Models and molecular mechanisms for trade‐offs in the context of metabolism

Accumulating evidence for trade‐offs involving metabolic traits has demonstrated their importance in the evolution of organisms. Metabolic models with different levels of complexity have already been considered when investigating mechanisms that explain various metabolic trade‐offs. Here we provide a systematic review of modelling approaches that have been used to study and explain trade‐offs between: (i) the kinetic properties of individual enzymes, (ii) rates of metabolic reactions, (iii) the rate and yield of metabolic pathways and networks, (iv) different metabolic objectives in single organisms and in metabolic communities, and (v) metabolic concentrations. In providing insights into the mechanisms underlying these five types of metabolic trade‐offs obtained from constraint‐based metabolic modelling, we emphasize the relationship of metabolic trade‐offs to the classical black box Y‐model that provides a conceptual explanation for resource acquisition–allocation trade‐offs. In addition, we identify several pressing concerns and offer a perspective for future research in the identification and manipulation of metabolic trade‐offs by relying on the toolbox provided by constraint‐based metabolic modelling for single organisms and microbial communities.


| INTRODUC TI ON
Biological systems are constrained by the environment, specified by the abiotic (i.e., availability of nutrients, temperature) and biotic factors.The exchange and transformation of nutrients into building blocks to sustain different biological functions is also constrained by physicochemical laws.These physicochemical laws apply at different scales and govern the properties of individual biochemical components (e.g., enzymes) and biochemical pathways as well as the functions of cellular networks (e.g., gene regulation, protein-protein interactions and metabolism), performance of organisms and their interactions into communities.The interplay between the factors imposed by the environment and physicochemical laws is at the core of trade-off situations, whereby one trait cannot be improved without compromising others.As a result, trade-offs impose limits on feasible trait values and can determine evolutionary paths and constraints, emphasizing the importance of measuring trade-offs and understanding their genetic and molecular underpinnings.
Depending on the experimental designs in which traits are scored, trade-offs can be identified by negative correlations (e.g., genotypic or phenotypic; Agrawal et al., 2010;Reznick, 1985;Roff & Fairbairn, 2007;Stearns, 1989).Moreover, trade-offs can be measured: (i) in a single individual, based on phenotypic correlations between trait values across multiple environments, or (ii) in a population of individuals, based on genetic correlations between traits measured in a single or multiple environments (for a systematic review and classification, see Laitinen & Nikoloski, 2022).Observed genetic correlations can be explained either by antagonistic pleiotropy, whereby a single gene controls two or more traits in opposing directions, or by linkage disequilibrium, where the genes controlling two traits in opposite directions are linked and inherited together due to proximity on the genome (Dingemanse & Dochtermann, 2013).Both of these explanations correspond to the genetic mechanisms underlying trade-offs, detected by negative genetic correlations.
Yet, the identification of molecular and cellular mechanisms that can explain negative phenotypic correlations remains underexplored, probably due to the different ways to measure trade-offs.In addition, there are gaps in our ability to predict the impact of different environments in the attenuation or strengthening of trade-offs.It has already been recognized that models of trade-offs that rely on the integration of cellular networks can provide the means to address these challenges (Ekkers et al., 2022;Mauro & Ghalambor, 2020;Stearns, 1989).To this end, metabolic modelling studies have already begun to address the problems of identifying, and explaining, metabolic trade-offs (Pfeiffer et al., 2001;Schuetz et al., 2012).
Metabolic networks consist of all biochemical reactions that contribute to the uptake and secretion of metabolites in and out of cells along with their transformations into building blocks of cellular structures to perform all functions sustaining life.Recent developments in the constraint-based metabolic modelling framework have contributed approaches to predict the (re)distribution and utilization of resources under different environments (Kerkhoven, 2022;Palsson & Yurkovich, 2022), facilitating mechanistic studies of trade-offs.This framework can readily consider physicochemical constraints related to molecular crowding, membrane enzyme occupancy and allocation of available resources (e.g., enzymes; Basan et al., 2015;Beg et al., 2007;Mori et al., 2016).As a result, the framework helps provide mechanistic explanation for three types of tradeoffs: trade-offs that result from allocation of resources, trade-offs in mortality due to duration of resource acquisition and trade-offs due to specialization to a particular environment (Angilletta et al., 2003).
Here, we provide a systematic review of studies that have used metabolic models to determine and study molecular mechanisms underlying trade-offs observed for metabolic traits at different organizational levels, from individual reactions and single pathways to entire networks (i.e., organisms) and interactions between them.
