• Mohammadjavad Meghrazi, 
• Sarah P. Otto

x

Abstract

Citation: Meghrazi M, Otto SP (2026) Bridging energy and proteome allocation models allows prediction of the resource cost of traits across environments. PLoS Biol 24(10): e3003995. https://doi.org/10.1371/journal.pbio.3003995

Academic Editor: Claudia Bank, Universität Bern: Universitat Bern, SWITZERLAND

Received: January 15, 2026; Accepted: August 28, 2026; Published: October 13, 2026

Copyright: © 2026 Meghrazi, Otto. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper and its Supporting information files. The code underlying the figures can be found at https://doi.org/10.5281/zenodo.22756291.

Funding: This research was supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant to S.P.O. (RGPIN-2022-03726; website: nserc-crsng.canada.ca). The funding agency played no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

To understand how evolution shapes organismal traits, it is necessary to consider both positive and negative fitness consequences of traits. While positive effects can result from improving a wide range of functions, one of the main sources of negative impacts is the resource cost of traits [1,2]. Due to physiological trade-offs, the diversion of resources to trait construction can reduce the reproduction rate, and this reduction can be defined as the resource cost of traits (hereafter, cost of traits) [2].

Various traits may require the allocation of different resources, making it challenging to convert their costs to a reduction in reproduction rate using a unified framework. For example, resistance to antibiotics in bacteria could be achieved by the production of new proteins (e.g., proteins that degrade the antibiotics) or energy expenditure (e.g., faster activity of pumps that remove antibiotics from the cytoplasm) [3]. Both of these traits are costly, but it is unclear how their costs, i.e., protein production and energy expenditure, can be compared given their different units. Moreover, the cost depends on the environment (e.g., temperature) and genetic background (e.g., temperature tolerance) of the cells in a manner that is not well understood [4]. As less costly traits are more likely to evolve, if we can convert the costs of different traits to a common unit and compare them, we can predict which traits are more likely to evolve, all else being equal. Moreover, understanding how the environment and genetic background affect the cost of traits will help us predict the conditions that facilitate the evolution of various traits, including resistance to stressors (e.g., antibiotics, extreme temperatures, and predation). This can have important implications for the management of the antimicrobial resistance crisis and, more broadly, understanding the evolutionary response of organisms to the increased prevalence of human-induced stressors.

Energy allocation and proteome allocation are the two main frameworks used to study the cost of traits, however, they have different underlying assumptions, and the connection between them is underexplored [5]. According to energy allocation models, the cell divides its limited energy budget between maintenance, growth, and different activities. The traits that directly or indirectly (i.e., by using metabolites) consume energy reduce energy allocation to growth and are thus costly. In this framework, the cost of a trait is thus translated into energy requirements and depends on the fraction of the cell’s energy budget consumed [2,5–7]. This is equivalent to assuming cells are energy-limited, i.e., they cannot increase their energy budget [5], and thus omitting the limitations imposed by the catalytic capacity of enzymes that convert metabolites and energy into biomass, i.e., biosynthetic limitations [8–10]. On the other hand, the proteome allocation framework (also known as ribosomal allocation) considers protein production to be the limiting process in cell reproduction, emphasizing limitations in translation. From this perspective, allocating proteins to a trait decreases the allocation to the protein production machinery, which increases the time required to reproduce the proteome of a cell [9–14]. While proteome allocation models investigate the connections between nutritional and translational limitations [9,10,15,16], their scope remains limited to proteins, and it is unclear how the cost of traits that require other macromolecules or energy can be modeled in this framework. Overall, energy provisioning and translation are specific cases of processing nutrients (provisioning) and converting them to macromolecules (biosynthesis), respectively (Fig 2). As there is ample evidence for limitations in both provisioning and biosynthesis [6,8,9,17], the unification of these two frameworks is necessary for a better understanding of the cost of traits and the constraints under which cells grow and reproduce.

