Combined use of empirical data and mathematical modelling to better estimate the microbial turnover of isotopically labelled carbon substrates in soil

The flow of carbon (C) through soil is inherently complex due to the many thousands of different chemical transformations occurring simultaneously within the soil microbial community. The accurate modelling of this C flow therefore represents a major challenge. In response to this, isotopic tracers (e.g. 13C, 14C) are commonly used to experimentally parameterise models describing the fate and residence time of individual C compounds within soil. In this study, we critically evaluated the combined use of experimental 14C labelling and mathematical modelling to estimate C turnover times in soil. We applied 14C-labelled alanine and glucose to an agricultural soil and simultaneously measured their loss from soil solution alongside the rate of microbial C immobilization and mineralization. Our results revealed that chloroform fumigation-extraction (CFE) cannot be used to reliably quantify the amount of isotopically labelled 13C/14C immobilised by the microbial biomass. This is due to uncertainty in the extraction efficiency values (kec) within the CFE methodology which are both substrate and incubation time dependent. Further, the traditional mineralization approach (i.e. measuring CO2 evolution) provided a poor estimate of substrate loss from soil solution and mainly reflected rates of internal microbial C metabolism after substrate uptake from the soil. Therefore, while isotope addition provides a simple mechanism for labelling the microbial biomass it provides limited information on the behaviour of the substrate itself. We used our experimental data to construct a new empirical model to describe the simultaneous flow of substrate-C between key C pools in soil. This model provided a superior estimate of microbial substrate use and microbial respiration flux in comparison to traditional first order kinetic modelling approaches. We also identify a range of fundamental problems associated with the modelling of isotopic-C in soil, including issues with variation in C partitioning within the community, model pool connectivity and variation in isotopic pool dilution, which make interpretation of any C isotopic flux data difficult. We conclude that while convenient, the use of isotopic data (13C, 14C, 15N) has many potential pitfalls necessitating a critical evaluation of both past and future studies. © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
Soil carbon (C) turnover is a fundamental process in ecosystem functioning. However, understanding the factors regulating soil C dynamics is difficult due to the many hundreds, or even thousands, of processes occurring simultaneously within soil microbial populations (Hanson et al., 2000;Kuzyakov, 2006). A key challenge in ecosystem science is to understand this inherent complexity.
Below-ground respiration represents the primary flux mediating the passage of terrestrial C back to the atmosphere (Van Hees et al., 2005). This flux is dominated by plant (autotrophic) and soil microbe (heterotrophic) respiration. Soil respiration is controlled by the activity of soil microbial communities, which are typically limited by the availability of labile C (Ald en et al., 2001;Glanville et al., 2012). Root exudation (Jones et al., 2009;Nguyen, 2009) and root and mycorrhizal hyphal turnover (Gill and Jackson, 2000;Wallander, 2006) are the main processes by which labile C enters the soil. Within this, low molecular weight (MW) C compounds are of particular importance because, despite representing <10% of total dissolved organic C (DOC), they are postulated to dominate soil CO 2 efflux, due to their rapid rate of production, uptake, assimilation and turnover by the soil microbial community (Jones et al., 2003;Van Hees et al., 2005. However, it is difficult to gain a detailed mechanistic understanding of how microbial communities mineralize these low MW C substrates as they can undergo a number of fates in soil which are difficult to experimentally separate (e.g. sorption processes and biological transformation).
To accurately model C substrate flow, isotopic tracers (e.g. 13 C, 14 C) are commonly used to experimentally model the fate and residence time of individual C compounds within soil. In particular, measuring isotopically labelled substrate-derived CO 2 provides an indication as to the speed at which C compounds are specifically biodegraded by the microbial biomass. Measurements of 13 C-or 14 C-substrate assimilation into the microbial biomass is routinely quantified using the chloroform-fumigation-extraction method of Vance et al. (1987). After extraction of the fumigated sample (e.g. in 0.5 M K 2 SO 4 ), a correction factor (k ec ) is applied to the data to account for the fraction of the biomass not soluble in the extractant (e.g. cell walls, insoluble protein; Wu et al., 1990;Joergensen et al., 2011). Correction factors (k ec ) applied to microbial biomass data using isotopic tracer studies often vary depending on soil type, soil depth, and rate of substrate addition (Bremer and van Kessel, 1990;Dictor et al., 1998). The k ec value is unlikely to remain constant when using 13 C/ 14 C pulse-labelled substrates, as the isotopic C becomes progressively transformed within the cells over time (i.e. moves out of the soluble cytosolic pool into insoluble components; Bremer and van Kessel, 1990). In addition, as contrasting C substrates may be partitioned differently within the cell, it is likely that the k ec values also vary significantly between compounds. However, most studies add much higher amounts of labelled substrate than the intrinsic isotopic concentration within the soil solution, therefore stimulating microbial growth and altering partitioning. Here, we directly evaluate the robustness of the k ec approach for tracing isotopically labelled compounds through the soil microbial pool over different chase periods, at native soil solution concentrations not inducing microbial growth.
Combining detailed experimental data with mathematical modelling approaches, can help provide insight into the mechanisms involved in complex processes such as soil C cycling (Roose and Schnepf, 2008). Applying kinetic equations to experimental data makes it possible to parameterise mathematical models, which can then be used to estimate future changes in soil C storage and other key ecosystem services.
First-order reaction kinetics have been used to build a variety of soil organic matter (SOM) decomposition models for over fifty years, mathematically modelling the turnover of multiple C pools within soils and sediments (Salter and Green, 1933;Berner, 1964;Middleburg, 1989;Ostle et al., 2009). First-order reaction kinetics are commonly used for modelling enzyme kinetics and are frequently used within multi-component models describing not only SOM turnover, but also large-scale soil nutrient dynamics (e.g. CENTURY, RothC, ECOSYS; Paul, 2007). Within, this study, we are interested in applying first-order kinetic reaction kinetics to explicitly consider C flows through the soil microbial biomass, focusing specifically on respiration of substrate-derived C from the soluble biomass pool. Commonly, research mainly considers substrate C decomposition by soil microorganisms, rather than specifically including C dynamics within the microbial biomass (Middleburg, 1989;Bosatta and Ågren, 1995;Manzoni et al., 2012). Some other models consider microbial growth and activity when investigating microbial biomass dynamics and apply second order Michaelis-Menton type kinetics. However, these studies often add enough substrate to change the concentration of the intrinsic soil solution pool and stimulate microbial growth (Whitmore, 1996;Nguyen and Guckert, 2001). Here, we directly label the soil solution so as to not induce a microbial growth response beyond that which is naturally occurring, to investigate how C is processed within the microbial biomass.
Following introduction of isotopically labelled low MW compounds into soil, a biphasic pattern of CO 2 evolution is often observed which is suggestive that C can be partitioned into two major compartments/pools (Saggar et al., 1996;Chotte et al., 1998;Boddy et al., 2007). To describe the microbial-driven substrate mineralization kinetics of these two pools, a double first-order exponential decay model is often applied to the experimental data (Boddy et al., 2007). Mineralization experiments using isotopically labelled substrates are conducted over varying time frames; from minutes (Hill et al., 2008;Fujii et al., 2010), days (Coody et al., 1986;Glanville et al., 2012), weeks and years (Simfukwe et al., 2011;Farrar et al., 2012).
However, a two pool model may be too simplistic to describe how some substrates are compartmentalized during their metabolism and a three pool model may describe data more accurately Farrar et al., 2012). The first pool of this model is attributed to a rapidly cycled microbial labile C pool (half-lives for this pool reported in minutes/hours; Hill et al., 2008;Farrar et al., 2012;Glanville et al., 2012). The second pool represents microbial structural or stored C (pool half-lives in days/weeks; Boddy et al., 2007;Farrar et al., 2012;Glanville et al., 2012) while the third, slowest turning over pool represents recalcitrant extracellular SOM (pool half-lives in weeks/months; Farrar et al., 2012). In addition, the duration of incubations may influence how pools are attributed to biological function. In particular, for long incubations (months/ years), the slowest pool may represent turnover of quasi-stable soil C rather than being representative of microbial C. Further, biological attribution of modelled pools is not only time and substrate specific, but very likely also depends on a range of soil chemical, physical and biological factors (Van Hees et al., 2005).
Even if attribution of biological function to the different pools is achievable, a further major caveat associated with using exponential decay models is the assumed lack of interaction between the discrete pools and the lack of experimental techniques to validate them. To address this issue, we have developed a new empirical model based on our measured pools and applied a set of new, independent equations which allow for interactions between pools.
The first aim of this study was to monitor the temporal dynamics of two contrasting 14 C-labelled C substrates through the microbial biomass within an agricultural soil. Secondly, we used this experiment to evaluate whether a single k ec value can be used reliably to assess the amount of substrate-14 C contained within the microbial biomass. Thirdly, we used the experimental data to test how the duration of the isotopic tracer experiment influences the choice of modelling approach. Fourthly, we attempted to validate components of the model. Lastly, we used our experimental data to develop a new integrated empirical model (based on measured C pools) which describes the different interactions of C between key pools in soil.

