We begin by formulating a simplified model of growth which follows the flow of mass from nutrients in the environment to biomass by building upon and extending the general logic of low-dimensional resource allocation models (Molenaar et al., 2009; Scott et al., 2010; Scott et al., 2014; Dai et al., 2016; Giordano et al., 2016). Specifically, we focus on the accumulation of protein biomass, as protein constitutes the majority of microbial dry mass (Churchward et al., 1982; Feijó Delgado et al., 2013) and peptide bond formation commonly accounts for ≈80% of the cellular energy budget (Stouthamer, 1973; Belliveau et al., 2021). Furthermore, low-dimensional allocation models utilize a simplified representation of the proteome where proteins can be categorized into only a few functional classes (Molenaar et al., 2009; Scott et al., 2014; Hui et al., 2015; Maitra and Dill, 2015; Dourado and Lercher, 2020). In this work, we consider proteins to be either ribosomal (i.e., a structural component of the ribosome, excluding ternary complex members like EF-Tu), metabolic (i.e., enzymes catalyzing synthesis of charged-tRNA molecules from environmental nutrients), or being involved in all other biological processes (e.g., lipid synthesis, DNA replication, energy generation, and chemotaxis) Molenaar et al., 2009; Scott et al., 2010; Scott et al., 2014; Hui et al., 2015; Figure 1—figure supplement 1; in Appendix 1 What makes the fraction of ‘other’ proteins?, we outline in more detail how individual protein species are partitioned between the ‘metabolic’ and ‘other’ sectors depending on their functional annotations. Simple allocation models further do not distinguish between different cells but only consider the overall turnover of nutrients and biomass. To this end, we explicitly consider a well-mixed batch culture growth as reference scenario where the nutrients are considered to be in abundance. This low-dimensional view of living matter may at first seem like an unfair approximation, ignoring the decades of work interrogating the multitudinous biochemical and biophysical processes of cell-homeostasis and growth (Macklin et al., 2020; Karr et al., 2012; Hui et al., 2015; Grigaitis et al., 2021; Noree et al., 2019). However, at least in nutrient replete conditions, many of these processes appear not to impose a fundamental limit on the rate of growth in the manner that protein synthesis does (Belliveau et al., 2021). In Appendix 1 The major simplifications of low-dimensional allocation models and why they might work we discuss this along with other simplifications in more detail.

To understand protein synthesis and biomass growth within the low-dimensional allocation framework, consider the flux diagram (Figure 1A, Molenaar et al., 2009; Giordano et al., 2016; Belliveau et al., 2021; Balakrishnan et al., 2021b; Scott et al., 2014) showing the masses of the three protein classes, precursors which are required for protein synthesis (including charged-tRNA molecules, free amino acids, cofactors, etc.), nutrients which are required for the synthesis of precursors, and the corresponding fluxes through the key biochemical processes (arrows). This diagram emphasizes that growth is autocatalytic in that the synthesis of ribosomes is undertaken by ribosomes which imposes a strict speed limit on growth (Dill et al., 2011; Belliveau et al., 2021; Kafri et al., 2016). While this may imply that the rate of growth monotonically increases with increasing ribosome abundance, it is important to remember that metabolic proteins are needed to supply the ribosomes with the precursors needed to form peptide bonds. Herein lies the crux of ribosomal allocation models: the abundance of ribosomes is constrained by the need to synthesize other proteins and growth is a result of how new protein synthesis is partitioned between ribosomal, metabolic, and other proteins. How is this partitioning determined, and how does it affect growth?

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To answer these questions, we must understand how these different fluxes interact at a quantitative level and thus must mathematize the biology underlying the boxes and arrows in Figure 1A. Taking inspiration from previous models of allocation (Molenaar et al., 2009; Scott et al., 2010; Scott et al., 2014; Giordano et al., 2016; Dourado and Lercher, 2020), we enumerate a minimal set of coupled differential equations which captures the flow of mass through metabolism and translation (Figure 1B, with the dimensions and value ranges of the parameters listed in Figure 1C and Supplementary file 1). While we present a step-by-step introduction of this model in ‘Methods,’ we here focus on a summary of the underlying biological intuition and implications of the approach.

We begin by codifying the assertion that protein synthesis is key in determining growth. The synthesis of new total protein mass $M$ depends on the total proteinaceous mass of ribosomes $M_{Rb}$ present in the system and their corresponding average translation rate $\gamma$ (Figure 1Bi). As ribosomes rely on precursors to work, it is reasonable to assert that this translation rate must be dependent on the concentration of precursors $c_{pc}$ such that $\gamma \equiv \gamma (c_{pc})$ (Scott et al., 2014; Giordano et al., 2016), for which a simple Michaelis–Menten relation is biochemically well motivated (Figure 1Bii). With changing precursor concentrations, the translation rate $\gamma$ varies between a maximum value ${\gamma }_{max}$, representing rapid synthesis, and a minimum value ${\gamma }_{min}$, representing the slowest achievable translation rate. In our model, this minimum rate ${\gamma }_{min}$ is zero and corresponds to the condition where there are no available precursors to support translation. The standing precursor concentration $c_{pc}$ is set by a combination of processes (Figure 1Biii), namely the production of new precursors through metabolism (synthesis), their degradation through translation (consumption), and their dilution as the total cell volume grows. The synthesis is driven by the abundance of metabolic proteins $M_{Mb}$ in the system and the speed by which they convert nutrients into novel precursors. As the metabolic networks at play are complex, low-dimensional allocation models describe the process of metabolism using an average metabolic rate $\nu$ in lieu of mathematicizing the network’s individual components. As such, the metabolic rate is difficult to directly measure but generally depends on the quality and concentration of nutrients in the environment (see below, Figure 1—figure supplement 2 and ‘Methods’). In the following, we focus on a growth regime in which nutrient concentrations are saturating. In such a scenario, metabolism operates at a nutrient-specific maximal metabolic rate $\nu \equiv {\nu }_{max}$. Finally, the relative magnitude of the ribosomal, metabolic, and ‘other’ protein masses is dictated by ${\varphi }_{Rb}$, ${\varphi }_{Mb}$, and ${\varphi }_O$, three allocation parameters which range between zero and one to describe the fraction of ribosomes being utilized in synthesizing the corresponding protein pools. Importantly, as ribosomes only translate one protein at a time, the allocation parameters follow the constraint ${\varphi }_{Rb}+{\varphi }_{Mb}+{\varphi }_O=1$ (Figure 1Biv). For readers familiar with allocation models, we emphasize that we here use ${\varphi }_X$ to denote allocation parameters rather than mass fractions, $M_X/M$; both quantities are only equivalent in the steady-state regime. Together, the introduced equations provide a full mathematicization of the mass flow diagram shown in Figure 1A.

