#### 3.1 Respiration and growth

The relationship of CUE to growth and respiration rates is nonlinear. On broad scales, faster biomass-specific growth rates generally increase CUE in a saturating fashion as it approaches the thermodynamic limit (CUE_{max}). This large-scale pattern masks several confounded factors. For example, productivity may be greater in relatively cold, nutrient-rich water bodies than in warmer oligotrophic ones. As a result, CUE decreases as temperature increases, but the trend can result from declining nutrient availability rather than a different temperature sensitivity for growth and respiration (Lopez-Urrutia & Moran 2007). This example indicates that both environmental constraints and resource supply need to be considered in interpreting CUE patterns and representing CUE in models.

For organisms, communities and ecosystems, the temperature sensitivity of respiration is largely determined by the activation energy of the respiratory electron transport system (~ 0.62 eV) (Brown *et al*. 2004; Yvon-Durocher *et al*. 2012). Within this pattern, the ‘apparent’ activation energy for microbial respiration may vary with local conditions because resource supply and other environmental factors affect responses to temperature (e.g. Ågren & Wetterstedt 2007; Wagai *et al*. 2013). For example, activation energies for the mineralisation of recalcitrant organic matter by microbial communities can be greater than 0.62 eV because more enzymatic steps are needed to transform organic carbon into CO_{2} (Sierra 2012). Ramirez *et al*. (2012) reported a value of ~ 0.85 eV for microbial respiration from 28 soils in unamended microcosms maintained for 1 year.

Because CUE represents the ratio of growth to assimilation rates, differences in the temperature sensitivity of these two components causes variations in CUE as a function of temperature. Generally, respiration increases more than growth as a function of temperature, so that CUE tends to decrease with temperature, in both soils and aquatic systems (Rivkin & Legendre 2001; Allison *et al*. 2010; Wetterstedt & Ågren 2011).

Other environmental drivers such as water availability in soils may also uncouple growth and respiration, causing shifts in CUE. In a short-term water stress event, for example, CUE increases as osmoregulatory solutes and storage compounds are accumulated (Uhlířová *et al*. 2005; Herron *et al*. 2009). However, in the long term CUE is reduced by repeated stress events, as the cumulative effects of the C costs for water stress responses become apparent (Tiemann & Billings 2011). Aquatic organisms may experience similar effects in response to fluctuating salinity conditions.

As resource supply or composition shifts, microorganisms, as individuals or communities, respond to changes in resource availability by altering the kinetics of enzyme-mediated assimilation pathways (Button 1993; Narang 1998; Hobbie & Hobbie 2012). Monod models describe enzyme-mediated uptake (I) as a saturating function of substrate or nutrient concentration (C), I = I_{max} C/(C + K_{s}), where I_{max} is the maximum uptake rate and the K_{s} is the half-saturation constant. In natural systems, the availability of substrates for uptake is generally linked to the activities of extracellular enzymes that deconstruct macromolecules. These activities are represented by Michaelis–Menten models, V = V_{max} S/(S + K_{m}), where V_{max} is the maximum reaction rate and the K_{m} is the half-saturation constant. Selective pressures to optimise uptake rate in relation to the resource costs of sustaining the uptake system are such that the Monod parameters C, I_{max} and K_{s} and the Michaelis–Menten parameters S, V_{max} and K_{m} are correlated, with C ≈ K_{s} and S ≈ K_{m} (Williams 1973; Lobry *et al*. 1992; Sinsabaugh & Follstad Shah 2010; Hobbie & Hobbie 2012). As a result, equilibrium values of I and V approach I_{max}/2 and V_{max}/2 and the ratio of growth rate (μ) to K_{s} remains relatively constant. A pulse of substrate that exceeds the concentration to which the community has adapted will result in an increase in uptake, which transiently uncouples catabolism and anabolism. An analogous effect is expected if there is a transient loss of a key substrate. As assays of microbial growth are often conducted over short-time intervals (one-to-few hours), a dynamic system with respect to substrate availability may show considerable variance in μ (and CUE). As the temporal scale expands to ecosystem models, CUE variation will attenuate, approaching the value of a steady-state system.

