNRS UMR 7625, 46 rue d’Ulm 75230 Paris cedex 05, France Received 30 March 2005; received in revised form 5 July 2005; accepted 20 July 2005 Available online 24 August 2005

Abstract Priming effects are expressions of complex interactions within soil microbial communities. Thus, we aimed at building a microbial population growth model which could deal with different substrates, resources and populations. Our model divides the decomposition/growth process at the population level in two stages, mimicking mechanisms taking place at molecular and cellular scales: (1) the first stage is a reversible process whereby microbial biomass capture their substrate to form a complex within definite proportions; (2) the second stage is the irreversible rate-limiting utilization of substrate per se. It is supposed to be a first order process with respect to the quantity of complex. We put these assumptions into equations using an analogy with chemical reactions at equilibrium. We show that this model (1) provides a mathematical formalism that bridges the gap between first order decay of substrates and Monod kinetics; (2) sets constraints on the possible combinations of microbial functional traits, yielding microbial strategies in agreement with observations; (3) allows to model both positive and negative priming effects, and more generally complex interactions between the various components of a soil system. This model is designed to be used as a kernel in any soil organic matter model. q 2005 Elsevier Ltd. All rights reserved. Keywords: Monod kinetics; First order kinetics; Stoichiometry; Colimitation; Multiple limitation; Maintenance; Priming effects

1. Introduction First order kinetics have long been the mainstay in soil organic matter models (McGill, 1996) because they are often good approximations of mass losses in litter bags. However, litter decomposition takes place in soils where, through microbial action, it is liable to interact with native soil organic matter decomposition. These interactions have been recently experimentally demonstrated using isotope tracing (for instance Wu et al., 1993; Fontaine et al., 2004b). In unlabeled soils, any change in unlabeled CO2 respiration after the addition of a labeled substrate has been termed a priming effect (Kuzyakov et al., 2000). There is an increasing number a studies which suggest that priming effects are ubiquitous, can be of quantitative importance (Kuzyakov et al., 2000; Fontaine et al., 2004a; Hamer and Marschner, 2005) and are very variable in intensity and in * Corresponding author. Tel.: C33 144 32 38 78; fax: C33 144 32 38 85. E-mail address: [email protected] (C. Neill).

0038-0717/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.soilbio.2005.07.007

direction (positive or negative, see also Hamer and Marschner, 2002). It seems that priming effects cannot be accounted for with linear effects and even that their interpretation may need to take into account antagonistic effects specific of different microbial functional groups (Bell et al., 2003; Fontaine et al., 2003; 2004b; Hamer and Marschner, 2005). Priming effects are perhaps the most conspicuous reason why one should want to see soil organic matter models based on a more mechanistic, microbially-driven treatment of decomposition, as already advocated by McGill (1996). But, because decomposition is driven by microbial growth, features such as microbial stoichiometric and maintenance requirements also have important consequences on soil organic matter dynamics. Recent models have introduced a number of microbial constraints (Gignoux et al., 2001; Schimel and Weintraub, 2003), but these attempts were not without parameterization troubles, especially for maintenance rates (Gignoux et al., 2001). Actually, it turns out that it is not so easy to introduce microbial growth as modeled by microbiologists in soil organic matter models. First, just as first order kinetics have been the mainstay in soil organic matter models, Monod model has been