As a result, we provide a first classification of metabolic trade-offs along with the metabolic modelling approaches used.In doing so, our goal is also to investigate the extent to which the classical Ymodel model of trade-offs (van Noordwijk & de Jong, 1986) (re) emerges as an explanation of metabolic trade-offs.We also discuss the possibility of using metabolic models to predict and manipulate metabolic trade-offs.

| THE Y-MODEL E XPL AIN S RE SOURCE ACQUIS ITION-ALLOC ATION TR ADE-OFFS
Trade-offs inferred from data on traits pave the way for identifying the mechanisms and developing models that can explain tradeoffs.The so-called Y-model of van Noordwijk and de Jong ( 1986) is a simple mathematical model that links trade-offs between traits to resource acquisition and resource allocation.The Y-model postulates that two (competing) traits, X 1 and X 2 , are full determined by the proportions, b and (1 − b), of the common resource, T, allocated to each of them, respectively (Figure 1).More specifically, Roff and Fairbairn (2007), when the traits are scored in a population of individuals, the covariance between the traits is given by where b denotes the mean proportion of resource allocated to trait X 1 and T denotes the mean resource.We note that the same derivation holds for determining the covariance over different environments in a single individual.Equation (1) shows that the sign of the covariance between the two traits depends not only on the variance and mean value of the acquisition, T, but also on the variance and mean value of the allocation, b.Further, with acquisition that is fixed, i.e., var(T) = 0, due to the completion of the two traits for the common resource, the Y-model results in negative phenotypic correlation (of −1) for the two traits in trade-off (Figure 1a).This situation leads to so-called absolute trade-offs (Hashemi et al., 2021).An example of an absolute tradeoff is that involving cell growth and lipid accumulation in microalgae (Mulgund, 2022).However, in the case when the acquisition of the common resource changes, i.e., var(T) ≠ 0, the two traits can exhibit negative as well as positive phenotypic correlations, characterizing a relative trade-off (Hashemi et al., 2022;Figure 1b).A typical example of a relative trade-off arises in allocation of total carbon assimilated by photosynthesis in microalgae and plants, which can be considered as the common resource, T, that varies with the environment; this The Y-model for resource acquisition-allocation trade-offs.Two scenarios with respect to a common resource, T, partitioned between two traits are possible: (a) variance of T is zero, whereby the traits show an absolute trade-off for any sample of points considered; (b) variance of T is positive, in which case, depending on the points sampled, both negative (blue line) and non-negative (red line) phenotypic correlations can be found.

| PRIN CIPLE S OF ME TABOLI C MODELLING WITH RE SOURCE ALLOC ATION CON S TR AINTS
Metabolism represents the entirety of biochemical reactions through which nutrients are imported from the environment and are transformed into building blocks to support growth, defence, reproduction and, ultimately, survival (Stitt et al., 2010).A quantitative property of a biochemical reaction is its rate or flux, denoting the speed at which it transforms the substrates into products (McMurry & Fay, 2015).In the simplest case, where a reaction is catalysed by a single enzyme, the flux depends on: (i) the abundance, E, of the enzyme, (ii) the turnover number or catalytic efficiency, k cat , of the enzyme, denoting the number of substrate molecules that each active site of the enzyme converts to product molecules per unit time, and (iii) the concentration of metabolites, x, acting as substrates and/ or effectors (e.g., allosteric regulators, inhibitors).The flux of a metabolic reaction can then be described as: where k denotes a vector of parameters (e.g., Michaelis-Menten constants, K m , denoting the concentration of the substrate at which the maximum rate of reaction is halved [all other concentrations maintained constant], equilibrium constants, K eq ), and (k, c) is a function that models the effect of metabolite concentration on the flux.
