Trade‐offs between the instantaneous growth rate and long‐term fitness: Consequences for microbial physiology and predictive computational models

Microbial systems biology has made enormous advances in relating microbial physiology to the underlying biochemistry and molecular biology. By meticulously studying model microorganisms, in particular Escherichia coli and Saccharomyces cerevisiae, increasingly comprehensive computational models predict metabolic fluxes, protein expression, and growth. The modeling rationale is that cells are constrained by a limited pool of resources that they allocate optimally to maximize fitness. As a consequence, the expression of particular proteins is at the expense of others, causing trade‐offs between cellular objectives such as instantaneous growth, stress tolerance, and capacity to adapt to new environments. While current computational models are remarkably predictive for E. coli and S. cerevisiae when grown in laboratory environments, this may not hold for other growth conditions and other microorganisms. In this contribution, we therefore discuss the relationship between the instantaneous growth rate, limited resources, and long‐term fitness. We discuss uses and limitations of current computational models, in particular for rapidly changing and adverse environments, and propose to classify microbial growth strategies based on Grimes's CSR framework.


INTRODUCTION
A key goal of microbial systems biology is to obtain a quantitative understanding of the physiology of microorganisms in terms of the underlying biochemistry and molecular biology. [1,2]Building upon the increasing availability of annotated genomes and suitable theoretical methods, computational models of microbial growth and metabolism can play a crucial role to address this challenge. [3,4][7][8] Genome-scale models and constraint-based analysis can, notwithstanding the caveats predictions based solely on knowledge of biochemical mechanisms by direct computation.Instead, many current computational approaches rely on the postulation of an objective function and its subsequent optimization.The objective function is a hypothesis about a fitness measure in terms of the (constrained) biochemical capacities of cells: predictions are not based solely on biochemical and biophysical mechanisms, but also on assumptions about how a cell ought to behave given that its phenotypic properties are the outcome of an evolutionary process.Here, systems biology connects to evolutionary biology.
In selecting a suitable objective or fitness function, a limitation of many current metabolic modeling frameworks is the assumption of a constant environment.The fitness of a genotype is then captured by a constant (per-capita or specific) growth rate, measurable within an instant and therefore termed the instantaneous growth rate.
Correspondingly, the most important objective function employed in current constraint-based metabolic modelling approaches is the maximization of the instantaneous growth rate.Although this is in line with evolutionary theory, [12,13] since the long-term fitness coincides with the instantaneous growth rate under constant conditions, predictions based on maximizing the instantaneous growth rate may be misleading or even erroneous for organisms that evolved in a dynamic, time-dependent, environment -a situation that is presumably true for the vast majority of microbes.

Despite this limitation, the predictions about the physiology and metabolic properties of common model organisms, in particular
Escherichia coli and Saccharomyces cerevisiae during batch or chemostat growth on different carbon sources are remarkably accurate, [9,10] indicating that the maximization of instantaneous growth rate is indeed a suitable objective function for these organisms under these conditions.This predictive success, however, does not necessarily imply that the predictive success translates to other microorganisms or different growth conditions.Multiple examples exist where the observed physiology of a microbe seems to be inconsistent with maximizing the instantaneous growth rate, ranging from the gram-positive bacterium Lactococcus lactis [14] to the chemolithoautotrophic archaeon Methanococcus maripaludis. [15]8] For these reasons, the choice of a suitable objective function in the context of constraint-based modeling is still subject to considerable debate.This debate, however, does not imply that the modeling method itself is flawed.There is a growing consensus that the underlying concepts, namely that protein expression by microbes is universally constrained by limited biosynthetic resources and that enzymatic rates obey similar biochemical constraints, are applicable across all microbial domains of life.In other words, constrained-based modeling and associated theoretical frameworks hold great promise.We are, however, struggling with consequences of the facts that microbial physiology reflects the (usually unknown) evolutionary history of the respective organism and that trade-offs that shape the adaptations to changing environments are still poorly understood.
Given the fact that microbes do not only invest resources in proteins that are required for instantaneous growth and maintenance, but also in energy and nutrient storage, as well as in proteins that may be beneficial in future environments, how should we deal with such anticipatory resource allocation?The observation that biosynthetic resources are limited, and cellular tasks generally improve when more protein is allocated to the respective tasks, implies a trade-off between cellular tasks. [19][22][23] But how cells precisely balance the resource investments in instantaneous growth, capacity for adaptation to future changes, and stress-tolerance, also depends on their evolutionary history which is often unknown to us.How to translate these trade-offs into our current modeling frameworks?And, finally, how to deal with processes that likewise involve investment of resources, and hence trade offs, but whose impact on the growth rate is difficult to quantify, such as resource investments into sensing, information processing, motility, and regulation?
In this review, we reflect on these open problems of current microbial system biology with a focus on E. coli as our main model microorganism.We discuss why this bacterium appears to manifest a phenotype that maximizes the instantaneous growth rate, even though it has evolved in a dynamic environment and is known to express anticipatory proteins to prepare for future environmental conditions.We discuss experimental studies that indicate optimal expression of metabolic proteins for growth, and trade-offs between the allocation of proteins for different cellular tasks.We discuss that sufficiently large single-genotype-derived populations of E. coli have been shown to contain different genotypes (differing only by a single mutation event) that likely exist in a mutation-selection equilibrium, such that the dominant genotype is environment dependent, and the population exhibits a combined fitness that may exceed that of any single genotype.We argue that we tend to study the growth-prioritizing genotype in our labs, adapted to particular carbon sources, which may explain why modeling methods based on maximizing the instantaneous growth rate give satisfactory predictions for E. coli, even though the assumed objective function does no encapsulate its full fitness strategy.
Finally, we conjecture that the physiologies of E. coli and S. cerevisiae may only be representative for a particular class of microorganisms.
Therefore, novel paradigms are required for other yet-to-be-identified classes that build upon an increased consideration of ecological and evolutionary concepts in systems biology.

LONG-TERM FITNESS EQUALS THE AVERAGE GROWTH RATE
According to evolutionary theory, the fitness of a microbial genotype during a certain time interval equals the logarithm of the relative increase (or decrease) of its abundance divided by the duration of the interval. [12,13]This fitness measure equals the average (specific or percapita) growth rate of the genotype during the time interval.In Box 1, we illustrate how short-and long-term fitness are related to the instantaneous and average growth rates, respectively.Microbial growth is most often studied under conditions of balanced growth.Balanced growth is a property of a microbial population and refers to the steady-state condition of an average cell. [1,24]Balanced growth occurs when environmental conditions are (approximately) constant and the number of cells increases exponentially for times much longer than the generation time.During balanced growth, the physiological properties of the average cell, including the concentrations of all intracellular molecules and their interconversion fluxes, remain constant.Hence, the instantaneous growth rate is constant, and corresponds to the specific (or per-capita) growth rate of the population.
When conditions change, the instantaneous growth rate becomes time-dependent.The instantaneous growth rate can then be interpreted as an instantaneous (or short-term) fitness, and, over longer time intervals, the average growth rate determines which genotype has the highest fitness.
In constant environments, we therefore expect that a genotype that maximizes its instantaneous growth rate emerges as the fittest.
Conversely, genotypes that grow in a changing environment and experience varying stresses and nutrient levels are expected to maximize their average growth rate, and hence long-term fitness, by balancing the biosynthetic resource demands for instantaneous growth, stresstolerance, and capacity for fast adaptation to new environments.Under such conditions, the expression of preparatory or anticipatory proteins is expected to increase the long-term fitness, even though their expression may reduce the instantaneous growth rate.The instantaneous growth rate is therefore not always an appropriate measure of fitness.
In randomly-changing environments, cell-to-cell heterogeneity in stress tolerance, adaptive potential and instantaneous growth rate can, in principle, also enhance the fitness of a genotype.For instance, in environments with varying potential stresses, it may be beneficial for a genotype to bet-hedge and diversify its phenotypic behaviors. [25,26]In environments with infrequent extinction-threatening conditions, phenotypic diversification, via the formation of persister-cells or spores, may be beneficial for long-term fitness. [27]us, the diversity of the microbial fitness strategies that we find in nature is expected to reflect the diversity of their environmental conditions.This observation lies at the heart of the problem we are addressing in this paper.Current analyses of resource allocation models have a bias towards describing an average cell in a constant environment.