Field site
Soil was obtained from a hyper-oceanic, freely draining, temperate agricultural grassland located in Abergwyngregyn, Gwynedd, North Wales (53 14 0 N, 4 1 0 W). The mean annual rainfall is 1250 mm and the mean annual soil temperature (at 10 cm depth) is 11 C. The soil is classified as a Dystric Eutrudepts (US Soil Taxonomy). The vegetation at the site consists of perennial ryegrass (Lolium perenne L.) and white clover (Trifolium repens L.) and is subject to periodic intensive sheep grazing (>5 ewe ha À1 ) and receives regular fertilizer addition (120 kg N ha À1 y À1 ). Due to the high root density in the grassland (0.35 ± 0.02 kg m À2 in the top 10 cm), all the sampled soil was classified as rhizosphere soil.

Soil characterisation
Three independent samples of soil (0e10 cm depth) were collected and immediately transferred to the laboratory for analysis ( Table 1). The soil passed freely through a 5 mm mesh enabling removal of stones, roots and macro-fauna and ensuring soil homogeneity (Jones and Willet, 2006). Soil water content was measured gravimetrically (24 h at 105 C). pH and electrical conductivity (EC) were determined in a 1:2.5 (w/v) soil-to-deionised water mixture. Soil solution was extracted within 12 h of soil collection from the field by the centrifugal-drainage method (3200 g, 15 min, 20 C; Giesler and Lundstr€ om, 1993). The extracted soil solutions were passed through a 0.22 mm filter and the soil solution stored in polypropylene vials at À20 C prior to use.
Dissolved organic C (DOC) and total dissolved N (TDN) in soil solution were determined using a TOC-TNV analyzer (Shimadzu Corp., Kyoto, Japan). Ammonium and nitrate were determined colorimetrically following the procedures of Mulvaney (1996) and Miranda et al. (2001) respectively. Dissolved organic nitrogen (DON) was calculated as the difference between TDN and dissolved inorganic N. Total free amino acids (TFAA) were determined by fluorescence using the procedure of Jones et al. (2002). Total free monosaccharides were determined following the 2,4,6-tripyridyls-triazine (TPTZ) spectroscopic method of Myklestad et al. (1997). Soil solution P was determined using the molybdate blue method of Murphy and Riley (1962). Total dissolved phenolics were determined following the Folin e Ciocalteu colorimetric method of Velioglu et al. (1998).

Preparation of isotopically labelled soil solution
Uniformly 14 C-labelled D-glucose (SigmaeAldrich Company Ltd., USA) and L-alanine (Amersham Biosciences, UK) were selected for this study as they represent two common root exudates which are known to be important in soil C cycling (Van Hees et al., 2005). Soil solution was extracted, as described above, and spiked with either 14 C-labelled glucose or L-alanine to give a specific activity of ca. 0.9 kBq ml À1 . In contrast to many previous studies using high C addition rates, our experiments were designed to replicate substrate concentrations and turnover rates naturally found in the soil. We assume that C flow in the soil was at quasi-steady state and that the concentration of substrate added (<10 nM) did not significantly alter the intrinsic soil solution alanine or glucose concentration and was insufficient to induce microbial growth above that occurring naturally. Then we followed the distribution of the 14 C-labelled substrates across five measureable pools over time i.e. the soil solution pool, the exchangeable soil pool, soluble and insoluble microbial pools, and the respired pool. At all times, the sum of the 14 C fractions for all pools was equal to 1 (100% of added label).

Microbial uptake/soil solution pool
The rate of microbial removal of 14 C-labelled substrate from soil solution was determined according to Hill et al. (2008). Briefly, 1.7 g of field-moist soil was placed in a 1.5 ml micro-centrifuge tube with a hole pierced through the bottom; this tube was then placed into another intact micro-centrifuge tube. 0.35 ml of 14 C-labelled soil solution containing either 14 C-glucose or 14 C-alanine was added to the soil surface. The soil solution infiltrated immediately through the soil column. Soil solution was collected via centrifugation (4000 g, 1 min, 20 C) after substrate incubation times of 0.5, 1,2,4,8,12,24,48,168 h and then weekly up to 7 weeks after initial 14 Clabelling. Soils were maintained at 20 C for the duration of the experiment and for all subsequent extractions. The volume of soil solution recovered was recorded and the solution subsequently mixed with Scintisafe 3 ® scintillation cocktail (Fisher Scientific, UK) and analysed using a Wallac 1404 liquid scintillation counter (Wallac EG&G, UK) to determine its 14 C concentration.