For constant allocation parameters (${\varphi }_{Rb}^{*},{\varphi }_{Mb}^{*}$), a steady-state regime emerges from this system of differential equation. Particularly, the precursor concentration is stationary in time ($c_{pc}=c_{pc}^{*}$), meaning the rate of synthesis is exactly equal to the rate of consumption and dilution. Furthermore, the translation rate $\gamma (c_{pc}^{*})$ is constant during steady-state growth and the mass abundances of ribosomes and metabolic proteins are equivalent to the corresponding allocation parameters, e.g. $\frac{M_{Rb}}{M}\equiv {\varphi }_{Rb}^{*}$. As a consequence, biomass is increasing exponentially $\frac{dM}{dt}=\lambda M$, with the growth rate $\lambda =\gamma (c_{pc}^{*}){\varphi }_{Rb}^{*}$. The emergence of a steady state and analytical solutions describing steady growth are further discussed in Figure 1—figure supplement 2 and Figure 1—figure supplement 3. Notably, dilution is important to obtain a steady state as has been highlighted previously by Giordano et al., 2016 and Dourado and Lercher, 2020 but is often neglected (Appendix Precursors concentrations and the importance of dilution by cell growth).

Figure 1D and E show how the steady-state growth rate $\lambda$ and translation rate $\gamma (c_{pc}^{*})$ are dependent on the allocation towards ribosomes ${\varphi }_{Rb}^{*}$. The figures also show the dependence on the metabolic rate ${\nu }_{max}$ which we here assert to be a proxy for the ‘quality’ of the nutrients in the environment (with increasing ${\nu }_{max}$, less metabolic proteins are required to obtain the same synthesis of precursors). The non-monotonic dependence of the steady-state growth rate on the ribosome allocation and the metabolic rate poses a critical question: What biological mechanisms determine the allocation towards ribosomes in a particular environment and what criteria must be met for the allocation to ensure efficient growth?

While cells might employ many different ways to regulate allocation, we here consider three specific allocation scenarios to illustrate the importance of allocation on growth. These candidate scenarios either strictly maintain the total ribosomal content (scenario I), maintain a high rate of translation (scenario II), or optimize the steady-state growth rate (scenario III). We derive analytical solutions for these scenarios (as has been previously performed for scenario III; Giordano et al., 2016; Dourado and Lercher, 2020; Figure 1F and ‘Methods’), and ultimately compare these predictions to observations with E. coli to show this organisms’ optimal allocation of resources.

The simplest and perhaps most näive regulatory scenario is one in which the allocation towards ribosomes is completely fixed and independent of the environmental conditions. This strategy (scenario I in Figure 1F, gray) represents a locked-in physiological state where a specific constant fraction of all proteins is ribosomal. This imposes a strict speed limit for growth when all ribosomes are translating close to their maximal rate, $\gamma (c_{pc}^{*})\approx {\gamma }_{max}$. If the fixed allocation is low (e.g., ${\varphi }_{Rb}^{(I)}=0.2$), then this speed limit could be reached at moderate metabolic rates.

A more complex regulatory scenario is one in which the allocation towards ribosomes is adjusted to prioritize the translation rate. This strategy (scenario II in Figure 1F, green) requires that the ribosomal allocation is adjusted such that a constant internal concentration of precursors $c_{pc}^{*}$ is maintained across environmental conditions, irrespective of the metabolic rate. In the case where this standing precursor concentration is large ($c_{pc}^{*}\gg K_M^{c_{pc}}$), all ribosomes will be translating close to their maximal rate.

The third and final regulatory scenario is one in which the allocation towards ribosomes is adjusted such that the steady-state growth rate is maximized. The analytical solution which describes this scenario (scenario III in Figure 1F) resembles previous analytical solutions by Giordano et al., 2016; Dourado and Lercher, 2020. More illustratively, the strategy can be thought of as one in which the allocation towards ribosomes is tuned across conditions such that the observed growth rate rests at the peak of the curves in Figure 1D. Notably, this does not imply that the translation rate is constantly high across conditions (as in scenario II). Rather, the translation rate is also adjusted and approaches its maximal value ${\gamma }_{max}$ only in very rich conditions (high metabolic rates). All allocation scenarios and their consequence on growth are discussed in further detail in Figure 1—figure supplement 4 and the corresponding interactive figure on the paper website (cremerlab.github.io/flux_parity).

Thus far, our modeling of microbial growth has remained ‘organism agnostic’ without pinning parameters to the specifics of any one microbe’s physiology. To probe the predictive power of this simple allocation model and test the plausibility of the three different strategies for regulation of ribosomal allocation, we performed a systematic and comprehensive survey of data from a vast array quantitative studies of the well-characterized bacterium E. coli. This analysis, consisting of 26 studies spanning 55 years of research (listed in Supplementary file 2 and as Figure 1—source data 1 and Figure 1—source data 2) using varied experimental methods, goes well beyond previous attempts to compare allocation models to data (Scott et al., 2010; Hui et al., 2015; Erickson et al., 2017; Giordano et al., 2016; Bosdriesz et al., 2015; Hu et al., 2020; Dourado and Lercher, 2020; Serbanescu et al., 2020; Hu et al., 2020; Roy et al., 2021; Maitra and Dill, 2015; Weiße et al., 2015).

These data, shown in Figure 1H and I (markers), present a highly consistent view of E. coli physiology where the allocation towards ribosomes (equivalent to ribosomal mass fraction in steady-state balanced growth) and the translation rate demonstrate a strong dependence on the steady-state growth rate in different carbon sources. The pronounced correlation between the allocation towards ribosomes and the steady-state growth rate immediately rules out scenario I, where allocation is constant, as a plausible regulatory strategy used by E. coli, regardless of its precise value. Similarly, the presence of a dependence of the translation speed on the growth rate rules out scenario II, where the translation rate is prioritized across growth rates and maintained at a constant value. The observed phenomenology for both the ribosomal allocation and the translation speed is only consistent with the logic of regulatory scenario III where the allocation towards ribosomes is tuned to optimize growth rate.

This logic is quantitatively confirmed when we compute the predicted dependencies of these quantities on the steady-state growth rate for the three scenarios diagrammed in Figure 1F based on literature values for key parameters (outlined in Supplementary file 1). Deviations from the prediction for scenario III are only evident for the ribosomal content at very slow steady growth ($\lambda \le 0.5$ hr^-1), which are hardly observed in any ecologically relevant conditions and can be attributed to additional biological and experimental factors, including protein degradation (Calabrese et al., 2021) and cultures which have not yet reached steady state, factors we discuss in Appendix 1 – Additional considerations relevant at slow growth. The inactivation of ribosomes is another such explanation, though a growth rate-independent inactive fraction is not sufficient to explain the observations, Appendix 1 —Inactive ribosomes.