RELATIONSHIP BETWEEN GROWTH RATE AND BIOSYNTHETIC RESOURCE ALLOCATION
Cells have finite biosynthetic resources for the synthesis of proteins, and are constrained by finite cellular machineries for nutrientassimilation, catabolism, transcription and translation, as well as by finite intracellular space, leading to a finite supply rate of macromolecular building blocks and energy, and, therefore, growth rate. [28,29]As

BOX 1: Long-term fitness and the instantaneous and average growth rates
We start with the instantaneous growth rate (t) of a microbial genotype, defined as the specific, or per-capita growth rate at a particular point in time, with N(t) denoting the abundance of the genotype at time t.The instantaneous growth rate depends on the environment via the (time-dependent) concentrations of nutrients and other effectors, such as temperature.Formally, the differential equation can be solved, giving rise to, with Here, F(T) denotes the average growth rate of the genotype and is equal to the time-average of the instantaneous growth rate over a time interval T. The fitness of a genotype is generally defined as F(T).When conditions are constant, (t) does not depend on time and F(T) = .Strictly speaking, an isogenic populations of cells will display phenotypic variation causing individual cells to have different growth rates.The growth rate of the genotype ((t)) is then the average of that of its phenotypes.When we study a population of microbial species in its natural environment then generally that population will be polyclonal and involve competing genotypes.
illustrated in Figure 1, each of these finite resources acts as a constraint on protein expression and can, in principle, limit it.
][32][33] This trade-off can be illustrated by the consequences of expressing proteins that are not required for instantaneous growth under the current conditions.Such expression is expected to reduce the instantaneous growth rate, as has indeed been confirmed experimentally by overexpressing −galactosidase (required for growth on lactose) during growth on glucose.The overexpression results in a reduction of the instantaneous growth rate. [28,34,35]ade offs between cellular tasks result from the fact that, in general, the capacity for a cellular task increases when more of its Cells have finite biosynthetic resources and each finite resource may act as a constraint on cellular functions and can, in principle, limit it.Shown are biosynthetic resource costs associated with DNA replication, transcription, translation, metabolism and growth.
facilitating proteins are expressed.This relationship is best illustrated by the dependence of a metabolic reaction on its catalyzing enzyme.
The maximal (forward) rate V max of a metabolic reaction is proportional to the enzyme concentration E and the (forward) catalytic rate constant k + cat , [36][37][38][39] V Hence, to catalyze a reaction with a rate v requires (at least) an enzyme concentration E ≥ v∕k + cat .Likewise, the capacity for growth depends on the allocation and amount of expressed proteins that are dedicated to transport, metabolism and the required transcriptional and translational machinery.The actual instantaneous growth rate then depends on the 'quality' and concentration of the available nutrients, as well as the biosynthetic investment in growth-associated proteins. [28]antitatively, the instantaneous growth rate equals the net protein synthesis rate (i.e., the net translation rate) divided by the cellular protein content. [28,29]Hence, the expression of proteins that are not directly involved in processes related to instantaneous growth, reduces the growth rate by contributing to the cellular protein content while not increasing the protein synthesis rate. [28]This relationship is detailed in Box 2.

THE EXPRESSION OF ANTICIPATORY PROTEINS REDUCES THE INSTANTANEOUS GROWTH RATE
The theoretical relationship outlined in Box 2 suggests that the synthesis of proteins not related to growth reduces instantaneous growth rate, and hence the short-term fitness.Accordingly, expression of anticipatory proteins, that is, proteins that provide stress tolerance or capacity to rapidly adapt to new environmental conditions, is expected to be at the expense of growth-related proteins, and hence reduces the instantaneous growth rate. [19]This trade-off was indeed illustrated experimentally by studies showing that E. coli is more stress tolerant Here, protein synthesis includes all processes directly required for the synthesis of proteins, including uptake of carbon, nitrogen and other nutrients, regeneration of energy equivalents (e.g., ATP), and the synthesis of amino-acids.
At balanced growth, the rates of each of these constituent processes are constant and balanced, and each rate is proportional to the concentration of the associated catalyzing protein or protein complex.Consequently, if the concentrations of all associated proteins are increased by a factor , all reaction rates, including the total protein synthesis rate v P , increase by a factor  (while the concentrations of all intracellular metabolites remain unaffected [40] ).More formally, the relationship v P ( P G ) =  v P (P G ) holds, with P G denoting the protein content directly contributing to growth.
If we denote the remaining protein content that is not associated with growth as P NG , with P T = P G + P NG , we can derive an expression how the growth rate decreases if the ratio  = P NG ∕P T of non-growth related proteins to total proteins increases.Let v P (P T ) denote the protein synthesis rate when all proteins are involved in growth, and v P (P G ) the protein synthesis rate with only P G involved in growth.Since P G = A nonzero ratio  incurs a linear reduction of the instantaneous growth rate that is proportional to .For an extended discussion see Bruggeman et al. (2020). [41]d adaptive at slow growth than at fast growth. [20,23]Under the latter condition, the majority of its expressed proteins have a metabolic function. [10] therefore emphasize that expression of anticipatory proteins is not necessarily evidence of suboptimal fitness, but only of suboptimal short-term fitness.To assess and test this hypothesis experimentally, however, is challenging, since the benefits of anticipatory protein expression are dependent on future states of the environment.Their expression constitutes a gamble (or a bet), as their benefits only materialize when the environment changes in accordance with the functions of the expressed anticipatory proteins.
Demonstrating the reverse, that is, the expression levels of proteins that are required under the current environmental conditions are such that the expression maximizes short-term fitness (i.e., the instantaneous growth rate) is more straightforward.44][45][46][47][48]

EXPRESSION OF METABOLIC ENZYMES OFTEN MAXIMISES THE INSTANTANEOUS GROWTH RATE
The expression of growth-related proteins or, more generally, any protein required for growth in the present environment, contributes to the instantaneous growth rate.Accordingly, for any required protein, an optimal expression level exists, such that below this level, the protein expression is limiting growth and the growth rate increases with increasing expression.Vice versa, expression above the optimal level has an excess biosynthetic resource demand at the expense of other proteins that are then limiting growth.This dependence of the growth rate on the expression of growth-related proteins was experimentally illustrated by Dekel et al. [35] using the lac operon in E. coli.Dekel et al. [35] also showed that laboratory evolution leads to the fixation of mutants with optimal expression of the lac operon.More generally, considerable experimental evidence exists for various microorganisms that shows optimal expression of metabolic proteins.44][45][46][47][48][49][50] For example, it was recently shown that E. coli expresses its F 1 F 0 -ATPase, a protein complex that uses the proton-motive force to regenerates ATP, at optimal levels across more than 25 conditions. [43]gure 2 exemplifies the effect of modulating F 1 F 0 -ATPase expression on the growth rate for growth on glucose, showing that the WT growth rate indeed corresponds to the growth rate of the mutant with optimal F 1 F 0 -ATPase expression.Keren et al. [44] studied the expression of over eighty different proteins in S. cerevisiae under a single growth condition, and showed that they too were optimally expressed.We note that suboptimal expression of a highly abundant protein, which metabolic proteins often are, typically has a greater fitness cost than suboptimal expression of a low abundant protein, such as a signalling protein or a transcription factor. [41]kewise, computational modeling and theory suggests that the expression of ribosomal proteins as a (linear) function of the growth rate can be explained as optimal protein expression to maximize the instantaneous growth rate, [51][52][53] consistent with results obtained with ribosomal gene knockouts. [54]Although less well studied, also growing on glucose.The growth rate of the wild type (red dots) is compared with the growth rate of a mutant strain with a isopropyl -d-1-thiogalactopyranoside (IPTG)-titratable ATPase.Titration with IPTG changes the expression level of the ATPase.There exists an optimal expression level that maximizes the instantaneous growth rate.At optimal expression, growth rate of the mutant equals that of the WT, supporting the hypothesis that expression of ATPase in the WT maximizes the instantaneous growth rate under this growth condition.Error bars indicate the standard deviation.Data sourced from, [43] see the original publication for experimental methods and analysis of 26 different carbon sources.
evidence of optimal expression of global regulators of metabolism exists, for example of cAMP [55] and ppGpp [56] (see appendices of both papers), studied in E. coli grown on glucose and ammonium.
We emphasize, however, that studies on optimal expression of metabolic proteins were typically performed under conditions of nutrient excess (e.g., in batch cultivation), reflecting the fact that titration of gene expression is experimentally much harder under conditions of nutrient limitation in chemostats.It therefore remains unclear whether expression levels of these proteins are also optimal at nutrient limitation or under dynamic conditions.At least one experiment provides indirect evidence of suboptimal expression of glycolytic proteins in S.
cerevisiae grown in a glucose-limited chemostat. [57]The authors found that during prolonged cultivation the expression of glycolytic proteins decreased, indicating initial excess.This finding suggests that S. cerevisiae has evolved a protein expression strategy tuned towards excess conditions rather than limiting conditions.Excess of glycolytic proteins is also consistent with anticipatory protein expression during nutrient limitation.Such behavior is in agreement with other studies that show anticipatory expression of metabolic proteins and proteins required for adaptation to new growth conditions during slow growth. [20,21,23,58,59]

