Without the loss of generality, the macrochemical catabolic reaction is formulated for one C-mol of OM. When OM does not contain any N or have high C:N ratios (N-limited), microorganisms immobilize N_Inorg from the environment. If this N_Inorg-source is not NH4+, then some of the electrons produced from the oxidation of the OM [Eq. (7)] must be allocated to the reduction of the N_Inorg-source. This allocation is often performed by intracellular electron carrier proteins, either NADP(H) or ferredoxins (Bloom, 2010). Therefore, the energetic cost of the reduction of the N_Inorg-source to ammonium must be accounted for in the catabolic reaction. The catabolic reaction is formulated by considering the half-reaction of oxidation of OM and reduction of EA and N_Inorg. To keep the formulation general, we separately consider the redox reactions for OM and EAs (top left in Figure 2).

The half-reaction of oxidation of the OM can be written as follows:

where OM_ox is the oxidized form of OM, 1/CN_OM is the stoichiometric coefficient for NH4+ released during the oxidation, γ_OM is the degree of reduction of the OM, and Δ_oxG_OM [kJ (C-mol OM)^–1] is the change in Gibbs energy of the oxidation reaction. Note that for all Gibbs energy changes, the subscript of Δ indicates the reaction and the subscript of G is the substance per unit of which the Gibbs energy is reported. For simplicity, we assume that OM_ox is always bicarbonate and catabolism does not produce any other organic product.

Next, the reduction half-reaction of a generic EA can be written as follows:

where γ_EA is the number of moles of electron received by the EA, EA_red is the reduced form of EA (e.g., H₂O in case of O₂), and ν_EAred is its stoichiometric coefficient; Δ_redG_EA [kJ (mol EA)^–1] is the change in Gibbs energy of the reduction reaction. A list of commonly occurring half-reactions of reduction of EAs is provided in Supplementary Table 1.

Next, the reduction of the N_Inorg-source to ammonium is given by

where γ_N and Δ_redG_N are, respectively, the number of moles of electrons accepted when reducing 1 N-mol of N_Inorg-source and the change in Gibbs energy of the reaction to reduce the N-source. For NH4+ as N_Inorg-source, no reduction would be required and γ_N = 0. The amount of N-source reduced to NH4+ depends on the microbial N-demand for growth, which in turn depends on the CUE and the C:N ratio of the OM. If x_N is the number of moles of N_Inorg reduced to NH4+, x_N × γ_N moles of electron per mol of OM must be transferred from the oxidation of OM to the reduction of the N-source in Eq. (9).

Finally, the catabolic reaction is obtained by adding the three half-reactions, Eqs. (7)–(9), and adjusting the stoichiometric coefficients so that the electrons are balanced (top center of Figure 2),

where xEA=γOM-xNγNγEA is the stoichiometric coefficient of the EA; Δ_catG_OM [kJ (C-mol OM)^–1] is the amount of Gibbs energy released by catabolizing 1 C-mol of OM for a given EA. Similar to Eq. (11), Δ_catG_OM is given by summing the Gibbs energy of the three half-reactions, Eqs. (7)–(9), multiplied by appropriate stoichiometric coefficients.

In principle, Δ_catG_OM can be calculated directly from Eq. (11) using Hess’s law, if the involved species and their chemical formulae are known. However, such information is often not known, as in the case of plant residues or soil OM. Using Eq. (12) is advantageous because the change in Gibbs energy of oxidation of OM can be estimated based on its degree of reduction γ_OM, as proposed by LaRowe and Van Cappellen (2011),

The empirical formulation of Δ_oxG_OM in Eq. (13) holds under standard conditions, therefore it needs to be modified under non-standard OM aqueous concentrations [for more details, see LaRowe et al. (2012) and Song et al. (2020)]. The γ_OM of 1 C-mol of OM is estimated based on the OM chemical formula as follows:

where n_i for i = C, H, N, or O are the numbers of thei_th element and z is the overall charge in the chemical formula of OM. While the actual values of γ_OM can be calculated from the chemical formula of OM, here we explore a range of values from 0 (least reduced, CO₂) to 8 (most reduced, CH₄), to cover the whole spectrum of organic compounds.

