## ABSTRACT

The biological fitness of non-interacting unicellular organisms in constant environments is given by their balanced growth rate, i.e., by the rate with which they replicate their biomass composition. Evolutionary optimization of this growth rate occurred under a set of physicochemical constraints, including mass conservation, reaction kinetics, and limits on dry mass per volume (cellular capacity). Mathematical models that account explicitly for these constraints are inevitably nonlinear, and their optimization has been restricted to small, non-realistic cell models. Here, we show that states of maximal balanced growth are elementary flux modes of a related flux balance problem, i.e., deactivating any active reaction makes steady-state growth impossible. For any balanced growth state that corresponds to an elementary flux mode of an arbitrarily sized model, we provide explicit expressions for individual protein concentrations, fluxes, and growth rate; all variables are uniquely determined by the concentrations of metabolites and total protein. We provide explicit and intuitively interpretable expressions for the marginal fitness costs and benefits of individual concentrations. At optimal balanced growth, the marginal net benefits of each metabolite concentration and of total protein concentration equal the marginal benefit of the cellular capacity. Based solely on physicochemical constraints, our work unveils fundamental quantitative principles of balanced cellular growth, quantifies the effect of cellular capacity on fitness, and leads to experimentally testable predictions.

## I. INTRODUCTION

The defining feature of live is self-replication. For non-interacting unicellular organisms in constant environments, the rate of this self-replication is equivalent to their evolutionary fitness [1]: fast-growing cells outcompete those growing more slowly. Accordingly, natural selection has optimized the cellular composition of many microbes for maximal balanced growth rate in specific environments [2, 3], i.e., for the fastest possible reproduction of all cellular components in proportion to their abundances [4].

The variables of this evolutionary optimization are the abundances of the cellular metabolites and of the proteins and ribonucleic acids that catalyze their conversion into biomass. The corresponding boundary conditions are provided by the environment and by physicochemical constraints, including mass conservation, the kinetics of enzymatic and spontaneous reactions, and capacity constraints that limit cellular concentrations [3, 5–8].

Molenaar et al. [5] proposed a coarse-grained, mathematical model of balanced growth encapsulating the most important physicochemical constraints and the activity of up to seven biochemical reactions. Numerical optimization resulted in predictions that recovered qualitatively the growth-rate dependencies of cellular ribosome content, of cell size, and of the emergence of over-flow metabolism. Similar to other constraint-based approaches [9–11], this modelling paradigm does not consider regulatory constraints, but instead assumes that protein regulation evolved to implement the optimal state.

Due to the inclusion of non-linear enzyme kinetics, maximizing the growth rate is inevitably a non-linear optimization problem; this may explain why no attempts have been made to extend this approach to more detailed biochemical whole-cell models. Instead, “toy models” of 1-3 reactions were solved analytically to gain further qualitative understanding of systems-level effects, such as the increased cellular investment into ribosomes at faster growth [6–8, 12] and optimal gene regulation strategies [3, 7].

Alternative modeling strategies, such as flux-balance analysis (FBA) [9, 13], resource balance analysis (RBA) [10], and ME (metabolism and expression) models [11], can be viewed as simplifications of the balanced growth scheme [14]. These models are also based on the idea of growth rate optimization in a steady-state solution space defined by physicochemical and environmental constraints. However, they consider only linear constraints, and they ignore the influence of metabolite concentrations on reaction kinetics [9, 13] or approximate their effects through growth-rate dependent phenomenological scaling laws [10, 11, 15].

Fig. 1A shows a simple model of balanced cellular growth, where a transporter protein imports a nutrient (G), which is converted into precursors for proteins (AA) through an enzymatic reaction, and where the precursors are converted by a “ribosome” protein into the three proteins used as catalysts. Fig. 1B shows a slightly more involved model with cofactors, while Fig. 1C sketches an arbitrarily complicated balanced growth model. The boundary conditions and the requirement of balanced exponential growth define the system’s solution space; maximizing the growth rate results in quantitative prediction of reaction fluxes and cellular concentrations (biomass composition). Here, we develop a theoretical framework for the analysis of such systems.

## II. MODELS OF BALANCED EXPONENTIAL GROWTH

Our model assumes that in balanced growth, the cell increases exponentially in size, while the concentrations of all cellular components remain constant. In particular, we assume that the number of different membrane constituents per cell volume (and thus membrane composition and surface/volume ratio) remain constant in a given environment [5]. We do not explicitly model cell division; thus, our model can also be interpreted as describing the growth of a population of cells, with the simplifying assumption that individual cells have the same molecular composition [7]. To satisfy these assumptions, the net production rate of each molecular constituent must balance its dilution by growth,
where *x* denotes the concentration of a given component and *μ* is the cellular growth rate [7].