Soil exchange phase/soil sorption pool
To determine the amount of 14 C-labelled substrates adsorbed to the soils solid phase 14 C-labelled glucose or alanine was added to soil as described above and the soil extracted using 0.5 M K 2 SO 4 . The amount of 14 C in the solution/exchange phase was measured at each time point (as above) by performing a 1:5 (w/v) soil-to-K 2 SO 4 (0.5 M) extraction in which the samples were shaken (30 min, 200 rev min À1 ), centrifuged (18,000 g, 5 min) and a 1 ml aliquot of the supernatant retained for 14 C analysis as described above.

Soluble microbial biomass extraction
To determine the amount of 14 C retained within the soluble fraction of the microbial biomass, 14 C-labelled glucose or alanine was added to the soil as described above. At the sampling times shown in Section 2.4, the soil was fumigated with chloroform (Vance et al., 1987) for 24 h in the dark at 20 C. After fumigation, the samples were extracted with 0.5 M K 2 SO 4 (1:5 w/v) as described above for the non-fumigated samples.

Microbial biomass not extracted by K 2 SO 4 /methanol extractable microbial biomass
To investigate the organic solvent extractable fraction of 14 Clabelled substrate still retained in the microbial biomass, soil remaining after CHCl 3 -fumigation was re-suspended in methanol (MeOH). Briefly, the excess K 2 SO 4 was removed from the soil and replaced with 100% methanol (1:5 w/v) and the soils shaken (30 min, 200 rev min À1 ), centrifuged (18,000 g, 5 min) and 1 ml of Table 1 Selected soil properties for the Eutric Cambisol soil used for the mineralization studies. Values represent means ± SEM (n ¼ 3).

Substrate-C mineralization
To measure the rate of substrate mineralization, 1.7 g of soil was placed in a 1.5 ml micro-centrifuge tube placed inside a sealable 50 ml polypropylene cylinder, and 0.35 ml of soil solution containing 14 C-labelled glucose or alanine added to the soil surface. After addition, a NaOH trap (1 ml, 1 M) was added and the cylinders sealed. NaOH traps were changed and their 14 C content determined as described in Section 2.4. To determine the amount of 14 C still remaining in the soil after 7 weeks, soil which had not been subjected to the extraction or fumigation steps described above, was combusted in an OX400 biological oxidiser (R.J. Harvey Instrument Corp., USA) and 14 CO 2 measured by scintillation counting after capture in Oxosol scintillant (National Diagnostics, Atlanta, GA, USA).

Traditional modelling approach for describing 14 C dynamics in soil
Traditionally, models describing C dynamics use first-order kinetics to obtain a series of decay constants that best represent substrate decomposition by microbial communities (Berner, 1964;Middleburg, 1989;Harakuk et al., 2015). These decay constants represent only a composite of the multitude of processes that occur simultaneously within the microbial "black box", and as such often overesimplify complex microbial processes (Allison and Martiny, 2008). To describe 14 C soil dynamics, often exponential decay models are used to estimate microbially-driven substrate mineralization rates (Chotte et al., 1998;Van Hees et al., 2005;Boddy et al., 2007). Nguyen and Guckert (2001) and Farrar et al. (2012) suggested that glucose mineralization follows a triphasic pattern which can be described by a triple exponential decay model: where f(t) is the amount of 14 C remaining in the soil at time t, a 1 , a 2 and a 3 describe the initial size of each respective pool, and k 1 , k 2 and k 3 correspond to the exponential decay constants for each respective pool. The first phase (k 1 ) is thought to correspond to 14 CO 2 efflux as substrates are immediately used for catabolic processes (i.e. respiration) (Boddy et al., 2007). The second pool (a 2 ) relates to a slower mineralization phase (k 2 ) attributable to C temporarily immobilized within the biomass (i.e. storage-C). The third pool (a 3 ) probably relates to the very slow breakdown (k 3 ) of microbial structural-C compounds and C chemically and/or physically protected in soil (Farrar et al., 2012). Alternatively, a first-order double exponential decay model (biphasic mineralization pattern) may be used to describe substrate mineralization: Where, a 4 and a 5 describe the size of each respective pool, and k 4 and k 5 correspond to the exponential decay constants for each mineralization phase. Here, the first pool (a 4 ) is thought to correspond to a C pool which is rapidly used (k 4 ) for catabolic processes (i.e. comparable to pool a 1 in the triphasic model). The second, more recalcitrant pool (a 5 ) probably constitutes the remaining C immobilized within the microbial biomass (i.e. used for cell growth and maintenance) or necromass which turns over much more slowly (k 5 ).
The half-life (t ½ ) for the first mineralizable pool from both models (a 1 and a 4 ) can be calculated as follows: Calculating the half-life for the slower second and third pools is probably unreliable as the double and triple models assume the different pools are unconnected (see Discussion).
Eqns. (1) and (2) cannot be solved explicitly for t and so the NewtoneRaphson algorithm was applied to both the double and triple exponential decay equations to calculate the time when the overall 14 C remaining in the soil is half the initial amount; from here on this is referred to as the substrate halving-time.
To describe the removal of 14 C-substrate from soil solution, a first-order model with asymptote was fitted to the data; where y 0 represents the maximum depletion of 14 C from the soil solution pool, a 6 describes initial size of the bioavailable soil solution 14 C pool and k 6 is the exponential rate constant describing the turnover rate of this second pool. A two pool model was also fitted to the experimental data describing the dynamics of the 0.5 M K 2 SO 4 extractable pool ( 14 C-substrate adsorbed to the soil phase). Here y 0 represents the maximum depletion of 14 C from the soil sorption pool, with a 6 replaced by a 7 to describe the size of the initial pool and k 6 replaced by k 7 for the exponential rate constant for this pool. The half-lives were calculated according to Eqn. (5), with a 7 and k 7 replacing a 6 and k 6 for the K 2 SO 4 extractable pool. It is worth noting that this equation calculates the overall half-life, however, when y 0 > a 6 or a 7 then the substrate will never be reduced to half and therefore Eqn. (5) would have no solution.
t ½ ¼ lnð2 Â a 6 =ða 6 À y 0 ÞÞ=k 6 (Eqn. 5) To describe changes in the K 2 SO 4 -soluble 14 C-labelled microbial biomass pool (CHCl 3 -fumigation) and the MeOH extractable (K 2 SO 4 -insoluble) microbial biomass pool, a first-order double exponential decay equation (Eqn. (2); two pool model) was fitted to the experimental data. Here, a 8 and a 10 likely represent the more soluble components of the microbial biomass (CHCl 3 -fumigation and K 2 SO 4 -insoluble extractions respectively). Pools a 9 and a 11 may describe the more insoluble microbial components (CHCl 3 -fumigation and K 2 SO 4 -insoluble extractions respectively) which require further breakdown from extra-cellular enzymes to become remobilised and before passing into the more soluble pool prior to microbial utilization. Rate constants for each pool are defined by k 8 , k 9 , k 10 and k 11 respectively.
CHCl 3 fumigation efficiency values (k ec ) were calculated as follows: where k ec is defined as the correction factor value to account for the amount of microbial biomass 14 C not extractable with K 2 SO 4 . F is the percent of 14 C-labelled substrate extracted from the soluble microbial biomass pool by CHCl 3 fumigation-extraction, R is the percent of 14 C-labelled substrate respired, S s is the percent of 14 Clabelled substrate remaining in soil solution. Finally, S is the amount of 14 C-labelled substrate sorbed to soil particles (after accounting for that recovered in soil solution). Here, we assume that there is minimal transfer of microbial-14 C to stable soil organic matter, particularly over the first few days of the experiment.