The change in Gibbs energy of the reduction reactions (Δ_redG_EA and Δ_redG_N, Supplementary Table 1) are calculated using standard Gibbs energy of formation listed in Supplementary Table 3. With this information, Δ_catG_OM can be estimated from the degree of reduction of the OM. An example of the calculations for the catabolism of glycine with NO3- as both EA and the N-source, and of glucose with O₂ as the EA and NO3- as the N-source, are provided in Supplementary Table 2.

Energy from catabolism is then transferred to the anabolism to support the synthesis of new biomass (center of Figure 2). Here, we formulate a generic reaction for the anabolism (biosynthesis) for 1 C-mol of microbial biomass. There are multiple ways of writing a macrochemical representation of an anabolic reaction (von Stockar et al., 2008; Battley, 2009; Kleerebezem and Van Loosdrecht, 2010). We follow Battley (2009), who simplifies Gibbs energy calculations by neglecting EA in anabolism. Briefly, 1 C-mol of biomass is formed from the same organic C used in catabolism and NH4+ as the N-source [other N-sources are reduced to NH4+ before being used, Eq. (9)]. Further, a degree of reduction balance is used to balance the electrons, and the imbalance of C is balanced by adding OM_ox(i.e., bicarbonate) to either side of the reaction. Thus, the anabolic reaction for an OM with given CN_OM can be written as follows:

where γ_B = 4.2 is the degree of reduction of the microbial biomass and Δ_anaG_B is the change in Gibbs energy of anabolism. Balancing N on both sides yields the stoichiometric coefficient of the N-source (i.e., NH4+),

The value of Δ_anaG_B [kJ (C-mol B)^–1] is estimated by writing the Gibbs energy balance of Eq. (15), as follows:

where Δ_CG_OM and Δ_CG_B are the changes in Gibbs energy of combustion for a given EA. Note that we have used combustion as the reference state so that only the organic C appears in Eq. (17).

To find Δ_CG_OM and Δ_CG_B, we can regard a catabolic reaction as analogous to a combustion reaction when a N_Inorg-source is not included. Therefore, for x_N = 0 and O₂ as the EA, Eq. (11) represents the complete combustion of an organic C compound to CO₂. Thus, from Eqs. (12) and (13), we obtain Δ_CG_OM and Δ_CG_B as follows:

Now, from Eqs. (17)–(19), we obtain Δ_anaG_B as follows:

Examples of the anabolic reaction on glycine and glucose are provided in Supplementary Table 2.

The overall metabolic reaction is obtained by summing the catabolic and anabolic reactions so that 1 C-mol of OM is used to form e C-mol of biomass (right of Figure 2). Thus, multiplying the catabolic reaction Eq. (11) by 1-eγBγOM and the anabolic reaction Eq. (15) by e, and summing these reactions, gives the overall metabolic reaction of microbial growth as follows:

In Eq. (22), Δ_rG_OM is the Gibbs energy change for the overall metabolic reaction, and ν_i are the stoichiometric coefficients of each reaction species (i = EA, EA_red, NH4+, N_Inorg). Equation (22) thus recovers the form of Eqs. (1) or (2), but now all stoichiometric coefficients can be determined as a function of the degrees of reduction of OM, EA, and biomass, and the C:N ratios of OM and biomass,

In Eqs. (23)–(26), x_N is not known, but it can be calculated by setting ν_NH4 = 0, because NH4+ used for microbial growth is already accounted for in the stoichiometric coefficient of the N-source. Thus, setting ν_NH4 = 0 yields the following:

The expressions for ν_EA and ν_N can now be simplified by inserting x_N in Eqs. (23) and (25), which are expressed as follows:

Equations (28) and (29) can be used to assess N-demand for microbial growth under oxic or anoxic conditions for any EA. Under C-limited, oxic conditions and assuming NH4+ is the N-source, the rate of N-uptake, ν_NU_OM, is the same as given by Manzoni et al. (2017). By setting Eq. (29) equal to zero, we can calculate the threshold C:N ratio of the OM at which no NH4+ is formed or immobilized. Above this C:N ratio—often called threshold elemental ratio (TER)—microbial growth becomes limited by the supply of organic N, which is expressed as follows:

If CN_OM < TER, then ν_N > 0 and net release of NH4+ occurs (i.e., net N mineralization), whereas if CN_OM > TER, then ν_N < 0 and net uptake of NH4+ occurs (i.e., net immobilization). Moreover, if the supply of N_Inorg, I_N, cannot sustain the required rate of immobilization, the microbial growth rate is determined by the rate of supply of the external N_Inorg-source; this condition is denoted as N-limitation (net required immobilization rate > I_N). Under N-limited conditions, microorganisms reduce the C-uptake rate, thereby decreasing their growth rate so that N-demand matches N_Inorg-availability (i.e., I_N). Mathematically, the reduction in U_OM is obtained from Eq. (6) as a function of the stoichiometry of N in the overall metabolic reaction, which in turn is a function of CN_OM and I_norm.

If EA and N_Inorg-source are the same (i.e., NO3-), such as during denitrification or DNRA, N is used for both oxidation of OM and microbial N-demand for growth; therefore, N-limitation would affect both uptake of OM and microbial growth. Thus, the total amount of N used [y_N in Eq. (2)] is given as the sum of ν_EA and ν_N, as follows:

Examples of the overall metabolic reaction on glycine and glucose are provided in Supplementary Table 2.

The change in Gibbs energy of the metabolic reaction, Δ_rG_OM [kJ (C-mol)^–1 OM], can be obtained by adding catabolic and anabolic reactions as done in Eq. (21), and can be written as follows:

Equation (32) can be used to estimate Δ_rG_OM knowing the CUE, or inversely it can be used to estimate CUE if Δ_rG_OM is known. It has been shown that the energy dissipated from microbial systems can be predicted by the degree of reduction of the OM, γ_OM (Heijnen and Dijken, 1992; Liu et al., 2007). However, these formulations are based on the Gibbs energy dissipated to produce 1 C-mol of biomass (Δ_rG_B); therefore, we divide Eq. (32) by CUE and obtain the following:

where ΔrGB=ΔrGOMe, Δ_catG_OM is given by Eq. (12), and Δ_anaG_B by Eq. (20). Simplifying the above equation to obtain CUE as a function of Gibbs energies yields the following:

Finally, the only remaining unknown in Eq. (34) is Δ_rG_B, which is given by Liu et al. (2007) as follows:

It is clear from Eqs. (12), (20), and (35) that the microbial CUE depends mainly on the degree of reduction of the OM and microbial biomass, and the type of EAs. Note that in Eq. (34), Δ_catG_OM is also a function of e, making Eq. (34) an implicit non-linear equation in e, which therefore needs to be solved numerically. An example of metabolic reaction on glycine and glucose is presented in Supplementary Table 2.

We assumed standard conditions for calculating Gibbs energy, so that the stoichiometry of the metabolic reactions is also representative of standard conditions, i.e., species concentrations are at 1 mol L^–1, pH 7, temperature of 298 K, and pressure of 1 bar. In a dynamic system, changing concentrations of the involved species, pH, or temperature would change the Gibbs energy of metabolic reactions, and thus also the stoichiometric coefficients of such reactions. Our framework can be generalized by including the effects of non-standard conditions as shown in previous work (Kleerebezem and Van Loosdrecht, 2010; LaRowe and Amend, 2015, 2016; Delattre et al., 2019).

To summarize the theory section, we started with a general description of microbial growth rate under C- or N-limited conditions (section “Microbial Growth Rate Under C- and N-Limitations”). The stoichiometric coefficients needed to calculate growth rate are estimated by splitting the overall metabolic reaction into catabolic and anabolic parts and considering their Gibbs energies (sections “Catabolism of Organic Matter” and “Anabolism of Microbial Biomass”). Finally, the stoichiometric coefficients of the metabolic reaction—mainly CUE (e)—and of N-uptake (v_N or y_N) were constrained using bioenergetics (sections “Overall Metabolic Reaction” and “Gibbs Energy Change of Metabolic Reaction and Carbon Use Efficiency”). Table 2 summarizes some simplifications of ν_EA, v_N, and y_N under specific conditions for the OM; e.g., when OM contains N or when EA and N_Inorg-source are both NO3-.