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the mainstay in microbiology (Kovarova-kovar and Egli, 1998). Unfortunately, these two models are incompatible with each other’s hidden assumptions. Monod model assumes that the microbial specific growth rate is ultimately limited, whereas the first order decay rate of the substrate is not. Indeed, writing that a biomass increment follows a Monod curve: db=dtZ mbðs=K C sÞ means that the first order substrate consumption rate (1/s)(ds/dt) can increase infinitely as biomass increases. First order kinetics with respect to the substrate yield just the opposite: if the substrate concentration is increased, the biomass could potentially grow infinitely fast. Therefore, these two models seem to be totally different in essence. Second, later microbial growth models in microbiology have mainly focused on detailed intracellular processes (Koch, 1997; Kovarova-kovar and Egli, 1998), and as such are not suitable for soil modeling. Third, most microbial growth models have been designed for suspended cultures growing on soluble substrates, whereas, in soils, insoluble substrates are predominant and might lead to different behaviors. Therefore, the aim of this work was to build a model at the population or at the community level able to reconcile the microbiologists’ insights with the soil organic matter decomposition process. It is not a soil organic matter model in itself, but is intended to be used as a microbial growth based kernel in any soil organic matter model. For that purpose, we kept it as simple as possible. It is based on a two-stage formulation of decomposition/growth, leading to two key assumptions regarding these stages. We give three qualitative applications of the model that makes it a potentially useful model. First, we show that the model does bridge the gap between first order decay and Monod kinetics. Second, we show that the model yields predictions about microbial physiology consistent with experimental evidence: this may help to refine microbial strategies. Finally, we show that the two simple assumptions of the model make complex interactions between substrates and microbial populations possible. In particular, we illustrate the ability of the model to predict positive as well as negative priming effects.

2. Model description 2.1. One population, one substrate Notations of variables and parameters are listed in Table 1. We will first consider one microbial population B and a single substrate S. We will denote abundances by small letters. We will express abundances in units of moles of carbon per kg of soil (C-moles). The model assumes that decomposition is driven by microbial growth, therefore: ds db K f dt dt The model splits the decomposition/growth process into two stages. The first one is a stage where microbial biomass

Table 1 Summary of the model parameters and variables Symbol

Meaning

B S X n m h Yc

Microbial biomass concentration Substrate concentration Complex formed by biomass and substrate Stoichiometric coefficient of the substrate in the complex First order constant of stage 2 in the model Carbon to nitrogen ratio of microbial biomass (mole basis) Carbon yield: number of units of b formed per mole of carbon uptaken Growth carbon yield: carbon yield if maintenance is set to zero Nitrogen yield: number of moles of nitrogen required to form one unit of biomass Instantaneous specific growth rate of microbes on a given substrate Instantaneous specific decay rate of substrate by a given microbial population Maximum specific growth rate of microbes Maximum first order decay rate of substrate Affinity of a given microbial population for a given substrate Turnover rate coefficient of biomass Maintenance energy coefficient of biomass

Ycmax Yn G D mmax kmax K mt mx

must capture enough resources before subsequent processing. When microbes have collected a piece of substrate, we will say that they form a complex together. The second stage is the subsequent utilization of complexed substrate to yield new biomass. Specifically, the model is based on two hypotheses. Hypothesis 1. The first stage is reversible and each unit of biomass must capture a definite number of units of substrate before entering stage 2. This definite number, denoted by the stoichiometric coefficient n, sets when resources are in sufficient amount for being processed through stage 2. At any time, we note x the quantity of biomass which has formed a complex with a quantity nx of substrate. We will call x the complexed fraction of biomass and bKx the free fraction. Likewise, nx will be called the complexed fraction of substrate and sKnx its free fraction. Hypothesis 2. The second stage is irreversible and ratelimiting of the whole process. It is a first order process with respect to x, and its first order constant will be denoted by m. Concretely speaking, substrates are either soluble or insoluble. The solubilization step is generally the irreversible, rate-limiting step of decomposition and growth on insoluble substrates (Lynd et al., 2002). Then, complexing an insoluble substrate simply means that microbial cells will get adsorbed on or attached to their substrate, such as bacteria on cellulose. Detachment may occur so that it is a reversible process. In contrast, for a soluble substrate which can be readily uptaken inside the cell, capturing it is presumably equivalent to absorbing it into the cell. Excretion of the substrate as is or as slightly transformed metabolites is the opposite process. This mechanism is known to happen and is called ‘overflow metabolism’ (Russell and Cook, 1995). The irreversible, rate-limiting

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step is presumably the utilization of intracellular molecules of substrate into metabolic pathways for biosynthesis. As a result, the complexed state of the substrate is a small metabolite. Then, the model gives biomass increments through: db Z mx dt

(1)