The mass conservation in chemical reaction networks such as in Fig. 1 is commonly described through a *stoichiometric matrix N*, where rows correspond to metabolites and each column describes the mass balance of one reaction, with negative entries for consumed and positive entries for produced metabolites [19]. Here, we will focus on matrices A that describe a network of active reactions, i.e., A is a sub-matrix of N that contains all columns *j* for reactions with flux *v _{j}* ≠ 0 and all rows for reactants

*i*involved in these reactions either as substrates or as products. Note that the activity of each reaction

*j*in A implies a positive concentration

*p*> 0 for the protein catalyzing

_{j}*j*; in our notation, catalytic proteins include not only enzymes, but also transporters and the ribosome.

For the development below, it will be convenient to express each concentration in units of mass concentration (mass per volume), which can be obtained from the corresponding number concentration by multiplying with the molecular mass. Accordingly, the entries of *A* are not stoichiometric coefficients but are mass fractions. We normalize the columns of *A* such that the negative entries (the mass concentration fraction consumed) sum to −1 and the positive entries (the mass concentration fraction produced) sum to +1; transport reactions do not have to be mass balanced, and thus one of these sums may have a smaller absolute value [19].

The mass conservation of each component in an active reaction network can then be stated in matrix notation as
where *a _{α}* is the mass concentration of reactant

*α*and

*P*is the total protein mass concentration, summed over all proteins

*j*,

The rate *v _{j}* of a biochemical reaction

*j*is the product of the concentration of its catalyzing protein

*p*and some kinetic function

_{j}*k*(

_{j}**) that depends on the concentrations**

*a**a*of active reactants,

_{α}We assume that the functional form of *k _{j}* (

**) and the (constant) kinetic parameters are known.**

*a**k*has units of [time]

_{j}^{−1}, and

*v*has units of [mass][volume]

_{j}^{−1}[time]

^{−1}.

*k*(

_{j}**) may depend on the mass concentrations of substrates, products, and other molecules**

*a**a*acting as inhibitors or activators. In the simplest case of reaction

_{α}*j*following irreversible Michaelis-Menten kinetics with a single substrate

*α*, with constant enzyme activity

*k*(in units of [time]

_{cat}^{−1}) and Michaelis constant

*K*(in units of [mass][volume]

_{m}^{−1}).

The final constraint considered here reflects the cellular requirement for a minimal amount of free water to facilitate diffusion [20, 21]. *E. coli* growth decreases when cellular free water content is reduced below standard conditions, eventually stopping altogether when free water disappears [21]. *E. coli* ‘s buoyant density [22] and cellular water content [21] depend only on external osmolarity; at fixed external osmolarity, buoyant density remained constant even when experimenters induced drastic changes in cell mass and macromolecular composition [23].

Accordingly, we assume here that the cellular dry weight per volume is limited to *ρ*, where *ρ* is determined by external osmolarity. We express this capacity constraint as

For simplicity of notation, we use the following conventions: {*α*} is the set of all reactants in the active stoichiometric matrix *A*, and *Σ _{α}* indicates that we sum overall

*α ∈*{

*α*}. We use corresponding notations for the sets of

*basis reactants*{

**β**} and

*dependent reactants*{

*γ*} (see below). For an overview over the symbols used in this manuscript, see

**Suppl. Table S2**.

Based on biophysical considerations, we could replace Eq.(6) with separate capacity constraints on the total volume concentration inside each cellular compartment [20] and on the total area occupied by non-lipid membrane components per membrane area [5, 24]. An even simpler capacity constraint imposed in most previous models [3, 5–8, 10–12] is to fix total protein concentration *P* to a constant value. However, it has been shown that *P* decreases with increasing growth rate [23, 25]. Thus, while a constant *P* allows to simplify the presentation, Eq.(6) provides a more meaningful constraint; moreover, Eq.(6) allows us to determine the costs and benefits of varying the total protein concentration.

Each cellular state, defined through the (element-wise) positive concentration vectors ** p** > 0 and

**> 0 (and the corresponding flux vector v uniquely defined by the concentrations), is a balanced growth state if and only if it satisfies Eqs. (2), (4), and (6); the set of all such states forms the solution space of balanced growth. Mass conservation (Eq.(2)) and reaction kinetics (Eq.(4)) relate reaction fluxes to the concentration vector [**

*a**P*,

**]**

*a*^{T}in two fundamentally different ways; below, we will exploit this fact to eliminate the flux variables and to derive explicit expressions for

*p*and for

_{j}*μ*.