New model to describe processing of labelled low MW C substrates within the microbial biomass
A new empirical model is proposed here which incorporates interaction pathways between different pools identified within the model (Fig. 1), which is something both the first-order double and triple exponential decay models fail to account for. For both alanine and glucose, microbial substrate decomposition was modelled over time using a new and independent set of equations, and outputs are expressed as a percentage of the total specific activity added to the soil.
We define five C pools based on the measured pools of 14 C, i.e. C l , the 14 C-labelled substrate in soil solution, C s , the 14 C-substrate adsorbed to the soil solid phase, C m,sol , soluble 14 C contained in the microbial biomass, C m,ins , insoluble 14 C contained in the microbial biomass, and C resp , respired 14 CO 2 .
We assume that there is flow of 14 C between those pools as shown in Fig. 1. Substrate added to the soil solution can either be adsorbed to the soils solid phase or it can be taken up into the soluble microbial biomass pool. There are fluxes in both directions to and from the soluble and insoluble microbial biomass pools. We assume that respiration can only occur from the soluble microbial biomass pool. The composition of the microbial community as well as the substrate quality can change over time; therefore, the respiration rate constant can also vary over time (Bosatta and Ågren, 1995;Manzoni et al., 2012). As such, we assume that the respiration rate constant is inversely proportional to time, with initially rapid respiration that slows down with time.
Conservation of mass leads to the following system of equations (Eqns. (7)e(17) are completely independent from Eqns. (1)e(6)): where k upt (day À1 ) is the uptake rate constant from soil solution into soluble biomass pool, k ins,f (day À1 ) is the forward reaction rate constant describing the flow of C from the soluble to insoluble biomass pool. k ins,b (day À1 ) is the backward reaction rate constant describing C flow from the insoluble to soluble biomass pool. k res (t) ¼ k res_c /(t þ b) is the time dependent respiration rate constant where the parameters k res_c (dimensionless), t is time (days) and b (days) determine the shape of this time dependence, k a (day À1 ) is the adsorption rate constant, and k d (day À1 ) is the desorption rate constant.

Parameter estimation
Parameters and 95% confidentiality bounds were determined using the Matlab function lsqcurvefit in three steps: 1. In an independent experiment on sterile soil (data not presented), we measured the kinetics of substrate sorption, based on methods described by Barber (1984), in order to determine the sorption rate constants k a and k d . As there are no microbes present in sterile soil, the two pools of interest are now only the liquid and sorbed phase. Assuming linear sorption, the equations are given by: This is a system of linear differential equations with constant coefficients and can be solved analytically. For initial conditions (C l , C s ) ¼ (100,0), the solutions are: Eqn. (14) was then fitted to the measured data in order to obtain the sorption rate constants k a and k d .
2. Moving back to non-sterile soil and assuming that the sorption rate constants will not change whether the soil is sterile or not, the uptake rate constant can be determined by fitting the following equations solely to the data from the C l and C s pools in the main experiment.
3. The remaining parameters k upt , k ins,f , k ins,b , k res_c , and b were determined by fitting the full model to the data of all pools of the main experiment.

Statistical and data analysis
All experiments were carried out in triplicate and data visually inspected for normality using quantileequantile plots (Crawley, 2007). All statistical procedures were carried out using the statistical package 'R' v 2.12.1 (2010), with P ¼ 0.05 used as the upper limit for statistical significance. Non-normally distributed data were log-transformed prior to analysis. A two-sample student's t-test was used to compare the different model parameters. Exponential decay curves were fitted to the experimental data using a least squares iterative model in SigmaPlot v12.3 (Systat Software Inc., USA). Dependency values for each model parameter were used to indicate whether the parameter values were strongly dependent on one another. To critically evaluate which decay model best described the experimental data, the following criteria were employed: An r 2 value of 0.90 was deemed acceptable for assessing the fit of the model to the experimental data. To check for model over-fitting, a dependency value cut-off of 0.98 was selected (Farrar et al., 2012). Further, a least-squares estimation (LSE) was calculated to evaluate the accuracy of the estimation of parameter values (Wolfe and Chinkes, 2005). To evaluate how increasing the number of data points in the mineralization experiment influenced the choice of kinetic model, we compared model fits using a minimum of 6, up to a maximum of 15 individual data points. Six was chosen as the minimum number because this represented the number of parameters in the triphasic model. Further, we investigated whether the pools/compartments assigned after fitting a kinetic model to the 14 CO 2 evolution data correspond to the different operationally defined measurements (chemical extracts) of the known mineralizable pools. This could provide biological justification of the chosen mathematical model.

Microbial uptake/soil solution pool
The depletion of 14 C-substrate from the soil solution was extremely rapid and the first order single exponential model with asymptote fitted well to the data ( Fig. 2a; r 2 > 0.99 for both substrates). After the initial sampling, levels of 14 C recovered in soil solution were low throughout the experiment for both substrates (Fig. 2a, Fig. S1). The half-life for both alanine and glucose in soil solution was estimated to be very short at 4.3 ± 0.3 min (Tables 2  and 3). The amount of 14 C recovered in the soil solution pool after 49 d was very low (0.09 ± 0.04 and 0.02 ± 0.008% of the total 14 C activity added for alanine and glucose respectively).

Soil exchange phase/soil sorption pool
After accounting for the 14 C recovered in soil solution, the amount of 14 C recovered in the 0.5 M K 2 SO 4 soil extracts was also very low (Fig. 2b, Fig. S1). This loss of 14 C from soil exchange sites was well described by a single exponential decay with asymptote model (r 2 > 0.89 for both substrates). The half-lives for substrates within this K 2 SO 4 extractable pool were very short being 0.5 ± 0.2 h and 1.2 ± 0.5 h for alanine and glucose respectively (Tables 2 and 3).