If all the biomass has complexed enough substrate, xZb, and microbes are exponentially growing at specific rate m. Conversely, if all the substrate is being complexed by biomass, then xZ(s/v), and, assuming for the moment that all complexed substrate is ultimately consumed, we have: ds ZKmnx ZKmns dt

(2)

Then, decomposition is first order with respect to the substrate abundance. But, the model allows for all possible values of x between those two extreme cases, which leads to deviations from exponential increase (for biomass) or decay (for the substrate). We can sum up the model for the decomposition(growth process of a single population on a single substrate through writing: B C nS4 X/ 2B C W

(3)

where W stands for waste (CO2, NH4 and organic wastes). The stoichiometric coefficient v is supposed to depend only on intensive properties of both biomass and substrate, not on abundances. Defining h as the constant carbon to nitrogen ratio of microbial biomass, Yc and Yn as the carbon and nitrogen yields of microbial population B with respect to substrate S (i.e. the number of moles of carbon and nitrogen necessary to build one C-mole of biomass), n as the nitrogen concentration of the substrate (carbon concentration is one since substrate abundance is expressed is C-moles), the stoichimetric coefficient n can be computed in order to fulfill the stoichiometric requirements of microbes:   1 1 n Z Max ; (4) Yc hYn n From this, if the carbon to nitrogen ratio of the substrate exceeds the threshold element ratio of microbes, hYn/Yc, it is the amount of nitrogen which sets v and there will be excess carbon. We assume that it will be respired through energy spilling metabolic pathways (Russell and Cook, 1995). Conversely, if carbon is limiting, excess nitrogen will be mineralized. There exist various possible formulations for computing x, knowing b and s. We will assume that the first stage reaches its thermodynamic equilibrium because the second one is a slow process. We use a simplified version of the mass action law. To the left hand side of the equation, we put the product of the free fractions because only those are liable to meet and increase the complexed fraction. To the right hand side of the equation we put the quantity of complexed fraction divided by an equilibrium constant, or

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affinity, K. This yields: ðbKxÞðsKnxÞ Z

x K

(5)

K has a dimension of 1/C-moles. We dropped the exponents usually found in mass action laws because consistency requires that the solution x of Eq. (5) should not depend on the unit which was arbitrarily chosen to express our state variables. The formulation above is invariant with respect to changes in units. Eq. (5) is a second order equation, whose solution is analytically straightforward and has been noted in other contexts (Koch, 1997; Baird and Emsley, 1999; Lynd et al., 2002). It is a simple equation for expressing that x is colimited by b and s (Fig. 1). 2.2. Several populations, several substrates The model can be generalized to the case where there are several microbial populations, (Bi)i%n, and substrates, (Sj)j%p. In this case, we write one Eq. (3) for every possible interaction between the considered entities. For instance, if we consider only pairwise interactions, we write: Bi C nij Sj 4 Xij / 2Bi C Wij

Fig. 1. Solution of colimitation equation with K/N and YcZ1 (top) and KZ1 and YcZ1 (bottom).

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and we introduce the affinities Kij and the first order rate constants mij Mass conservation and mass action laws yield a system of coupled equations whose unknowns are the xij: ! ! X X xij bi K xij nij xij Z (6) sj K K ij j i where it is apparent that each population competes with every other for common substrates and that substrates also ‘compete’ with each other for being complexed by any given population. We can also consider reactions involving more than one substrate. The line of reasoning is the same, the only difficulty here lying in the calculation of stoichiometric coefficients. For instance, let us consider a possible reaction between biomass B on the one hand, and substrates Sj and Sk on the other. We propose that we set stoichiometric coefficients so that they satisfy the two constraints: Yc ðnj C nk ÞR 1

hYn ðnj nj C nk nk ÞR 1

which express that there must be enough carbon and nitrogen in the complex to build one C-mole of biomass. A solution (nj, nk) with equality in both constraints exists only if the two substrates are such that one has a carbon to nitrogen ratio higher than the threshold element ratio of the population whereas the second substrate has a carbon to nitrogen ratio lower than this threshold.