We develop a macromolecule allocation framework that considers provisioning and biosynthesis simultaneously to quantify how each limits cell growth. Using this model, we quantify the cost of allocating different resources to traits and the effect of environmental conditions and genetic background on this cost. Notably, we show that the cost of traits that require macromolecule production is higher in environments favorable for growth. However, the cost of traits that require energy expenditure or a higher degradation rate of harmful proteins depends on the extent to which provisioning and biosynthesis limit growth, which in turn depends on genotype and environment. Factors that increase the speed of provisioning and make it less limiting (e.g., higher food quality) decrease the cost of energy expenditure and increase the cost of degradation, while factors that increase the speed of biosynthesis (e.g., faster ribosomes) have the opposite effect. Moreover, our model makes novel predictions about the effect of degradation, recycling, and energetic demands for maintenance on the relationship between the fraction of ribosomes in the proteome and growth rate (also known as bacterial growth laws; explained in Box 1), particularly during slow growth.

Model

To quantify the rate of reproduction (i.e., biomass production), we develop a three-compartment model similar to Scott and colleagues (2010) (described in Box 1) but with three differences [10]. First, we include all macromolecules in our model rather than just proteins. Second, the speed of the biosynthesis machinery is allowed to vary. Third, we incorporate degradation, recycling, and energetic demands for maintenance.

We divide cell macromolecules into three fractions based on their metabolic role. The macromolecules that uptake food and nutrients and convert them into substrates (i.e., energy stored in different forms, and building blocks of macromolecules such as amino acids, nucleotides, etc.) make up the provisioning fraction (). The macromolecules that synthesize new macromolecules using substrates comprise the biosynthesis fraction (). The remaining macromolecules are grouped into the structural fraction () (Fig 2). While these fractions are structurally similar to the ones defined by Scott and colleagues (2010), we broaden the scope of macromolecules considered in the model beyond proteins to capture cell contents more completely. See more details on the components of each fraction in Fig 2 caption and S1 Appendix.

We assume that the structural fraction is fixed, but the cell can optimize the ratio allocated to provisioning versus biosynthesis to maximize biomass production. Previous studies have suggested molecular mechanisms for top-down regulation of proteome allocation between nutritional and biosynthetic pathways [32,33], which motivates our analysis of the impacts on growth rate.

Three processes affect the flux of substrates supplied to the substrate pool. (1) The provisioning fraction supplies substrates proportional to its mass at the rate , where is the mass of the provisioning machinery and (provisioning efficiency) is the substrate supply rate per unit mass of the provisioning machinery. (2) Maintenance demands consume substrates (mainly in the form of energy) at a constant rate equal to , where is the substrate demand for maintenance per unit mass of cell macromolecules and is the total mass of cell macromolecules. (3) Recycling of degraded macromolecules supplies substrates proportional to the degradation rate and the efficiency of substrate recycling at the rate , where D is the degradation rate per unit of macromolecule mass and c (recycling efficiency) is the fraction of substrates recycled after degradation. Overall, substrates are supplied to the substrate pool at the rate , where . The biosynthesis fraction consumes substrates proportional to the mass and speed of biosynthesis machinery at the rate , where is the mass of biosynthesis machinery, (biosynthesis efficiency) is the rate of substrate consumption per unit mass of biosynthetic machinery when saturated, and is the biosynthesis saturation level, which depends on the steady-state substrate concentration. As the steady-state substrate concentration depends on the biosynthesis fraction, we model the biosynthesis saturation level directly as a function of the biosynthesis fraction rather than the steady-state substrate concentration. Lastly, substrates are allocated to new cells in proportion to the birth rate and the steady-state substrate concentration (dilution due to growth, see S1 Appendix for more details). Table 1 summarizes the notation.