Soluble microbial biomass extraction
The CHCl 3 fumigation-extractable microbial biomass pool contained a larger amount of 14 C in the glucose treatment in comparison to alanine throughout the experiment (P < 0.05 for all time points, except 48 h where P > 0.05; Fig. S1). The amount of 14 C recovered by CHCl 3 fumigation-extraction was well described by a first order double exponential decay model ( Fig. 2c; r 2 ¼ 0.97 for both substrates). The first modelled pool (a 8 ) for glucose had almost a third more 14 C allocated to it compared to alanine (P < 0.01; Tables 2 and 3). However, the rate constant (k 8 ) and half-life describing loss of 14 C from this pool showed no difference between substrates (P > 0.05).

CHCl 3 fumigation-extraction efficiency (k ec ) values
When including all measured 14 C pools, approximately 50% of the added 14 C remained unrecovered for both substrates (Fig. 3ab). Over 49 d, the amount of unrecovered 14 C stayed relatively constant in the glucose treatment (ca. 50%) but showed a progressive decline in the alanine treatment (Fig. 3ab). If the widely accepted extraction efficiency factor was used (k ec ¼ 0.45; Vance et al., 1987), the calculated total 14 C in the biomass after 24 h, increased from approximately 50% to 80% and 84% for alanine and glucose respectively. From our data, however, we empirically calculated the k ec values to be 0.35 and 0.36 ± 0.01 for alanine and glucose respectively at 24 h (Fig. 3cd). These k ec values decreased over time and after 49 d, the values had fallen to 0.21 and 0.28 ± 0.01 for alanine and glucose respectively (Fig. 3cd).
3.5. Microbial biomass not soluble in K 2 SO 4 /Methanol extractable microbial biomass More 14 C was recovered by MeOH extraction in the glucose treatment in comparison to alanine over 49 d (P < 0.05; Fig. 2d). The recovery of 14 C in the MeOH extraction was well described by a first order double exponential decay model ( Fig. 2d; r 2 > 0.90 for both substrates). The MeOH-extractable microbial biomass fraction did not differ statistically between substrates, constituting 2.88 ± 0.28% and 4.82 ± 0.75% of the non-mineralized pool, obtained after biological oxidation for alanine and glucose respectively after 49 d.

14 C-Substrate mineralization
More 14 C-alanine was recovered as 14 CO 2 compared to glucose (P < 0.05; Fig. 2ef). The mineralization data for both alanine and glucose fitted well to both double and triple exponential decay models with dependency levels for all parameters well below the critical 0.98 cut-off threshold (Fig. 2ef, Table 4). Overall, however, the double exponential decay model gave a poorer fit to the experimental data over the entire dataset. It is worth noting that, irrespective of the number of data points used in the analysis, both substrates showed strong model fits to the experimental data for both double and triple exponential decay models (r 2 > 0.995). Dependency values for all model parameters, again for both double and triple exponential decay models, were highest at the beginning of the experiment and declined over time (with inclusion of greater numbers of data points). Overall, however, the dependency values were significantly higher for the triple decay model than for those Table 2 Model parameters describing the size and turnover of the extractable 14 C pool for alanine over time. The models are described by single, double or triple exponential decay equations (see Materials and Methods for further details). Only the single decay equations were fitted with an asymptote (asym.). Values represent means ± SEM (n ¼ 3), n.a. indicates not applicable, letters and values in brackets denote individual model parameters. Pools were assigned to each extractable 14 C pool, with the NaOH traps representing a respired pool (triple exponential decay model only), the centrifuged soil solution represents the labile soil solution pool, K 2 SO 4 extracts are ascribed to the substrate held on the exchangeable phase (sorption pool), CHCl 3 -fumigation and MeOH extracts represent the soluble and more insoluble pool of the microbial biomass pool respectively.
Extraction method and associated model e Alanine  Table 3 Model parameters describing the size and turnover of the extractable 14 C pool for glucose over time. The models are described by single, double or triple exponential decay equations (see Materials and Methods for further details). Only the single decay equations were fitted with an asymptote (asym.). Values represent means ± SEM (n ¼ 3), letters and values in brackets denote individual model parameters. Pools were assigned to each extractable 14 C pool, with the NaOH traps representing a respired pool, the centrifuged soil solution represents the labile soil solution pool, K 2 SO 4 extracts are ascribed to substrate held on the exchangeable phase (sorption pool), CHCl 3 -fumigation and MeOH extracts represent the soluble and more insoluble pool of the microbial biomass pool respectively.  ) 0.73 ± 0.14 (k 1 ) 11.1 ± 0.40 (k 6 ) 0.46 ± 0.33 (k 7 ) 0.03 ± 3.0 Â 10 À3 (k 8 ) 0.10 ± 0.03 (k 10 ) Pool 1 half-life (d) 0.04 ± 0.01 0.003 ± 9.5 Â 10 À5 0.05 ± 0.01 1.15 ± 0.11 0.34 ± 0.09 Pool 2 e (%) 7.79 ± 0.46 (a 2 ) 22.8 ± 0.99 (a 9 ) 3.45 ± 0.11 (a 11 ) Pool 2 rate constant e (h À1 ) 2.2 Â 10 À2 ± 4.6 Â 10 À4 (k 2 ) 2.3 Â 10 À4 ± 6.7 Â 10 À5 (k 9 ) 1.7 Â 10 À4 ± 3.3 Â 10 À5 (k 11 ) Pool 2 half-life e (d)  For alanine, 14 C allocation to the first mineralizable pool showed no difference between the double and triple exponential models (i.e. pools a 1 and a 4 ; P > 0.05; Tables 2 and S1 for triple and double models respectively). However, the double exponential decay model produced half-lives twice as long as those for the triple decay model for this first pool (P < 0.01; Tables 2 and S1 for double and triple models respectively), with values of 0.10 ± 0.01 d and 0.04 ± 0.002 d for the triple and double models respectively. For 14 C-glucose, attribution of 14 C to pools a 1 and a 4 was similar and differed by only 3% with the double exponential decay model having higher values (P < 0.05; Tables 3 and S1 for triple and double models respectively). Half-lives for these pools were three times longer for the double exponential decay model (0.12 ± 0.01 d) in comparison to the triple exponential model (0.04 ± 0.01 d; P < 0.01; Tables 3 and S1 for triple and double models respectively).
Pool sizes for the second mineralizable pool (a 2 and a 5 for the triple and double model respectively) differed for both alanine and glucose (P < 0.001; Tables 2 and 3 and S1) between the two models. Values from the triple exponential decay model for alanine were approximately 60% lower than those for the double exponential decay model. Glucose showed greater disparity between the two models with the three pool model showing approximately 75% less apportionment to this pool in comparison to the two pool model. Rate constants for this pool (k 2 and k 5 for the three and two pool model respectively) were two-orders of magnitude shorter for the three pool model than the two pool model for both alanine and glucose (P < 0.01).
We also compared model parameters from the third mineralizable pool (a 3 and k 3 ) taken from the triple exponential model, with parameters from the second pool from the double model (a 5 and k 2 ). Results showed no significant difference (P > 0.05) between any model parameters or pool half-life.
Using the NewtoneRaphson algorithm, total substrate halvingtime was determined for each 14 C-substrate for both models. Overall, there were no differences in total substrate halving-time between the different models (P > 0.05). Alanine-derived C had a total substrate halving-time of 52.7 ± 9.6 d and 62.7 ± 16.0 d for the two and three pool models respectively. Total substrate halvingtime for glucose-derived C was calculated as 151.3 ± 33.8 d and 154.4 ± 54.1 d for the two and three pool models respectively.
After combustion of the soil at the end experiment, we calculated that 64.8 ± 2.2% of the alanine 14 C remained unmineralized compared to 64.5 ± 9.6% for glucose. This was not significantly different to what we estimated to be remaining after accounting for the amount captured as 14 CO 2 from the exponential decay models (51.9 ± 4.8% and 69.5 ± 3.1% activity remaining in soil, for alanine and glucose respectively; P > 0.05).