3. Applications 3.1. Bridging the gap between microbial kinetics and decomposition kinetics First, we will consider pure microbial cultures on single substrates. In the case where the affinity K/N, the value of x tends towards the smallest of b and ns, leading to first order kinetics with respect to either b or s. If K is finite, x departs from its maximum value and the model predicts deviations from first order kinetics. Specifically, we can consider three important cases: – When bOO(s/n), i.e. substrate is limiting, approximating bKx with b in Eq. (5) leads to

a reverse Michaelis-Menten formalism: xZ ðbsÞ= ðð1=KÞC nbÞ. – When s/nOOb, i.e. biomass is limiting, approximating sKnx with s in Eq. (5) leads to Monod kinetics: xZ ðbsÞ=ðð1=KÞC sÞ. – When the substrate is soluble, and if we assume that the complexed substrate is intracellular as mentioned earlier, then the residual measurable concentration of substrate, sobs, (in, for instance, the soil solution) is sKnx. Substituting for sobs, in Eq. (5), we get Monod kinetics again, with respect to sobs: xZ ðbsobs Þ=ðð1=KÞC sobs Þ. These three particular cases confer to the model a solid, well-documented basis, on a large spectrum of values for b and s. In addition, the model extends decomposition formalisms and microbial kinetics, and bridges the gap between them. In light of the model, each of these formalisms can be seen as valid approximations only at opposite ends of the range spanned by the amounts of biomass and substrate: this is no surprise that Monod kinetics and first order kinetics cannot be applied at the same time. Besides, the model formalism features a continuous colimitation between b and s, without any need to implement threshold values. Therefore, the model appears as a very natural extension and generalization of well-known formalisms. It also brings an important supplementary constraint. From Eqs. (1) and (2), it follows that m, the first order rate constant of the second stage is at the same time the maximum value of two specific rates: the instantaneous specific decay rate, dZKð1=sÞðds=dtÞ, and the instantaneous specific growth rate,gZ ð1=bÞðdb=dtÞ (note that here specific does not refer to the same denominator). We examined data from continuous cultures of aerobic and anaerobic strains on cellulose to test this (Table 2, from Lynd et al., 2002). Within any given experiment, the maximum of g assessed as the maximum sustainable dilution rate and the mean value of d calculated from linear regressions over all dilution rates are quite close to each other. Usually, decomposition models assign to any substrate a maximum specific decay rate, say kmax Conversely, let us introduce the absolute maximum specific growth rate, mmax, of a given microbial population (where absolute means that

Table 2 Kinetic parameters for microbial cellulose utilization. Specific decomposition rates of cellulose are not necessarily maximum ones. Organism

Substrate, cultivation mode, temperature

Maximum net specific growth rate (hK1)

Specific decay rate d (hK1)

Clostridium cellulolyticum, anaerobic Clostridium thermocellum, anaerobic Fibrobacter succinogenes, anaerobic Ruminococcus albus, anaerobic Ruminococcus flavefaciens, anaerobic Trichoderma reesei, aerobic

MN301, continuous, 34 8C Avicel, continuous, 60 8C Sigmacell 20, continous, 39 8C Avicel, continuous, n.a. Sigmacell 20, continuous, 39 8C Ball milled cellulose, continuous, 30 8C

0.09 0.17 0.076 0.095 0.1 0.08a

0.05 0.16 0.07 0.05 0.08 0.1

a

Here we reported the maximum observed in the original paper (Peitersen, 1977), contrary to Lynd et al who reported an extrapolated value.