The need to allocate cellular resources to optimise the acquisition of multiple essential nutrients imposes additional limits on microbial growth and growth efficiency (Chen & Christensen 1985; Zinn *et al*. 2004; Cherif & Loreau 2007; Danger *et al*. 2008; Franklin *et al*. 2011). Sinsabaugh & Follstad Shah (2012) proposed a community growth model that incorporates the co-limiting effects of multiple resource acquisition:

- (1)

where S_{i} and K_{Si} are, respectively, the concentration and half-saturation constant of resource *i*. The premise of the model is that the acquisition of multiple resources by a microbial community is neither wholly independent nor fully integrated across the constituent populations. Because nutrient assimilation is a saturating function, growth increases sublinearly with environmental nutrient concentration, approaching an asymptote (μ_{max}) that represents the maximum capacity of a cell. At the community scale, biomass increases may allow growth to continue rising until pressed by another limit. CUE may increase with growth rate, if fixed maintenance and respiratory costs per unit biomass decline as a fraction of energy and material income. Alternatively, growth limitation caused by limited availability of an essential non-carbon element may decouple growth from respiration, decreasing CUE.

#### 3.2 Stoichiometry

The elemental C, N and P composition of microbial biomass varies narrowly relative to environmental variation in C, N and P availability. The mean C : N ratios for microbial biomass in soils, plankton and aquatic ecosystems are 8.6, 6.6 and 8.3 respectively; C : P ratios are more variable with means of 60, 106 and 166 (Cleveland & Liptzin 2007; Sterner *et al*. 2008; Manzoni *et al*. 2010; Sistla & Schimel 2012). Variation within and across systems is about twofold for C : N and threefold for C : P (Sardens *et al*. 2012). These stoichiometric requirements for biomass production force microbial communities to adapt their foraging strategies to the available substrates, which affects rates of growth and respiration.

In stoichiometric theory, nutrient-limited growth occurs when the availability of an essential element (E) relative to carbon (C : E) falls below the critical ratio or threshold element ratio (TER) required for optimum growth. The relationship between TER_{C : E} and CUE is commonly defined by

- (2)

where A_{E} is the assimilation efficiency of element E and B_{C : E} is the C : E ratio of biomass (Frost *et al*. 2006; Manzoni & Porporato 2009). This definition is typically adopted in litter and soil biogeochemical models with the assumption that A_{E} = 1 (Bosatta & Staaf 1982). In contrast, Doi *et al*. (2010) define TER_{C : E} as

- (3)

where GE^{max} _{E} and GE^{max} _{C} are maximum growth efficiencies with respect to C and E. These definitions highlight a confusing issue in the literature regarding the interpretation of TER. Studies of ectothermic animals focus on variation in assimilation efficiency, which is calculated as a fraction of ingestion, and biomass composition as the principal determinants of TER because CUE is relatively constant. For osmotrophic microbial communities, assimilation efficiency is a problematic concept, given that ingestion and assimilation are not distinct processes, and CUE can vary considerably, implying that TER has a similar variance.

Moreover, A_{E} may also vary with the physical structure of the environment. In soils, for example, pore-scale spatial heterogeneities in substrate availability and stoichiometry may cause transfer of nutrients between microbial populations in different patches. At larger scales, that is, soil core, these transfers may manifest as lower nutrient assimilation efficiency (A_{E} < 1) (Manzoni *et al*. 2008).

Traditionally, the TER_{C : N} for terrestrial microbial communities is considered something close to a constant with a value of 20–25, based on empirical studies that measure the critical transition in organic matter decomposition from net N immobilisation to net N mineralisation (Berg & McClaugherty 2003). However, these studies focused on the mineralisation of plant residues with relatively low C : N ratios. An expanded analysis that includes litter types with C : N ratios ranging from 10 to 1000 (i.e. including conifer litter and wood) suggests that TER_{C : N} does scale with the initial litter C : N ratio, implying that CUE decreases with increasing litter C : N (Fig. 1b). This analysis is based on the assumptions that the CUE and microbial composition are constant through time, and that B_{C : N} does not depend on substrate quality. As a result, a single value of TER is obtained for each substrate type. A more recent study including variability in B_{C : N} and thus TER, however, shows similar (albeit weaker) patterns (Ågren *et al*. 2013).