Figure 4 shows how the normalized microbial growth rate varies under C-limitation vs. N-limitation under oxic conditions. At high N-availability (high values of I_norm), microbes are C-limited and the normalized growth rate only depends on the CUE. In turn, the CUE increases with a higher degree of reduction of the OM, causing the growth rate to increase with γ_OM (lines with different colors), as also shown in Figure 3. In contrast, at low N-availability (low values of I_norm), microbes are N-limited and their growth is constrained by I_norminstead of CUE. As a consequence, N-limited growth is independent of the degree of reduction of the OM (Table 2). When microbes are supplied with an N_Inorg-source other than NH4+, the energetic cost for NO3- reduction to NH4+ reduces the CUE [dashed vs. solid lines in Figure 4]. Even if Figure 4 shows the normalized growth rate under oxic conditions, the same dependence of the growth rate on N_Inorg-availability and γ_OM occurs for any EA also under anoxic conditions, although the lines shift depending on the metabolic pathway of N, as shown next.

Figure 5 shows the transition of microbial growth rate from C- to energy- to N-limitation under anoxic conditions when NO3- is used both as EA and N_Inorg-source via DNRA (solid curves) or denitrification (dashed curves) pathway. The normalized growth rate for both pathways increases linearly at low values of γ_OM, attains a maximum, and then decreases non-linearly for high values of γ_OM (Figure 5A). Under C-limited conditions, the growth rate is higher for the denitrification as compared to the DNRA pathway for all values of γ_OM (cf. dashed vs. solid brown curves in Figure 5A). Conditions transition to N-limitation at high γ_OM when I_norm < 1.5 (colored curves). To clarify where this transition occurs, Figure 5B shows the full responses of G_{C, norm} (only C-limitation; black curves) and G_{N, norm} (only N-limitation; colored curves) to changes in γ_OM. For a given level of I_norm, the N-limited branch of the growth curve is initially flat, then decreases with increasing γ_OM. This is because under N-limited conditions, CUE per unit of total NO3-utilized (e/y_N) decreases as γ_OMincreases for reduced compunds (γ_OM > 4.7). Changing the level of external N_Inorg (I_norm) simply re-scales the growth rate under N-limited conditions because G_{N, norm} = (e/y_N)I_norm [Eq. (4)]. Under N-limitation and in contrast to C-limited conditions, the growth rate is higher for DNRA as compared to the denitrification pathway (cf. solid vs. dashed blue curves in Figure 5A, enlarged view provided in Supplementary Figure 1).

The microbial growth rate for given γ_OM (curves with different colors in Figure 6) and availability of N_Inorg (NH4+; solid vs. dashed curves) is stable at low OM C:N ratio (CN_OM) when organic N is abundant (C- and energy-limitations). Under these conditions, the microbial growth rate also increases with increasing γ_OM and it does not depend on N-availability (as shown in Figure 3). In contrast, under N-limited conditions, the growth rate depends on both γ_OM and CN_OM (Table 2). As CN_OM increases, the supply of organic N for microbial growth decreases, up to the point where growth becomes limited by the supply of N_Inorg. Under these conditions, the growth rate is reduced to match the N_Inorg supply (Table 2); therefore, the growth rate also decreases with decreasing I_norm (solid vs. dashed curves). Contrary to the negative effect of low γ_OM under C-limitation, low γ_OM reduces the effects of N-limitation by forcing microbes to grow at a slower rate, which lowers their N-demand.

The different EAs shift the relative position of energy-, C-, and N-limitation regions in the space of CN_OM and γ_OM (Figure 7). In the case of O₂ and Fe³⁺ (goethite or ferrihydrite) as EAs, the growth rate varies as in Figure 6, i.e., it is maximum for OM with high γ_OM and low CN_OM (C-limited region), decreases with decreasing γ_OM (energy-limited region), and is lowest at high values of CN_OM (N-limited region). In all these cases the growth rate decreases only slightly when OM is highly reduced. In the case of SO42- as the EA (see also the orange curve in Figure 3), the growth rate at a given CN_OM first increases with increasing γ_OM, reaches its maximum values for γ_OM close to 4 and then decreases. For lower values of γ_OM, the growth rate decreases as CN_OM increases because of transition from C- to N-limited conditions, similar to the behavior under other EA, whereas at high γ_OM N-limitation does not occur even at CN_OM as high as 1,000 C-mol (N-mol)⁻¹. As a result, the energy-limited region shifts from left [Figure 7, panels (A–C)] where it is caused by low γ_OM, to the right in panel D where it is caused by high |Δ_rG_B| (see Supplementary Figure 2 for Δ_rG_B as function of γ_OM).