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the value is taken over all ossibly observed specific growth rates, whatever the substrate). What are the relationships between m, mmax and kmax? m must satisfy m%mmax and m%kmax. If mZmmax the rate-limiting step is independent of the nature of the substrate sustaining growth; then it must be a late step of biomass synthesis. Conversely, if mZkmax, the rate-limiting step of the substrate consumption is independent of the microbial population actually growing on it. So it must be an early step of substrate degradation. Since, the rate-limiting step either pertains to the degradation process of substrate into metabolites or to the process of biomass synthesis from metabolites, we conclude that mZMin(mmax, kmax). The model predicts that any given microbial population may have different maximum specific growth rates m depending on the substrate it grows on, which is commonly observed in continuous cultures. Furthermore, if kmax%mmax, as is presumably the case for insoluble substrates, the model predicts that mZkmax. Microbial populations growing on similar insoluble substrates at similar temperatures should display similar maximum specific growth rates. Data from Table 2 support this conclusion. Note that the strains differ in their mmax: for instance the reported mmax for Clostridium cellulolyticum is 0.17 hK1 at 34 8C on cellobiose (Guedon et al., 1999), whereas Fibrobacter succinogenes can grow as fast as 0.38 hK1 at 39 8C on ball-milled filter paper (Lynd et al., 2002). Finally, one might be surprised at the fact that anaerobic and aerobic bacteria may grow at similar specific rates, despite the fact that the former have three times lower yields than the latter on cellulose. It is true that anaerobic bacteria will consume three times as much cellulose as aerobic ones at similar dilution rates. But, the prediction on the maximum m of g and d is a different one and does not depend on yields.

3.2. Maintenance and microbial strategies Here, we will still consider a single microbial population B growing on a substrate S. We will examine the effects of maintenance respiration (which we distinguish from biomass turnover), as forecast by the model. We define exogenous maintenance as an exogenous substrate consumption, which is respired to provide energy for maintenance related functions; endogenous maintenance means that maintenance related functions are carried out using energy derived from recycling of preexisting biomass. It requires an autolytic mechanism. In the model, endogenous maintenance does not have any effect upon decomposition/growth other than decreasing microbial biomass. But exogenous maintenance has entirely different consequences. For a microbial population B with an exogenous maintenance, some of the carbon contained in the complexed substrate must be assigned to maintenance. Let Ycmax denote the growth yield, i.e. the yield that would be observed had B have no maintenance requirements.

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From mass conservation we must have: nxR

x C mx b Ycmax

(7)

where mx stands for the maintenance coefficient. Hypothesis 1 tells us that, should too many substrate be complexed by b, it could be released, as is or as an intermediate metabolite. But, it sets a definite maximum to the amount of substrate that can be complexed by one unit of biomass. As a result, the model predicts that there is a lower limit to the complexed fraction of biomass, which enables growth. Rearranging Eq. (7) yields: x 1 mx R b m ðnK1=Ycmax Þ This minimum fraction is inversely proportional to the maximum specific growth rate, m, so exogenous maintenance must have been selected only for those microbes which are able to grow fast (high mmax)on labile substrates (high kmax). Otherwise, it would more than frequently impede growth. Therefore, the model suggests that exogenous maintenance should be associated with r strategists, and endogenous maintenance with K strategists. It is presumably among slow growing Basidiomycetes or Ascomycetes that K strategists are to be found. These fungi constantly grow and forage for food throughout the soil. This seems only possible with some endogenous maintenance. Fungi are known to be able to autolyse their older hyphae and reallocate nutrients over long distances, to the tip of growing hyphae (Jennings and Lysek, 1999). In contrast, most bacteria do not autolyse (Koch, 1997) and display an exogenous maintenance (Russell and Cook, 1995). The model also predicts that neither growth nor maintenance will take place below a threshold amount of complexed biomass. This implies that maintenance energy ensures growth related functions, which may be cut down when no growth is possible. Reported important maintenance functions in bacteria are indeed growth related: wall lysis so that the cell volume may expand (Mitchell and Moyle, 1956), ions fluxes to maintain the membranne potential, which many bacteria let decrease as soon as exogenous substrates are depleted (Russell and Cook, 1995). Also, Button (1985) showed that there was a threshold for substrate uptake in seawater. According to Russell and Cook (1995), in bacteria, ‘maintenance should be used only to define growth when most of the cells in the population are capable of growing’. When no growth is possible, bacterial cells can display a quiescent but neither sporulated nor encysted state (Koch, 1997). In this state, they can slowly catabolize small energy reserves, a process which has been termed endogenous metabolism but which should not be confused with endogenous maintenance. Its rate is much lower than that of maintenance energy and it ‘should be defined as a state when no net growth is possible’ (Russell and Cook, 1995). Thus, in bacteria,