We assume the macromolecule production rate determines the rate of cell reproduction. Therefore, the birth rate (b) equals the net macromolecule production rate per unit of macromolecule mass, which equals the biosynthesis rate minus macromolecule degradation rate divided by the mass of all macromolecules. Assuming biosynthesis rate equals the rate of substrate consumption by the biosynthesis fraction (i.e., no mass lost during biosynthesis), the birth rate equals:

(1)

Assuming the degradation rate is independent of macromolecular allocation, the allocation that maximizes will maximize the birth rate. However, there is a trade-off between the biosynthesis fraction () and biosynthesis saturation level (). As the biosynthesis fraction increases, there are fewer provisioning enzymes to supply each biosynthetic enzyme, and the biosynthesis saturation level decreases at an accelerating rate. Fig 3a shows a graphical representation of this trade-off.

To proceed further, we must specify the nature of the trade-off, . In S1 Appendix, we add substrate dynamics to the model, which allows us to explicitly model the trade-off between generating substrates via provisioning or using them via biosynthesis. In the main text, however, we approximate the trade-off function, which allows us to derive interpretable analytical expressions for the optimal allocation. Specifically, we assume that the biosynthesis saturation level rises linearly with the substrate supply rate () when the supply rate is small compared to the maximum rate of macromolecule production (), but approaches an asymptote of one as substrate supply rate increases:

(2)

(notice that cancels out). When substrates are supplied into the substrate pool at a rate much faster than the maximum rate of macromolecule production (), the ratio of substrates to macromolecules in the cell is high, and the biosynthetic enzymes become saturated (). On the other hand, when provisioning is limiting and the supply of substrates (including energy) is exhausted (), the saturation level of biosynthetic enzymes approaches zero, and biosynthesis halts. Equation (2) is one of the main differences between our model and Scott and colleagues (2010), which assumes the saturation level of ribosomes is constant.

Under some parameter combinations, equation (2) can result in an unrealistically high rate of substrate production by the provisioning fraction, relative to the rate of substrate use by the biosynthesis fraction, which leads to a high concentration of substrates in the cytoplasm. This is, however, an artifact of the approximation used in equation (2). Explicitly tracking substrate concentrations leads to more realistic substrate concentrations (S1 Appendix). Nevertheless, the results obtained are qualitatively similar, except where noted.

By construction, this model has greatly simplified the features of a cell (Fig 2) in order to model growth rate under different conditions. Nevertheless, we highlight several limitations of this simplified model that could profitably be relaxed in future work. First, we hold the structural fraction constant, which may not be accurate, e.g., if altering the allocation between provisioning and biosynthesis alters cell size or shape in a manner that affects the surface area to volume ratio [35–37]. Second, we treat the stoichiometric composition of the provisioning, biosynthesis, and structural fractions as constant and similar, whereas their stoichiometric composition may vary in a manner that affects the optimal allocation. For example, if an element that is disproportionately found in the biosynthesis fraction, such as phosphorous, is scarce, the cell benefits less from allocating macromolecules to the biosynthesis fraction because the scarcity of that element ultimately limits the return on this investment [38] (discussed in greater detail in S1 Appendix). Finally, we assume that the cells are exponentially growing under constant environmental conditions. As environmental fluctuations can affect the optimal allocation [39–41], our results should be treated with caution in changing environments.

Results

Discussion

The trade-off between allocation to biosynthesis and provisioning is a universal optimization problem all organisms face. By modeling this trade-off generally, the model we explore transitions smoothly between different factors limiting growth. This model can predict the fitness surface, i.e., the birth rate under different allocations (Fig 3), and the growth of an organism not at the optimum, as well as predicting optimal allocation patterns (Figs 4 and 5). Using this model, we predict the effect of environmental change and genetic background on the cost of various traits (Fig 4). Notably, we predict that the cost of traits that involve the production of a new macromolecule depends on the overall speed of provisioning and biosynthesis (point I). On the other hand, the cost of traits that require energy allocation (point II) or increased degradation (point III) depends on how limiting provisioning and biosynthesis are. Furthermore, we showed that the energetic demands of maintenance, degradation, and recycling affect the shape of bacterial growth laws, particularly during slow growth (Fig 5). Overall, we argue that combining provisioning and biosynthetic limitations provides a broader framework to think about the costs of traits than just focusing on energy or proteome allocation.