Metabolic reactions do not operate in isolation but jointly affect the temporal change of metabolite concentrations.A metabolic reaction can be described by the stoichiometry of its substrates and products; these are gathered in a vector with negative entries denoting the molarity of substrates and positive entries the molarity of products.Collecting the reaction vectors from a given metabolic network yields a stoichiometric matrix, denoted by N, with as many rows as there are metabolites and as many columns as there are reactions (Figure 2a).The change of metabolite concentrations over time can then be modelled as dc dt = Nv, where v gathers the fluxes of all reactions in the modelled metabolic network.Reactions can be divided into exchange and intracellular based on whether they metabolic modelling framework.The constraint-based modelling framework requires a stoichiometric matrix N as input, comprising the stoichiometry with which metabolites enter as substrates or products of modelled reactions.(a) A toy metabolic network with four metabolites and five reactions is presented, along with its stoichiometric matrix.The panel includes the mathematical description of the flux of reaction r 2 in terms of Michaelis-Menten kinetics with parameters k cat,2 and K m .(b) The core of the constraint-based modelling framework is given by flux balance analysis, whereby the flux through the biomass reaction, v bio , modelling growth is optimized under steady-state constraints, flux capacities and other constraints (depending on the "flavour" of the extension).(c) Consideration of protein constraints that require integration of turnover numbers, k ij cat , for every reaction and enzyme pair.The resulting protein-constrained models can provide predictions about enzyme usage.(d) Integration of translation and transcription constraints, allowing for varying biomass reactions, leads to different variants of resource allocation models.et al., 2002).However, intracellular fluxes are more difficult to quantify, and require setting up isotope labelling experiments and measurement of metabolite labelling patterns, which are then fitted to a metabolic model (Antoniewicz, 2015;Basler et al., 2018), using atom mappings that trace the transfer or atoms between molecules in a reaction (Huß et al., 2022).Therefore, isotope labelling experiments are currently too laborious to allow estimation of fluxes in a population of individuals from a given species, rendering it infeasible to dissect trade-offs at the level of reaction and pathway activities (i.e., fluxes).
As a result, other computational approaches have been developed to predict fluxes in the constraint-based modelling framework based on the assumption that an organism optimizes a cellular task (e.g., biomass growth or yield) under a set of physicochemical constraints (Figure 2b;O'Brien et al., 2015).This is the essence of flux balance analysis (FBA), which provides an efficient means to predict fluxes and specific growth rates (Orth et al., 2010).To this end, FBA requires that an artificial reaction is included in the stoichiometric matrix N; this so-called biomass reaction is parameterized by the biomass composition and specifies the amount of biomass components drawn to create 1 g dry weight (see r bio in Figure 2a).Maximizing flux through this artificial reaction with certain available inputs then corresponds to the maximum specific growth rate supported by the network in particular growth conditions.As a result, the physicochemical constraints can be divided into hard, involving mass-and charge-balancing as well as reaction (ir)reversibility due to thermodynamic laws, and soft, including specification of a biomass-reaction composition and compound uptake/excretion rates that can change under different growth conditions (Feist & Palsson, 2010).The same constraints can be used to determine the maximum rate of production of a particular compound from a specified nutrient, from where product yield can be calculated.
Since fluxes depend on enzyme abundances, see Equation ( 2), the classical FBA approach has been extended to include constraints that bound the flux through a reaction by available enzyme abundances and catalytic rates of reactions under the constraints of a total enzyme (protein) content (Figure 2c; Adadi et al., 2012;Beg et al., 2007;Bekiaris & Klamt, 2020;Domenzain et al., 2022;Sánchez et al., 2017;Wendering & Nikoloski, 2022).This has led to the development of so-called enzyme-constrained models for several model organisms (for review see de Becker et al., 2022).
Expanding further on these approaches to also consider the synthesis of proteins, with the translation apparatus (i.e., ribosomes), protein degradation, DNA replication and RNA transcription, has led to three approaches: the so-called resource allocation models (Goelzer et al., 2011), metabolism and macromolecular expression models (Waldherr et al., 2015), and dynamic enzyme-cost FBA (Figure 2d; Waldherr et al., 2015).Interestingly, like coarser versions (Mori et al., 2016), these approaches include assumptions about maintaining the concentration/content for some of the modelled components fixed (i.e., constant).

| MOLECUL AR MECHANIS MS OF ME TABOLI C TR ADE-OFFS AT D IFFERENT SC ALE S
Metabolic networks provide a rich structure to study trade-offs.
Metabolic trade-offs can occur at different scales that exist or emerge at the level of individual enzymes, subnetwork of reactions or the network as a whole (Figure 3).In the following, we classify the metabolic trade-offs based on the traits involved, namely: (i) kinetic properties of individual enzymes (Figure 3a), (ii) rates of metabolic reactions (Figure 3b), (iii) rate and yield of metabolic pathways and networks (Figure 3c), (iv) different metabolic objectives in single organisms and in metabolic communities (Figure 3d), and (v) metabolic concentrations (Figure 4).We also highlight examples of metabolic trade-offs at these different levels and establish connections with the Y-model, discussed in Section 2, above.