The overall patterns in normalized growth rate or CUE with the degree of reduction and type of N_Inorg-source for microbial growth are similar to those described in other studies (Roels, 1980b; Heijnen et al., 1992; Smeaton and Van Cappellen, 2018). When microorganisms catabolize OM using high energy yielding EA such as O₂ or NO3-, their growth rate increases with the increasing energy content of the OM utilized, i.e., γ_OM (Figure 3). Garayburu-Caruso et al. (2020) showed decreased respiration rates under oxic conditions for more reduced OM (high γ_OM). Based on our framework, we interpreted their observations as low respiration rate coupled with high CUE (thus, high growth rate) when more reduced compounds are decomposed. However, if the catabolism is performed using low energy yielding EA (low values of |−Δ_redG_EA|), the growth rate increases with increasing γ_OM at first, but then decreases to a value close to zero for more reduced OM (Figure 3). The key to understanding this pattern is the variation of Δ_catG_OM and Δ_oxG_OM with γ_OM (Supplementary Figure 2). Δ_oxG_OM (black curve, Supplementary Figure 2) is negative for γ_OM < 1.88 and positive for γ_OM > 1.88; therefore, for γ_OM < 1.88 a low energy yielding EA would result in overall negative Δ_catG_OM (blue or orange curve Supplementary Figure 2A). This means that the catabolic reaction is still feasible, although with lower CUE because of the low magnitude of Δ_catG_OM. For γ_OM > 1.88, Δ_oxG_OM is positive, so that a low energy yielding EA could result in overall positive Δ_catG_OM values representing a non-spontaneous reaction, which means that the catabolic reaction is not feasible and uptake of OM stops. Such a reaction would lead to CUE = 0 in Figure 3. Moreover, for intermediate values of |−Δ_redG_EA| (e.g., EAs such as Fe³, SO42-), the growth rate or CUE decreases with γ_OM above γ_OM > 4.7 (Figure 3) because microbes dissipate Gibbs energy (|Δ_rG_B| from the overall metabolic reaction) faster than it is produced from the catabolic reaction with increasing γ_OM (Supplementary Figure 2, panel C). However, this is not the case when O₂ is the EA as CUE remains a monotonically increasing function of γ_OM. Thus, purely C-limited conditions are only attained when the substrate is reduced and oxygen is the EA; in other conditions, the energy-limitation reduces microbial growth, even though the reaction rates still scale with substrate C content (C-energy co-limitation).

Our results show an inhibition effect of reduced OM (high γ_OM) on growth when catabolism is coupled with low energy yielding EAs such as Fe³⁺ or SO42-, as an outcome of decreased growth rate caused by low CUE (Figures 3, 7). Under energy-limited environments, if microbial growth is not limited by the supply of C, then our framework would predict higher respiration for more reduced compounds caused by low CUE. For example, when sulfate reducing bacteria are grown in batch or chemostat with ethanol, acetate, or lactate as substrates, more hydrogen sulfide (a proxy for respiration) was produced with ethanol compared to the other two substrates (White and Gadd, 1996). This is expected because ethanol produces more electrons compared to acetate and lactate per C mol; therefore, more SO42-is used, resulting in higher respiration. Further, Zheng et al. (2019) showed that as O₂ becomes limiting, growth is mainly controlled by CUE, and respiration remained unchanged since the supply of dissolved organic C did not change during their incubation experiment (Zheng et al., 2019). However, in some natural environments such as marine sediments or deep soil, the respiration rate of reduced OM can decrease under energy-limitation imposed by low energy yielding EA (Jin and Bethke, 2007; LaRowe et al., 2012; Boye et al., 2017; Keiluweit et al., 2017). This decrease in metabolic rates (respiration rate) is explained using a thermodynamic factor that decreases as the energy produced from catabolism decreases. Such natural environments are often limited in the availability of C as well, so the microbial metabolism is restricted to maintenance functions (or basal power requirement). In other words, microbes are under a physiological survival state without significant growth, while in our framework, we focus on respiration processes that are coupled to growth.