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maintenance appears to be a truly growth related function and, just as growth, requires a minimum amount of exogenous substrate. We can also see that the stoichiometric coefficient v should be as large as possible to lower the amount of complexed biomass required to trigger activity. From Eq. (4), Yc should be as small as possible. So the microbial threshold element ratio is shifted up by exogenous maintenance whereas that of microbes with an endogenous maintenance should be comparatively lower. This is consistent with (1) an adaptation of r strategists as consumers of nitrogen depleted substrates and fungal K strategists as consumers of nitrogen-rich humus; (2) a compensation of r strategists (bacteria and sugar fungi) having a nitrogen-rich biomass (h%6) when fungal K strategists display higher carbon to nitrogen ratios (10%h%15). 3.3. Modeling priming effects We will now consider a system with one microbial population B and two substrates S1 and S2. S1 is the native substrate of population B until S2 is added to the system. S2 is labeled so that one can track the origin of the CO2 respired by B. Any change in S1 mineralization by B after addition of S2 has been termed a ‘priming effect’, where we allow for positive as well as negative effects. We will assume that B has an endogenous maintenance. Now, the model predicts that before the addition of S2, a certain amount of b, say x1, will be complexed to S1. If the affinity K1 of B for S1 is finite, then x1 will be smaller than b and s1/v1. Because the first stage is reversible (Hypothesis 1), B can dissociate from S1 upon addition of S2, especially if B has a high affinity K2 for S2 or if S2 is added in substantial amounts. This will result in a negative priming effect. But, the addition of S2 will presumably enhance microbial growth, and the newly formed biomass may in turn complex again with S1, especially if all S2 is already being complexed or when S2 becomes exhausted. Since, x1 did not reach its maximum value before S2 addition, an increase in b may result in a higher x1 than what was initially the case. Then the model will predict a positive priming effect. We can illustrate the behavior of the model numerically. We considered a single fungal population whose characteristics were those of slow growing K strategists (Baath, 2001; Henn et al., 2002, see Table 3). Quantities were set according to Fontaine et al. (2004b), except for mineral nitrogen, considered as non-limiting. Note that due to poor affinity, the initial specific decay rate of organic matter approximated 2!10K5 hK1 (half life: 4 a). Because microbes had a mmax of 0.04 hK1, more labile substrates than cellulose, i.e. kmaxR0.05 hK1, would not be decomposed by them at higher rates than mmax. We computed the dynamics of these experiments using the following system of ordinary differential equations:

Table 3 Parameters values for the simulation of priming effects Parameter symbol

Values

b mmax mt mx Yc s1 kmax1 Ks1 s2 kmax2 Ks2

0.04 C-mol kgK1, soil 0.04 hK1 0.001 hK1 0.25 0.4 0.8 C-mol kgK1 soil 0.01 hK1 0.05 0.04 C-mol kgK1 0.05/0.01/0-005 hK1 1000

db Z m1 x1 C m2 x2 Kmt b dt

(8)

ds1 x ZKm1 1 C ð1Kmx Þmt b dt yc

(9)

ds2 x ZKm2 2 dt yc

(10)

dw ð1Kyc Þ Z ðm1 x1 C m2 x2 Þ C mx mt b dt yc

(11)