| Metabolic trade-offs between the kinetic properties of individual enzymes
Enzymes can be described by several kinetic parameters and functions thereof, including: the turnover number, k cat , apparent binding constant, K M , denoting substrate affinity, apparent second-order rate, , and substrate specificity (also called accuracy) given by the ratio between the rates of the primary (P) and alternative (A) substrate, i.e., (see Section 3 for definitions of k cat and K M ).Since natural selection and physicochemical laws constrain these kinetic parameters, several trade-offs have been postulated and studied for the properties of individual enzymes.These include the catalytic rate-substrate affinity, catalytic rate-substrate specificity, accuracy-affinity trade-offs as well as the trade-off between catalytic rates of multifunctional enzymes and the enzyme stabilityactivity trade-off.
Trade-offs between the enzyme parameters can be explained by: (i) coupling of the parameters due to constraints as proposed in the Y-model, whereby knowledge of one parameter fully determines the other, or (ii) selection that improves the properties closer to the limits imposed by the kinetic mechanism, forcing negative correlations between the parameters (Flamholz et al., 2019).We will illustrate some of these trade-offs with the example of the most abundant enzyme in the biosphere and one of the best-characterized enzymes, RuBisCO, which catalyses the carboxylation and oxygenation of ribulose-1,5-bisphosphate (RuBP).
The catalytic rate-substrate affinity trade-off indicates that enzymes with higher catalytic rate, k cat , have smaller substrate affinity K M (Figure 3a; Nakamura et al., 2014).It has been suggested that the trade-off between the catalytic rate of RuBisCO carboxylation and affinity for CO 2 leads to trade-offs (quantified by negative correlations) between catalytic rate and substrate specificity (Tcherkez et al., 2006).However, this explanation for the catalytic rate-substrate specificity trade-off fully neglects the RuBisCO oxygenase reaction and the possible coupling between the two RuBisCO functions that compete for the same substrate.In fact, modelling has shown that constraints embedded in the catalytic mechanism enforce trade-offs between CO 2 specificity and the maximum carboxylation rate (Flamholz et al., 2019).However, by using data from more photosynthesizing organisms (Flamholz et al., 2019) as well as accounting for phylogenetic relationships (Bouvier et al., 2021), the claim about this trade-off in RuBisCO parameters has been considerably weakened.Finally, it seems that the new evidence does not support the suspected catalytic rate-substrate affinity trade-off (see figure 4 in Flamholz et al., 2019).
An accuracy-rate trade-off refers to the scenario in which increased enzyme accuracy leads to lower enzyme rate.Tawfik (2014) proposed two mechanisms of discrimination (within the primary active site of an enzyme) to explain the accuracy-rate trade-off, namely ground-state discrimination, which affects K M , and transition-state discrimination, which affects k cat .A better understanding of the accuracy-rate trade-off can help us the limits in engineering enzymes with improved properties.function (Kaltenbach et al., 2016).This has been supported by the observed orders of magnitude difference in rates between the primary and side reactions of promiscuous enzymes.However, a recent study has found evidence against this assumption, suggesting that the presence of weak trade-offs in rates of promiscuous enzymes give rise to generalist enzymes (Küken et al., 2021(Küken et al., , 2022)).

| Trade-offs between rates of metabolic reactions
Trade-offs can also arise between the rates (i.e., fluxes) of metabolic reactions in a network.Trade-offs among fluxes of metabolic reactions arise due to the structure of metabolic networks, whereby metabolites participate in multiple reactions and enzymes can catalyse several reactions, leading to competition for these metabolic resources.One way to determine fluxes that are in trade-off employing the FBA modelling framework (see above, Figure 2b) is to use flux sampling by imposing realistic constraints on growth and nutrient uptake (Haraldsdóttir et al., 2017).The samples of fluxes can be used to identify pairs of reactions that show negative correlations that then indicate the presence of trade-offs (Figure 3b).However, this approach cannot be easily expanded to identify trade-offs that involve more than two reactions.
In an attempt to address this problem, recently a constraintbased approach has been proposed to identify all absolute tradeoffs between steady-state reaction rates (Hashemi et al., 2021).
