Our overall goal is to determine the growth-optimizing proteome allocation for the core translation factors. Conceptually, varying tlF concentrations has two opposing effects on cell proliferation. At the biochemical level, high tlF expression can facilitate growth by allowing more efficient usage of ribosomes. At the systems level, increased tlF expression can nonetheless limit growth by reducing the number of ribosomes and other proteins that can be produced. The tradeoffs between various tlFs and ribosomes create a multidimensional optimization problem.

We solve this multidimensional problem by treating translation as a dynamical system, in which ribosomes cycle through initiation, elongation, and termination. The resulting flux drives cell growth. During steady-state growth, every interlocked step of the translation cycle must have the same ribosome flux that is specified by the growth rate. We show that at the growth optimum, concentrations for distinct tlFs can be solved independently. The resulting analytical solutions can be expressed in terms of the growth rate and simple biophysical parameters.

To describe the biochemical effects of tlF concentrations on cell growth, we first introduce a coarse-grained translation cycle time ${\mathrm{\tau }}_{tl}$, or the time it takes for a ribosome to complete a typical cycle of protein synthesis (Figure 1A), which consists of three sequential steps: initiation ('$ini$'), elongation ('$el$'), and termination ('$ter$'). Each of these steps is catalyzed by multiple tlFs. The full translation cycle time is then sum of ribosome transit times at the three steps (${\mathrm{\tau }}_{tl}={\mathrm{\tau }}_{ini}+{\mathrm{\tau }}_{el}+{\mathrm{\tau }}_{ter}$), whose dependence on individual tlF concentrations can be quantitatively described through mass action kinetic schemes (schematically depicted in Figure 1A, see Appendices 2, 3, and 4 for details and examples below). We express tlF concentrations in units of proteome fractions (dry mass fraction of a specified protein to the full proteome), denoted by $\varphi$ (Scott et al., 2010) (Materials and methods, section Conversion between concentration and proteome fraction). Using this notation, the translation cycle time ${\mathrm{\tau }}_{tl}$ is a decreasing function of various tlFs concentrations ($\left\{{\varphi }_{tlF,i}\right\}$).

In addition to its dependency on tlF concentrations, the translation cycle time provides a bridge between the cell growth rate and ribosome concentration. In steady-state growth (Monod, 1949; Scott et al., 2010; Dai et al., 2016), the growth rates of cells and of their protein content (total number of proteins) must be identical, denoted here as λ, as a result of the constant average cellular composition. The protein content grows at a rate determined by the flux of active ribosomes completing the translation cycle, that is $N_{ribo}^{act}/{\mathrm{\tau }}_{tl}$, where $N_{ribo}^{act}$ is the number of active ribosomes per cell, divided by the total number of proteins $N_P$ per cell: $\lambda =N_{ribo}^{act}/{\mathrm{\tau }}_{tl}{N}_P$. Active ribosomes are defined as those functionally engaged in, and cycling through, the initiation, elongation, and termination reactions of peptide synthesis. Rescaling to the total mass fraction (Materials and methods, section Conversion between concentration and proteome fraction) of proteome for active ribosomes (${\varphi }_{ribo}^{act}$) yields

where ${\mathrm{\ell }}_{ribo}$ is the number of amino acids in ribosomal proteins and $⟨\mathrm{\ell }⟩$ is the average number of codons per protein, weighted by expression levels (Materials and methods, section Average number of codons per protein: $⟨\mathrm{\ell }⟩$). The rescaling factor (${\mathrm{\ell }}_{ribo}/⟨\mathrm{\ell }⟩\approx 7300/200=36.5$) is approximately constant across growth conditions (Matrials and methods, section Average number of codons per protein: $⟨\mathrm{\ell }⟩$). This equation establishes how tlF concentrations affect the growth rate biochemically via ${\mathrm{\tau }}_{tl}$.