We expand proteome allocation models to incorporate all macromolecules rather than just proteins, and suggest that the relationship between the ribosome fraction and growth rate is a specific case of the more general relationship between the biosynthesis fraction and growth rate. This agrees with previous findings that other biosynthetic enzymes, such as RNA polymerases and cell envelope biosynthesis enzymes, increase as growth accelerates [17,21,22,34,36,42]). We thus demonstrate that the bacterial growth laws can be explained without assuming that protein production or any other specific process is the limiting step in growth, suggesting such patterns could be prevalent across a wide range of biological systems. Our work thus complements previous studies that investigated the limitations imposed by transcription, cell envelope synthesis, and other processes on growth [17,36,43], without making the assumption that protein production is the main limiting step in growth [30,10,13–15,30]. Overall, by incorporating all macromolecules into the proteome allocation theory, our work facilitates further synthesis between this theory and other theories of growth that are not limited to proteins, such as the metabolic theory of ecology (which is related to the energy allocation framework) [7,44–47]. Moreover, incorporating explicitly how the optimal allocation depends on the availability of particular nutrients (e.g., nitrogen and phosphorous) might allow further synthesis with ecological stoichiometry theory, which is a promising direction for future studies [48–50].

Our model shows that a framework focusing only on energy allocation will overestimate the cost of traits unless growth is limited primarily by resource provisioning (). Although some energy allocation models account for the opportunity cost of metabolites used for biomass production, they assume the total energy (or carbon) budget is limited [5], which is equivalent to assuming provisioning is limiting. If, instead, the rate of biosynthesis limits cell reproduction (), then the energetic cost of a trait does not proportionately affect the rate of reproduction. For instance, in E.coli and yeast, ribosome-associated proteins comprise between 10% (slow growth) to 40% (fast growth) of the proteome, while the allocation to glycolysis and energy production varies between 10% (fast growth) to 30% (slow growth) [22,23]. Notable allocation to ribosomal proteins suggests a high level of biosynthesis limitation, particularly during fast growth (or biosynthesis inhibition). To clarify this point, consider a biosynthesis-limited cell () that allocates half of its substrates (or energy) to a trait. According to the energy allocation framework, this trait should halve the cell growth rate [2,5,6], however, our model predicts a smaller cost. This is because the cell can increase its smaller provisioning fraction to supply the trait, with minimal impact on the biosynthesis fraction, which limits growth rate. Therefore, it is necessary to determine to what extent provisioning limits growth (e.g., using the RNA-to-Protein ratio or the fraction of ribosomes) when using the energy-allocation framework to determine the cost of traits. Nevertheless, our model predicts that when the degradation rate is low, the cost of traits scales linearly with their energy or macromolecule requirements (i.e., doubling the energy requirement doubles the cost), in agreement with energy allocation models [2,6,7].

While our model provides a unified framework to understand the costs of different traits and patterns observed (the bacterial growth laws) when growing rapidly, our model does not fully reproduce the shape of growth laws during slow growth (compare curvatures of Figs 1a and 5a). This discrepancy suggests that some of our assumptions might be less accurate during slow growth (e.g., constant structural fraction and similar stoichiometric composition of different fractions) or that additional processes are playing an important role. For example, it has been suggested that the degradation rate increases during slow growth, which could explain the concave-up shape of the growth laws during slow growth [29](but a more recent study disagrees [51]). Alternatively, limitations in cell surface area could play a role [52,53], or cells may regulate the ribosome fractions in a manner that is non-optimal [33]. It has also been suggested that cells have evolved to grow in fluctuating environments and maintain more ribosomes than needed during starvation, which is beneficial upon increased food availability [19,23,39,43]. Our model supports the last explanation by showing that when food quality is very low, having a higher than optimal biosynthesis fraction results in a relatively small reduction in birth rate (i.e., the fitness surface is asymmetric; Fig 3c). Moreover, the regulation of gene expression is noisy in bacteria, and there is considerable heterogeneity in the allocation among individual cells [54]. As the fitness surface is assymetric (Fig 3c), selection acting on this heterogeneity over time or among cells may have resulted in a higher-than-optimal average biosynthesis fraction in the population [55,56]. Nevertheless, a recent study has argued that previous empirical studies might have overestimated the ribosome fraction during slow growth by not giving cells enough time to reach steady state before measurement [14], casting doubt on whether growth laws are concave-up during slow growth.