It relies on the observation that imposing constraints on nutrient uptake may force some fluxes to be robust (i.e., be constant) irrespective of how the rest of the environmental factors change.Such robust fluxes can be readily identified by flux variability analysis (Mahadevan & Schilling, 2003).More specifically, the approach extends the Y-model by identifying non-negative linear combinations of steady-state reaction rates that amount to the rate of a reaction of fixed flux.This approach has been further extended to enumerate all relative trade-offs between steady-state reaction rates that amount to biomass growth (Hashemi et al., 2022).For instance, for the metabolic network in Figure 4a, it must hold at steady state that dc 2 dt = v 1 + v 6 − v 2 = 0, indicating that the presence of a relative trade-off between reactions r 1 and r 6 or an absolute trade-off if the flux of reaction r 2 is robust (Figure 4b).
We note that the identification of the trade-offs with the two constraint-based approaches allows insights in the space of alternative flux distributions and relationships of more than two fluxes in this space.In addition, due to the specification of trade-offs as nonnegative linear combinations of fluxes that amount to an objective (e.g., biomass growth), these approaches have also pointed out a relationship between inherent trade-offs and metabolic engineering strategies aimed at increasing the objective (Hashemi et al., 2022).This follows from the observation that an increase in the flux for any of the reactions in a relative trade-off must increase the flux of the reaction that determines the trade-off (Figure 4b).

| Trade-off between rate and yield of metabolic pathways
All approaches discussed in the previous section are formulated in the context of FBA.However, as noted in the enzyme-constrained and resource allocation models (Figure 2c,d), each flux is (upper) bounded by the product of enzyme abundance and turnover number.Since total enzyme content is bounded (or even fixed, as the resource in the Y-model), trade-offs between fluxes can arise due to the strategies used for enzyme allocation.
One prominent metabolic trade-off is between the rate and yield of a metabolic pathway (e.g., glycolysis, tricarboxylic acid [TCA] cycle) that synthesizes a product of interest.The rate is defined as the moles of product generated per unit of time, while the yield is defined as moles of product per mole of substrate used.For heterotrophic organisms, there is a robust trade-off between adenosine triphosphate (ATP) rate and yield solely due to thermodynamic constraints, since the free energy difference between substrate and product, seen as the resource in the Y-model, can be partitioned between ATP production and ATP usage to drive the reactions (Pfeiffer et al., 2001).
In the case when the product can be synthesized by multiple pathways, with alternative rates and yields, the question arises: Are pathways of higher rate but lower yield preferred over those of lower rate but higher yield in optimizing the growth rate of the biological system?Trade-offs between growth rate and pathway yield indicate that fast-growing cells employ low-yield metabolic pathways (Pfeiffer et al., 2001).Indeed, in fast-growing cells, including bacteria, fungi, and mammalian cells, such a trade-off underlies overflow metabolism (so-called Warburg effect) referring to the seemingly wasteful strategy in which cells use fermentation instead of the more efficient respiration to generate energy (Basan et al., 2015); similarly, in Saccharomyces cerevisiae, the crabtree effect (de Deken, 1966;Herbert Grace Crabtree, 1929) denotes the trade-off between ethanol production under aerobic conditions at high external glucose concentrations, instead of producing biomass via the TCA cycle.
Coarse-grained modelling has shown that the trade-off between growth rate and yield reflects a trade-off between metabolically (substrate) efficient pathways, which produce more product for each substrate molecule but with higher demand for enzymes, and enzyme efficient pathways, which produce less product for each substrate molecule but have higher catalytic efficiency (Molenaar et al., 2009;Figure 3c).This trade-off has been explained by: (i) competition for membrane space between substrate transporter and respiratory chain, and (ii) allocation of proteins, which can be partitioned into sectors whose content scales linearly (positively or negatively) with growth (Hui et al., 2015).The latter has led to a coarse-grained metabolic model (Basan et al., 2015) and a refined resource allocation model (Mori et al., 2016(Mori et al., , 2019) ) to explain overflow metabolism in Escherichia coli.The crabtree effect in S. cerevisiae has similarly been investigated using enzyme-constrained models to identify enzymes with greatest control on the metabolic trade-off (Nilsson & Nielsen, 2016).
The growth rate-yield trade-off has also been studied using elementary flux modes in combination with kinetic models to determine enzyme abundances (Wortel et al., 2018).Their study showed that the growth rate-yield trade-off is environment-dependent.