Importantly, the agreement between theory and observations works with a minimal number of parameters and does not require the inclusion of fitting parameters. All fixed model parameters such as the maximum translation rate ${\gamma }_{max}$ and the Michaelis–Menten constant for translation $K_M^{c_{pc}}$ have distinct biological meaning and can be either directly measured or inferred from data (Supplementary file 1). Furthermore, we discuss the necessity of other parameters such as the ‘other protein sector’ ${\varphi }_O$ (Appendix 1— What makes the fraction of 'other' proteins?), its degeneracy with the maximum metabolic rate ${\nu }_{max}$, and inclusion of ribosome inactivation and minimal ribosome content (Appendix 1— Inactive ribosomes). We, furthermore, provide an interactive figure on the paper website (cremerlab.github.io/flux_parity) where the parametric sensitivity of these regulatory scenarios and the agreement/disagreement with data can be directly explored. Notably there is no combination of parameter values that would allow scenario I or II to adequately describe both the ribosomal allocation and the translation speed as a function of growth rate. These findings are in line with a recent higher-dimensional modeling study (Hu et al., 2020), which, based on the optimization of a reaction network with >200 components, rationalized the variation in translation speed with growth as a manifestation of efficient protein synthesis. Together, these results confirm that scenario III can accurately describe observations over a very broad range of conditions, in strong support of the popular but often questioned presumption that E. coli optimally tunes its ribosomal content to promote fast growth (Giordano et al., 2016; Bosdriesz et al., 2015; Towbin et al., 2017).

In Appendix 1 – Application of the model to Saccharomyces cerevisiae, we present a similar analysis for yeast, which, in line with previous studies (Metzl-Raz et al., 2017; Xia et al., 2021; Paulo et al., 2015; Paulo et al., 2016; Kostinski and Reuveni, 2021), suggests that this eukaryote likely follows a similar optimal allocation strategy, although data for ribosomal content and the translation rate is scarce. The strong correlation between ribosome content and growth rate has further been reported for other microbial organisms in line with an optimal allocation (Karpinets et al., 2006; Jahn et al., 2018; Zavřel et al., 2019; Jahn et al., 2021), though the absence of translation rate measurements precludes confirmation. An interesting exception is the methanogenic archaeon Methanococcus maripaludis, which appears to maintain constant allocation, in agreement with scenario I (Müller et al., 2021). The presented analysis thus suggests that E. coli and possibly many other microbes closely follow an optimal ribosome allocation behavior to support efficient growth. Moreover, the good agreement between experiments and data establishes that a simple low-dimensional allocation model can describe growth with notable quantitative accuracy. However, this begs the question: how do cells coordinate their complex machinery to ensure optimal allocation?

To optimize the steady-state growth rate, cells must have some means of coordinating the flow of mass through metabolism and protein synthesis. In the ribosomal allocation model, this reduces to a regulatory mechanism in which the allocation parameters (${\varphi }_{Rb}$ and ${\varphi }_{Mb}$) are dynamically adjusted such that the metabolic flux to provide new precursors ($\nu (c_{nt}){\varphi }_{Mb}$) and translational flux to make new proteins ($\gamma (c_{pc}^{*}){\varphi }_{Rb}$, equivalent to the steady-state growth rate $\lambda$) are not only equal, but are mutually maximized. Such regulation therefore requires a mechanism by which both the metabolic and translational flux can be simultaneously sensed.

Thus far, we have referred to the end product of metabolism as ambiguous ‘precursors’ which are used by ribosomes to create new proteins. In reality, these precursors are tRNAs charged with their cognate amino acids. One can think of metabolism as a two-step process where (i) an amino acid is synthesized from environmental nutrients and (ii) an amino acid is attached to the appropriate uncharged-tRNA. As we assume that nutrients are in excess in the environment, we make the approximation that nutrients in the environment are saturating such that $c_{nt}\gg K_M^{c_{nt}}$ and the metabolic rate $\nu$ now depends solely on the concentration of uncharged-tRNAs $\nu ({\text{tRNA}}^u)$. This enforces some level of regulation of metabolism; if the uncharged tRNA concentration is too low, the rate of metabolism slows and does not add to the already large pool of charged tRNA. But when charged-tRNA is available, translation occurs at a rate $\gamma ({\text{tRNA}}^c)$, forming new protein biomass and converting a charged-tRNA back to an uncharged state. This process is shown by gray arrows in Figure 2A.

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To describe the state-dependent adjustment of the allocation parameters (${\varphi }_{Rb}$ and ${\varphi }_{Mb}$), we further include in this feedback loop a regulatory system we term a ‘flux-parity regulator’ (Figure 2A, red), which controls the allocation parameters in response to relative changes in the concentrations of the two tRNA species. Together, the arrows in Figure 2 represent a more fine-grained view of a proteinaceous self replicating system, yet maintains much of the structural minimalism of the simple ribosomal allocation model without requiring explicit consideration of different types of amino acids (Bosdriesz et al., 2015), inclusion of their myriad synthesis pathways (Hu et al., 2020), or reliance on observed phenomenology (Wu et al., 2022).

The boxes and arrows of Figure 2A can be mathematized to arrive at a handful of ordinary differential equations (Figure 2B) structurally similar to those in Figure 1B. At the center of this model is the ansatz that the ribosomal allocation ${\varphi }_{Rb}$ is dependent on the ratio of charged- and uncharged-tRNA pools and has the form

where the ratio $\frac{{\text{tRNA}}^c}{{\text{tRNA}}^u}$ represents the ‘charging balance’ of the tRNA and $\tau$ is a dimensionless sensitivity parameter which defines the charging balance at which the allocation towards ribosomes is half-maximal. Additionally, we make the assertion that the synthesis rate of new uncharged-tRNA via transcription $\kappa$ is coregulated with ribosomal proteins (Skjold et al., 1973; Dong et al., 1996) and has a similar form of

where ${\kappa }_{max}$ represents the maximal rate of tRNA transcription relative to the total biomass.

Numerical integration of this system of equations reveals that the flux-parity regulation is capable of optimizing the allocation towards ribosomes, ${\varphi }_{Rb}$, such that the metabolic and translation fluxes are mutually maximized (Figure 2D), thus achieving optimal allocation. Importantly, the optimal behavior inherent to this regulatory mechanism can be attained across a wide range of parameter values for the charging sensitivity $\tau$ and the transcription rate ${\kappa }_{max}$, the two key parameters of flux-parity regulation (Figure 2C). Moreover, the emergent optimal behavior of this regulatory scheme occurs across conditions without the need for any fine-tuning between the flux-parity parameters and other parameters. For example, the control of allocation via flux-parity regulation matches the optimal allocation (scenario III above) when varying the metabolic rate ${\nu }_{max}$ (Figure 2E and Appendix 1 – Parameter dependence of the flux-parity model).

The theoretical analysis presented in Figure 2 suggests that a flux-parity regulatory mechanism may be a simple way to ensure optimal ribosomal allocation that is robust to variation in the key model parameters. To test if such a scheme may be implemented in E. coli, we compared the behavior of the steady-state flux-parity regulatory circuit within physiological parameter regimes to steady-state measurements of ribosomal allocation and the translation rate as a function of the growth rate (Figure 3A and B). Remarkably, the predicted steady-state behavior of the flux-parity regulatory circuit describes the observed data with the same quantitative accuracy as the optimal behavior defined by scenario III, as indicated by the overlapping red and blue lines, respectively.

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Collated measurements of relative ppGpp concentrations.

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Collated measurements of excess protein mass fractions.