## III. CELLULAR STATE DEFINED BY THE CONCENTRATION VARIABLES

Our first aim is to derive a simple mathematical description of the solution space of balanced growth, with an emphasis on optimal states (i.e, states of maximal balanced growth rate *μ*). Let A be the active stoichiometric (sub-)matrix, ** v*** the flux vector, and

***:= [**

*x**P**,

***]**

*a*^{T}the concentration vector of such an optimal balanced growth state. In

**SI text VIII A**, we use results on constrained FBA problems [26, 27] to show that

*** is an**

*v**elementary flux mode*(EfM) [28] of a related FBA problem defined by

*A*together with a constant biomass vector

***.**

*x*The derivations below assume that *A* has full column rank. This will be the case if *A* is the active matrix of an EFM for any constant biomass vector *x* [29]; in particular, this is true for the active matrix of optimal balanced growth states (**SI text VIII A**; see also Ref. [30]).

*A* may have more rows than columns. It is convenient to decompose the linear system of equations represented by Eq.(2) into two parts, rearranging the rows of *A* into matrices *B*, *C* such that *#rows*(*B*) = *rank*(*B*) = *rank*(*A*),
where ** b** and

**are the reactant concentration vectors corresponding to the rows of**

*c**B*and

*C*, respectively.

*B*is identical to the reduced stoichiometric matrix in Ref. [31]. The relationship between

*A*and

*B*,

*C*can be understood in terms of Matroid theory, where the rows of

*B*form a

*basis*for the

*matroid*spanned by the rows of

*A*, and the set of rows of

*C*is the

*closure*for the set of rows of

*B*. If the choice for the partitioning of

*A*into

*B*and

*C*is not unique, some partitionings may be pathological and should be avoided (

**SI text VIIIB**).

*B* is a square matrix of full rank, so there is always a unique inverse *I*:= *B*^{−1}. Multiplying both sides of Eq.(7) by *I* from the left, we obtain

*I _{ji}* quantifies the proportion of flux

*j*invested into the dilution of component

*i*, and we thus name

*I*the

*investment*(or

*dilution*) matrix (see Fig. 1 for examples). In contrast to the stoichiometric matrix

*A*, which describes local mass balances (Eq.(2)),

*I*describes the structural allocation of reaction fluxes into the production of cellular components diluted by growth, and thus carries global, systems-level information.

By substituting *v* in Eq.(8) with Eq.(9), we obtain
where we defined the *dependence* matrix *D*:= *CI*. *D* describes the linear dependence of the *dependent concentrations* ** c** on

*P*and

**; it is identical to the link matrix in Ref. [31].**

*b*When *A* is not square, *B* includes a proper subset of the rows in *A*, and thus *B* on its own is not mass balanced. The “missing” mass fluxes are balancing ** c**, and hence the flux investment into c is already accounted for by Eq.(9).

We are now in a position to express growth rate as an explicit function of the concentrations [*P*, ** a**]

^{T}. As

*k*(

_{j}**) ≠ 0, we can use the kinetic equations (4) to express the individual protein concentrations as**

*a**p*=

_{j}*v*/

_{j}*k*(

_{j}**). Inserting**

*a**v*from the investment equation (9) gives

_{j}Substituting these expressions into the total protein sum, Eq.(3), we obtain

Below, we simplify the notation by writing *k _{j}*:=

*k*(

_{j}**). Solving for**

*a**μ*results in the

*growth equation*

Thus, if the active matrix *A* of a balanced growth state is full rank, there are unique and explicit mathematical solutions for ** p, v**, and

*μ*. In particular, this is the case for optimal states, as well as for all other states whose active matrix is the active matrix of an EFM for any constant biomass

**. If all**

*x**p*are positive (Eq.(11)), the corresponding cellular state is a balanced growth state; otherwise, no balanced growth is possible at these concentrations.