Relationship between the size and turnover of soil solution, microbial biomass and 14 CO 2 pools
Model parameters describing the depletion of substrate from soil solution, showed no agreement with rate constants describing the turnover of the first mineralization pool (a 1 ) from the triple exponential decay model for both substrates (P < 0.001). Half-lives for the soil solution pool were much shorter (0.003 ± 0.0002 d) for both substrates, than for the first pool (a 1 ) from the triple decay model (0.04 ± 0.01 d; P < 0.01).
We also compared model parameters obtained from combining results from soluble microbial biomass extractions (not corrected for k ec ) and the MeOH extractable pool, and applied a double exponential model to the data. These two pools were combined because they both represent extractable fractions of the microbial biomass. Model outputs were compared to those obtained from the triple exponential decay model describing 14 CO 2 evolution.
The rate constants describing the turnover of the first mineralization pool for alanine only, showed no difference when compared to the first pool of the combined biomass pools (P > 0.05), but this  ) 51.9 ± 2.1 0.013 k res_c (À) 0.11 ± 0.03 6.42 k a (d À1 ) n/a n/a k d (d À1 ) n/a n/a b (d) 0.01 ± 0.01 n/a most likely reflects the high variability seen within the rate constants associated with the combined biomass pools for this substrate. Rate constants from the second and third mineralizable pools of the triple exponential decay model were significantly different to parameters from the first phase from the combined biomass pools, suggesting no relationship between these parameters (P < 0.05). Glucose showed the opposite trend to alanine, with the rate constant associated with the first mineralizable pool from the triple exponential decay model being strongly different (P < 0.001) compared to the combined biomass rate constant from the first pool. However, the second and third pools from the triple exponential decay model, showed no differences when comparing rate constants from the combined biomass model (P > 0.05).
3.8. Results from the new integrated C flow model Fig. 4 shows that the new integrated C flow model fitted well to our experimental data (r 2 ¼ 0.98 for alanine and 0.97 for glucose). We see that sorption of the substrate to the soils solid phase only plays a minor part and does not limit respiration. Glucose does not sorb at all, and the buffer power for alanine (k a /k d ¼ 0.06) is very small. Most of the 14 C-labelled substrate is rapidly taken up by the microbial biomass with an uptake rate constant that is an order of magnitude larger than the initial respiration rate constant (with k upt ¼ 246 d À1 and 256 d À1 for alanine and glucose respectively, compared to k res_t ¼ 15.8 d À1 and 9 d À1 for alanine and glucose respectively; Table 5). Correspondingly, a large fraction of the 14 C is immobilised in the microbial biomass for the full duration of the experiment. For both glucose and alanine, the soluble and insoluble biomass pools tend towards an equilibrium, where approximately twice as much 14 C-labelled substrate was partitioned to insoluble biomass as determined by the ratios between k ins,f and k ins,b . A full list of respiration rate constants (k res_t ) from the new model are presented in supplementary online material (Table S2). Our new model shows two distinct phases in CO 2 efflux, with a rapid initial respiration phase between 0 and 1 d after addition of the labelled substrates. After 1 d, the rate coefficients (k res_t ) remain fairly constant.

Microbial uptake and sorption of the added substrates
The loss of both substrates from soil solution was extremely rapid with half-lives < 0.5 h. This suggests a very high C flux through the soil solution, particularly in comparison to the subsequent flux of substrate-derived C through the biomass. In agreement with Hill et al. (2008Hill et al. ( , 2012, a very simple model successfully described the microbial uptake of substrate from soil solution. Of the 14 C-substrate added to the soil, sorption to the solid phase only accounted maximally for <4% of the total (<1% after 1 h), indicating that this process had minimal influence on the microbial uptake of glucose or alanine (Fischer et al., 2010).

Estimating microbial C assimilation
Almost all of the added 14 C was rapidly taken up by the microbial community (>99% after 1 h). CHCl 3 fumigation-extraction and an additional methanol extraction, however, failed to recover all of the 14 C present in the biomass. The incomplete recovery of 14 C from the biomass supports the need to apply a k ec correction factor if a full isotopic mass balance is required. The method we used to calculate k ec makes the assumption that all of the 14 C not accounted for in soluble soil pools or respiratory losses, was located in the microbial biomass. In our opinion, this assumption is likely to be valid for the early part of the experiment. After longer incubation times, incorporation of a small amount of 14 C into non-microbial SOM pools may lead to a slight underestimation of k ec (Jenkinson et al., 2004). Nevertheless, it is clear even from the first 24 h of incubation that the k ec value was both critically dependent upon substrate incubation time and appeared substrate specific. This is accordance with estimates made following much higher substrate addition rates (Bremer and van Kessel, 1990). We therefore conclude that the use of k ec correction factors in isotopic tracer studies is frequently invalid and their future use should be interpreted with considerable caution. In our opinion, the argument will also hold true for the equivalent use of k ec values in other isotopic (e.g. 13 C and 15 N) tracer studies.

