The model is formulated using Onsager's reciprocal relations (Onsager, 1931b): 1J1=l+cf2X1-cfX2,2J2=-cfX1+c+ϵX2,3TdSedt=J1X1+J2X2. In the above model, l, f, c and ϵ are constant parameters (whose definitions depend on the application), X1 and X2 are forces, J1 is the supply flux into the system under the influence of force X1, and J2 is the output flux with force X2. The sum of products of fluxes and forces contributes to the heat dissipation of the system as measured by the entropy production rate dSe/dt multiplied with temperature T in Eq. (3). Note that Eq. (3) follows directly from the first law of thermodynamics (aka energy conservation), so that given the supply power P1=J1X1, and the heat dissipation rate TdSedt, the useful output power is P2=-J2X2. Consequently, the system's energy use efficiency (η) is 4η=P2P1=1+lcf2-p1-lpcf2-ϵcl+cf2pcf2-1, where p=P1/cfX12. Further, we note that 5p=P1cfX12=1+lcf2-X2fX1<1+lcf2, so that η may vary from being negative to a positive maximum value, depending on the value of terms in [] . Particularly, when η is plotted against p, we obtain the non-monotonic tradeoff curve (Fig. 1).

Odum and Pinkerton (1955) (hereafter OP1955) used the above model to analyze (1) Atwood's machine (e.g., Monteiro et al., 2015), (2) a water wheel turning a grindstone, (3) one battery charging another battery, (4) a thermocouple running an electric motor, (5) a thermal diffusion engine, (6) the metabolism of a pseudo-organism (with no self-repair), (7) food capture by an organism for its maintenance and growth, (8) a model of photosynthesis, (9) primary production in a self-sustaining climax community, and (10) growth and maintenance of a civilization. Among them, the 7th example is closest to the biological growth we discuss here. Moreover, the fact that these equations can describe so many types of systems suggests what can be inferred from these equations for the energy use efficiency is a universal feature that is guaranteed by nonequilibrium thermodynamics. Consequently, the 7th example and the subject of our analysis – biological growth – should have the same characteristics in terms of energy use. Following the interpretation by OP1955, in this example, J1 describes the rate of food supply, X1 corresponds to the energy drop in metabolism of captured food units (thus X1 is equivalent to Gibbs free energy, e.g. that of glucose, representing the energy quality of the substrate taken from the environment), J2 represents the rate of effective food use for biomass synthesis, X2 is the energy drop inherent in the metabolism of a unit of food (thus X2 is analogous to the Gibbs free energy of reserve biomass in the DEB models), l designates the basal metabolism spent in self-repair (as paid by X1 in the original OP1955 formulation), c is related to the effectiveness of food concentrating method, and f equals to X2/X1, measuring the metabolic efficiency of the food capture process. However, considering that biological organisms may keep respiring in the absence of food uptake (aka when J1=0; e.g., Dawes and Ribbons, 1964), it is more reasonable to pay the basal metabolism with X2, so that l=0 while ϵ>0. Nonetheless, this change does not alter the non-monotonic tradeoff between efficiency η and the input power P1 (Fig. 1). Moreover, in the OP1955 model, all input power P1 goes to output power P2 and waste, thus there is no change in storage energy density (aka X2). Therefore, the implied non-monotonic power-yield tradeoff corresponds to exponential biological growth, as discussed below.

We next analyze six biological growth models: (1) Pirt model (Pirt, 1965), (2) Compromise model (Beeftink et al., 1990), (3) Modified Droop (mDroop) model (Thingstad, 1987), (4) Variable Internal Storage (VIS) model modified from Thornley's plant model (Thornley, 1972), (5) the standard dynamic energy budget (sDEB) model (Kooijman, 2009), and (6) the modified dynamic energy budget (mDEB) model (Tang and Riley, 2025). Both the Pirt and Compromise models consider a single biomass pool, and treat growth and maintenance as independent processes, having the same priority in using the substrate. As biological growth in the Pirt model and Compromise model is directly driven by substrate uptake, they are in the category of source-driven growth models (Fatichi et al., 2014). The other four models assume biomass consists of structural and reserve biomass. Among them, the VIS, sDEB and mDEB models drive structural biomass growth using reserve biomass, thus they are in the category of sink-driven models (Fatichi et al., 2014). Following Thingstad (1987), the mDroop model has the most parameters, and uses the total biomass to compute the growth rate (through the total-biomass-based substrate quota, even though it does have reserve biomass), thus it is a sink-driven model only apparently. For the Pirt and Compromise models under general conditions, their equations stay the same as in Table 1. The general forms of the other four models differ from those (for exponential growth) in Table 1 and are detailed (in Sects. S1 to S4) in the Supplement.