OPTIMALITY OF METABOLIC PROTEIN EXPRESSION IS CONDITIONAL ON THE FRACTION OF BIOSYNTHETIC RESOURCES ALLOCATED TO METABOLISM
The observation that, at least under condition of nutrient excess, the expression levels of metabolic enzymes often seem to maximize the instantaneous growth rate does not imply that the respective microorganism has attained the maximal instantaneous growth rate that is 'evolutionary possible' under these conditions.Experimentally observed optimal expression only implies that metabolic protein concentrations are optimally tuned to the current condition, and that any change in their expression will reduce the instantaneous growth rate.
In other words, these proteins do not show any under-or overcapacity.
The fact that the instantaneous growth rate is not at its theoretical maximum is indicated by the expression of growth rate-decreasing anticipatory proteins even under excess nutrient conditions, as well as by the existence of adaptive mutations that can enhance the instantaneous growth rate.Adaptive laboratory evolution experiments of E.
coli on various carbon sources (e.g., acetate, glycerol, pyruvate, alphaketoglutarate [60][61][62] ) indicate that the instantaneous growth rate can generally be increased by genetic mutations.Surprisingly, it is often observed that the evolved strain then has, compared to the ancestral strain, a reduced growth rate on other carbon sources.For example, the growth rate of E. coli on glycerol can increase significantly during prolonged serial-dilution experiments, but these genetic adaptations occur at the expense of the growth rate on glucose and other carbon sources. [61]A similar phenomenon was observed with S. cerevisiae: adaptation to galactose was at the expense of the growth rate on glucose. [63]It is unclear what the precise origins of these 'coevolved maladaptations' are, but they may indicate that the regulatory machinery of E. coli is limited by trade offs in its metabolic regulatory machinery that are contingent on particular conditions. [64,65]e fact that E. coli and other microorganisms can still increase their growth rate and exhibit anticipatory protein expression even under excess nutrient conditions is sometimes interpreted as evidence of 'suboptimality' , with the implication that their physiology may therefore not be predictable using optimization-based computational methods.But if so, how should experimental observations of optimal metabolic protein expression then be interpreted?We argue that a solution to this apparent contradiction is that optimal expression of metabolic proteins should be considered with respect to the fraction of proteins allocated to metabolism, rather than the entire protein pool of the cell.From this perspective, the expression of metabolic proteins may be suboptimal relative to all biosynthetic resources of a cell but optimal with respect to the pool of biosynthetic resources allocated to growth.This view is also consistent with the predictive success of optimization-based computational methods without implying that the assumed optimization objective fully encapsulates the long-term fitness strategy of the organism.
Accordingly, many current computational models of cellular resource allocation and growth contain a fraction of proteins that are not related to growth, typically denoted as quota (Q) proteins, [10,28,59,66,67] whose expression can (currently) not be predicted or explained by a modeling formalism that maximizes the instantaneous growth rate.The optimization of the remaining protein allocation is then carried out with respect to a known Q fraction.The Q fraction includes structural proteins, as well as proteins related to stress response, motility, capacity for adaptation and other cellular cellular functions that are not directly associated with biosynthesis and growth. [28,66,67]We emphasize that the Q fraction is not strictly defined, its size and composition is likely to change under different growth conditions, and its composition may also be subject to stochastic, bet-hedging processes.

MODEL PREDICTIONS OF PROTEIN EXPRESSION ARE OFTEN IN LINE WITH EXPERIMENTS
Concomitant with experimental advances to understand microbial physiology, computational models have advanced to a stage that predictions can be made of protein expression, metabolic fluxes and growth rate under particular growth conditions, in particular balanced growth, nutrient excess in batch cultures, and nutrient limitations in chemostat cultures.Improving upon earlier genome-scale metabolic reconstructions and flux balance analysis (FBA), current models of microbial metabolism and growth allow us to incorporate constraints that arise from enzyme biochemistry, transcription, translation and cellular space. [11]Examples of constraint-based methods based on the concept of (genome-scale) cellular resource allocation are Metabolism and macromolecular Expression (ME) models, [10,68] as well as, mathematically largely equivalent, resource-balance analysis (RBA), [69,70] in the following collectively referred to as RBA/ME-type models.
RBA/ME-type models retain many of the advantageous properties of FBA.In particular, they can be solved efficiently using methods from linear programming, and do not require extensive knowledge of enzyme kinetic parameters.Different from conventional FBA, however, RBA/ME-type models explicitly consider the constraints on cellular reaction rates by protein expression and enzyme kinetics.
That is, within RBA/ME-type models, cellular processes are limited by the abundances of the respective catalyzing macromolecules, such as enzymes for metabolic reactions and ribosomes for protein translation.
RBA/ME-type models are formulated as constraint-based autocatalytic resource-allocation problems, such that biosynthesis and protein translation require the presence of catalytic macromolecules that are themselves the product of these processes.The computational objective is to maximize the instantaneous growth rate in a given (typically constant) environment by optimal allocation of catalytic macromolecules.Hence, while conventional FBA assumes a known and constant biomass composition and maximizes the biomass production rate given upper bounds on nutrient uptake rates, RBA/ME-type models explicitly predict the instantaneous growth rate, as well as the optimal allocation of biosynthetic resources that give rise to the maximal growth rate.A brief synopsis of RBA/ME-type models is provided in Box 3.
][73] These methods implement the optimization of the average growth rate over a defined time interval, and therefore allow us to asses the maximal long-term fitness over the respective time interval.
While RBA/ME-models do not consider the detailed kinetics of data.RBA/ME-models have been constructed for T. maritima, [68] E.
coli, [10] as well as for S. cerevisiae, [9] and a set of other microorganisms, such as L. lactis, [74] B. subtilis, [70] and the cyanobacterium Synechococcus sp.PCC 7942. [17]ecifically, the ME model developed by O'Brien et al. for E. coli [10] was the first model at genome scale that was able to successfully predict changes in yield and metabolism as a function of glucose availability, with good quantitative agreement at glucose excess -where conventional FBA ('M-models') fails.Within the study, only the onset of acetate secretion was slightly too high, but this prediction is dependent on specific, tweakable, parameters.For S. cerevisiae, a RBA/ME-type model was recently published, termed 'proteome-constrained' model (pcYeast) to emphasize the compartmentation of protein pools in a eukoryotic cell, each with their own constrained size.This model explores the wealth of quantitative data of yeast physiology and biochemistry to accurately predict changes in yield and the onset of overflow metabolism. [9]In a RBA/ME model of L. lactis, proteome constraints were combined with measured flux constraints to identify which of these constraints limit in silico growth.Even though the model did not correctly predict all observed fluxes, these identified limitations (known as reduced costs in linear programming) correctly predicted adaptive changes upon laboratory evolution experiments. [74]ken together, these results suggest that, at least for E. coli and S.
cerevisiae, as well as several other model organisms, the experimentally observed metabolic behavior under the studied conditions can indeed be reasonably well described by RBA/ME-type models, and therefore can be understood from the perspective of instantaneous growth-rate maximization.
Some of the growth-related properties predicted by RBA/ME-type models can also be illustrated using a coarse-grained models of microbial growth.An example is detailed in Box 4, the implications for our understanding of microbial physiology are summarized in Figure 4.