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While the flux-parity regulation scheme appears to accurately describe the behavior of E. coli, how are metabolic and translational fluxes sensed at a mechanistic level? Many bacteria, including E. coli, utilize the small molecule guanosine tetraphosphate (ppGpp) as a molecular indicator of amino acid limitation and has been experimentally shown to regulate ribosomal, metabolic, and tRNA genes through many routes, including directly binding RNA polymerase (Magnusson et al., 2005; Anderson et al., 2021; Potrykus and Cashel, 2008; Potrykus et al., 2011; Imholz et al., 2020) and plays an important role in other cellular processes, including cell size control (Büke et al., 2022). Mechanistically, ppGpp levels are enzymatically controlled depending on the metabolic state of the cell, with synthesis being triggered upon binding of an uncharged-tRNA into an actively translating ribosome. While many molecular details of this regulation remain unclear (Magnusson et al., 2005; Anderson et al., 2021; Potrykus and Cashel, 2008; Wu et al., 2022), the behavior of ppGpp meets all of the criteria of a flux-parity regulator. Rather than explicitly mathematicizing the biochemical dynamics of ppGpp synthesis and degradation, as has been undertaken previously (Bosdriesz et al., 2015; Giordano et al., 2016; Wu et al., 2022), we model the concentration of ppGpp being inversely proportional to the charging balance,

encompassing the fact that processes beyond allocation use ppGpp as an effector molecule. This ratio, mathematically equivalent to the odds of a ribosome binding an uncharged-tRNA relative to binding a charged-tRNA, is one example of a biochemically motivated ansatz that can be considered (‘Methods’) and provides a relative measure of the metabolic and translational fluxes.

With this approach, the amount of ppGpp present at low growth rates, and therefore low ribosomal allocation, should be significantly larger than at fast growth rates where ribosomal allocation is larger and charged-tRNA are in abundant supply. While our model cannot make predictions of the absolute ppGpp concentration, we can compute the relative ppGpp concentration to a reference state [ppGpp]₀ as

To test this, we compiled and rescaled ppGpp measurements of E. coli across a range of growth rates from various literature sources (Figure 3C and Figure 3—source data 1). The quantitative agreement between the scaling predicted by Equation 4 and the experimental measurements strongly suggests that ppGpp assumes the role of a flux sensor and enforces optimal allocation through the discussed flux-parity mechanism.

We find that the flux-parity allocation model is extremely versatile and allows us to quantitatively describe aspects of microbial growth in and out of steady state and under various physiological stresses and external perturbations with the same core set of parameters. Here, we demonstrate this versatility by comparing predictions to data for four particular examples using the same self-consistent set of parameters we have used thus far (Supplementary file 1). First, we examine the influence of translation-targeting antibiotics like chloramphenicol (Figure 3D) on steady-state growth in different growth media (Scott et al., 2010; Dai et al., 2016). By incorporating a mathematical description of ribosome inactivation via binding to chloramphenicol (described in ‘Methods’), we find that the flux-parity allocation model quantitatively predicts the change in steady-state growth and ribosomal content with increasing chloramphenicol concentration (Figure 3E, red shades). Furthermore, the effect on the translation speed is qualitatively captured (Figure 3F, red shades). The ability of the flux-parity allocation model to describe these effects without readjustment of the model and its core parameters is notable and provides a mechanistic rationale for previously established phenomenological relations (Scott et al., 2010; Dai et al., 2016).

As a second perturbation, we consider the burden of excess protein synthesis by examining the expression of synthetic genes (Figure 3G). A decrease in growth rate results when cells are forced to synthesize different amounts of the lactose cleaving enzyme $\beta$-galactosidase in different media lacking lactose (Figure 3H, red shades). The flux-parity allocation model (dashed lines) quantitatively predicts the change in growth rate with the measured fraction of $\beta$-galactosidase without further fitting (‘Methods’). The trends for different media (red shades) quantitatively collapse onto a single line (Figure 3I and Figure 3—source data 2) when comparing changes in relative growth rates, a relation which is also captured by the model (dashed black line) and is independent of the overexpressed protein (symbols). This collapse, whose functional form is derived in ‘Methods,’ demonstrates that the flux-parity allocation model is able to describe excess protein synthesis in general, rather than at molecule- or media-specific level.

As the flux-parity regulatory circuit responds to changes in the metabolic and translational fluxes, it can be used to explore behavior in changing conditions. Consider a configuration where the starting conditions of a culture are tuned such that the ribosomal allocation ${\varphi }_{Rb}$, the tRNA charging balance ${\text{tRNA}}^c/{\text{tRNA}}^u$, and the ribosome content $M_{Rb}/M$ are set to be above or below the appropriate level for steady-state growth in the environment (Figure 4A). As the culture grows, the observed ribosomal content $M_{Rb}/M$ is steadily adjusted until the steady-state level is met where it directly matches the optimal allocation (Figure 4B). This adaptation of the ribosomal content is controlled by dynamic adjustment of the allocation parameters via the flux-parity regulatory circuit (Figure 4C). To further test the flux-parity allocation model, we examine how accurately this system can predict growth behavior under nutritional shifts (Figure 4D–F) and the entry to starvation (Figure 4G–I).

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We first consider a nutrient shift where externally supplied low-quality nutrients are instantaneously exchanged with rich nutrients. Figure 4E shows three examples of such nutritional upshifts (markers), all of which are well described by the flux-parity allocation theory (dashed lines). The precise values of the growth rates before, during, and after the shift will depend on the specific carbon sources involved. However, by relating the growth rates before and immediately after the shift to the total shift magnitude (as shown in Korem Kohanim et al., 2018), one can collapse a large collection of data onto a single curve (Figure 4F, markers). The collapse emerges naturally from the model (dashed line) when decomposing the metabolic sector into needed and non-needed components (‘Methods’), demonstrating that the flux-parity allocation model is able to quantitatively describe nutritional upshifts at a fundamental level.

Finally, we consider the growth dynamics during the onset of starvation, another non-steady-state phenomenon (Figure 4G–I). Figure 4H shows the growth of batch cultures where glucose is provided as the sole carbon source in different limiting concentrations (Bren et al., 2013) (markers). The cessation of growth coincides with a rapid, ppGpp-mediated increase in expression of metabolic proteins (Magnusson et al., 2005; Dennis et al., 2004). Bren et al., 2013 demonstrated that expression from a glucose-specific metabolic promoter (PtsG) rapidly, yet temporarily, increases with the peak occurring at the moment where growth abruptly stops (Figure 4I, solid gray lines). The flux-parity allocation model again predicts this behavior (Figure 4I, red lines) without additional fitting (‘Methods’), cementing the ability of the model to describe growth far from steady state.

Here we present a step-by-step derivation of the low-dimensional allocation model we use to describe bacterial growth. We provide additional biological motivation for its construction and highlight the different assumptions and simplifications invoked. To maintain consistency with the literature, we largely follow the notational scheme introduced by Scott et al., 2014 and define each symbol as it is introduced.