We compared the six models' predictions of CUE dynamics for (1) an exponentially growing cell population under constant carbon inputs using equations in Table 1, (2) 50-year simulations using equations from Sect. S5 in the Supplement that mimic microbe-driven soil carbon dynamics under monthly varying but annually constant carbon inputs (with a rate of 350 gC m⁻² yr⁻¹; red line in Fig. S1a), and (3) 50-year simulations that further consider the influence of smoothed daily variable temperatures (red line in Fig. S1c) on related temperature-dependent kinetic parameters. For models used in the 50-year simulations of soil carbon dynamics, we detailed their mathematical formulations in Sect. S5 and parameterizations in Sect. S6 in the Supplement, and gave here only a high-level description. Specifically, each of the six models describes the dynamics of one microbial population that feeds on one fast C pool and one slow C pool. Note that the use of fast and slow pools here simply means the former carbon substrate is kinetically more favorable than the latter to the microbes, and they do not have the same meaning as in microbe-implicit models, which are directly associated with turnover times (Parton et al., 1988). We formulated the substrate uptake based on the argument in (Tang et al., 2024), i.e. a form that resembles the competitive Michaelis-Menten kinetics as simplified from the equilibrium chemistry approximation (ECA) kinetics (Tang and Riley, 2013). In these models, the fast and slow C pools increase due to external input, and all microbial biomass from mortality is added to the slow pool C. Additionally, since the Pirt model, Thornley's model and the sDEB have also been adopted for modeling plant growth (Thornley, 1972; Russo et al., 2022; Thornley, 2011), the assertions we draw below can also be more generally extended to plants.

For an exponentially growing population of biological cells, predictions by these six models can be sorted into three groups: (1) the Pirt and Compromise models predict that (both structural and total biomass) CUE increases asymptotically with increasing specific growth rate; (2) the mDroop model predicts that the total biomass CUE is insensitive to specific growth rate, while the structural biomass CUE decreases with increasing specific growth rate; and (3) the VIS, sDEB, and mDEB models predict that structural biomass CUE first increases then decreases with specific growth rate, while the total biomass CUE increases asymptotically with specific growth rate (Fig. 2). Particularly, the increasing total biomass CUE as a function of specific growth rate aligns well with empirical measurements (e.g., Zheng et al., 2019; Hu et al., 2025; Collado et al., 2014), and the Pirt and Compromise models were specifically created to capture this feature observed in batch experiments (Pirt, 1965; Beeftink et al., 1990). However, the Compromise model is argued to be more realistic than the Pirt model by allowing negative growth resulting from biomass degradation under low to no substrate conditions (Beeftink et al., 1990; Wang and Post, 2012), while the Pirt model forces the growth rate to be zero under the no-supply substrate condition.

Among the six models, predicted relationships between total biomass CUE and specific growth rate can also be organized in three groups (Fig. 3): (a) the Pirt and Compromise models predict essentially identical, almost linearly increasing, relationships (Fig. 3a); (b) the mDroop model predicts a curve that gradually closes on itself (Fig. 3b); and (c) the VIS, sDEB, and mDEB models predict relationships look like ellipses that are slanting slightly up and to the right, and those by the two DEB models are nearly overlapping each other (Fig. 3c). (The predicted soil carbon dynamics are presented in Fig. S2. Even though the temporal variability of fast pool C, slow pool C, and total biomass are different but arguably quite consistent among all six models by our choice to share as many parameters as possible among the models.)

Since the Pirt and Compromise models each have only one biomass pool, their predicted microbes have only one CUE for each specific growth rate (Fig. 3a). In contrast, the representation of reserve biomass makes the other four models predict multiple total biomass CUE values for each specific growth rate (Fig. 3b, c), which gradually converge to a closed curve due to repeating growth pattern (Fig. S2d) driven by the temporal response of reserve biomass to the fixed seasonal pattern of carbon input (Figs. S1b and S2a).

When the structural biomass CUE is analyzed with respect to the specific growth rate (Fig. 4), the mDroop model predicts this relationship to have a left-leaning asymmetric oval shape (Fig. 4a). Meanwhile, the VIS, sDEB, and mDEB models all predict the relationship to be of more vertically aligned asymmetric oval shapes (Fig. 4b, c, d). In all, just like that for the total biomass CUE, these four models predict that there is not a one-to-one relationship between structural biomass CUE and specific growth rate. Rather, the relationship traces in curves that gradually close on themselves as enabled by the temporal response of reserve biomass to the constant temperature and monthly varying but annually constant carbon input (Figs. S1b and S2a).