We note that Equation 1 is a generalized form of the bacterial growth law that relates the mass fraction of elongating ribosomes to growth rate ($\lambda =\frac{{\varphi }_{ribo}^{el}}{{\mathrm{\tau }}_{el}}\frac{⟨\mathrm{\ell }⟩}{{\mathrm{\ell }}_{ribo}}=\gamma {\varphi }_{ribo}^{el}$, where γ is a rescaled translation elongation rate and ${\varphi }_{ribo}^{el}$ is the proteome fraction of actively translating ribosomes [Scott et al., 2010; Dai et al., 2016; Scott et al., 2014]). This classic growth law was derived by considering the steady-state flux of peptide bond formation by elongating ribosomes, whereas our model focuses on the flux of ribosomes that traverse the entire translation cycle, thereby allowing us to consider the effects of translation factors and ribosomes engaged in additional steps (initiation, elongation, and termination). For each step, Equation 1 can be extended to show that the growth rate is similarly proportional to the mass fraction of the corresponding ribosomes divided by the transit time at that step (Materials and methods, section Equality of ribosome flux in steady-state).

Steady-state growth thus imposes the requirement that the growth rate be inversely proportional to the translation cycle time and proportional to the number of active ribosomes engaged in the translation cycle (Equation 1). Inactive ribosomes, comprised of assembly intermediates, hibernating ribosomes, or otherwise non-functional ribosomes, have been found to constitute a small fraction (≈5%) of the total ribosome pool for fast growth (Lindahl, 1975; Dai et al., 2016). Based on Equation 1, both increasing ribosome concentration and increasing tlF concentrations (which decreases ${\displaystyle {\tau }_{tl}}$) can accelerate growth. However, production of ribosomes and tlFs is subject to competition under a limited proteomic space, which we consider next.

To model the production cost tradeoff between tlFs and ribosomes, we integrate the flux-based formulation above with a proteomic constraint. Assuming that components of the translation machinery together accounts for a fixed fraction of proteome, that is, the ‘translation sector’ ${\varphi }_{tl}$ (denoted ${\varphi }_R$ in the context of growth laws [Scott et al., 2010]), the proteome fraction for active ribosomes is related to the proteome fraction for translation factors via

Equations 1 and 2, together with to the kinetic schemes for each step of the translation cycle, constitute the core of our model. Combining the biochemical effects (Equation 1) and the systems-level constraints (Equation 2) on tlFs, we arrive at a self-contained relationship between growth and tlF concentrations:

 where we explicitly express ${\mathrm{\tau }}_{tl}$ as a function of ${\varphi }_{tlF,i}$ to reflect the dependence of ribosome transit times on translation factor abundances. The above relationship (Equation 3) allows us to ask: what is the stoichiometry of tlFs, or partitioning of the translation sector, that maximizes the growth rate (Figure 1A)?

The condition for the optimal TF abundances, that is, the set of ${\varphi }_{tlF,i}$ that satisfies ${\left(\partial \lambda /\partial {\varphi }_{tlF,i}\right)}^{*}=0$, can be obtained by considering the ${\varphi }_{tlF,i}$ as independent variables and taking the derivative of Equation 3 with respect to a specified tlF abundance. Under the assumptions that the translation sector (${\varphi }_{tl}$) and the proteome fraction for inactive ribosomes (${\varphi }_{ribo}^{inact}$) are both fixed in a given external nutrient condition, this yields

where the asterisk refers to the growth optimum within our model, that is, ${\left(\partial \lambda /\partial {\varphi }_{tlF,i}\right)}^{*}=0$. Hence, under this framework, the tlF abundances are growth-optimized when the sensitivity of the translation cycle time to changing the considered tlF abundance ($\partial {\mathrm{\tau }}_{tl}/\partial {\varphi }_{tlF,i}$) reaches a value determined solely by the growth rate and protein size factors. We emphasize that the derivative above corresponds to a perturbation scenario in which the tlF abundance is changed while maintaining fixed the total proteomic resources to the translation sector, as prescribed by our optimization procedure. As such, it does not correspond an actual perturbation easily realizable experimentally.