The rate of protein synthesis is determined by two quantities: the total number of ribosomes $N_{Rb}$ and the speed $v_{tl}$ at which they are translating. The latter depends on the concentration of precursors needed for peptide bond formation, such as tRNAs, free amino acids, and energy sources like ATP and GTP. Taking the speed $v_{tl}$ as a function of the concentration of the collective precursor pool $c_{pc}$, the increase in protein biomass $M$ follows as

There exists a maximal speed at which ribosomes can operate, $v_{tl}^{max}$, that is reached under optimal conditions when precursors are highly abundant, in E. coli approximately 20 amino acids (AA)/second (s) (Forchhammer and Lindahl, 1971). Conversely, the translation speed falls when precursor concentrations $c_{pc}$ get sufficiently small. Simple biochemical considerations support a Michaelis–Menten relation (Ehrenberg and Kurland, 1984; Klumpp et al., 2013; Belliveau et al., 2021) as good approximation of this behavior with the specific form

where $K_M^{c_{pc}}$ is a Michaelis–Menten constant with the maximum speed $v_{tl}^{max}$ only observed for $c_{pc}\gg K_M^{c_{pc}}$. The number of ribosomes $N_{Rb}$ can be approximated given knowledge of the total mass of ribosomal proteins $M_{Rb}$ and the proteinaceous mass of a single ribosome $m_{Rb}$ via $N_{Rb}\approx M_{Rb}/m_{Rb}$ (more details in Appendix 1 Estimating the number of ribosomes within the cell). The increase in protein biomass (Equation 5) is thus

The translation rate$\gamma (c_{pc})\equiv v_{tl}(c_{pc})/m_{Rb}$ describes the rate at which ribosomes generate new protein.

The maximal translation rate ${\gamma }_{max}\equiv v_{tl}^{max}/m_{Rb}$ imposes a firm upper limit (Dill et al., 2011; Belliveau et al., 2021; Kafri et al., 2016) of how rapidly biomass can accumulate, unrealistically assuming the system would consist of only ribosomes translating at maximum rate. Notably, however, this upper limit is not much faster than the fastest growth observed, highlighting the importance of protein synthesis in defining the timescale of growth. For example, the maximal translation rate for E. coli is ≈ 10 hr^-1 and thus only ≈4 times higher than the growth rates in rich LB media ($\lambda \approx 2.5$ hr^-1). Including the synthesis of rRNA, another major component of the cellular dry mass, lowers this theoretical limit only marginally (Kostinski and Reuveni, 2020), further supporting our sole consideration of protein synthesis in defining growth. The difference between measured growth rates and the theoretical limits can be mostly attributed to the synthesis of metabolic proteins which generate the precursors required for protein synthesis, which we consider next.

Microbial cells are generally capable of synthesizing precursors from nutrients available in the environment, such as sugars or organic acids. This synthesis is undertaken by a diverse array of metabolic proteins ranging from those which transport nutrients across the cell membrane, to the enzymes involved in energy generation (such as those of fermentation or respiration), and the enzymes providing the building blocks for protein synthesis (such as those involved in the synthesis of amino acids). While these enzymes vary in their abundance and kinetics, we group them all into single set of metabolic proteins with a mass $M_{Mb}$ which cooperate to synthesize the collective pool of precursors from nutrients required for protein synthesis. We make the approximation that these metabolic proteins generate precursors at an effective metabolic rate $\nu$. In general, this rate depends on the concentration of nutrients $c_{nt}$ in the environment. This relation is canonically described by a Monod (Michaelis–Menten) relation

where ${\nu }_{max}$ is the maximum metabolic rate describing how fast the metabolic proteins can synthesize precursors, and $K_M^{c_{nt}}$ is the Monod constant describing the concentration below which nutrient utilization slows (Monod, 1949). Novel precursors are thus supplied with a total rate of $\nu (c_{nt})M_{Mb}$ and consumed via protein synthesis at a rate $\gamma (c_{pc})M_{Rb}$. Translation relies on precursors and, as introduced above, the translation rate $\gamma (c_{pc})M_{Rb}$ thus depends on the concentration of precursors in the cell, $c_{pc}$. As we do not explicitly model cell division, we here approximate this cellular concentration as the relative mass abundance of precursors to total protein biomass. This approximation is justified by the observation that cellular mass density and total protein content is approximately constant across a wide range of conditions (Belliveau et al., 2021; Martínez-Salas et al., 1981; Kubitschek et al., 1983). The dynamics of precursor concentration follows from the balance of synthesis, consumption, and dilution as the total biomass grows:

While the dilution term is often assumed to be negligible, this term is critical to describe growth and derive analytical expressions. Furthermore, we note that the precursor concentration is defined such that the consumption of one precursor yields the addition of one amino acid to the biomass $M$. As we measure proteins in units of amino acids, there is thus no conversion factor needed when describing the consumption of precursors by protein synthesis.

The introduced dynamics simplifies when the nutrient concentration in the environment $c_{nt}$ well exceeds the Monod constant $K_M^{c_{nt}}$ as $\nu (c_{nt})$ simplifies to ${\nu }_{max}$. Steady growth for which biomass increases exponentially readily emerges. This is the scenario we focus on in in the first half of this work. It should be noted, however, that biologically such a scenario can only be realized temporarily as the nutrient supply required by the exponentially growing biomass can only be sustained by the environment for a limited amount of time. In general, the nutrient levels vary.

The synthesis of novel precursors relies on the availability of nutrients which changes depending on the environment. In Figure 1—figure supplement 2, we consider specifically a ‘batch culture’ scenario in which nutrients are provided only at the beginning of growth and are never replenished. Therefore, growth of the culture continues until all of the nutrients have been consumed. The concentration of nutrients in the environment is thus given as

where $Y$ is the yield coefficient which describes how many nutrient molecules are needed to produce one unit of precursors.

As final step of the model definition, we must describe how cells direct their protein synthesis towards making ribosomes, metabolic proteins, or all other proteins that make up the cell (colored arrows in Figure 1A). We do so by introducing three allocation parameters ${\varphi }_{Rb}$, ${\varphi }_{Mb}$, and ${\varphi }_O$ (such that ${\varphi }_{Rb}+{\varphi }_{Mb}+{\varphi }_O=1$) which define how novel protein synthesis is partitioned among these categories:

These equations are summarized in Figure 1B and Figure 1, Figure 1—figure supplement 2 and define the accumulation of biomass, from nutrient uptake to protein synthesis.

In addition to maintaining the total macromolecular densities, cells also maintain an approximately constant protein density (Bremer and Dennis, 2008). This observation allows for a major simplification when formulating the allocation model, namely the approximation of concentrations as relative mass abundances. The rate $\gamma$ at which ribosomes can synthesize protein is dependent on the abundance of precursors, $c_{pc}$, in the cell. To compute the concentration and/or density in typical units (e.g. µM, or mass/volume), we would require some measure of the total cellular volume, $V_{cell}$, such that the concentration follows

with $M_{pc}$ denoting the total mass of the precursor pool. By making the experimentally supported assertion that the protein density $\rho$ is constant, we can say that

where $M$ is the total protein biomass. Thus, the total cellular volume $V_{cell}$ can be computed as

Plugging this result into Equation 12, we arrive at the approximation

In this work, we neglect $\rho$ as a multiplicative constant and treat $c_{pc}$ as being dimensionless. We direct the reader to Scott et al., 2010 and Milo, 2013 for a further discussion of the conversion between concentration and relative abundance.