OVEREXPRESSION AND RAPID ADAPTATION TO NEW NUTRIENT ENVIRONMENTS
As noted above, experimental studies of protein expression in E. coli and other microorganisms often reveal expression of proteins that either fulfill no functional role in the current environment, or whose expression exceeds the current requirement for growth (overcapacity).Experimental evidence further suggests that such expression can serve anticipatory purposes, in particular under conditions of nutrient limitation.In particular, it has been shown that E. coli is more adaptive to new growth conditions at slow growth. [20,21,23,58,67]Likewise, experiments show that fast growing cells have a longer adaptation time (lag phase) when abruptly shifted to a new carbon source compared to slow growing cells. [23]Data-driven comparisons using RBA/ME-type models also indicate low substrate saturation of metabolic enzymes at slow growth for E. coli and S. cerevisiae in glucose-limited chemostat cultures. [10,59,78]pported by theoretical analysis, [23,58] these findings suggest that overexpression, as well as expression of currently unneeded proteins, can serve as a 'strategic reserve' that is kept by cells to be capable to adapt to a sudden improvement in growth conditions.Indeed, simulation experiments show a strong asymmetry when switching to new nutrient environments: cells that are adapted to a low-nutrient environment will (initially) perform poorly in a high-nutrient environment.
Vice versa, cells that are adapted to a high-nutrient environment will only have a slightly reduced instantaneous growth rate when growing under low-nutrient conditions (compared to perfectly adapted cells), but will have a large advantage when nutrient levels increase. [73]is asymmetry can also be illustrated using the coarse-grained model described in Box 4.  4E, F).For organisms that evolved in feast-famine environments, overcapacity and the expression of anticipatory proteins during slow growth has only a small impact on the absolute value of the (momentary) instantaneous growth rate, but can provide a major

BOX 3: A brief synopsis of FBA and RBA/ME-type models
Genome-scale metabolic reconstructions (GSMRs) aim to provide a comprehensive account of the metabolic capabilities of a microbial organism. [11]Based on an available genome sequence, and typically augmented by extensive manual curation, a GSMR describes the stoichiometry of all reactions that may take place within a cell, including enzyme-catalyzed metabolic reactions, as well as transport processes and non-catalyzes reactions (e.g., diffusion and spontaneous degradation of metabolites).A GSMR can be summarized by a stoichiometric matrix N, together with the respective gene-protein-reaction associations.
Assuming spatial homogeneity ('the microbial cell as well stirred reactor') the dynamics of metabolite concentrations x are described by the product of stoichiometric matrix N and the vector of reaction rates v(x, k), In general, the vector of reaction rates v(x, k) consists of nonlinear functions that depend on reactant concentrations x and parameters k.At steady-state, however, the balance equation N v = 0 holds and constrains the solution space for the unknown reaction rates.That is, instead of the set of nonlinear ordinary differential equations, constraint-based analysis, such as FBA, [75] only considers the linear constraints on the flux vector v imposed by the balance equation.Since GSMRs typically consist of more reaction rates than metabolites, the balance equation is under-determined.Therefore, FBA makes use of an objective function, typically the synthesis of biomass (biomass objective function, BOF), to obtain predictions for the reactions rates.The canonical form of FBA is max v BOF s.t.Nv = 0, (mass balance) where v BOF = c T v denotes a (linear) objective function, and v min and v max denote upper and lower bounds on the reaction rates, respectively.The impact of constraints on the feasible flux space is illustrated in Figure 3.
RBA/ME-type models augment the stoichiometry with the transcriptional and translational machinery, [68,69] and introduce capacity constraints on reaction rates that depend on the respective catalyzing macromolecules.An example of a capacity constraint is given by Equation (4).More generally, for irreversible reactions, capacity constraints can be formulated as an inequality C v ≤ x.The mass balance equation explicitly considers dilution of molecules by growth. [41,76]The canonical form of RBA/ME-type models is max x min ≤ x ≤ x max , (optional concentration bounds) where the vector of concentrations includes metabolites as well as macromolecules, such as enzymes and ribosomes (measured in mol per gram dry mass).Since all fluxes and concentrations are measured relative to cellular dry mass, the density constraint, with elements of  given in gram dry mass per mol, ensures normalization (i.e., the summed dry mass of all compounds in one gram dry mass equals one gram).
We note that we require no formal distinction between metabolites and catalyzing macromolecules.Further limiting constraints can be implemented.For example a limited membrane space or a limited volume of a cellular compartment results in additional constraints of the form  T R x ≤ R, where the vector  T R defines the amount of the shared resource that one mol compound requires and R denotes the total available resource (per gram dry weight).
For any given growth rate , the feasibility of a RBA/ME-type model can be solved by linear programming.The maximal growth rate  is obtain by bisection, that is, we determine the maximal growth rate  for which the resource-allocation is feasible.70] Continued RBA/ME-type models are a major advancement over conventional FBA, since they allow us to predict protein allocation and hence (aspects of) the cellular composition as a function of environmental conditions.As noted in the main text, predictions are typically conditional on a known Q fraction.That is, investment into proteins (or other compounds) that serve no catalytic function in the modeled environment must be enforced by an optional lower bound on the respective concentrations.We further emphasize that the formulation of constraints may have significant impact on model predictions.Box 4 provides a small coarse-grained growth model that illustrates the utility of resource-allocation models to describe properties of microbial growth.increase of the instantaneous growth rate under feast conditions, and hence a major increase in the average growth rate and long term fitness.This reasoning does not hold for organisms that constitutively live under low nutrient conditions (obligate oligotrophs).
We therefore argue, consistent with current literature, [23,58] that microorganisms that evolved in repeated famine-and-feast cycles are likely to exhibit an opportunistic growth-prioritizing phenotype characterized by anticipatory protein expression and overcapacity of growth-related proteins under low nutrient conditions.

USE AND MISUSE OF THE TERM ENZYME OVERCAPACITY
A prerequisite to analyze potential overcapacity, trade-offs and (sub)optimality of protein expression is that the concept of 'overcapacity' is well defined.Unfortunately, some commonly used definitions, in particular in the context of RBA/ME-type models, can be misleading.
The problem arises when overcapacity is defined solely in terms of an actual reaction rate compared to the maximal enzymatic capacity V max .
An enzyme can have a catalytic rate far below the maximal rate V max even though it is optimally expressed and exhibits no true overcapacity.
To illustrate why the ratio v∕V max is not a suitable measure for overcapacity, we consider the canonical description of the rate v of an enzyme-catalyzed reaction, [5,38,39] The rate of an enzyme is a product of three factors: a reactantsaturation factor f kin (x) that depends on reactant concentrations (and may also include regulatory terms), a thermodynamic factor RT that describes the displacement of the reaction from thermodynamic equilibrium, and the maximal reaction rate, V max = k + cat ⋅ E, that depends on the catalytic rate constant k + cat and the enzyme concentration E. The saturation factor is bounded from above, f kin (x) < 1, and the reaction is considered to be 'saturated' with respect to reactants at values of f kin (x) close to one.Likewise, considering only the forward direction, the thermodynamic term g thd < 1 is also bounded from above, and attains a value close to 1 for reactions that operate far from equilibrium (ΔG ′ ≪ −RT).Vice versa, reactions that operate close to thermodynamic equilibrium have a thermodynamic driving force close to zero (g thd ≈ 0).Since there is no a priori or empirical reason to assume that evolutionary optimization of enzyme expression will only, or even primarily, result in reactions that are fully saturated and operate far from equilibrium, we must expect reactions to exhibit rates below their maximal rate V max .A prime example is the glyceraldehyde-3phosphate dehydrogenase (GAPDH), an enzyme involved in glycolysis and gluconeogenesis.Its thermodynamically preferred direction favors gluconeogenesis, hence it typically operates close to thermodynamic equilibrium during glycolysis.Under such conditions, GAPDH necessarily exhibits an actual rate that is far below the maximal rate because of thermodynamic constraints.From this fact, however, we cannot conclude that the enzyme exhibits excess capacity.A similar reasoning applies to the kinetic saturation term.Enzyme saturation increases with high substrate concentrations and low product concentrations (relative to their respective Michaelis-Menten constants).But since products are also the substrates of subsequent reactions, their concentrations can neither be arbitrarily small, nor arbitrarily large, resulting in a saturation f kin (x) < 1 for almost all reactions.
Thus, a definition of 'overcapacity' solely in terms of actual versus maximal rate may (erroneously) results in claims of suboptimal protein expression as soon as enzymes operate below saturation.We therefore suggest to refer to 'overcapacity' only if the actual expression level exceeds the optimal expression level.That is, 'overcapacity' implies that a reduced expression of the respective enzyme will not result in a decrease of the reaction rate (and hence growth rate).Since we rarely can assess or measure optimal expression levels (which usually require titration experiments discussed above), the conclusion that enzymes have overcapacity from comparing actual versus maximal rates is not warranted.
The issue can also be illustrated using the coarse-grained model described in Box 4. Figure 6A shows the actual uptake flux of the external nutrient together with the maximal capacity V max as a function of the (relative) concentration of the external nutrient.Figure 6B shows the saturation function f(a x ) = v∕V max as a function of the growth rate .For most external nutrient concentrations, the uptake flux is far below the maximal capacity, despite the fact that the enzyme expression was obtained by maximizing the instantaneous growth rate, and any reduction of the expression would result in a decrease of the instantaneous growth rate (as demonstrated in Figure 4E).