Although Equation 3 and the resulting optimization conditions (Equation 4, one for every tlF) corresponds to a coupled nonlinear system of multiple ${\varphi }_{tlF,i}$, substantial decoupling occurs at the optimal growth rate. In this situation, most ${\varphi }_{tlF,i}$ are only connected through the resulting growth rate. The optimization problem is then further simplified by the fact that the translation cycle consists of sequential and largely independent steps. The translation cycle time ${\mathrm{\tau }}_{tl}$ corresponds to the sum of the coarse-grained initiation, elongation, and termination times, that is, ${\mathrm{\tau }}_{tl}={\mathrm{\tau }}_{ini}+{\mathrm{\tau }}_{el}+{\mathrm{\tau }}_{ter}$. Given that each tlF is involved in a specific molecular step, the sensitivity matrix of these times to tlF concentration is sparse: ${\left(\partial {\mathrm{\tau }}_j/\partial {\varphi }_{tlF,i}\right)}^{*}=0$ for most combinations of ${\mathrm{\tau }}_j$ and ${\varphi }_{tlF,i}$. This lack of ‘cross-reactivity’ expresses that, for example, the initiation time ${\mathrm{\tau }}_{ini}$ is unaffected by the tRNA synthetase concentration. This sparsity only occurs at the optimal expression levels, as the transit times typically depend on the growth rate (see an example in section Non binding-limited regime [one stop codon]) and $\partial \lambda /\partial {\varphi }_{tlF,i}\ne 0$ away from the optimum. The optimum condition for factor $i$ then simplifies to:

where $j$ above denotes the translation step(s) that tlF_i participates in. This leads to simplifications that allow the system to be solved analytically in most cases: instead of solving the full system at once, individual reactions within the translation cycle can be considered in isolation. The resulting optimal concentrations are connected via the growth rate ${\lambda }^{*}$. Interestingly, the optimal stoichiometry among most tlFs is independent of ${\lambda }^{*}$ if the reactions are in the binding-limited regime, as we show below.

We first illustrate the process of solving for the optimal tlF concentration for the relatively simple case of translation termination. The principles used here and the form of solutions provide conceptual guideposts for solving other steps of the translation cycle.

In bacteria, translation termination (Bertram et al., 2001) consists of two distinct, sequential steps: (1) stop codon recognition and peptidyl-tRNA hydrolysis catalyzed by class I peptide chain release factors RF1 and RF2, followed by (2) dissociation of ribosomal subunits from the mRNA, that is, ribosome recycling, catalyzed by RF4. We do not explicitly consider the additional factors (e.g. RF3 and EF-G) due to their lack of conservation or because they are non-limiting for this specific step (Appendix 2, section Omitted molecular details). RF1 and RF2 have the same molecular functions but recognize different stop codons (Scolnick et al., 1968): RF1 recognizes stops UAA and UAG, whereas RF2 recognizes UAA and UGA. For simplicity, we describe here a scenario where RF1 and RF2 have no specificity towards the three stop codons, which allows us to combine them in a single factor (denoted RFI). The model is readily generalized, with similar results, to the case of the two RFs with their specificity towards the three stop codons (Appendix 2, section Full three stop codons model).

Under a coarse-grained description, the total ribosome transit time at termination ${\mathrm{\tau }}_{ter}$ can be decomposed into a sum of peptide release time and ribosome recycling time. In the treatment below, we consider a regime of binding-limited reactions for simplicity (rapid catalytic rate). A full model with catalytic components can also be solved analytically (Appendix 2, section Non binding-limited regime (one stop codon), Figure 2A). In the binding-limited regime ($k_{cat}\to \mathrm{\infty }$), the peptide release time and ribosome recycling time are inversely proportional to the corresponding tlF concentrations:

where the association rate constants $k_{on}^i$ are rescaled by the factor’s sizes in proteome fraction units (Materials and methods, section Conversion between concentration and proteome fraction). The above expression constitutes the solution of the mass action scheme for termination, connecting factor abundances to termination time.

Figure 2

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The termination time (Equation 6) can then be directly substituted into the optimality condition (Equation 5) and solved in terms of ${\lambda }^{*}$:

If the reactions are not binding-limited, an additional catalytic term $\propto {\lambda }^{*}/k_{cat}$ is added to the minimally required levels above (Appendix 2, section Non binding-limited regime [one stop codon]). The square-root dependence in the optimal RF concentrations emerges from the ${\varphi }_i^{-1}$ dependence of ${\mathrm{\tau }}_i$, for example, for ribosome recycling ${\mathrm{\tau }}_{recyc}\propto {\varphi }_{RF4}^{-1}$, which becomes ${({\varphi }_i^{*})}^{-2}$ upon taking the derivative in the optimality condition (Equation 5). The square root is then obtained by solving for ${\varphi }_i^{*}$. A similar square-root dependence has been noted in optimization of the ternary complex and tRNA abundances (Ehrenberg and Kurland, 1984; Berg and Kurland, 1997). Analysis of tlF expression across slower growth conditions supports the derived square root dependence (Figure 4—figure supplement 2). As a result of the square-root, the optimal RF concentrations are weakly affected by biophysical properties such as the association rate constants and protein sizes. In the binding-limited regime above, the ratio of the optimal concentrations between RFI and RF4 is independent of the growth rate and only depends on the kinetics of binding.