Critical evaluation of the modelling approach
Our aim was to identify a kinetic model which accurately describes the dynamics of CO 2 evolution arising from microbial uptake, assimilation and mineralization of C substrates. Mineralization of simple C compounds is frequently assumed to follow a biphasic pattern (Saggar et al., 1996;Chotte et al., 1998;Van Hees et al., 2005). Our results confirmed that a two pool model passed the specified statistical criteria and gave a reasonable fit to the experimental data, at least in the short term (<48 h). However, after 48 h and certainly at 49 d, this model was clearly too simple and a three pool model proved far superior, supporting the previous findings of Nguyen and Guckert (2001) and Farrar et al. (2012).
A good mathematical fit, however, does not imply that the model has any biological relevance and ascribing biological functionality to different pools is fraught with potential error. It is therefore critical that the results are not over interpreted. The triple exponential decay model for substrate mineralization, infers there are three distinct mineralizable pools into which substrate C is partitioned once it is added to the soil and subsequently taken up into the cell. Fundamentally, the model assumes (i) that these pools are progressively depleted independently of each other, and (ii) that the isotopic dilution is the same for all pools (Fig. 5a). In soil, however, these assumptions are almost certainly not valid for the following reasons (with specific reference to the three pool model): a) The model assumes that C partitioning and assimilation rate is similar throughout the community. Although the kinetics of the transporters driving substrate uptake from soil may be similar within a diverse microbial community (Dijkstra et al., 2011), the expression of individual metabolic pathways within the cell undoubtedly varies depending on both differences in underlying physiology (e.g. eukaryote vs. prokaryote) and local microsite conditions. Given such complexity, it is surprising that simple models describe data as well as they do. However, complexity represents a problem when assigning function to discrete pools. In addition, microbial process rates will vary across the community. This is supported by studies showing temporal variation in 13 Csubstrate assimilation into PLFAs by different microbial groups (Dungait et al., 2011;Tavi et al., 2013). The implications of this are that the turnover times of individual organisms for the process represented by pool a 1 (energy production), for example, may start to overlap with those of pool a 2 (short-term C storage) (Fig. 5b). This will lead to poor parameter estimates for the size and turnover rate of pools a 1 and a 2 .
b) The model assumes that each pool describes a known suite of distinct biological processes. We fully accept that pools a 1 , a 2 , and a 3 comprise a multitude of different processes operating simultaneously within the community. Pool a 1 appears functionally quite distinct (rapid use of substrate for energy production). However, large uncertainty exists in the separation of pools a 2 and a 3 (as evidenced from the statistical criteria). It is clear that more pools and processes could be included. However, the resulting outputs would be largely meaningless on statistical grounds (data not presented). There is therefore a high degree of uncertainty associated with the parameter estimates for pools a 2 and a 3 and their associated rate constants and they should be interpreted with caution. The interpretation of the results would also benefit from ascribing a function to these pools. Combined for both substrates, the CHCl 3 -fumigation-extraction (CFE) results, respiratory pool a 2 has a half-life similar to the first soluble CFE pool (t ½ 1.4 vs. 1.0 d), whilst respiratory pool a 3 has a similar half-life to the second soluble CFE pool (t ½ 172 vs. 107 d) and the second MeOH extractable pool (t ½ ¼ 135 d). Thus respiratory pool a 3 probably reflects the turnover of both soluble (e.g. core metabolic enzymes) and insoluble structural elements (e.g. lipids, peptidoglycan) within the microorganisms upon death. Respiratory pool a 2 may therefore reflect a short-lived energy storage pool. It should, however, be considered that the attribution of pools to biological function may change with the duration of incubations. For instance, in long incubations (e.g. >49 d) the pool turning over most slowly may become more representative of slow turnover of quasi-stable soil C than of microbial C. However, this could be substrate specific and influenced by soil physical, chemical and biological factors (Van Hees et al., 2005).
c) The model assumes that the different C pools are depleted independently of each other. Based on the discussion above it is likely that the isotopic label contained in pool a 2 must pass back through a 1 before it is mineralized (Fig. 5c). This therefore underestimates the importance of pool a 1 and possibly its turnover rate depending on the pool linkages.
d) The model assumes the same rate of isotopic dilution within the three pools. Unfortunately, we know little about the size of the pools the isotopic label enters. However, we hypothesize that pool a 1 is very small in size and receives a disproportionately high level of isotopic labelling, overemphasizing its importance relative to pools a 2 and a 3 . This is supported by the low concentration of respiratory metabolites in the cell relative to the rate of flux. In contrast, pool a 3 represents the structural components of the cell and will likely receive low isotopic enrichment (Fig. 5d). Despite its slow turnover, it may actually represent a much larger C flux. This problem is exacerbated when the added isotopically-labelled substrate has a low concentration relative to the C content of the microbial community (as performed here). While higher pulse additions may give better uniform enrichment across the three pools (as the isotope is used for generating new cells where all components are labelled), this will likely induce microbial growth, cause changes in community structure and thus poorly reflect natural C cycling processes.
e) The model assumes that C turnover takes place within the same organism. It is probable that pools a 1 and a 2 are turned over in response to general cell maintenance, however, pool a 3 may be turned over by a combination of mechanisms including: (i) direct consumption and breakdown by grazers (e.g. protozoa), and (ii) cell death and breakdown via extracellular enzymes releasing soluble products which are re-assimilated by neighbouring organisms. In both scenarios, a large proportion of the C may again enter a second set of long-lived C pools (Fig. 5e). While the amount of C allocated to pool a 3 will be correct, the rate constant describing its turnover will be vastly underestimated.
f) The model outputs are only as good as the experimental data. We clearly show that the duration and number of data points taken from the experiment strongly influenced the final model parameter estimates. In our view, the first mineralization phase of the two and three pool models (i.e. a 1 and a 4 ) represents the same flux and therefore their parameter estimates should be identical. It is reassuring that these rate constants do agree when interpreted appropriately (i.e. two pool model for experiments <48 h in length and three pool model for experiments >48 h: Figs. S2, S3, S4 & S5).