In the first section of this work, we present several analytical relations pertinent to steady-state growth. These relations follow from the simple allocation model and describe (i) how the growth rate depends on model parameters (Figure 1C) and (ii) how ribosome content depends on other model parameters for the three different regulation scenarios we discuss (Figure 1F). Here, we introduce a step-by-step derivation of these expressions.

We begin with deriving an expression for the steady-state growth rate $\lambda$ which is similar to previous approaches taken by Giordano et al., 2016 and Dourado and Lercher, 2020. As discussed in Figure 1—figure supplement 2, steady-state conditions are satisfied when two conditions are met. First, the dynamics of the precursor concentration is constant (i.e., $\frac{d{c}_{pc}}{dt}=0$) and the composition of the proteome matches the allocation parameters (i.e., $\frac{M_{Rb}^{*}}{M^{*}}={\varphi }_{Rb}^{*}$ and $\frac{M_{Mb}^{*}}{M^{*}}={\varphi }_{Mb}^{*}$). Furthermore, we assume that in steady-state growth, the concentration of nutrients in the environment is saturating ($c_{nt}\gg K_M^{c_{nt}}$), meaning that $\nu (c_{nt})\approx {\nu }_{max}$. With these conditions satisfied, we can rewrite Equation 9 as

where $c_{pc}^{*}$ is the steady-state precursor concentration.

Noting that in steady-state conditions the total biomass increases exponentially at a rate $\lambda \equiv \gamma (c_{pc}){\varphi }_{Rb}^{*}$, Equation 16 can be simplified to

We can therefore solve for the steady-state precursor concentration $c_{pc}^{*}$ to yield

Assuming a Michaelis–Menten form for the translation rate $\gamma (c_{pc}^{*})$, we can now define it as a function of the growth rate $\lambda$ as

Knowing that the growth rate $\lambda \equiv \gamma (c_{p{c}^{*}}){\varphi }_{Rb}^{*}$, and ${\varphi }_{Mb}^{*}=1-{\varphi }_{Rb}^{*}-{\varphi }_O^{*}$, we say that

This can be algebraically manipulated to yield a quadratic equation of the form

which has one positive root of

For notational simplicity, we can define the maximum metabolic output and the maximum translational output as ${\displaystyle \mathsf{N}={\nu }_{max}(1-{\varphi }_{Rb}-{\varphi }_O)}$ and $\mathrm{\Gamma }={\gamma }_{max}{\varphi }_{Rb}$, respectively, and substitute them into Equation 22 to generate

Here we expand upon and derive the equations defining the flux-parity allocation model shown schematically in Figure 2A and explore its dependence on parameter values.

In Equation 33, we made the assumption that the concentration of ppGpp was inversely proportional to the charging balance of the tRNA pools. We put this forward as an ansatz with the motivation that the degree of tRNA charging should be related to the amount of ppGpp synthesized. However, there are other ansatzes that could be made relating the amount of ppGpp to the individual concentrations of the tRNAs, or other ratiometric definitions.

To test how sensitive our findings are to the particular ansatz used, we considered other ways in which the ppGpp concentration could be related to the tRNA pools. There is strong biochemical evidence that a primary route of ppGpp synthsesis is via the enzyme RelA, which becomes active when associated to a ‘stalled’ ribosome—one that is bound to an uncharged tRNA—though some details remain enigmatic. In manner similar to other works (Giordano et al., 2016; Wu et al., 2022; Bosdriesz et al., 2015), we can assert that the amount of ppGpp is proportional to the abundance of stalled ribosomes. Mathematically, we can define the ppGpp concentration as being proportional to the probability of a ribosome binding an uncharged tRNA. Assuming that the tRNA concentration (of both charged and uncharged forms) is sufficiently high that all ribosomes are complexed with a tRNA, this equates to

where $tRN{A}^c$ and $tRN{A}^u$ represent the absolute concentrations of charged and uncharged species, respectively. If the ppGpp concentration is inversely proportional to the allocation towards ribosomes, we can similarly make the argument that the ribosomal allocation ${\varphi }_{Rb}$ will be proportional to the probability of a ribosome being bound to a charged-tRNA,

This equation mechanistically operates in a similar way as Equation 34—the allocation towards ribosomes depends on the relative amounts of charged- and uncharged-tRNAs. In the extreme limit where the total concentration of tRNA is fixed (for which there is conflicting evidence; Dong et al., 1996; Skjold et al., 1973; Bremer and Dennis, 2008; Bosdriesz et al., 2015; Giordano et al., 2016), Equation 41 and Equation 34 are mathematically equivalent. However, the predicted scaling dependence of ppGpp takes a different form.

In the main text, we noted that the concentration of ppGpp relative to a reference growth rate $\frac{[ppGpp]}{{[ppGpp]}_0}$ is equivalent to the inverse ratio of the charging balances. Under the ansatz that the ppGpp concentration is depending on the uncharged-tRNA binding probability, this relation takes the form

where the subscript 0 denotes the reference state value. This distinction, coupled with experimental measurements of the relative ppGpp concentrations, allows us to test the validity of the two assumed forms for ${\varphi }_{Rb}$.

Figure 3—figure supplement 1 shows the predictive capacity of these two ansatzes with the simple binding (Equation 41) and ratiometric (Equation 33) predictions shown in solid-blue and dashed-red lines, respectively. While both of these assumptions are capable of predicting the scaling of the ribosome content and translation speed with quantitative equivalence, there is a distinct difference in the predicted behavior of the relative ppGpp concentrations. The simple binding ansatz predicts a significantly shallower dependence on the growth rate than is observed in the data and in the ratiometric prediction. Thus, it appears that relating [ppGpp] to the ratio of uncharged- to charged-tRNA concentrations accurately captures the behavior of E. coli, though there remain gaps in our understanding of this relationship at a biochemical level.

This work leverages a large collection of data, primarily from E. coli, to evaluate the accuracy of our model in describing biological phenomena. These data come from a range of studies spanning around 50 years of measurements from different groups and different geographical locations. Collecting and curating this large data set required the manual transcribing of data from papers as well as various standardization steps to ensure that measurements were truly comparable between studies, as is outlined in Supplementary file 2.

For proper referencing and attribution, we list the data sources here as follows: Albertson and Nyström, 1994; Baracchini and Bremer, 1988; Basan et al., 2015; Bentley et al., 1990; Bremer and Dennis, 2008; Bren et al., 2013; Brunschede et al., 1977; Büke et al., 2022; Buckstein et al., 2008; Coffman et al., 1971; Dai et al., 2016; Dalbow and Young, 1975 ; Dong et al., 1995; Erickson et al., 2017; Forchhammer and Lindahl, 1971; Gausing, 1972; Hernandez and Bremer, 1990; Hernandez and Bremer, 1993; Imholz et al., 2020; Kepes and Beguin, 1966; Korem Kohanim et al., 2018; Lacroute and Stent, 1968; Lazzarini et al., 1971; Li et al., 2014; Li et al., 2018;; Mori et al., 2017; Morris and Hansen, 1973; Oldewurtel et al., 2021; Panlilio et al., 2020; Pedersen, 1984; Ryals et al., 1982; Sarubbi et al., 1988; Schmidt et al., 2016; Schleif, 1967; Schleif et al., 1973; Scott et al., 2010; Si et al., 2017; Skjold et al., 1973Sloan and Urban, 1976; Sokawa et al., 1975; Wu et al., 2022; You et al., 2013; Young and Bremer, 1976; Zhu and Dai, 2019; Bonven and Gulløv, 1979; Lacroute, 1973; Metzl-Raz et al., 2017; Paulo et al., 2015; Paulo et al., 2016; Riba et al., 2019; Siwiak and Zielenkiewicz, 2010; Waldron and Lacroute, 1975; Xia et al., 2021; Rohatgi, 2021.