_{j}## IV. MARGINAL FITNESS CONTRIBUTIONS OF CELLULAR CONCENTRATIONS

In biological systems, costs and benefits should be expressed in terms of fitness effects. In situations where fitness *f* is determined exclusively by growth rate, a small change in growth rate *δμ* translates into a corresponding change in relative fitness (**SI text VIII C**) of

Accordingly, we define the *direct marginal net benefit* of the concentration *x _{i}* with

*i*∈ {

*P*,

*β*} (i.e.,

*x*∈ {

_{i}*P*,

*b*{) as the relative change in growth rate due to a small change in

_{β}*x*[32],

_{i}While we assume that the original state before the change in *x _{i}* respected the capacity constraint Eq.(6), we ignore the capacity constraint for the perturbed state in these definitions (i.e., we allow capacity to “adjust” to the change in

*x*and in any dependent concentrations

_{i}*c*).

_{γ}For *i* ∈ {*P*, *β*}, let us define

From the growth equation (13), it follows that and with

The summands in the denominator of the growth equation (13) can be expressed as *p _{j}/μ* =

*v*/(

_{j}*μk*) (Eq.(11)). Accordingly, quantifies the pro-portional increase of

_{j}*p*to help offset the increased dilution of component

_{j}*i*, and we thus call this the marginal (relative)

*production cost*incurred by the system via protein

*p*. If

_{j}*I*and

_{ji}*k*are both positive, then is also positive, i.e., it decreases fitness. The production costs are global, systems-level effects, quantified through the investment matrix

_{j}*I*.

Conversely, quantifies the proportion of protein *p _{j}* “saved” due to the change in kinetics associated with an increase in

*b*[33]. The benefit will generally be positive if

_{β}*β*is a substrate of reaction

*j*. We thus call the marginal (relative)

*kinetic benefit*of reactant

*β*to reaction

*j*. The kinetic benefit is a purely local effect, as it is non-zero only for reactants that affect the kinetics of reaction

*j*. Because fluxes are proportional to the concentrations of the proteins catalyzing the corresponding reactions, the marginal benefit of total protein in terms of relative fitness is simply

*P*

^{−1}.

Combining the two relationships for production cost and kinetic benefit, we see that the direct net benefit is the reduction of the protein fraction *p _{j}*/

*P*at constant

*μ*facilitated by the increase in

*x*,

The definition of accounts for the production costs of dependent reactants *c _{γ}* (as these are embedded in

*I*), but ignores their kinetic benefits. In analogy to Eq.(19), we define these as with the marginal (relative)

*kinetic benefit*of reactant

*γ*to reaction

*j*

We can now use the above relations to define the *total marginal net benefit η _{i}* of concentration

*i*∈ {

*P*,

*β*} as the relative change in growth rate due to a small change in

*x*and the resulting change in the concentrations of its dependent reactants

_{i}*c*, where the second equality follows from Eqs.(23), (10), (13) and definitions (21), (22).

_{γ}A change *δx _{i}* of

*x*(

_{i}*i*∈ {

*P*,

*β*}) causes a correlated change of each dependent concentration

*δc*=

_{γ}*D*(Eq.(10)). Thus, a change by

_{γi}δx_{i}*δx*results in a total change of the utilization of cellular capacity by

_{i}*κ*, with the

_{i}δx_{i}*capacity factor*

## V. OPTIMAL GROWTH AND THE BALANCE OF MARGINAL NET BENEFITS

At maximal growth rate, the cellular components will utilize the full cellular capacity *ρ* to saturate enzymes with their substrates, and thus the constraint in Eq.(6) will be active, that is, the inequality will become an equality.

To derive necessary conditions for any optimal balanced growth state at constant cellular capacity *ρ*, we use the method of Lagrange multipliers, which quantify the importance of the capacity constraint, Eq.(6), and of the constraints for the dependent reactants, Eq.(10), for the maximization of the objective function. The Lagrangian is a function of *P*, ** a**, and

*ρ*(

**SI text VIIID**).

Note that instead of using Lagrange multipliers, one could express the total protein concentration (Eq.(6)) and the dependent reactant concentrations (Eq.(10)) in terms of *ρ* and of the independent reactant concentrations ** b**. Substituting the resulting expressions in the growth equation (13) would result in an objective function that depends only on

*ρ*and

**, and that is constrained only by the requirement of positive concentrations. While this would lead to the same balance equations (26) as derived in the Lagrange multiplier framework, this formulation misses important insights that can be derived from the Lagrange multipliers themselves.**

*b*The maximal balanced growth rate *μ** will be a function of the cellular capacity *ρ*. In analogy to the marginal net benefits of cellular components, we define the *marginal benefit* of the cellular capacity as the fitness increase facilitated by a small increase in *ρ*,
where the second equality follows from the envelope theorem [34].