Furthermore, Worrall et al. (2018) showed that the Gibbs energy of formation of particulate OM decreases with depth in peatlands, which implies that the Gibbs energy of combustion (assuming O₂ as EA) increases with depth. Worrall et al. (2018) related this observation to the accumulation of OM in peatlands, as OM becomes “thermodynamically inhibited” for microbial uptake and can thus remain in the system. In Figure 3, we showed a similar effect. With increasing depth, oxygen-availability decreases, other EAs become available, and when OM oxidation is coupled with low energy yielding EAs, Gibbs energy produced from catabolism decreases even if OM is composed of labile C such as glucose. As a result of energy-limitation, growth rate and CUE decrease with depth. Therefore, our approach provides an alternative explanation to Worrall et al. (2018) results, in which the thermodynamic factor proposed by Jin and Bethke (2007) was used to explain thermodynamic inhibition.

Understanding the interplay between C- and energy-limitations requires considering all components of bioenergetic regulation of OM decomposition, because the degree of reduction of OM (and thus its chemical nature) controls decomposition together with the coupled half-reaction of the EA reduction. The latter might constrain microbial growth even on energy-rich OM.

The metabolism of NO3- varies across microorganisms and with environmental conditions (Lin and Stewart, 1997; Kraft et al., 2014; Kuypers et al., 2018). For example, CUE and growth rate are decreased if NO3- is used instead of NH4+ as the sole N-source (Stouthamer, 1977; Wray et al., 1996) (Figure 4). Our model explains this decrease with the energetic cost of NO3- reduction, providing a complementary explanation to the inhibition of NO3--uptake in the presence of NH4+(Kobayashi and Ishimoto, 1973; Rice and Tiedje, 1989; Polcyn and Luciński, 2003). Moreover, the transition from N- to energy-limitation (or vice versa) depends on different processes under oxic and anoxic conditions. Under oxic conditions, the energy-limitation caused by a low degree of reduction of the OM alleviates N-limitation when N-availability decreases. In fact, the transition from energy to N-limitation occurs at lower N-availability when γ_OM decreases, because N-demand at low γ_OM is also lower (Figures 4, 6). Thus, under oxic conditions, shifts in N-demand driven by energy availability define the transition from energy- to N-limitation.

Under anoxic conditions, when NO3- is used as both EA and N_Inorg-source for biomass, microorganisms compete for NO3- reduction via denitrification or DNRA pathway. As a result, the N metabolic pathway (e.g., denitrification vs. DNRA), by determining the N-demand for catabolic and anabolic processes, controls how the transition between energy- and N-limitations occurs (Figure 5).

On the one hand, denitrification may cause N-limitation by removing N from the system, whereas DNRA simply reduces NO3- to NH4+. In fact, under N-limited conditions, denitrifiers need to reduce an additional amount of NO3- to NH4+ to meet their N-demand for growth compared to the DNRA pathway, which produces excess NH4+ through catabolism. Therefore, when NO3--availability is low, microbes performing DNRA would outcompete denitrifiers, as DNRA allows higher growth rate than denitrification for all values of γ_OM (Figure 5B, solid vs. dashed curves). On the other hand, denitrification produces more Gibbs energy per electron transferred to oxidation of OM compared to DNRA, which results in overall a higher Gibbs energy of catabolism for 1 C-mol of OM (Supplementary Table 1). Therefore, with abundant NO3-, microbial CUE for denitrification is higher than for the DNRA pathway (Figure 5B, dashed vs. solid black curves). As a result, microbial N-demand is always met, and growth is limited by the energy produced from catabolism; hence, denitrifiers have a higher growth rate than microbes performing DNRA for all values of γ_OM (Figure 5A, dashed vs. solid curves for I_norm = 1.5).