Over the past several decades, theoretical studies have introduced a huge variety of mathematical models to better understand how microbes grow. These range from simple phenomenological relations, first introduced by Verhulst, 1845 and Monod, 1942 to describe exponential growth and nutrient consumption, to more recent high-dimensional models which explicitly consider hundreds of cellular processes involved in biomass synthesis and growth. Here, we summarize different modeling approaches which integrate considerations of protein synthesis and metabolism to rationalize bacterial growth and quickly compare the construction and results to the model presented in the main text.

As outlined in the main text, the idea that protein synthesis constitutes a major limitation of microbial growth has a storied history. With an ever-improving characterization of both the composition and the major biochemical processes undertaken by microbes, it has become increasingly clear that the auto-catalytic nature of protein synthesis (i.e., ‘ribosomes making ribosomes’) and the corresponding allocation of ribosomal activity towards different proteins are of paramount importance. Therefore, it is imperative to consider protein synthesis and this allocation when modeling cell growth (Hernandez and Bremer, 1993; Koch, 1988). Over the years, very different approaches have been introduced to model protein synthesis and growth. To provide an overview, we roughly distinguish between higher-dimensional, coarse-grained, and low-dimensional approaches in the coming sections.

Higher-dimensional models build on the genetic annotations and reaction networks in E. coli to specifically account for hundreds (or even thousands) of molecular reactions. An important class of these are holistic cell-growth models which include thousands of reactions involved in protein synthesis (Karr et al., 2012; Macklin et al., 2020). These studies can describe a number of observations, including the concentration of different metabolites, as well as the relation between gene expression and the abundances of thousands of proteins. However, as it is difficult to estimate the many parameters involved, these models are typically constrained to only a few reference growth conditions like growth on glucose, often with moderate uncertainty. Moreover, the integration of hundreds of different reactions often counteracts the development of a more intuitive understanding of microbial growth and necessitates computationally costly analyses.

A second class of models are coarse grained models which substantially simplify cellular life, often by focusing on a subset of biochemical processes in a less detailed manner, yet still requiring dozens to hundreds of parameters. One example is the framework introduced by Weiße et al., 2015 to analyze the relation between gene expression (transcription), protein synthesis (translation), and growth across varied conditions. Compared to the aforementioned holistic growth models, this is a lower-dimensional approach with only a few different types of reactions. A more recent example of different scope is the work by Hu et al., 2020. The authors formulated a modeling framework which focuses on protein synthesis and explicitly considers 274 reactions driven by equally many enzymes. The authors then studied how the allocation of protein synthesis towards the different enzymes affects growth. Numerical simulations show that an allocation behavior to reach optimal growth does not imply that the speed with which ribosomes translate is fast. Rather, optimal allocation and efficient growth is often reached even when translation speeds are substantially lower than the maximum in some conditions. We arrive at a similar conclusion and discuss this important point in the main-text when we introduce different regulatory ‘scenarios.’

Different in scope to these ‘medium-dimensional’ coarse-grained approaches are those we truly call ‘low-dimensional’ which utilize only a few parameters, typically less than 10. In a now seminal work, Molenaar et al., 2009 introduced such a low-dimensional approach to model protein synthesis and the growth-rate dependent switch between different metabolic pathways. Central to this model is a consideration of how protein synthesis resources are partitioned among four different protein classes, including ribosomal and metabolic proteins. Through enumerating coupled differential equations that relate the partitioning to growth rate, their model predicts, for example, that ribosome content needs to scale linearly with growth rate to ensure efficient growth, though comparison with experimental data is absent.

Another early and significant low-dimensional modeling approach was introduced by Scott et al., 2014 only a few years later. This work is similar in spirit to that of Molenaar et al., but adds a deeper motivation of the low-dimensional allocation approach by building on the authors’ previous experimental study (Scott et al., 2010) which introduced novel phenomenological relations describing ribosome content and its dependence on growth rate. In particular, the authors discuss how feedback in the regulation of ribosomes and metabolic proteins (end-product inhibition and supply-driven activation) can lead to the observed scaling of ribosome content with growth rate. As such, the article highlights the important question of global regulation beyond the control of single genes. However, the authors rationalize this regulatory scenario as being necessary to ensure a stable steady-state regime. This arises in part due to the authors’ assumption that the dilution of metabolic precursors is negligible, a process we believe is physiologically critical to consider. Consequentially, the regulatory scheme which underlies the coordination of ribosomes and metabolic proteins is quite different to what other low-dimensional models have proposed and what we present here (see also Section 3 of this supplement).

Following studies by Molenaar et al. and Scott et al., other low-dimensional allocation models have been introduced with various extensions and modifications. The models presented by Maitra and Dill, 2015 and Giordano et al., 2016, for example, follow a similar low-dimensional description as Scott et al. The explicit consideration of precursor dilution allowed them to derive analytical solutions describing steady state growth, similar to those we present in this work. Dourado and Lercher, 2020 recently extended these results by providing analytical solutions for a generalized allocation model with the solution presented for a low-dimensional limit being similar to that of Giordano et al., 2016 and what we used as the starting point for our analysis of optimal allocation.

Several studies also extended the allocation framework to more explicitly model the global regulation that may be at play. In particular, several studies have investigated how the global regulator guanosine tetraphosphate (ppGpp) may tune the allocation of ribosomes to obtain optimal growth across conditions, specifically in E. coli (Giordano et al., 2016; Bosdriesz et al., 2015). These models often take a quite fine-grained view of the kinetics of ppGpp synthesis, its degradation, its dependence on stalled ribosomes, and even its mechanism of regulation. In some cases, particular details of the kinetics of transcriptional initiation of ribosomal RNA genes is explicitly considered (Bosdriesz et al., 2015). A commonality between these models is (once again) an enumeration of a handful of coupled ordinary differential equations, though their precise functional forms are unique. These theoretical analyses suggested that the ppGpp-mediated regulation feedback can robustly tune ribosome content with encountered conditions to support optimal growth in steady conditions. However, the evaluation of these models and particularly the chosen approaches to relate tRNA charging levels (for which there is contradictory data; Skjold et al., 1973; Bremer and Dennis, 2008) to ppGpp concentrations remain limited as only a cursory and largely qualitative comparison with data is presented. In this work, we provide an unprecedented quantitative comparison between available experimental data and our own low-dimensional very coarse-grained approach to model tRNA charging and ppGpp regulation. The excellent match between theory and observation (Figure 3C) confirms the important role of tRNA charging and ppGpp in mediating allocation and growth as highlighted by Giordano et al. and Bosdriez et al.