where w stands for the respired CO2. The concentrations x1 and x 2 were determined from the system 6 after linearization. Linearization was a good approximation and allowed a straightforward numerical computation. We plotted the fraction of CO2 respired from either substrates (Fig. 2). All substrates induce a negative priming effect at first. The more recalcitrant the substrate, the longer lasting the negative priming. When a substrate is exhausted, the microbes shift to native organic matter, thus inducing a positive priming effect. A number of studies have conclusively shown the existence of positive and negative priming effects in soils, whether by adding carboneous or nitrogenous substrates (reviewed by Kuzyakov et al., 2000). The hypotheses listed in Kuzyakov et al. (2000) correspond to the mechanisms predicted by the model, i.e. preferential substrate utilization for negative primings and biomass increase for positive ones. The results of our simulation find some support in the experiments of Hamer and Marschner (2002, 2005) who found that catechol, a phenolic compound, was more liable to induce negative priming effects on lignin, peat and soils than more labile substrates. The quantitative importance of priming effects in natural systems not only depends on the quality of incoming substrates but also on the time scale considered and on the frequency of substrate inflow. From our simulation, we may conjecture that with a periodic inflow of S2 of 14 days or so, S1 mineralization will be dramatically altered in the long term compared to a situation where no interaction is taken into account. Note, though, that here we have considered only one microbial population

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Fig. 2. Unlabeled soil respiration rates and microbial biomass after addition of a labeled substrate. Labeled respiration rates are not shown as they featured classical single peaked curves. The solid line represents the control treatment (no addition). Other lines feature variables after addition of three types of substrates: substrate 21 has kmaxZ0.05 hK1; substrate 22 has kmax 0.01 hK1; and substrate 23 has kmaxZ0.005 hK1.

and that antagonistic effects of the quality and of the frequency of incoming substrates on the resulting priming are to be expected, especially if one considers microbial competition (Fontaine et al., 2003).

4. Discussion 4.1. Comparison with other microbial growth models Although Monod kinetics have remained the mainstay in microbial physiology and ecology, a number of attempts have been made to carry out more detailed and mechanistic models. Some models have tried to solve for growth fluxes by considering the whole set of enzymatic reactions along the metabolic chain (Metabolic control analysis, Kacser and Burns, 1973). This approach cannot be reasonably extended to a population or community level. Others have focused on the analysis of just the initial steps bringing substrates into

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the metabolism of the cell, leading to a two-stage uptake formulation: reversible uptake followed by irreversible enzymatic utilization. Independent derivation of the same second order equation giving the uptake flux of one cell were made by different workers (reviewed by Koch, 1997), and this equation is essentially the same as the solution of Eq. (5), except that it has been used at the cellular level: instead of using the total quantity of biomass, they have used the total quantity of substrate carriers present at the surface of the cell, and no stoichiometric coefficient were introduced. These models proved more accurate than the Monod model (Koch, 1997). Our model really follows the same line of reasoning, but we applied the corresponding mechanisms at the population level, and we extended it to insoluble substrates, and other substrates where the ratelimiting step is located uphill uptake along the chain. Therefore, the first reversible step is not necessarily uptake. But it is most important that it be reversible, because this feature allows growth regulation by microbes. The two-stage mechanism hypothesized here also resembles that of a single enzymatic reaction, but our model differs from the concept of Synthesizing Unit of Kooijman (2000). Kooijman considers that intermediate states are at steady-state and do not build up, so he advocates for the use of fluxes rather than states. We consider states rather than fluxes because we assumed that the intermediate state just before the rate-limiting step does build up. Furthermore, he also argues that enzymes do not dissociate with their substrates, whereas it is a key assumption of our model. Logically then, the mathematical formulations of the two models are quite different. 4.2. Comparison with other soil organic matter model kernels Although first order kinetics have remained the mainstay in soil organic matter models, some important aspects of microbial physiology and growth have progressively been included in recent models. The first one is microbial stoichiometry. Thresholds have been introduced to decide whether carbon or nitrogen, or possibly another nutrient, was limiting for organic matter decomposition. But usually these thresholds do not alter the decay rates of soil organic matter pools, they only limit microbial growth through reduced substrate utilization efficiency (Franko, 1996; Li, 1996; Molina, 1996; Schimel and Weintraub, 2003 but see Gignoux et al., 2001). Our model also introduces such a threshold in Eq. (4), but the substrate C to N ratio not only affect substrate utilization efficiencies but also their decay rates, since an increase in n will decrease the quantity of complex, hence the specific decay rate d. Also the fate of excess nutrients differs between models. This is an important issue given that it might represent a significant flux. Excess nitrogen is usually mineralized, not stored. We assumed the same for excess carbon because bacteria have limited ability for storage and accumulation of intracellular