BOX 4: Example: a coarse-grained growth model
To illustrate the properties obtained from RBA/ME-type models, we consider a coarse-grained growth model.The model consists of four intracellular variables: an extracellular nutrient a x is taken up by a metabolic protein M and converted into the intracellular nutrient a which serves as a precursor for the translation of three proteins M, R, and Q by the ribosome R. The Q protein denotes a quota protein with no catalytic function (but subject to an optional lower bound on its concentration).The model is shown in Figure 4A.
The mass-balance constraint is, where n X denotes the size of protein X in units of a, and  X denotes the respective translation rate.
The rate of nutrient uptake is constrained by the availability of extracellular nutrient and the concentration of the metabolic protein M, The translation rates are constrained by the concentration of ribosome R, where k cat,R denotes the elongation rate of R per unit of a. Hence, the capacity constraints are, , where f(a x ) = a x ∕(K M + a x ) refers to the saturation of the nutrient uptake reaction.Neither the metabolite a nor protein Q have a catalytic function and impose no functional constraint on the reaction rates, hence the corresponding rows in the capacity matrix C are zero.
Together with the density constraint and a (possible) lower bound on Q, the system is fully defined, and a solution can be obtained by maximizing the instantaneous growth rate .
The model, despite its simplicity, recapitulates many of the properties also observed for genome-scale RBA/ME-type models.Key properties obtained from the model are discussed below and illustrated in Figure 4.
• Figure 4B shows the instantaneous growth rate  obtained by maximization of  as a function of the (relative) extracellular nutrient availability a x ∕K M .The growth rate is an emergent property of the model, its function is consistent with a Monod equation  = a x  max ∕(K A + a x ) with a half-saturation constant K A = 0.5 (dimensionless, measured relative to a x ∕K M ) and a maximal growth rate The organism-specific half-saturation constant K A is lower than the half-saturation constant K M of the transporter reaction, as also observed experimentally at least for S. cerevisiae. [77] Figures 4C shows the change in the mass fractions Φ of intracellular proteins for two different external nutrient concentrations.
Figure 4D shows the mass fractions as a function of the (nutrient-limited) instantaneous growth rate.The results recapitulate the growth laws of microbial physiology: the mass fraction Φ R of ribosomes and Φ M of metabolic protein are linear functions of the growth rate.The (approximately) linear dependency is also observed experimentally, and is obtained here as a consequence of growth rate maximization.Within the model, the mass fraction Φ Q = 0.5 of the quota protein Q is given as a parameter.Here, Φ Q is kept constant but, if the respective data are available, may also be implemented as a decreasing function of growth rate.
The coarse-grained model is also useful to illustrate the consequences of under-or overcapacity of proteins, as well as the consequences of expression of proteins not required for growth under the current conditions (anticipatory proteins, Q fraction).

Continued
• Figure 4E shows the instantaneous growth rate as a function of the mass fraction Φ M of the metabolic protein M.There exists an optimal expression level, such that expression of M below this optimal mass fraction limits growth, whereas expression above the optimum values unnecessarily requires resources at the cost of the expression of other proteins (here R).The plot recapitulates the titration experiments shown in Figure 2.
• Figure 4F illustrates the decrease of the (maximal) instantaneous growth rate with increasing expression of the quota protein Q (mass fraction Φ Q ).We note that, in accordance with theory (Box 2), the relative reduction in growth rate as a function of Φ Q is constant and does not depend on the external nutrient level.The absolute reduction of the instantaneous growth rate, however, is larger for higher nutrient levels (and hence higher instantaneous growth rates).
• Figure 4G shows the (maximal) instantaneous growth rate as a function of (relative) nutrient availability for two different values of Φ Q .
As noted above, the absolute reduction of the growth rate is again larger for higher growth rates.
Figures 4F and G suggest that for microbes that evolved and grow under varying environmental conditions (feast-famine environments) the expression of Q proteins under conditions of slow growth only has a comparatively small impact on the average growth rate.Since the impact of many stress factors does not necessarily depend on the instantaneous growth rate, the potential fitness benefit of expressing Q proteins therefore increases under slow growth conditions.We therefore hypothesize that for organisms that evolved in feast-famine environments, the potential benefits on long term fitness of the expression of Q proteins during slow growth can outweigh the (momentary or short term) reduction of the instantaneous growth rate, whereas under high nutrient level the expression of Q proteins would entail a strong reduction of the average growth rate.We note that this reasoning does not hold for organisms that constitutively live under low nutrient conditions (obligate oligotrophs).In this case, the relative reduction in growth rate determines fitness.The issue can also be resolved using detailed kinetic models of the respective pathways or organisms.For example, for S. cerevisiae, a detailed kinetic model of glycolysis was used to compute the minimal expression level of the glycolytic enzymes given the observed flux and boundary concentrations (of glucose and ethanol). [79]It was concluded that only at glucose excess conditions, but not at glucose-limited conditions, the expression of the enzymes was close to the computed optimal level.This approach is more indirect and requires detailed kinetic information that is not readily available for most organisms and pathways.

Parameters used: k
We note, however, that while low saturation of metabolic reactions does not necessarily indicate suboptimal expression (and hence does not necessarily constitute true overcapacity), a low saturation can indeed facilitate faster adaptation to a rapid increase in nutrient concentration, and hence can fulfill a similar functional role as anticipatory protein expression.Both concepts are not mutually exclusive.For example, the experimentally observed 'overshoot metabolism' , that is, the rapid uptake of phosphorus by microalgae after a period of phosphorus starvation, [80] can be explained by increased protein investment in uptake capacity during the starved state that translates into an excess capacity immediately after a nutrient upshift. [73]Similar, adding maltose to yeast cells that are adapted to maltose-limited growth, results in excessive uptake of maltose, in this case leading to a loss of cell viability. [81]

FITNESS-ENHANCING ADAPTATION-AND GROWTH-PRIORITIZING GENETIC VARIANTS LIKELY CO-EXIST IN LARGE ENOUGH E. Coli P-OPULATIONS
Above we discussed theoretical arguments as well as experimental evidence that suggests E. coli sacrifices its potential for rapid adaptation to new conditions (such as new nutrients and stresses) for instantaneous growth rate under conditions of nutrient excess.This growth-rate prioritizing behavior, however, may enhance the risk of extinction (or outcompetition) when condition rapidly change. [20,21,23,30,82]We already noted that single adaptive mutations exist that can increase the instantaneous growth rate of E. coli on new carbon sources. [61]][85] Since mutations are an inevitable byproduct of DNA replication, different genotypes will coexist in a mutation-selection equilibrium as long as the cell population is large enough.Given a mutation frequency of ∼ 1 × 10 −3 mutations per genome [86] and a genome size of about 5 million base pairs, a population size of ∼ 10 9 is required for all single point mutations to exist within the population-numbers that are easily attainable in a lab environment but also in the gut (estimated by BioNumbers at ∼ 10 8 cells per gram gut).