Our model can predict the effect of environmental quality on the type of traits more likely to evolve (points I–IV, Fig 4). All else being equal, a trait is more likely to evolve under conditions where it is less costly. Our model predicts that the cost of structural traits that require macromolecule allocation is the lowest in low-quality environments, where growth is already slow due to low food quality or slow biosynthesis (point I). On the other hand, the cost of energy allocation (which increases maintenance demands) is lowest when biosynthesis is limiting (due to high food quality or slow biosynthesis, point II). Alternatively, the cost of degradation is minimal when recycling is efficient and provisioning is limiting (due to low food quality or fast biosynthesis, point III). Lastly, the cost of slowing down provisioning (or biosynthesis) is the least when it is not limiting and the structural fraction is large (point IV). Furthermore, our model can make qualitative predictions about the likelihood of stress tolerance evolution in a specific environment based on the types of possible mutations. For example, if becoming resistant to a stressor is only possible by traits that require energy allocation, we predict resistance to evolve more readily in environments where food quality is high or biosynthesis is inhibited.

Our work highlights that to understand the effect of environmental quality on the cost of traits, multiple dimensions of environmental quality should be considered simultaneously. For example, although decreasing food quality (i.e., provisioning efficiency) increases the cost of energy-consuming traits, inhibiting biosynthesis (e.g., by antibiotics) has the opposite effect (point II). The same applies to the effect of genetic background on the cost of traits. A recent study did not find any meaningful relationship between the cost of traits and environmental quality, which may be because the environments differed in many factors with confounding effects on provisioning and biosynthesis efficiency [4]. We recommend that future studies determine to what degree provisioning and biosynthesis are limiting growth in different environments (using RNA-to-protein ratio as a potential proxy), and compare the cost of traits in similar environments that only differ in provisioning or biosynthesis efficiency. While these efficiencies are abstract quantities that cannot be directly measured, they can be inferred from the slope of the relationship between the biosynthesis fraction and the birth rate during fast growth (e.g., a slope of in Fig 5a).

Biomass production is a fundamental process required for the growth and reproduction of all living organisms. Our model may thus shed light on the trade-off between allocation to provisioning, biosynthesis, and structure at other levels of organization. For example, the prevalence of structural adaptations in desert-living plants (e.g., thorns) might be partially explained by the lower cost of structural traits in low-quality environments (or slow life-histories) for growth (point I). Wherever biomass production is maximized, insights from this model might be applicable.

Acknowledgments

We would like to thank David Anderson, Dan Kehila, Mary O’Connor, Steven Frank, Matthew Scott, Penelope Kahn, Anna Bazzicalupo, Prompt Suathim, Eully Ao, Alireza Tafreshi, Caleigh Charlabois, Gabriel Greenberg-Pines, William Ou, Jawad Sakarchi, Christopher Klausmeier, Christian Landry, Rees Kassen, Katie Marshall, Amy Forsythe, Asad Hassan, Bryn Whiley, Kaleigh Davis, Keila Stark, Nicola Love, Erga Peter, Nicole Knight, Fruin Pow, and many members of the Biodiversity Research Center for discussions and feedback.

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