It is important to emphasise that the scaling of TER_{C : E} and C : E is sublinear, implying (1) that A_{N} (eqn (2)) may not be a constant and (2) changes in CUE do not fully compensate for the stoichiometric imbalance between litter and decomposer biomass. This gap leads to N immobilisation when litter C : N ratio is high. In other terms, CUE can be greater than predicted by the bulk C : N ratio of the litter either because nutrients are translocated from the surrounding environment (net immobilisation) or opportunistic microorganisms are selectively targeting low C : E substrates within the litter matrix (Allison 2005; Bastian *et al*. 2009). Nonetheless, as litter decomposition progresses N immobilisation and community CUE generally increase, and TER declines until the critical value 20–25 is reached. What remains unclear is whether microbial community CUE continues to increase (and TER_{C : N} decrease) as the C : N ratio of soil organic matter moves below the N immobilisation-mineralisation threshold. If TER_{C : N} does not decrease, CUE is expected to decline as C increasingly becomes the limiting resource. Complicating this issue is the increasing recalcitrance of the residual organic matter to decomposition, which is associated with slower microbial growth, and presumably lower CUE.

The Doi *et al*. (2010) formulation, which defines TER as the element ratio corresponding to maximum growth efficiencies (eqn (3)), circumvents the problem of defining assimilation efficiency for osmotrophs. The ratio GE^{max} _{N}/GE^{max} _{C} appears to have a narrow range of variation with a mean value of approximately 1.4 (Herron *et al*. 2009; Jones *et al*. 2009; Doi *et al*. 2010; Zeglin *et al*. 2012). If so, eqn (3) predicts that TER_{C : N} is directly proportional to B_{C : N}, rather than a function of CUE as represented in eqn (2). As a result, the TER_{C : N} values predicted by eqn (3) vary more narrowly than those predicted by eqn (2). The Doi *et al*. (2010) formulation more closely approximates traditional ecological conceptions of the role of nutrient availability in regulating microbial community metabolism. From this perspective, maximal growth efficiency is predicted when the environmental availabilities of all essential elements are at their TER, or when nutrient concentrations are great enough to saturate available uptake capacity, conditions that are rare in natural environments.

While TER values are useful indicators of relative nutrient limitation, CUE can be more directly modelled as a function of nutrient and substrate availabilities. To evaluate the relationship between CUE and substrate C : E, Manzoni *et al*. (2010) developed a model describing remaining C and N during litter decomposition. The model explicitly accounts for CUE (*e* in their notation) and can be used to estimate the value of CUE through nonlinear fitting of the C and N data. The estimated CUE, averaged over the course of decomposition, declined with increasing initial litter C : N and C : P ratios (Fig. 1a), providing a scaling relationship linking substrate stoichiometry to CUE. This declining pattern also leads to increasing TER values as the C : N and C : P of the litter widens. For litter with an initial C : N ratio of 50–70 (global averages for initial litter C : N and C : P ratios are 57 and 1217 on a mass basis, McGroddy *et al*. 2004), the CUE for decomposition is predicted to be about half the maximum CUE achieved with high-nutrient substrates, approximately 0.3 (Fig. 1a).

Sinsabaugh & Follstad Shah (2012) presented a stoichiometric model that relates ecoenzymatic activities (EEA), biomass composition and environmental nutrient concentrations to the CUE of heterotrophic microbial communities.

- (4)

where CUE _{max} is set at 0.60; S_{C : N} = B_{C : N}/L_{C :N} ·1/EEA_{C : N} and S_{C : P} = B_{C : P}/L_{C : P}·1/EEA_{C : P}; L_{C : N} and L_{C : P} are the elemental C : N and C : P ratios of labile organic matter; K_{C : N} and K_{C : P} are half-saturation constants; EEA_{C : N} = BG/(LAP + NAG); EEA_{C : P} = BG/AP where BG, AP, NAG and LAP are the potential activities of β-1,4-glucosidase, acid (alkaline) phosphatase, β-1,4-N-acetylglucosaminidase and leucine aminopeptidase respectively. These indicator enzymes generate assimilable nutrients from the principal organic sources of C, N and P (β-linked glucans, protein and aminopolysaccharides, and phosphoesters respectively). A_{P} and A_{N} are assimilation efficiencies for P and N. B_{C : P} and B_{C : N} are the elemental C : P and C : N ratios of microbial biomass. The parameters S_{C : N} and S_{C : P} are scalar measures of resource availability for microbial growth based on the composition of available organic matter and the relative distribution of EEA. The model is a saturating function that predicts community CUE as a geometric mean of N and P supply relative to C. Using global EEA data sets, the model yields similar estimates of mean CUE for terrestrial soils, freshwater sediments and plankton (0.29, 0.27, 0.28, respectively, approximately CUE_{max}/2) even though relative nutrient availabilities, EEA and biomass composition vary across these systems (Sinsabaugh & Follstad Shah 2012).