F I G U R E 5
Overcapacity and rapid adaptation to new nutrient environments.Shown is the maximal instantaneous growth rate for strains adapted to an external nutrient concentration of a x ∕K M = 0.1 and a x ∕K M = 10.0, respectively.The growth penalty for misadaptation not symmetric.In a low nutrient environment, an organisms adapted to a low nutrient environment only slightly outperform organisms adapted to a high nutrient environment but exhibit a strong reduction of growth rate in a high nutrient environment.Vice versa, cells adapted to a high nutrient environment exhibit a only slightly reduced growth rate under low nutrient conditions, but have a major growth advantage under high nutrient conditions.Since long term fitness is determined by the average growth rate, the observed asymmetry supports the hypothesis that organisms that evolved in repeated famine-and-feast cycle are likely to exhibit an opportunistic growth-prioritizing phenotype.Simulations were carried out using the coarse-grained model described in Box 4.
These observations raise the question whether E. coli can reduce its risk of extinction, that is a byproduct of its growth-rate prioritizing behavior, by forming populations that contain co-existing genetic variants, which are each adapted to different future conditions.This hypothesis was first formulated by Ferenci and Maharajan, [87,88] but has not yet been put into our modern systems biology context.As yet, the emphasis in systems biology has been mostly on phenotypic heterogeneity and diversification. [25,26,89]ltiple lines of evidence support Ferenci and Maharajan's hypothesis.Firstly, a laboratory-evolution experiment in a chemostat (a rather stringent and simple niche) illustrated a rapid emergence of biodiversity into genetic variants of E. coli with different phenotypic characteristics that stably co-existed. [85]Secondly, mutations in transcriptional machinery, for example.sigma factors (D and S) and RNA polymerase, have been found that can turn growth-rate prioritizing cells into cells that prioritize capacity for adaptation. [62,83,84,90]ch mutation that affect key regulatory genes and 'globally' rebalance cellular allocation [62] are also often found in wild isolates of E. coli. [90]Thirdly, trade-offs between instantaneous growth and stress tolerance as a (mechanistic) consequence of sigma factors competing for RNA polymerase acquisition (a limited resource) were already proposed by Nyström, [19] and also subsequently experimentally observed. [20,21,83,91]comparison of wild E. coli strains indicates that the main controller of the generalized stress response, the sigma factor S, is often inacti-F I G U R E 6 Optimal expression and reaction saturation in the coarse-grained RBA/ME-type model.(A) Shown is the maximal rate V max = k cat,M M of the uptake reaction, as well as the actual uptake rate v M as a function of the (relative) external nutrient concentration a x ∕K M .The reaction operates below saturation even though expression of the protein M is obtained by maximization of the instantaneous growth rate (hence no 'suboptimality').(B) The saturation function f(a x ) = v∕V max as a function of the growth rate.While the model only considers the saturation of the uptake reactions, we expect a similar relationship to also hold for intracellular reactions.vated by point mutations during conditions of slow growth, when slow growth is due to nutrient limitation rather than stress.[83,84,90,92,93] This finding suggests that E. coli generally 'associates' slow growth with stresses rather than with nutrient limitation.Moreover, Utrilla et al. [62] have identified a number of E. coli mutants, found during adaptive laboratory evolution experiments at excess nutrient conditions, which displayed adaptive mutations in RNA polymerase that increased the instantaneous growth rate in constant conditions, but reduced the capacity for adaptation to new conditions, including stresses.Thus, it is likely that in large-enough populations of E. coli neighboring genotypes co-exist together with the best-adapted and (currently) most abundant genotype, each adapted to different future conditions.
Co-existing genotypes imply that a single-genotype view on fitness may be too limited, and we should rather consider the inclusive fitness of a set of genotypes. [94]These genotypes then occur at environmentspecific frequencies that are determined by their relative growth rates.
Since, no individual genotype can overcome the cellular-task trade offs that result from finite biosynthetic resources and no genotype can reduce its mutation rate to zero, such co-existence may be a very resourceful solution of evolution.A telling experimental example is the case described by Maharjan et al. [85] already noted above: they found that a clonal population of E. coli growing in glucose-limited chemostat 'radiated' into at least five distinct, co-existing genotypes with qualitatively different properties, coexisting in the chemostat, each prepared for different possible futures.Since we generally grow E. coli in excess concentrations of nutrients that support fast growth, we tend to study the growth-rate prioritizing genotypes in experiments.Only during nutrient limitation and slow growth rates are the other genotypes present at large enough, and therefore experimentally identifiable, fractions.

E. Coli and S. Cerevisiae REPRESENT A SPECIFIC CLASS OF MICROBES WHOSE FITNESS STRATEGY IS NOT NECESSARILY REPRESENTATIVE OF OTHER SPECIES
When focusing on the growth-prioritizing genotype, constrained maximization of the instantaneous growth rate often results in successful prediction of growth properties for model organisms, such as E. coli and S. cerevisiae.As discussed above, however, this predictive success does not necessarily imply a similar predictive success for other growth conditions or other microorganisms.Rather, we hypothesize that E. coli and S. cerevisiae, as well as other model organisms, belong to a specific class of microbes whose (primary) growth strategies prioritize rapid opportunistic growth over other cellular functions-a fact that makes them highly suitable for laboratory studies and growth under excess nutrient conditions as often encountered in laboratory experiments.Indeed, there are many known instances where optimization of the (instantaneous) growth rate fails to predict the growth properties of a given microbe.Examples include phototrophic microorganisms, such as microalgae and cyanobacteria, that have to coordinate their metabolism according to the diurnal rhythm of light availability, and exhibit anticipatory synthesis of storage compounds that are required for survival at night. [18]Such periodic changes in the nutrient environment already require us to go beyond the instantaneous growth rates as an optimization objective, and require us to consider the average growth rate over at least one diurnal period to obtain a meaningful proxy for fitness. [16,17]more fundamental challenge to the growth maximization paradigm, however, are organisms that grow slowly despite apparent optimal growth conditions and excess nutrients.Such organisms, known as obligate oligotrophs, persist in poor nutritional environments, but are not necessarily able to persist for extended periods in nutrient rich environments. [95]In this case, the failure to grow rapidly in nutrient rich environments is not a direct consequence of biophysical trade-offs and limited resources.Rather, the failure to grow in nutrient rich environments reflects the evolved growth strategy and hence the evolutionary history of these organisms.Obligate oligotrophs typically lack the suitable regulatory mechanisms that would allow them to take advantage of the growth opportunities in rich media -and may instead even be hampered by, for example, growth imbalances or the osmotic challenges that result from nutrient rich media. [95]e existence of evolved growth strategies like obligate oligotrophy begs the question under which conditions, and for which classes of microbes, constrained maximization of the instantaneous growth rate is a suitable objective function to predict and explain growth properties.To what extent, and under which conditions, do the optimization strategies implemented within RBA/ME-type models reflect actual microbial growth strategies?To answer this question, it is useful to consider a classification of microbial fitness strategies.