A necessary condition for optimal balanced growth is that all partial derivatives of with respect to the concentrations (*P,b _{β},c_{γ}*) and to the Lagrange multipliers (λ

_{ρ}, λ

_{γ}) are zero. As detailed in

**SI text VIIID**, this leads to the balance equations

The optimal state is perfectly balanced: the marginal net benefit of each independent cellular concentration *x _{i}* equals the marginal benefit of the cellular capacity, scaled by

*κ*to account for its total utilization of cellular capacity. If

_{i}*i*does not have any dependent reactants (∀

_{γ}

*D*= 0) or if

_{γi}*A*=

*B*, then the balance equation simplifies to .

Eq.(26) states that if the dry weight density *ρ* would be allowed to increase by a small amount, such as 1*μg*/*l*, then the marginal fitness gain that could be achieved by increasing protein concentration (plus dependent concentrations) by this amount is identical to that achieved by increasing the concentration of any reactant *β* (plus its dependent concentrations) by the same amount. In hindsight, this should not be surprising: if the marginal net benefit of concentration *x _{i}* (scaled by

*κ*) was lower than that of

_{i}*x*, growth rate could be increased by increasing

_{i′}*x*at the expense of

_{i′}*x*.

_{i}Eq.(26) together with Eq.(6) describes a system of *m* + 2 equations for *m* + 2 unknowns (with *m* the number of basis reactants *β*; **SI text VIIID**). Any state of optimal growth must satisfy these equations. In realistic cellular systems, this set of equations has a finite number of discrete solutions. Thus, solving the non-linear optimization problem described by Eq.(2) and the corresponding constraints may potentially be accelerated by instead searching for the solution of the balance equations. If the optimization problem is convex, the conditions given by Eq.(26) are necessary and sufficient, and the solution is unique.

## VI. QUANTITATIVE PREDICTIONS

A fully parameterized genome-scale balanced growth model could be used to predict all cellular concentrations at maximal growth rate. However, kinetic constants are currently lacking for many reactions even in the best studied model organisms [35]. We can still make quantitative predictions for the ribosome if we consider simplified models where the ribosome produces proteins from a single substrate, a generic ternary complex *AA* not consumed by other reactions (such as used in Ref. [36] and shown in Fig. 1A). In the balance equation , the costs on both sides largely cancel each other, as all reactions that contribute directly or indirectly to *AA* production contribute proportionally to protein production. The only cost not cancelled is (see, e.g., the balance equation for reactant “AA” in Fig.1A). Thus, only the kinetics of the ribosome *k _{R}* remain relevant, which we obtain from Ref. [36]. Using the mass balance of proteins,

*v*=

_{R}*μP*, we thus predict the optimal protein fraction of actively translating ribosomes,

*ϕ*=

_{R}*p*, at each growth rate (

_{R}/P**SI text VIIIE**), where

*r*is the mass fraction of the ribosome made up of protein, and

_{P}*k*,

_{cat}*K*are the kinetic parameters [36, 37].

_{m}Despite the simplicity of this model, the predicted *ϕ _{R}* is in good agreement with experimental values [16, 17] (Fig. 1D, solid red line). An approximation that ignores the dilution of intermediates (production costs) and bases its predictions only on the capacity

*ρ*(Fig. 1D, grey dashed line) results in less accurate predictions. However, this last approximation becomes better at lower growth rates, where the dilution of intermediates

*μa*becomes less and less important, and the optimal concentrations are increasingly determined by the capacity constraint. This finding explains why the assumption of a minimal utilization of cellular capacity by individual catalysts and their substrates provides a good approximation for the relationship between their concentrations [18].

_{α}To get a rough quantitative estimate of the marginal net benefits *η*, let us consider the simplest model of a complete cell, consisting of only a transport protein and the ribosome [3, 6] (Suppl. Fig. VIIIF). Based on the experimentally observed protein fraction of total dry weight *P/ρ* = 0.54 in *E. coli* [21], we estimate *ρ η _{ρ}* = 0.69 (

**SI text VIIIF**).

*ρ η*quantifies the relative change in the maximal growth rate

_{ρ}*μ** resulting from a small, relative change in

*ρ*. Thus, we estimate that a decrease in cellular dry weight density

*ρ*of 1% would lead to a 0.69% decrease in growth rate, indicating that the capacity constraint is indeed highly biologically significant.