Assuming for simplicity that only N limits decomposition, eqn (4) can be reduced to CUE/CUE_{max} = (1 + K_{C : N}·EEA_{C : N}·L_{C : N}/B_{C : N})^{−1}. Fitting this simplified expression to the CUE measured in soils and estimated from the stoichiometric model by Manzoni *et al*. (2010) yields a numerical value for the term K_{C : N}·EEA_{C : N}/B_{C : N} = 0.0155 (solid line in Fig. 1), which corresponds to a CUE/CUE_{max} value of 0.82 for soils and litter. Using the mean soil values for these parameters given by Sinsabaugh & Follstad Shah (2012), K_{C : N} = 0.5, EEA_{C : N} = 1.434, B_{C : N} = 8.6, their predicted value for CUE/CUE_{max} is 0.46. The latter estimate implies that the mean value of A_{N} for soils (eqn (2)) is approximately 0.5, rather than the generally assumed value of 1. This lower estimate may be reasonable for soils considering the net mineralisation of N and the competition for mineral nitrogen by plants and dissimilatory microbial processes. If A_{N} decreases as the C : N ratio of organic matter narrows, then TER_{C : N} is less variable than eqn (2) otherwise predicts, bringing the formulations of TER_{C : N} in eqns (2), (3) into congruity. Indeed, the equations yield similar values for the mean TER_{C : N} of soils (14.3 for eqn (2), 12.1 for eqn (3)), assuming mean B_{C : N} = 8.6, A_{N} = 0.5, CUE = 0.3, and GE^{max} _{E}/GE^{max} _{C} = 1.4. These TER_{C : N} estimates approximate the mean C : N ratio of soil organic matter (14.3 ± 0.5 SE, Cleveland & Liptzin 2007).

A simpler model of CUE can be constructed by assuming that all the carbon taken up by microbes that cannot be used for growth at a given B_{C : N} due to limited N availability is mineralised through overflow respiration (Moorhead *et al*. 2012; Manzoni & Porporato 2009; Schimel & Weintraub 2003). Assuming A_{N} = 1 and neglecting maintenance respiration, CUE is equal to CUE_{max} when L_{C : N} < A_{N}B_{C : N}/CUE_{max} (~ 14.3 for B_{C : N} = 8.6, CUE_{max} = 0.6, A_{N} = 1) and equal to A_{N}B_{C : N}/L_{C : N} when L_{C : N} > A_{N}B_{C : N}/CUE_{max}:

- (5)

where L_{C : N} is the substrate C : N ratio as in eqn (4). If, for example, A_{N} is 0.5 as eqn (4) implies, eqn (5) predicts that CUE at L_{C : N} = 14.3 (the mean C : N ratio of SOM) equals CUE_{max}/2 with CUE = CUE_{max} at L_{C : N} < 7.1. If A_{N} remains constant at 1.0, CUE_{max}/2 occurs at L_{C : N} ~ 28, which approximately corresponds to the ecological transition from net N immobilisation to net N mineralisation. This minimal model does not consider net immobilisation of mineral N as a mechanism to compensate for large substrate C : N ratios, which increase the sensitivity of CUE to changes in organic matter C : N. Because it neglects N immobilisation, this minimal model underestimates CUE when B_{C : N} is assumed equal to a reasonable value of 10 for litter (note the bias in the dashed line in Fig. 1a). Nevertheless, it predicts that CUE is inversely related to L_{C : N} when the C : N ratio is wide, consistent with empirical data. Fitting eqn (5) to the estimated CUE (dashed line Fig. 1) yields a value for the term A_{N}B_{C : N} (the only fitting parameter) near 15. If TER_{C : N} is also approximately 15, then CUE by soil microbial communities should approach CUE_{max}, which is consistent with empirical estimates. If TER_{C : N} is 28, then the CUE of soil microbial communities approximates CUE_{max}/2.

All three stoichiometric models (eqns (2), (4) and (5)) highlight the key problem of resolving the relationship between A_{N}, TER_{C : N} and CUE. If A_{N} for soil microbial communities averages 0.5, as eqn (4) predicts based on EEA, then the mean CUE for soils is approximately CUE_{max}/2, consistent with predictions and measurements for aquatic ecosystems. If A_{N} = 1, then the mean CUE of soil microbial communities approaches CUE_{max}, consistent with measurements for soil ecosystems. In the next section, we argue that the apparent discrepancy between model predictions and experimental measurements is likely the result of differences in the methodologies used to estimate CUE in aquatic and terrestrial ecosystems, rather than a fundamental difference in microbial community metabolism.