A CLASSIFICATION OF MICROBIAL GROWTH STRATEGIES BASED ON GRIME'S CSR FRAMEWORK
In the context of microbial ecology, the most well known conceptual framework to classify growth strategies is r/K-selection.The theory of r/K selection distinguishes between selection for rapid growth (r) versus selection for competition for limited resources (K).The former is typically advantageous in fluctuating and unpredictable environments, whereas the latter is advantageous in stable environments with population densities that are close to the carrying capacity of the respective environment.In our context, rapid opportunistic growth and growthprioritizing phenotypes, such as exhibited by E. coli and S. cerevisiae, represent an r-strategy, whereas obligate oligotrophs represent a Kstrategy.We note that microbes must not necessarily conform to a single strategy but may exhibit a combination of both strategies.
In our view, a more refined classification of microbial growth strategies can be obtained by adopting Grime's competitor-stress-ruderal (CSR) framework. [96,97]The CSR framework was originally developed for classification of vegetation and plant growth, but was recently also applied to microbial ecology. [98,99]In the following, we propose a slightly modified framework that reflects the concept of resource allocation in microbial systems biology.Following Grime, [96,97] and as illustrated in Figure 7, we consider three primary microbial life strategies: the growth-prioritizing strategy (R), the competitive resource acquisition strategy (C), and the stress-tolerance strategy (S).
The stress-tolerance (S) strategy is expected to emerge in stable, predictable but very challenging environments, characterized by, for example, high salinity, high acidity, high temperatures, or extended periods of drought.To persist in such environments requires microbes to invest significant resources into stress tolerance strategies, such as synthesis of osmolytes or repair proteins to maintain cellular integrity.
These investments are at the expense of the instantaneous growth rate and result in slow growth.In principle, the S-strategy can be readily predicted using RBA/ME-type models.A challenge, however, is that we currently often lack quantitative information about the cellular cost, trade-offs, and benefits related to stress tolerance, and hence cellular investment into stress tolerance is typically included within the Q fraction (and hence provided as a parameter), and not predicted by including the respective constraints and benefits as trade-offs within the model.Within stable but very challenging environments, we also do not expect significant investment into anticipatory proteins (notwithstanding periodic cycles that may require us to consider the average growth rate over a certain time intervals).
The competitive resource acquisition strategy (C-strategy) is likewise associated with predictable and stable ('settled') environments.

F I G U R E 7
A framework for microbial growth strategies, based on Grime's CSR triangle.Microbes may exhibit a stress-tolerance strategy (S), characterized by high investment into stress-related proteins, a competitive resource acquisition strategy (C), characterized by high investment into nutrient acquisition and high growth yield, or a growth-prioritizing strategy (R), characterized by high growth rate, even at the expense of yield, high investment into growth-related proteins, as well as expression of anticipatory proteins and overcapacity in times of nutrient scarcity.Organisms may adopt mixed strategies or may switch between strategies.
While the definition is not restricted to a particular nutrient environment, we conjecture that such environments are typically oligotrophic: a 'settled' and stable environment with high nutrient availability would be inherently unstable and inevitable give rise to an increasing population density and decreasing nutrient availability.Example of environments associated with the C-strategy are the global oligotrophic oceans.To persist in such environments requires microbes to prioritize nutrient uptake and growth yield over growth rate.We expect that C-strategies in stable oligotrophic environments are characterized by significant investments into nutrient acquisition, for example highaffinity transporters and the synthesis of siderophores-metabolically costly molecules for the uptake of iron.Different to the S-strategy, the required protein machinery is directly related to metabolism and growth, and hence represents processes and trade-offs that are already represented within RBA/ME-type models.Similar to the Sstrategy, and notwithstanding short-term periodic cycles, we do not expect significant investment into anticipatory proteins.Therefore we expect that RBA/ME-type models are well suited to describe the growth of microbes that exhibit the C-strategy, as long as predictions are restricted to their native environment.In particular, the respective growth and protein expression strategies are typically specific for the respective native environment, and organisms that evolved a Cstrategy cannot be expected to maximize their instantaneous growth rate in a different, nutrient-rich, environment.
Finally, the growth-prioritizing strategy (R) is associated with growth in fluctuating feast-famine environments with unpredictable nutrient availability.Persisting in such environments requires rapid opportunistic growth whenever environmental conditions allow.
Organisms that exhibit an R-strategy can be expected to adapt to different habitats and are likely to survive and grow also in environments that are different from the environment they originally evolved in, making them suitable for, for example, cultivation in a laboratory.While these organisms encounter and persist in extended periods of nutrient starvation, they do not specialize or thrive under such condition.Given the asymmetry in 'growth opportunities' illustrated in the previous sections (Figure 5), we expect R-strategists to not (fully) adapt to low growth conditions, but rather to persist and prepare until conditions improve-resulting in (true) overcapacity of growth-related proteins.
Hence, we expect RBA/ME-type models to be highly predictive when describing the 'feast' phase of the R-strategy, but to perform less successful when aiming to describe the 'famine' phase.In particular, we emphasize that, even though environmental conditions may seem similar, the physiological response of an R-strategist during famine is fundamentally different from the phenotype of a C-strategist in an oligotrophic environment.While a C-strategist specializes in low nutrient conditions, an R-strategist exhibits persistence and preparedness, and hence increased expression in anticipatory and stress-related proteins.
This view is also consistent with the observation, discussed in the previous section, that for E. coli slow growth seems to be associated with 'stress' rather than nutrient limitation.
We conjecture that E. coli and S. cerevisiae are prime examples of Rstrategist-and the predictive success of RBA/ME-type models under conditions of nutrient excess is a consequence of this fact.However, we again note that, as for r/K-selection, organisms are not necessarily confined to a single strategy but may exhibit a continuum of strategies.Switches between strategies may be the result of point mutations in key regulatory genes or other mechanisms.The latter are expected to also incur a cellular cost in terms of the required sensing-anddecision machinery that is currently not represented within RBA/ME type models.
We suggest that the classification discussed above and illustrated in Figure 7, while likely not exhaustive, can guide under which conditions, and for which classes of organisms, we can expect current modeling approaches based on RBA/ME-type models to provide reasonable predictions of an organism's phenotype.

CHALLENGES FOR PREDICTIVE MODELING IN MICROBIAL SYSTEMS BIOLOGY
Constraint-based modeling based on optimal allocation of limited cellular resources holds great promise.Going forward, and taking into account the caveats discussed in the previous sections, we outline some of the major challenges for predictive modeling in microbial systems biology.
Firstly, many current modeling frameworks based on optimal resource allocation do not yet incorporate kinetic properties of enzymes.While mechanistic predictive whole-cell models have already been constructed for selected model organisms, in particular M. genitalium [100] and E. coli, [101] albeit still subject to significant caveats, fully kinetic large-scale descriptions of microbial growth are unlikely to be feasible within the near future.Lack of information about kinetic parameters also affects the parameterization of RBA/ME-type models which can be sensitive to kinetic parameters (in particular catalytic rate constants).Improved estimations of kinetic parameters, increasingly also taking protein structure into account, will likely improve model predictions. [102,103]condly, current constraint-based analyses are predominantly focused on stationary (balanced) growth conditions.As a consequence, current constraint-based models struggle to predict anticipatory gene expression, and predictions do not reflect growth opportunities and trade-offs that result from preparing for better growth conditions.][73] However, these are, as yet, typically computationally expensive and not widely applied.Different from dynamic FBA (dFBA, [104] ), such dynamic framework explicitly consider the average fitness, in the sense of Box 1, over an extended period of time.Time-dependent frameworks give rise to novel insights into microbial physiology.For example, the analysis of phototrophic metabolism over a full diurnal cycle using a genome-scale RBA/MEtype model of cyanobacterial growth [17] exhibits time-dependent metabolic partitioning during the day and anticipatory synthesis of glycogen as a storage compound during the light period-in agreement with experimental data.Using a similar dynamic resource allocation framework, da Silva et al. [105] studied polyphosphate accumulating organisms (PAOs) in an environment where oxygen is periodically unavailable, and demonstrate an emergent succession of metabolic phenotypes, and how storage metabolism allows for different tradeoffs between growth yield, robustness and competitiveness.We expect that expanding the use of time-dependent constraint-based analysis will allow us to better understand principles of optimal resource allocation in variable environments.Time-dependent frameworks explicitly consider the average growth rate, and hence long-term fitness, and therefore will allow us to better understand the trade-offs that determine metabolic strategies in the sense of Figure 7.
A third challenge in current constraint-based analysis relates to unknown trade-offs in protein expression.As discussed above, proteins involved in processes not directly related to cellular growth, such as information processing, motility, stress tolerance, or damage repair, are currently often summarily included with the Q fraction of protein expression.Predictions are then carried out with respect to an assumed or measured Q fraction.We envision that modeling approaches can be refined such that an increasing number of processes can be incorporated into quantitative resource-allocation frameworks.For example, the expression of osmolytes, which can be a substantial fraction of cellular dry mass, is a direct consequence of the osmotic pressure of the medium, hence their synthesis and benefits can be included as a quantitative trade-off into constraintbased analysis.In general, incorporating processes into a quantitative resource-allocation framework requires us to know the synthesis cost of the respective compounds, as well as a description of their benefits in quantitative terms.While the former is often readily available, the latter is still often difficult to assess.08] We expect that the inclusion of such processes and trade-offs will result in an increasingly accurate prediction of cellular phenotypes.
A fourth challenge is to incorporate phenotypic heterogeneity and genetic variation in current constraint-based analysis.As outlined above, bet-hedging and population-derived properties are likely major aspects to understand microbial optimality beyond the laboratory environment.While computational modeling platforms to consider microbial growth in time and space are available. [109]the consequences and implications for cellular resource-allocation are still poorly understood. [110]As a simple approach, efforts to include metabolic heterogeneity into the analysis of RBA/ME-type models could also borrow concepts from frameworks such as EnsembleFBA, [111] that is, simulating an ensemble of (slightly different) models rather than a single instance.
A fifth challenge is to gather quantitative experimental evidence related to long-term fitness also for non model organisms.In addition to meticulous experimental studies concerning individual tradeoffs and constraints, novel methods, such as CRISPR interference screens, [112] allow us to uncover (hypotheses) about trade offs between the instantaneous growth rate and other properties on a genome-scale, and are increasingly applicable to non model organisms.
Likewise, adaptive laboratory evolution (ALEs) is a promising experimental method to use in combination with resource allocation models.
For instance, the constraints that limit growth rate in the models might be associated with adaptive, growth-rate-enhancing mutations that relieve those constraints. [74]Also, as discussed, mutations leading to genotypes that either have an enhanced or reduced preparatoryprotein expression strategy have been identified and rationalized in terms of models. [62]nally, computational models should increasingly aim to also incorporate limiting nutrients other than the primary carbon sources, such as nitrogen, sulfur, phosphorus and transition metals. [113]In particular in the context of microbial ecosystems ecology, elements other than carbon play a crucial role in shaping the physiology of microorganisms, [114] and ultimately determine the productivity of ecosystems. [115]