Between these two extreme cases of NO3-poor vs. NO3- rich conditions, the microbial growth rate of two pathways varies depending on γ_OM (Figure 5). At low values of γ_OM, growth is energy-limited and since denitrification produces more Gibbs energy, denitrifiers are likely dominant. In contrast, at high values of γ_OM, N-limitation becomes the controlling factor, so that microbes performing DNRA would dominate. With labile OM, such as glucose or acetate (both with degree of reduction 4), our theory would predict dominance of the DNRA pathway, because of higher growth rate at low nitrate concentration compared to dominance of denitrification pathway at high nitrate concentration (see Figure 5A). This prediction is similar to observations from field, lab or modeling studies (Kraft et al., 2014; van den Berg et al., 2016; Putz et al., 2018). However, most experimental studies ignore the role of OM degree of reduction as C is provided in labile form (e.g., glucose or acetate), and N_Inorg-availability is manipulated by adding NO3- or NH4+ (van den Berg et al., 2016; Putz et al., 2018). Therefore, our results from Figures 4, 5 could be used to generate hypotheses or explain empirical results where the dominance of microbial communities is assessed by altering the quality of added substrates (its degree of reduction) under varying N_Inorg-availability. For example, we predicted higher denitrifier growth rate feeding on oxidized OM, but higher growth rate of microbes performing DNRA feeding on reduced OM under moderate nitrate-availability (Figure 5A, green lines).

When all three limitations—C, N, and energy—are considered, the overall patterns in microbial growth rate remain similar to those described in previous sections. Under oxic conditions, the effects of N-limitation on growth rate are decoupled from those arising under C- and energy-limitations (Figures 6, 7A). Under C-limitation, the growth rate is mainly determined by the CUE, which is controlled by γ_OM, whereas under N-limitation, the growth rate is determined by the N-imbalance, which is controlled by the C:N ratio of OM and N_Inorg-availability. The transition point between C- and N-limitations occurs at progressively higher N-availability as the N-demand increases with more reduced OM. Garayburu-Caruso et al. (2020) showed that observed respiration rates under C excess (high concentration of C-, N-limited) conditions were controlled by N-availability, whereas under C-limited (low concentration of C) conditions, they were controlled by the degree of reduction of OM, indicating bioenergetic regulation of OM decomposition only under C-limitation; however, this is only valid for oxic conditions. When the growth rate is analyzed in the CN_OM−-γ_OM space, contrasting patterns emerged with the type of EA utilized [compare panels (A) and (D) in Figure 7]. For example, the dominating factors, C-, N-, or energy-limitation, controlling the growth rates are switched as the energy content of OM increases (γ_OM). This is explained by noting that when catabolism is coupled with low energy yielding EAs, the overall energy obtained from catabolism is very small; therefore, decreasing the growth rate as CUE is decreased (Figure 3).

We tested how energy-limitation would affect the growth rate, if microbes were to respire more rather than reducing OM-uptake during N-limitation. In fact, under N-limited conditions, the growth rate defined by Eq. (6) assumes that the growth is reduced by decreasing OM-uptake rate through the stoichiometric coefficients for N (Table 2), similar to the “N inhibition” mechanism described by Manzoni and Porporato (2009). This mechanism represents a downregulation of extracellular enzyme production. However, other strategies for microbial growth under N-limitation can have a different effect on the growth rate (Manzoni et al., 2021). One putative mechanism is overflow respiration or exudation of excess C, which suggests that under N-limitation, microbes do not reduce the uptake of OM; instead, they remove extra C by overflow respiration (Sistla et al., 2012; Wild et al., 2014; Wutzler et al., 2017), higher investment in extracellular enzymes, or possibly exuding more. This is mathematically achieved by reducing microbial CUE at a constant uptake rate of OM (Manzoni et al., 2017). We tested whether this mechanism would affect growth rate when varying γ_OM, CN_OM, and availability of different EAs in Supplementary Figures 3, 4. As expected, the variation of growth rate with γ_OM under C-limited conditions remains the same as shown in Figures 6, 7. However, under N-limitation, the growth rate is much higher when overflow respiration is performed, because the supply of N from the OM is not lowered (compare Supplementary Figure 3 vs. Figure 6, and Supplementary Figure 4 vs. Figure 7). This metabolic regulation would thus appear to be “optimal” (Manzoni et al., 2017) because it allows higher growth rate compared to reducing the substrate uptake rate. As CN_OM increases under N-limitation, the CUE itself decreases, so that all growth curves converge to a single curve for a given N_Inorg-availability and regardless of γ_OM.