When the temperature effect on kinetic parameters is further considered (on top of the 50-year monthly variable but annually constant carbon input), we find much richer variation of the relationship between total biomass CUE and specific growth rate. Still, the predicted relationships can be organized into the same three groups: (a) those by the Pirt and Compromise models (Fig. 5a), which look like a fishhook, and no longer overlap, (b) that by the mDroop model (Fig. 5b), which looks like a thread with an open knot, and (c) those by the VIS, sDEB, and mDEB models (Fig. 5c), which look like entangled threads. Among these three groups, the relationship predicted by the mDroop model has the largest spread. In particular, the mDroop model predicts significant occurrence of negative CUE, where excessive maintenance cost leads to the loss of biomass in the presence of active carbon substrate uptake. Meanwhile, predictions by the VIS, sDEB, and mDEB models show a much tighter spread, and, due to the chosen parameter combinations, do not involve negative CUE (Fig. 5c) as also found for the Pirt and Compromise models (Fig. 5a). Nonetheless, by considering temporally varying carbon inputs and temperatures, one specific growth rate can correspond to many values of CUE. However, the curves still gradually close on themselves due to the temporal response of reserve biomass to the fixed seasonal pattern of carbon input and temperatures.

When the relationship between total biomass CUE and temperature is analyzed, the patterns obtained are even more complex (Fig. 6). (Note that temperature was varied daily, and repeated over the 50-year simulation period.) The Pirt and Compromise models both show a pattern of larger spread at low and high temperatures, and a general pattern that first increases and then decreases with temperature (Fig. 6a). The relationships predicted by the VIS, sDEB, and mDEB models have a much tighter pattern and smaller variation (Fig. 6c), a phenomenon that is consistent with observations that biological organisms evolutionarily develop reserve dynamics to stabilize performance (thereby less variable CUE) in the presence of fluctuating environmental conditions (Kooijman, 2009). Predictions by the mDroop model show the largest variation: although the CUE-temperature curve is well-defined, its spread are largest under low temperature, followed by intermediate and high temperatures with a cross over at around 22.5 °C (Fig. 6b). Overall, all models predict that there are multiple CUE values almost at any given temperature.

It has long been recognized that CUE has a strong control on carbon and nutrient cycling through biological organisms and their host systems. However, many existing models treat microbial CUE as either a constant or a function depending statically on its controlling factors (Allison et al., 2010; Wang et al., 2021; He et al., 2015; Wieder et al., 2015). Our analysis here suggests that due to the indirect connection between maintenance respiration and substrate uptake, under time-varying conditions (like the varying carbon input and temperature considered here), there is not likely a static relationship between CUE and its controlling factors or specific growth rate. Note, in systems that couple multiple groups of microorganisms (Graham et al., 2007), we expect that CUE dynamics could even be chaotic. Therefore, depending on the sampling time and location (e.g. the line segments indicated by blue or red arrows in Fig. 4), as well as the temporal averaging method involved in analysis (Tang and Riley, 2015), we may find CUE to have various trends with respect to the controlling variable being investigated (He et al., 2024). While it is not investigated here how much the simulated soil organic matter dynamics will change if the CUE is otherwise parameterized with a static function, based on results from Tang and Riley (2015), we expect the change will be significant. For test, when we drive the models with monthly and annually variable carbon input and non-smoothed daily variable temperatures, the spread of CUE curves becomes much wider (Fig. S5). In all, we contend that CUE should be resolved dynamically by explicitly representing the acquired carbon use for processes other than assimilation. This representation has been conceptually adopted in plant growth models, and soil biogeochemical models should also adopt it to ensure the mechanistic coherency of the overall ecosystem models. However, as we argue below, current plant growth models are generally formulated inappropriately.

In the literature, due to their simple mathematical formulation and relatively good performance, the source-driven Pirt and Compromise models have been used much more frequently than the other four models in modeling microbes and plants. For instance, the land component (ELM) of the Energy Exascale Earth Model (Zhu et al., 2019) represents plant growth in a form that resembles the Compromise model, where new structural growth is driven by the residual gross primary productivity flux after subtracting the carbon requirement for maintenance and storage-replenishment. This source-driven approach is also adopted for plant growth by the BiomE model (Weng et al., 2019) and the FATES model (Knox et al., 2024). Sierra et al. (2022) criticized that this source-driven approach may lead to overly fast plant carbon turnover as indicated by a modeled too young radiocarbon signal in plant respiration compared to observations. Indeed, these source-driven models assume that carbon storage is negligible, so that biological growth is dictated by carbon supply. For plants, this approach contrasts with the existence of nonstructural carbon, which leads to observed phenomenon like delayed growth under nutrient limitation (Li et al., 2021; Boussadia et al., 2010), or coarse root growth during the dormant winter (Marchand et al., 2025). In the relative demand configuration of ELM, in order to match observed nighttime root growth (when carbon supply from photosynthesis is zero), its source-driven approach forces the model to introduce a carbon storage pool that may often (unrealistically) become negative at night and be replenished by new photosynthates in the following daytime (Burrows et al., 2020). When these source-driven models are extended to include nitrogen and phosphorus regulation of plant growth, they may be forced to adopt the law of the minimum (Yang et al., 2014), making predictions that contradict the often-observed multiple nutrients co-limited plant growth (Fay et al., 2015). Additionally, the source-driven models may also be forced to adopt the carbon overflow mechanism under nutrient limitation and forbid luxury nutrient uptake under carbon limitation (Jarrell and Beverly, 1981), both artificially accelerating the carbon cycling. When microbes are also modeled with the source-driven approach, such as in the traditional CENTURY-like models (Koven et al., 2013; Parton et al., 1988), or the more recent MIMICS and Millennial models (Wieder et al., 2014; Abramoff et al., 2018), by induction, they will also predict too-fast soil carbon cycling unless being compensated by physically unrealistic model parameter values.