*ρ* changes when external osmolarity is modified [21]. is the slope of the log-scale plot of *μ* vs. *ρ* at different external osmolarities. Increases in cellular dry weight density may have strong effects on diffusion and may hence change the kinetic constants. In contrast, reductions in *ρ* due to decreased external osmolarity are within the scope of our model, which assumes constant parameters for ** k**(

**). The very limited available experimental data (three data points from Ref. [38], Fig. VIIIF) suggests**

*a**ρ η*≈ 0.66, close to our rough estimate from the minimal cell model.

_{ρ}## VII. CONCLUSIONS

Our derivations are based on the insight that for any balanced growth state that corresponds to an EFM of the related FBA problem, the inverse *I* of the active stoichiometric matrix (or a basis thereof) contains global, systems-level information on the contribution of individual fluxes to the production of cellular constituents diluted by growth. Purely through structural constraints, this leads to an explicit dependence of reaction fluxes on the concentrations of the cellular constituents, scaled linearly by the growth rate (Eq.(9)). We combine this description with the complementary kinetic dependence of fluxes on concentrations. This allows us to provide explicit expressions for the individual protein concentrations ** p** and fluxes

**and for the growth rate**

*v**μ*as functions of arbitrary (positive) concentrations [

*P*,

**]**

*b*^{T}, and it provides the framework from which we derive the balance equations for the marginal net benefits of cellular concentrations at optimal growth.

Previous work has emphasized the central role of proteins in the cellular economy [3, 5–8, 10–12]. Whereas total protein mass concentration in real biological systems is indeed much higher than the mass concentration of any other cellular constituent *a _{α}*, the balance equations show that at optimal growth, their marginal net benefits are in fact equal, emphasizing the importance of explicitly accounting for all cellular constituents.

To make the presentation concise, our equations assume (i) that all proteins contribute to growth by acting as catalysts or transporters; (ii) that there is a 1-to-1 correspondence between proteins and reactions; (iii) that proteins are not used as reactants; and (iv) that all catalysts are proteins. It is straight-forward to remove these simplifications. E.g., assumption (i) can be removed by adding a sector of non-growth related proteins [15, 39] with concentration *Q* to the r.h.s. of Eq.(3); to remove assumption (iv), we can add different RNA species as cellular components and introduce reactions that combine proteins and RNA into molecular machines such as the ribosome.

An equation analogous to Eq.(9) can be formulated for non-growing cells (or cellular subsystems) that are instead optimized for the production of specific molecules, as is the case for many cell types in multicellular organisms. Instead of multiplying a dilution term, the production matrix *I* would then multiply a weighted *production vector* representing the desired output. The same strategy might be used to accelerate FBA solutions in a given EFM: FBA studies cellular growth, but assumes a fixed biomass composition [*P*, ** a**]

^{T}.

In principle, exploitation of the balance equations (Eq.(26)) may allow the numerical optimization even for cellular systems of realistic size, encompassing hundreds of protein and reactant species. One remaining obstacle to the accurate formulation of such models, though, is the current incompleteness of the kinetic constants needed to parametrize the functions ** k**(

**) [35]. The need for the development of high-throughput methods to systematically ascertain these parameters has been recognized [35]; in the meantime, methods from artificial intelligence may provide reasonable approximations [40].**

*a*As an alternative to genome-scale models, the balanced growth theory developed here could be applied to coarsegrained cellular models of increasing complexity, parameterized from experimental data. One would start from minimal models with two reactions [3, 6], proceeding to models comprising the six previously described sectors of the cellular economy [39] and beyond [41].

Our work extends the ad-hoc optimizations of toy models [3, 5–8] into a full-fledged theory of balanced growth. We show that the balanced growth framework allows general, quantitative insights into cellular resource allocation and physiology, as exemplified by the growth and balance equations. Application and further development of this theory may foster an enhanced theoretical understanding of how physicochemical constraints determine the fitness costs and benefits of cellular organization. Moreover, the explicit expressions for the (marginal) costs and benefits of cellular concentrations in terms of fitness provide a rigorous framework for analyzing the cellular economy. We anticipate that this approach will prove fruitful in the interpretation of natural and laboratory evolution, and in optimizing the design of synthetic biological systems.

## ACKNOWLEDGEMENTS

We thank Johannes Berg, Oliver Ebenhöh, Xiao-Pan Hu, Terry Hwa, Michael Lässig, Wolfram Liebermeister, and Deniz Sezer for discussions. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through grants IRTG 1525, CRC 680, CRC 1310, and, under Germany’s Excellence Strategy, through grant EXC 2048/1 (Project ID: 390686111).

## Footnotes

* Main text + SI