CONCLUDING REMARKS
Our main theme within this review is that constraint-based optimization and RBA/ME-type models have been remarkably successful in predicting growth properties of common microbial model organisms.While this predictive success is still subject to several caveats and, as yet, primarily holds for particular growth conditions and specific classes of organisms, the respective computational methods hold great promise.
The application of constraint-based optimization is predicated on the fact that all organisms are subject to a universal set of physical, biophysical and biochemical constraints that are applicable across all domains of life.The observed physiology and growth strategies of microbes, however, then reflects the (often insufficiently understood) evolutionary history of the respective organism and are governed by the specific trade-offs that emerge in the environment the organism evolved in.According to this view, microbial diversity is not a consequence of different constraints, but a consequence of trade-offs subject to universal constraints.
While the predictive success of constraint-based optimization currently primarily holds for a certain class of model organism and particular growth conditions, we expect that maximizing longterm fitness, in accordance with evolutionary theory, is a suitable objective function to understand microbial physiology.Current fallacies in the interpretation of optimization results can often be attributed to the use of instantaneous growth rate as a proxy for fitness.
We further suggest that growth-rate maximization should be considered with respect to the fraction of proteins whose trade-offs are explicitly described within the model, rather than the entire protein pool of the cell.Currently, such quantitative trade-offs are primarily known for proteins directly allocated to metabolism and (instantaneous) growth.Hence we observe that expression of metabolic proteins may be suboptimal relative to all biosynthetic resources of a cell but optimal with respect to the pool of biosynthetic resources allocated to growth.
From a practical perspective, we must also emphasize that predictions based on the assumption of optimality can serve as a 'gold standard' and, rather than necessarily suggesting 'suboptimality' when found to be incorrect, can often point us to missing or incorrectly described costs and benefits of proteins expression.
Finally, the examples, applications, and caveats discussed in this review indicate the far reaching consequences of limited biosynthetic resources for the long-term fitness strategies of bacteria.Understanding such strategies requires the integration of a range of concepts, including phenotypic heterogeneity and genetic variation in our theoretical description to acquiring experimental evidence about the quantitative trade-offs between the instantaneous growth rate and long-term fitness.Our current computational methods, while already successful in many specific application, must be further tailored in these directions.

BOX 2 :
The instantaneous growth rate and cellular protein contentThe instantaneous growth rate decreases with increasing expression of non growth-related proteins.Under conditions of balanced growth, and neglecting protein degradation, the rate v P of protein synthesis (measured in gram proteins per gram cellular dry mass per time) must match the growthinduced dilution of the total protein content P T (measured in gram protein per gram cellular dry mass).Hence, using v P =  P T , the instantaneous growth rate at balanced growth is given as, enzymes and still neglect several important aspects of cellular physiology, many prediction are in excellent agreement with experimental F I G U R E 3 FBA puts constraints on the feasible flux space.Starting with the mass balance constraint, N v = 0, fluxes are further constraint by upper and lower bounds (green dashed lines).To obtain predictions for the reactions rates, FBA makes use of a (linear) objective function, typically the synthesis of biomass v BOF = c T v (red line).Variations in the environment are represented by different flux bounds.In RBA/ME-type models, individual flux bounds are replaced by trade-offs in resource allocation, such that increasing the capacity in one flux necessarily decreases the capacity in other fluxes.Variations in the environment are represented by different weights of the capacity constraints.

Figure 5
shows the instantaneous growth rate of the model as a function of external nutrient concentrations for cells adapted to high and low nutrient conditions, F I G U R E 4 A coarse-grained model of microbial growth.(A) The model consists of 4 intracellular variables: 3 proteins complexes (M, R, Q) are translated from an intracellular nutrient a. (B) Using the maximization of the instantaneous growth rate as a proxy for fitness, we obtain the (maximal) growth rate  as a function of the (relative) external nutrient concentration.(C) Cellular protein mass fractions for two different external nutrient concentrations.(D) The change in the cellular protein mass fractions as a function of the instantaneous growth rate .Growth rate maximization gives rise to (optimal) protein mass fractions that are linear functions of the growth rate .(E) The instantaneous growth rate  as a function of the mass fraction Φ M .An optimal expression level of M exists (indicated by the red dot), such that expression of the metabolic protein M maximizes the growth rate.The figure recapitulates the experimental results shown in Figure 2. (F) Linear reduction of the instantaneous growth rate as a function of an increasing mass fraction Φ Q of non growth-related quota proteins (Q fraction).Shown are results for three different nutrient environments.(G) The (maximal) instantaneous growth rate as a function of (relative) nutrient availability for two different values of Φ Q .The absolute value of the reduction of the instantaneous growth rate due to the expression of Q proteins is higher at high growth rates.a x ∕K M = 10.0 and a x ∕K M = 0.1, respectively.Under low nutrient conditions, (mis-)adaptation to high nutrient conditions implies excess ribosomes at the expense of metabolic proteins, and hence a reduced growth rate.After an upshift of nutrient levels, however, these excess ribosomes provide a major growth advantage, compared to adaptation to low nutrients.Vice versa, adaptation to low nutrient condition will (slightly) increase the instantaneous growth rate under these conditions, but at the cost of a major disadvantage after a sudden upshift of nutrient levels.The predicted asymmetry suggests that adaptation to low nutrient conditions only provides a competitive advantage if the low nutrient condition persist over an extended period of time.For organisms that evolved and grow in fluctuating (feast-famine) environments, and hence experience periods of high nutrient availability, adaptation to high nutrient environments even under (momentary) low nutrient condition, and thus being prepared for rapid growth when conditions improve, has a competitive advantage.The argument inferred fromFigure 5 is similar to the argument already discussed in the context of non growth-related proteins (Q fraction, Figures molecular weight of a is  a = 180 gDM∕mol.The concentration of the external nutrient is measured relative to K M .Unless otherwise noted the mass fraction of the protein Q is Φ Q = 0.5.Parameters are arbitrary but chosen such that results are within biologically plausible ranges.