It has long been advocated that plant growth is better modeled using the sink-driven approach, where photosynthesis product is first stored as nonstructural carbon, which is then mobilized to drive the growth of plant organs (Fatichi et al., 2014; Fourcaud et al., 2008). Our analysis here suggests that the Pirt and Compromise models, although treating growth as driven directly by substrate uptake, are able to produce very rich variation in microbial CUE when analyzed with respect to either specific growth rates or temperatures (Figs. 5a and 6a). Interestingly but not surprisingly, their predicted CUE patterns vary more widely than those predicted by the VIS, sDEB, and mDEB models (Figs. 5c and 6c), where structural biomass growth is sink-driven as fueled by reserve biomass, which is evolutionarily developed by biological organisms to stabilize their metabolic performance in the face of environmental fluctuations (Kooijman, 2009). From the greater CUE variation by the Pirt and Compromise models, we deduce that, due to the missing of buffering effect from the reserve biomass, the source-driven models may very likely overestimate the plant and microbial response to environmental change, such as elevated CO₂, nutrient limitation or warming. (Some support is shown in Figs. S4 and S5, which show that by increasing the variability of carbon input and temperature, the source-driven models respond stronger.) Among the four sink-driven models, the VIS model considers maintenance respiration and growth as two-parallel processes of equal priority, while the sDEB and mDEB models regard maintenance respiration to have priority over growth. Thus, at the cellular level, the VIS model seems to be mechanistically less reasonable by triggering earlier death through maintenance and growth competition. The mDroop model represents biological growth as sink-driven only apparently (because it computes the carbon quota based on total biomass, rather than reserve biomass, to drive the growth), and predicted transient CUE dynamics that are quite different from the three truly sink-driven models (Figs. 5 and 6, and also Figs. S4 and S5). We believe that predictions by the mDroop model are likely to be more easily falsified empirically.

Our analysis indicates that only the sink-driven models (i.e. the VIS model and two DEB models) are able to capture the feature that structural biomass CUE first increases and decreases with specific growth rate as inferred from the synthesis of empirical observations (Lipson, 2015). Nonetheless, when exponential growth is considered, the VIS, sDEB, and mDEB models are found to be more consistent with the Odum-Pinkerton model (Odum and Pinkerton, 1955) that is formulated using non-equilibrium thermodynamics. Our numerical analysis here shows that the VIS, sDEB and mDEB models are equally good (as evidenced by their very similar performance for all cases analyzed here), because maintenance respiration is usually a small fraction of total respiration. As the VIS model was initially proposed to model plant growth (Thornley, 1972), we hypothesize that the sDEB and mDEB models would be comparably good for modeling plant growth. (Some prototype applications of the sDEB model to plant growth are in Russo et al. 2022; Kooijman, 2009.) However, if we consider previous comparisons between the sDEB and mDEB models for microbes (Tang and Riley, 2025), then the mDEB model has better mechanistic coherency with the first law of thermodynamics, and is numerically much easier to incorporate multiple nutrients co-limited growth. Therefore, mDEB model seems to be a better formulation for representing biological growth under general conditions. In particular, the mDEB model will enable the growth of plant shoots and roots to be modeled separately, while the two are modulated with shoot-plant exchange of water, carbon, and nutrients. Although the resultant model may appear to be more sophisticated, counter-intuitively (as we observed in EcoSIM), it will be better constrained by using more directly measurable parameters (e.g. nutrient kinetic parameters, root morphological size and hydraulic resistance) and state and flux variables (e.g. phloem transport rate of carbon and nitrogen, root biomass and morphology profile). For instance, the coarse root dynamics can then be better represented, while the source-driven models forbid coarse root growth in the absence of photosynthesis, a structural deficiency unlikely to be compensated by parameter tuning.