As a side note, the expression for termination time ${\mathrm{\tau }}_{ter}$ in Equation 6 must be modified in a regime where ribosomes are frequently queued upstream of stop codons. This would occur if the termination rate were slow and approached initiation rates on mRNAs (Bergmann and Lodish, 1979; Lalanne et al., 2021). In this regime, queues of ribosomes at stop codons would incur an additional time to terminate. In a general description, the resulting additional termination time can be absorbed in a queuing factor ${\displaystyle \mathcal{𝒬}:{\tau }_{ter}^{full}:={\tau }_{ter}\mathcal{𝒬}({\tau }_{ter})}$ (Appendix 1 for derivation and discussion). The resulting nonlinearity would forbid the decoupling in the optimization procedure between RFI and RF4. Although absolute rates of termination are difficult to measure in vivo, translation on mRNAs is generally thought to be limited at the initiation step (Laursen et al., 2005), and consistently, ribosome queuing at stop codons in bacteria is not usually observed (except under severe perturbations, e.g. Kavčič et al., 2020; Baggett et al., 2017; Mangano et al., 2020; Saito et al., 2020; Lalanne et al., 2021). In the physiological regime of fast termination, the queuing factor converges to 1, yielding simple solutions that depend only on biophysical parameters (Equations 7).

The optimal tlF concentrations (e.g. Equation 7) can also be intuitively derived from another viewpoint. For each reaction in the translation cycle, we can define an effective proteome fraction allocated to that process, combining the proteome fractions of the corresponding tlF and the ribosomes waiting at that specific step. As an example, for the case of peptide chain release factor (RFI) just treated, the effective proteome fraction includes the release factors and ribosomes with completed peptides waiting at stop codons (dashed box in Figure 2A), that is, ${\varphi }_{RFI}^{eff}:={\varphi }_{RFI}+{\varphi }_{ribo}^{stop}$. This effective proteome fraction corresponds to the total proteomic space associated to a tlF in the context of the translation cycle.

During steady-state growth, the concentration of ribosomes waiting at any specific step of the translation cycle is equal to the total active ribosome concentration multiplied by the ratio of the transit time of that step to the full cycle: for example, here ${\varphi }_{ribo}^{stop}=\frac{{\mathrm{\tau }}_{stop}}{{\mathrm{\tau }}_{tl}}{\varphi }_{ribo}^{act}$, where ${\mathrm{\tau }}_{stop}=1/(k_{on}^{RFI}{\varphi }_{RFI})$ is the time to arrival of RFI. Using Equation 1 for ${\varphi }_{ribo}^{act}$, the effective proteome fraction satisfies:

In the last line, we used the inequality of arithmetic and geometric means ($a+b\ge 2\sqrt{ab}$) to obtain the minimum of the effective proteome fraction. The equality holds when the two proteome fractions are equal (${\varphi }_{RFI}={\varphi }_{ribo}^{stop}$), which provides the solution for optimal ${\varphi }_{RFI}$:

Hence, we recover Equation 7 by minimizing the effective proteome fraction allocated to a given process in the translation cycle (the above argument applies to the optimal free concentration in the non-binding limited regime, see Appendix 2, section Non binding-limited regime (one stop codon) for an example). From this perspective, optimization of the translation apparatus balances the production cost of the enzyme of interest with the improved efficiency of a having less ribosomes idle at that step, Figure 2B. The optimal abundance in our model corresponds to a point of equipartition: the proteome fraction of free cognate factors equals the proteome fraction of ribosomes waiting at the corresponding step (Figure 2B).