Along these lines, a negative correlation between growth rate and yield has been observed for high growth rates, but positive correlations are found for low growth rate-resulting in a bell-shaped curve for the growth rate-yield relationship (Lipson, 2015).The curve has been recently explained by considering another trade-off, between glucose uptake and yield, arising in the context of a resource allocation model (Basan et al., 2015).Lastly, experiments have shown that within-population diversity provides a sensitive test to check the existence of the growth rate-yield trade-off (Novak et al., 2006).

| Trade-offs between multiple metabolic objectives
The trade-offs discussed in the sections above deal with models in which a single objective-specific growth rate (i.e., biomass growth)-is assumed to be optimized.However, depending on the evolutionary pressure experienced by an organism growing in specific environments, biomass growth may not be the only objective optimized.This holds particularly for communities of uni-and multicellular organisms (e.g., interactions between above-and belowground organs in plants; Budinich et al., 2017;Diener et al., 2020).
Optimization of multiple objectives (under certain constrains) is related to the concept of the Pareto frontier (or surface), whereby an increase in the value of one objective for any point on the frontier results in a decrease of the value of any other objective (Nagrath et al., 2007; Figure 3d).The Pareto frontier in the context of FBA can be readily determined by well-established approaches for multiobjective linear programming (Nagrath et al., 2010;Oh et al., 2009), and has been applied in the design of organisms with specific production capabilities (Byrne et al., 2012;Sendín et al., 2010).By comparing fluxes from multiple experiments in E. coli with the outcome of optimizing 54 cellular objectives, constraint-based modelling has been used to show that these fluxes are most congruent with optimization of three objectives, namely: ATP yield, biomass yield and sum of fluxes.The distance of the experimentally determined fluxes from the Pareto surface was used as a measure of congruence.This approach demonstrated that cellular fluxes are governed by the trade-off between optimality under a given condition and minimal adjustments between conditions (Schuetz et al., 2012).A recent study has uncovered a trade-off between growth rate and adaptability in E. coli, Bacillus subtilis and Saccharomyces cerevisiae (Basan et al., 2020).A similar study was conducted for Rhodopseudomonas palustris, a Gram-negative purple nonsulphur bacterium capable of growing phototrophically on various carbon sources, by testing eight metabolic functions.The results demonstrated that the optimization of carbon efficiency, biomass and ATP production govern the flux profiles under photoautotrophic conditions (Navid et al., 2019).Approaches based on multi-objective analyses have also been extended to study changes in metabolite concentrations by using the sensitivity of objective functions to the change of steady-state constraints for particular metabolites (Sajitz-Hermstein & Nikoloski, 2016).
Trade-offs are particularly relevant when modelling the growth of microbial communities (Larsen et al., 2012;Figure 3d).Three strategies have been used to address this problem: (i) creation of a community-wide objective function that is to be optimized (with or without consideration of regularization; Diener et al., 2020), assuming that all microbes grow by the same rate (following the balanced growth hypothesis; Bekiaris & Klamt, 2020); (ii) construction of a Pareto frontier by considering the objectives of all individual microbes simultaneously (Budinich et al., 2017); or (iii) multilevel extension that has facilitated the consideration of multiple objectives to model the (dynamics of) microbial communities (Zomorrodi et al., 2014;Zomorrodi & Maranas, 2012).While these multiobjective approaches can dissect the trade-offs in realistic microbial communities, they require substantial computational time due to the size of the considered problems and are currently limited to a handful of species.There is still considerable gap in our understanding of how the metabolites produced by one and consumed by another species affect or overcome metabolic trade-off at different levels.
With respect to trade-offs in microbial communities, there has been a recent body of work focusing on explaining trade-offs and including them in models for community growth by identifying microbial life history strategies for an organism (Malik et al., 2019).Life history strategies represent sets of mutually correlated traits, such that different strategies are favoured under different environmental conditions.For instance, diverting cellular resources towards production of enzymes excreted from a microbe to degrade resources to nutrients acquired for growth is in trade-off with growth yield (Sinsabaugh et al., 2013) at the time of enzyme synthesis.The same holds for diverting resources towards stress tolerance, which is then in trade-off with growth yield as well as resource acquisition.Such a perspective relies on the fuzzy division of cellular activities into those that contribute to growth and those that support maintenance; as a result, resource acquisition (e.g., extracellular enzyme production to facilitate nutrient availability) and stress responses (e.g., repair mechanisms, osmotic balance, defence) can be regarded as parts of maintenance costs.The allocation of cellular resources to different life history strategies (or tasks) is then reflected in the redistribution of metabolic fluxes at the level of both individual microbes and the entire community (Hall et al., 2018).Following these developments, recent studies have attempted to reconcile the high-yield, resource acquisition, stress-tolerance (Y-A-S) trade-offs in life history strategies for different environmental cues by dividing the environments across two axes-resources (abundant vs. limited) and stress (presence vs. absence; Malik et al., 2019).It would be of particular interest to investigate if and how these life history strategies emerge from the flux redistribution in metabolic models of microbial communities as well as to determine their realization under different modelled environments.