New model interpretation and comparison with triple exponential decay model
Models looking at C mineralization dynamics by the microbial biomass similar in style to our new model have been published (e.g. Nguyen and Guckert, 2001). However, they generally fail to include a feedback component between the different pools. They also alter the intrinsic soil solution pool by relatively high rates of substrate addition, which may stimulate microbial growth (Whitmore, 1996;Nguyen and Guckert, 2001;Blagodatsky et al., 2010). We have developed a new integrated model, using our empirical data to address the issue of microbial processing of C within discrete pools with a level of connectivity between the different pools at conditions not inducing microbial growth. This is something which cannot be achieved using simple exponential decay models (i.e. Eqns. (1) and (2)), therefore, we applied a set of new, independent equations to account for feedback between measured pools. Initially, we experimented with fitting a constant respiratory rate constant to our data within our new model, as used in a short 60 min incubation by Nguyen and Guckert (2001). However, this produced unacceptable fits to our data over our longer 49 d incubation, with the model failing to account for the slower dynamics observed after 2 days. Therefore, we had two values for the respiration rate constant; a larger one which represented well the early dynamics and a smaller one to best describe the later dynamics. Interestingly, Blagodatsky et al. (2010) also observed a similar phenomenon that the first three days of their 50 day investigation could not be well represented by the model and concluded that their simplistic model was not able Fig. 5. Schematic representation of the different ways in which isotopically-labelled 14 C can flow through the microbial biomass. See Section 4.3 for more detail on the characteristics of the different models. Panel (A) shows the flow of isotopic C assuming that the C enters the microbial biomass and is partitioned into three discrete pools which are turned over independently. Panel (B) shows the distribution of rate constant values across the microbial community in a situation where all the community metabolises the C at similar rates leading to a clear separation of the kinetic pools a 1 and a 2 (Panel B1) or at different rates leading to poor resolution of the two kinetic pools (Panel B2). Panel (C) shows a representation of the model where the three pools (a 1 , a 2 and a 3 ) are not independent but where the pools are interconnected with all C leaving via pool a 1 . Panel (D) shows the situation where the intrinsic pools have different C contents leading to differences in isotope pool dilution. Panel (E) shows the situation where the originally labelled cell is predated by another microorganisms leading to C leaving pool a 3 and entering a new set of multiple pools in a different organism. These situations are not exhaustive and accurate description of isotope flow may be best described by a composite of different situations (e.g. Panels C, D and E).
to capture the rapid glucose transport in microbial cells. Consequently, we investigated the idea that changes in the chemical nature of the introduced isotopic C tracer can result in a timevarying respiration rate constant (Bosatta and Ågren, 1995;Manzoni et al., 2012). Introducing a time-dependent respiration constant strongly improved the fit. Although we are aware that variation in the rate constant may actually reflect the dynamics of unmeasured pools upstream of respiration per se, using this approach, we could accurately reproduce the rapid initial phase of C accumulation in the biomass and respiration that we observed with our experimental data. The amount of substrate-C contained within the biomass is constrained by uptake (input) versus respiration rate (output). The model replicated the gradual accumulation of 14 Clabelled substrate in the microbial biomass due to the uptake rate constant being approximately twice as high as the initial respiration rate constant. Outputs from the model, suggest that the rate constants obtained from the triple exponential model, underestimate the respiration flux, as implied by the higher k-values from this model. This is not surprising given the simplicity of exponential decay models which attempt to describe a complex set of simultaneously occurring processes (as described above). Despite the lack of absolute agreement, the results are not that dissimilar from the triple exponential model, being within the same order of magnitude. For example, for alanine the first value of k res_t is 0.66 h À1 , which is in the same order of magnitude as the rate constant of the first pool (k 1 ) from the triple exponential decay model 0.77 h À1 . The end value of k res_t and the rate constant of the third pool (k 3 ) from the triple exponential model are also in the same order, with both at 0.0002 h À1 . The rate constant of the second pool (k 2 ) of the triple exponential decay model can also be found within the range of k res_t from the new model, after approximately 0.5 d. Values for glucose are also similar. The first k res_t value is 0.38 h À1 , compared with 0.73 h À1 for the first pool (k 1 ) from the triple exponential decay model. The final k res_t value again matches well with 0.0001 h À1 , compared to 0.0002 h À1 for the third pool (k 3 ) from the triple exponential decay model. The rate constant for the second pool (k 2 ) of the triple exponential decay model can also be found within the k res_t ranges from the new model, after approximately 0.25 d. This implies that outputs from a triple exponential model may still be useful in estimating respiration, but that interpretation of the biological meaning of the different pools needs to be made with caution. Our new model is more realistic in terms of specific microbial pools and their interactions. It also explicitly considers the microbial biomass whereas the triple exponential model does not.

Recommendations for model interpretation
Despite the caveats identified above, we can make clear recommendations on the use and interpretation of the modelling approaches for evaluating the turnover of simple C substrates in soil: 1. At realistic substrate concentrations, the turnover of the fast respiratory pool of exponential decay models (a 1 ) does not approximate depletion from the soil solution. The use of k 1 to estimate substrate depletion would underestimate substrate turnover rates in soil solution by 20-fold. If the flux of C compounds through soil solution is required, then an alternative approach is required (e.g. depletion approach described here and in Hill et al., 2008).
2. The choice of model should clearly match the duration of the experiment to avoid either under-or over-fitting of the data. Essentially, the longer the incubation time the more complex the model required. For example, if applying an exponential decay model, short-term incubations (<48 h) should use a two pool model while longer-term incubations should use a three pool model.
3. There is a high degree of confidence in the assessment of the size and rate of turnover for pool a 1 of exponential decay models which allows direct comparison of different experimental treatments. In contrast, problems in isotopic pool dilution and pool connectivity make the values obtained for pool a 3 extremely uncertain. If used at all, these values will only ever provide crude estimates of C turnover within the biomass.
4. Our new integrated model provides a more realistic estimate of microbial substrate turnover encompassing some of the more complex interactions between different C pools, rather than using simpler exponential decay models. In addition, it may also be changed/extended if required, i.e. if there are other data or new information to include.
5. Chloroform-fumigation extraction represents a poor technique for assessing the amount of isotope contained in the microbial biomass. The incomplete extraction of the biomass and the continually changing k ec values makes the data essentially uninterpretable.

Conclusions
Isotope pulse labelling is widely employed both in the laboratory and field to investigate the flow of C and N through soil ecosystems. In this study, we critically evaluated the combined use of experimental 14 C labelling and mathematical modelling to estimate C turnover times in soil. We conclude that the technique is frequently used inappropriately, particularly when using the chloroform fumigation-extraction technique to quantify the amount of 13 C/ 14 C immobilised by the microbial biomass. Further, the traditional mineralization approach provides poor estimates of substrate loss from soil solution with the model outputs mostly reflecting rates of internal C metabolism within the microbial cells. Choice of the exponential decay model which best describes substrate-C turnover in soil is influenced by a range of factors including (1) substrate type, (2) the duration of the experiment, (3) whether the allocated C pools have any biological meaning, and (4) whether the model can be validated. Therefore, while isotope addition provides a simple mechanism for labelling the microbial biomass, it provides limited information on the behaviour of the substrate itself. For longer term isotope labelling studies (>48 h) a three pool model is most appropriate to describe C mineralization and here we ascribe gross functions to these three pools which will aid in future interpretation. However, due to problems of pool connectivity and isotopic pool dilution it is difficult to use the model quantitatively to describe C loss rates from soil.
Our new empirical data provides a much better and more realistic estimate of microbial substrate use and in turn respiration flux from the microbial biomass. Respiration rate constants from the new model were remarkably similar to outputs from the triple exponential model implying that exponential models may still useful in estimating respiration, but that interpretation of the biological meaning of the different pools needs to be made with caution. Recent advances in metabolomic technology are likely to offer new opportunities for understanding the fate of individual substrates in soil particularly when used in combination with other techniques such as transcriptomics and stable isotope proteomics.
Harriet Orr for their assistance with the manuscript and figures. Andrea Schnepf thanks the Austrian Science Fund FWF (Grant No.: V220-N13).