We next consider a more complex step of the translation cycle – elongation – and demonstrate that the optimality criterion (Equation 5) can similarly provide simple analytical solutions in the physiologically relevant regime. Translation elongation involves multiple interlocked cycles (one for each chemical species) and enzymes (EF-Tu, EF-G, EF-Ts, aminoacyl-tRNA synthetases (aaRS), and more). Our simplified kinetic scheme for translation elongation is shown in Figure 3A: charged tRNAs are brought to ribosomes through a ternary complex (TC), corresponding to a bound tRNA and EF-Tu. Following tRNA delivery and GTP hydrolysis, EF-Tu is released from the ribosome, and nucleotide exchange factor EF-Ts recycles EF-Tu back into the active pool, after which EF-Tu can bind a charged tRNA again and form another TC. At the ribosome, translocation to the next codon is catalyzed by EF-G, followed by release of uncharged tRNAs. Aminoacyl-tRNA synthetases then charge tRNAs to complete the elongation cycle.

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Figure 3—source code 1

Source code to obtain panel (B) can be found in the associated scripts submitted with this work.

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To reduce the complexity due to different tRNA isoacceptors and aaRSs, we self-consistently coarse-grained the translation elongation cycle to have a single codon (derived in Appendix 3, section Coarse-grained one-codon model). The resulting model harbors a single effective species for tRNA, aaRSs, and TCs, respectively. A rescaling factor ($1/n_{aa}\approx 1/20$, estimated in section Estimation of coarse-grained rates) arises in the procedure to decrease the rates of codon specific reactions and can be attached to either the respective rate constants or chemical species concentrations. In our formulation, we choose to rescale the association rate constants such that the coarse-grained abundance for each effective species corresponds to the sum over all individual codon-specific components. For example, ${\varphi }_{aaRS}$ in our coarse-grained model corresponds to the summed proteome fraction of all aaRSs in the cell, and its association rate constant with the total tRNAs is rescaled by a factor of $1/n_{aa}$.

As a result of this choice of rescaling within our coarse-grained model, there are two classes of reactions in the elongation cycle that are distinguished by different kinetics: those that were codon specific (scaled by $1/n_{aa}$) and those that are not. Codon-specific reactions, for example, aaRS binding to cognate tRNAs and TC binding to cognate codons, are coarse-grained into one-codon reactions with reduced association rate constants (marked by # in Figure 3A). By contrast, codon-agnostic reactions do not incur such a rescaling and are thus much faster. We refer to this as a separation of timescale between the two classes of reactions (codon-specific vs. codon-agnostic), and note that this is not a reflection of slower underlying microscopic bimolecular reaction rates, but rather a result of our choice of variable in the coarse-graining.

Similar to translation termination, the factor-dependent ribosome transit time through a single codon (${\mathrm{\tau }}_{aa}$) is comprised of two steps, corresponding to binding of the TC and EF-G, respectively (formal derivation and non binding-limited regime in Appendix 3, section Coarse-grained translation elongation time):

The coarse-grained factor-dependent portion of the total translation elongation time in our model is then given by the single codon time above multiplied by the average number of codons per protein, that is, $⟨\mathrm{\ell }⟩{\mathrm{\tau }}_{aa}$. As discussed above, the rescaling of the TC association rate constant by $n_{aa}^{-1}$ arises as a result of our coarse-graining to a one-codon model (Appendix C, section C.1 Coarse-grained one-codon model). Note that the ternary complex concentration, ${\varphi }_{TC}$, is a nonlinear function of the concentrations of all elongation factors (including ${\varphi }_G$).

Despite the complexity of ${\mathrm{\tau }}_{aa}$ as a function of the ${\varphi }_{tlF,i}$, the fact that all fluxes are equal in steady-state allows several steps to be isolated and solved separately (EF-Ts and EF-G, greyed out in Figure 3A, respectively solved in Appendix C, sections C.3.3 Optimal EF-Ts abundance and C.3.4 Optimal EF-G abundance). For example, the approximate binding-limited solution for optimal EF-G concentration parallels that for termination factors:

Importantly, the optimum for EF-G is larger than the optimum for RFs by a factor $\sqrt{⟨\mathrm{\ell }⟩}$, reflecting that the typical translation cycle to produce a protein requires $⟨\mathrm{\ell }⟩$ steps catalyzed by EF-G and only one step for RFs (i.e. $⟨\mathrm{\ell }⟩{\mathrm{\tau }}_{aa}$ enters the optimality condition, Equation 5, in contrast to ${\mathrm{\tau }}_{ter}$ which is not multiplied by a scaling factor). The square root dependence arises here for the same reason as in the case of translation termination (derivative of ${\varphi }^{-1}$).