| Trade-offs between metabolite concentrations
Metabolic networks must respect mass-balance constraints that lead to conservation laws.Based on the structure of the metabolic network, one can derive (non-negative) linear combinations of compound concentrations that are preserved irrespective of whether the network is transient or stationary (Schuster & Höfer, 1991).
Conservation laws point at dependence of rows of the stoichiometric matrix; thus, they lie in the left null space of the stoichiometric matrix, and can be identified by efficient linear algebraic approaches.To improve interpretability, convex bases (composed of non-negative linear combinations of metabolite pools) for the left null space have also been proposed and applied to large-scale metabolic networks (Famili & Palsson, 2003;Schuster & Höfer, 1991).These can further be extended to determine conserved moieties (i.e., elements) by using atom transition maps (Haraldsdóttir & Fleming, 2016).It is clear that identification of conservation laws and moieties is integral to the study of trade-offs in metabolic networks, since they provide the interplay between constraints from the environment and network structure.The conservation laws, represented as non-negative linear combinations of metabolite pools, represent multivariate extension of the Y-model.More specifically, these laws amount to acquisition of particular resources available for interconversion by the reactions in the metabolic network.For instance, for the metabolic network in Figure 4c, it must hold that the sum of the concentrations, c 3 + c 4 + c 5 , is constant since the summation of the corresponding differential equations amounts to zero.As a result, under a constant environment, any increase in sum c 3 + c 4 will have to be balanced by a decrease in the concentration c 5 , indicating a trade-off.

| PER S PEC TIVE S
Trade-offs are widely studied in different areas of biology since they can be observed across different levels of organization (Roff & Fairbairn, 2007;Stearns, 1989)-between characteristics of individual cellular components (e.g., parameters of enzymes), rates of reactions and activities of pathways, as well as between whole-organism traits and functions, including growth, defence and reproduction.
The concept of trade-offs has also been implicated in understanding how whole-organism traits and functions shape the dynamics of ecosystems (Kneitel & Chase, 2004) and biodiversity.First, the identified absolute metabolic trade-offs mentioned above result from imposing fixed cellular resources.However, organisms have evolved mechanisms to ensure the robustness of concentrations of individual components or combinations thereof in the context of cellular networks (Karp et al., 2012;Shinar & Feinberg, 2010).These results pave the way for establishing necessary and sufficient conditions for the existence of absolute tradeoffs in large-scale metabolic networks.
Second, while the concept of identification of trade-offs from omics data has caught considerable attention (Shoval et al., 2012), the problem of predicting which trade-offs arise under specific conditions remains poorly understood.A recent study has made progress in this direction by making use of the knowledge of metabolic network structure to predict which functions interact antagonistically or agonistically to facilitate adaptation of Lactococcus cremoris to a specific carbon source (Ekkers et al., 2022).Specifically, the study predicted and tested the hypothesis that adaptation to galactose and fructose, as carbon sources, will target pathway branch points and reactions in glycolysis-reactions that need adjustment in comparison to the pre-adapted growth on glucose.We envisage that approaches for the discovery of metabolic network motifs and modules (Küken et al., 2021(Küken et al., , 2022) ) may provide a suitable direction to expand this line of research.This idea is of interest, since the presence of modules may decouple parts of the network, thus allowing control and manipulation of trade-offs-the main focus of the third research area.