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metabolites could even be poisonous for them (Russell and Cook, 1995; Koch, 1997). An increasing number of studies have pointed out that excess energy was spilled through futile cycles (reviewed by Russell and Cook, 1995). A second feature of microbially mediated decomposition is maintenance. In light of the model and given the current debate regarding the relative functional importance of r versus K strategists, or bacteria versus fungi, maintenance type may be of particular relevance. To date, there seems to be an important bias regarding maintenance. First, to our knowledge, all models or experimental settings consider only one possible maintenance type, either exo-or endogenous. We think that this might be a serious shortcoming if the two actually occur at the same time in any soil. Second, modeling works are biased towards endogenous maintenance and experimental settings towards exogenous maintenance. Very few of the models we have examined explicitly assume that maintenance is exogenous or is a growth related process (Gignoux et al., 2001; Schimel and Weintraub, 2003). On the contrary, many models account for maintenance as a constant fraction of biomass turnover (Powlson et al., 1996). This amounts to assume an endogenous maintenance, although it is not always stated so (authors sometimes refer to cannibalism process by other microbes). This bias may be due to the fact that endogenous maintenance is far simpler to implement in a model. In contrast, experimental settings usually consider that maintenance is exogenous, may be because fast-growing bacteria have been much more studied than slow growing fungi. For instance, the fact that Anderson and Domsch (1985) sought to measure maintenance rates through glucose amendment, and also that this led to a technique for measuring and analyzing microbial biomass (Anderson and Domsch, 1978; Stenstro¨ m et al., 2001) is of significance. Several studies have shown that in some cases, the primed CO2 came from native microbial biomass. Earlier hypotheses were that this apparent priming was due to an enhancement in microbial turnover or death and their subsequent mineralization (Dalenberg and Jager, 1989; Wu et al., 1993). Then later authors have suggested that it was due to endogenous metabolism in bacteria (De Nobili et al., 2001; Bell et al., 2003). We suggest that it might as well be the consequence of an increase in fungal biomass, hence an increase in their endogenous maintenance respiration. This is all the more plausible that continuously foraging fungi could indeed be those that stay alert all the time in soils and could preempt scarce resources from perhaps not dormant, but certainly more demanding bacteria. Finally, as for biomass-dependence of decomposition itself, none of the models we have been able to examine so far (1) departs from an irreversible process which is an increasing function of biomass; (2) introduces the supplementary constraint mZMin(mmax, kmax) The latter constraint is a not so intuitive prediction of our model. But, in practice, when it comes to model the decomposition

of recalcitrant substrates, it amounts to taking mZkmax. However, it might be important to consider when one studies microbial competition for the decomposition and uptake of a labile substrate. The former feature of other models is what, in our view, makes them unable to model negative priming effects, not to mention a continuous transition between positive and negative priming effects, as a function of all the abundances of all the substrates and the microbial populations considered in a given soil system. 4.3. Conclusion Our model rests on two assumptions from which result a series of qualitative predictions that find some experimental support in the literature. There exist various possible mathematical formulations of the model but it is its description of the mechanisms involved in the decomposition/growth processes that matters. As such, the model can be applied as a kernel in any decomposition model which provides some implementation of other aspects not addressed here such as organic matter compartmentalization, fate of microbial turnover, change in quality as decomposition proceeds. In a forthcoming paper, we test the model against data from cellulose continuous cultures (Neill and Gignoux, 2005). Current work is being carried on to parameterize and test the model on data from priming effects experiments.

Acknowledgements We thank Se´bastien Barot, Mehdi Che´rif, Se´bastien Fontaine, Ge´rard Lacroix and two anonymous reviewers for helpful comments on an earlier version of the manuscript. This work was supported by the French government, ministry of Research as a part of the Continental Biosphere National Program (PNBC), and from the GlobalSav project, funded by the ‘ACI Ecologie quantitative’.

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