Notably, however, we choose a different ansatz in relating tRNA charging levels to ppGpp which we find crucial to describe the diverse growth phenomena in changing conditions (see Section 8). Recently, Wu et al., 2022 also presented a phenomenologically guided modeling approach to further understand the relation between ribosome content and growth rate, with a focus on the possible role of ppGpp in regulating ribosome activity in addition to expression. We think the derived picture is substantially different from that present in our work. Particularly, the authors contend that a condition-dependent translation rate is at odds with the principle of optimal allocation and cannot be understood without explicit consideration of an inactive ribosome pool. Following our analysis, including a condition-dependent translation rate strongly supports the observed relation between ribosome content and growth rate, strengthening a picture of optimal allocation without requiring an inactive pool of ribosomes for the majority of growth conditions. We discuss the topic of inactive ribosomes further in Appendix 1 - Inactive ribosomes.

Low-dimensional allocation models have further been used to study growth beyond exponential growth in steady conditions. Particularly, Erickson et al., 2017 formulated an allocation model to analyze growth when the abundance of the major growth supporting carbon source rapidly changes (up- and downshifts). Their approach utilizes (and enforces) linear phenomenological relations describing ribosomal and metabolic protein content observed during steady-state growth. As such, the study can describe certain transitions quite well, particularly the up- and downshift from growth on gluconate/glucose to growth on succinate. Korem Kohanim et al., 2018 used a low-dimensional modeling approach to analyze the rapid increase in growth when conditions improve (nutritional upshift). The modeling approach particularly allowed them to explain the rapid increase of growth rates, with the increase depending on observed steady-state growth rates before and after the shift. Mori et al., 2017 also utilized phenomenological relations and a low-dimensional allocation model to explore the possible benefits of maintaining a reserve of inactive ribosomes. Balakrishnan et al., 2021a further extended the allocation modeling approach to analyze the physiological origin of long lag times which frequently emerge during diauxic growth on different carbon sources. These works illustrate that there are many approaches one can take toward rationalizing growth out of steady state, each with different assumptions and behaviors in various limits.

Our approach and the extensive comparison between data and theory we present confirm the overall logic presented by these studies: growth transitions are largely determined by the way cells coordinate translation and gene expression during the shift. However, our approach is different in scope as the core of our model is the tRNA-dependent control of allocation as fundamental regulation scheme, rather than relying on linear phenomenological relations to describe the ‘regulatory function.’ This allows us to rationalize the emergence of many growth behaviors without assuming specific relations between growth rate and physiological quantities as a starting point. Furthermore, we also caution against the common assumption in these models that the offset in the approximately linear relation between steady growth rate and ribosome content represents inactive ribosomes which changes the overall perception of how cells operate to efficiently grow in steady conditions and when environments change (see discussion in Appendix 1 - Inactive ribosomes)

More recently, low-dimensional allocation models have further been extended to investigate additional aspects of cell physiology beyond biomass accumulation and growth. For example, Roy et al., 2021 explicitly considered the relationship between protein and RNA synthesis and the interesting question how different autocatalytic cycles like those involved in protein and RNA synthesis have to be coupled to promote growth. We discuss in Appendix more specifically the role of RNA synthesis and why we refrained from modeling it explicitly. Additional studies have also extended the allocation framework to explicitly consider cell-size control and proteins involved in division (Serbanescu et al., 2020; Bertaux et al., 2020); however, a consistent description of growth and cell size is still missing and part of ongoing research efforts.

Cells are highly complex entities containing thousands of molecular players which interact in myriad ways varying over space and time. The allocation models boldly simplifies this reality. Some of the most important simplifications are:

1. All cells as being treated as identical and encountering the same condition, including the nutrients which they consume to build new cellular material.

2. Cells are not treated individually but the overall biomass is considered. As such, the approach ignores considerations of cell division and variations in cell physiology throughout the cell cycle.

3. Rather than considering the abundance of thousands of unique protein species within a cell, allocation models take a coarse-grained view of the composition of the proteome where proteins are pooled together. We pool specifically proteins being either ribosomal (i.e., synthesizing new protein from precursors such as charged-tRNA), metabolic (i.e., synthesizing precursors from nutrients), or being involved in other biological processes required for cellular growth and survival Figure 1—figure supplement 1.

Here we summarize our view of why some of the low-dimensional allocation models can develop such a predictive power despite all these major simplifications.

As mentioned, the low-dimensional allocation framework is completely ignorant to (i) the spatial arrangement of processes across the cell and (ii) neglects the existence of cells as a whole. Instead, the description only considers the change of total protein biomass of the system (i.e., culture) over time. Despite this objectively major simplification, the model quantitatively predicts growth phenotypes across a broad range of conditions, as we presented in the main text. Remarkably, this is accomplished with one core set of parameters (such as the maximal speed of translation) which remains fixed across conditions. This is a surprising result given that cell size and the spatiotemporal arrangement of cellular components are known to be highly dependent on conditions, which also should affect major model parameters (such as the maximal speed of translation). The finding that the model can nevertheless capture observations across conditions thus suggests to us that many cellular processes, including those involved in cell size control, are highly coordinated by the cell such that translation and metabolism work efficiently and can be captured by simple rate equations; it is the complexity of the cellular machinery which allows us to formulate a simple but predictive model of growth.

To further illustrate this point, we consider specifically the density of macromolecules within the cell which strongly informs the rates of myriad biochemical reactions. A density that is too large, for example, will strongly hamper diffusion and thus slow many reaction rates, while a low density generally reduces binding rates (Delarue et al., 2018; van den Berg et al., 2017). Cells are thus expected to maintain macro-molecular density ranges within narrow ranges to operate efficiently (van den Berg et al., 2017). Experimental studies support this idea. For example, mass densities in E. coli appear stay within narrow ranges across growth conditions (Oldewurtel et al., 2021). Complex biophysical processes appear to also be in place to control the spatial arrangement of macromolecules, such as the strong mutual-exclusivity of DNA and ribosomes within the cytoplasm (Gray et al., 2019). If it were not for these processes, macromolecular densities and thus major cellular processes (like translation) would vary tremendously with growth conditions and a model based on simple rate equations would have very limited predictive power.

In the main text, we consider that the translation rate $\gamma$ is dependent on the concentration of precursors $c_{pc}$. The tug-of-war between the metabolic processes (synthesizing precursors) and protein synthetic processes (consuming precursors) is what determines this value. With the protein density of cells being approximately constant , we consider the precursors concentration as the mass of precursors per total protein mass, $c_{pc}$. We describe the dynamics of this precursor concentration $c_{pc}$ as

where $M$, $M_{Mb}$, and $M_{Rb}$ denote the masses of the total protein, metabolic protein, and ribosomal protein pools, respectively. The latter term in the above equation denotes the decrease in the precursor concentration due to the increase in biomass; that is, this term considers the decrease in precursor concentration due to dilution upon a growing cell volume. Through a change-of-variables from $m_{pc}$ to $c_{pc}$, it mathematically follows that