Last, the constraint-based modelling framework has been particularly useful in the design of rational strategies for biotechnological application, such as increasing the production of strains of interest (Maranas & Zomorrodi, 2016).It is therefore conceivable that this framework for modelling of metabolic networks can also be used to design strategies to overcome trade-offs.This is a very important problem, particularly in crop research, where trade-offs between growth and defence, between yield components, and between stability of yield and yield itself have great effects for crop performance (Laitinen & Nikoloski, 2019).One way to address this problem is to break the dependence between the metabolic traits in a trade-off by modifying the metabolic network (e.g., inserting a non-native reaction) or regulation at specific points of the metabolic network (Küken & Nikoloski, 2019).For instance, this can be achieved by rendering some of the traits to be robust, in connection with the first research area, highlighted above.Irrespective of the strategy followed, such network modifications may lead to new metabolic trade-offs, and hence further research is required to understand the limits and benefits of manipulating metabolic trade-offs in the context of improving traits of interest.
then partitioned between growth, respiration and nonstructural components.More specifically, according to Equation (1), negative correlations arise whenever var(T) is smaller than the sum of the variances of the traits.Therefore, the Y-model emphasizes that the application of association measures can mask the underlying allocation trade-off that results in the observed phenotypes.It is trivial to expand the Y-model with a multiplicative relationship between the resource and traits (e.g., X 1 X 2 = T), in which case the trade-offs are explored for log-transformed trait values.While the Y-model has provided a conceptual starting point to understand trade-offs among diverse traits, the system whose traits are considered in the Y-model is treated as a black box.As a result, the Y-model does not provide insights into the molecular mechanisms that shape the trade-offs.Recent advances in metabolic modelling have opened up the black box to expand on the explanations of trade-offs provided by the Y-model.
or secretion of metabolites.For instance, for the toy network in Figure2a, the reactions r in , r out and r bio are exchange reactions, while r 1 and r 2 are intracellular reactions.Monitoring the change of extracellular metabolite concentrations over time can be readily used to estimate exchange reaction fluxes(Mahadevan

F
I G U R E 3 Classification of metabolic trade-offs.(a) Trade-offs between kinetic properties of individual enzymes, such as turnover number, k cat , and substrate affinity, K M .(b) Trade-offs between fluxes in a metabolic network.Shown are the trade-offs between steadystate flux distributions for fluxes, v 1 and v 2 , which are in absolute trade-off when the growth, modelled by the flux, v bio , through the biomass reaction is fixed.(c) Illustration of substrate-efficient and enzyme-efficient pathways, as an explanation for the rate-yield trade-off.Thicker lines denote reactions with enzymes with higher turnover numbers.(d) Trade-offs between metabolic objectives represented by a Pareto frontier.All points denote feasible flux distributions for a metabolic network associated with particular values for the two objectives (shown on the axes).A point (x′, y′) is dominated by another (x, y) if x′ > x implies that y′ > y or if y′ > y implies that x′ > x.The Pareto frontier is composed of all points that are not dominated by others.The Pareto frontier for the constellation of points is given by those on the blue line.
Illustration of absolute and relative flux trade-offs as well as conservation laws.(a) A metabolic network composed of seven metabolites and eight reactions (six internal and two exchange).(b) Illustration of relative tradeoffs with respect to reactions r 2 and r 6 .(c) Non-negative conservation law for the metabolic network in panel (a).
From an evolutionary perspective, it has been hypothesized that an enzyme cannot jointly optimize performance at both high and low temperatures due to a trade-off in stability and activity.However, evidence warrants caution against the existence of such a general assertion(Miller, 2017).Further, it has been assumed that adaptation towards new enzyme function for promiscuous enzymes(Leveson- Gower et al., 2019) involves a trade-off with the original (primary) Here, we have provided a systematic review of trade-offs arising in the context of metabolic networks.In doing so, we aimed to establish a relationship between the classic model for resource acquisition-allocation trade-offs-the so-called Y-model-and metabolic trade-offs.Interestingly, metabolic trade-offs have been studied in the context of different constraint-based modelling frameworks by considering only the structure (and stoichiometry) of the network, additional constraints arising from the available enzymes (and their catalytic efficiency), as well as consideration of the translation machinery.At the core of all trade-offs is the problem of partitioning limited resources (i.e., enzyme and metabolite pools) across metabolic reactions to facilitate multiple objectives ensuring organism' survival.On the other hand, trade-offs may arise due to simultaneous optimization of multiple objectives, which then shape the intracellular fluxes.Based on the overview of existing metabolic modelling results, we propose that the future research efforts focused on understanding of metabolic trade-offs focus on three areas, namely: (i) specifying the interconnection between (absolute) trade-offs and the robustness of metabolic traits, (ii) the prediction of trade-offs arising in specified environments, and (iii) the design of strategies to overcome trade-offs, with an emphasis on studying the implications of the strategies.