In contrast to EF-G and EF-Ts, EF-Tu and aaRS cannot a priori be treated in isolation because the TC is composed of both EF-Tu and charged tRNAs. Still, the separation of timescales within our coarse-grained model (see Appendix C, section Interpretation of the sharp separation between aaRS and EF-Tu limited regimes) simplifies the solution considerably. Indeed, rapid binding of charged tRNAs to EF-Tu leads to either component being limiting for ternary complex concentration in most of the aaRS/EF-Tu expression space, leading to two clearly delineated regimes (Figure 3B). In one regime, charged tRNAs are limiting (low aaRS), whereas EF-Tu is limiting in the other (low EF-Tu). These regimes are separated by a narrow transition region, whose sharpness is a reflection of the smallness of the rate rescaling parameter $n_{aa}^{-1}$ (see Appendix 3, section Interpretation of the sharp separation between aaRS and EF-Tu limited regimes). We term the focal region separating the two regimes in the aaRS/EF-Tu expression space the 'transition line’ (see 1 for derivation and additional details).

The transition line corresponds to conditions in which EF-Tu and aaRS are co-limiting for TC concentration. In the EF-Tu limited region, increasing aaRS abundance does not increase ternary complex concentration: since all EF-Tu proteins are already bound to charged tRNAs, increasing tRNA charging cannot further increase TC concentration. Conversely, in the aatRNA limited region, increasing EF-Tu abundance does not increase TC concentration: since all charged tRNAs are already bound by EF-Tu, increasing EF-Tu concentration does not alleviate the requirement for more charged tRNAs. Given that the optimality condition requires non-zero increase in ternary complex concentration with increasing factor abundance (Equation 5 using ${\mathrm{\tau }}_{aa}$ from Equation 10), the optimal EF-Tu and aaRS abundances must be on the transition line.

Which point on the transition line corresponds to the optimum? Note that inside the EF-Tu limited region, the ternary complex concentration is entirely set by the total EF-Tu concentration: ${\varphi }_{TC}\approx {\varphi }_{Tu}$ (since most EF-Tu proteins are bound by charged tRNAs, Figure 3—figure supplement 1). As an approximation resulting from the narrow range of transition region (Figure 3 and Figure 3—figure supplement 1), we assume that the EF-Tu limited regime solution ${\varphi }_{TC}\approx {\varphi }_{Tu}$ holds up to very close to the transition line. Replacing ${\varphi }_{TC}$ by ${\varphi }_{Tu}$ in the elongation time Equation 10 and substituting in the optimality condition (Equation 5), the approximate optimal abundance for EF-Tu (the full solution includes additional terms from the EF-Ts cycle, section Optimal EF-Tu and aaRS abundances) can then be obtained in the same way as for translation termination factors:

Importantly, compared to the solution for EF-G, the above is multiplied by an additional factor of $\sqrt{n_{aa}}$. This contribution arises from the rescaling of the association rate for the ternary complex to the ribosome in our coarse-grained one-codon model, increasing the requirement on EF-Tu abundance.

From the necessity for the combined EF-Tu and aaRS solution to fall on the transition line, the approximate solution for the optimal aminoacyl-tRNA synthetase abundance is then the intersection (yellow star in Figure 3B) of the transition line with the EF-Tu-only solution described above (dashed blue line in Figure 3B, derivation of solution in Box 1).

For the above derivation to be valid, the total number of tRNAs in the cell must be sufficient to accommodate all ribosomes (about two per ribosome, A- and P-sites) and binding to all EF-Tu (about gt₄ per ribosome based on endogenous expression stoichiometry [Li et al., 2014; Lalanne et al., 2018]). The number of tRNAs per ribosomes in the cell should thus be at least 6×. Remarkably, estimates of this ratio in the cell suggest that this is barely the case (between 6 and 7 tRNAs/ribosome at fast growth [Dong et al., 1996]). Although our model treats the total tRNA abundance as a measured parameter and omits its selective pressure (see Hu et al., 2020 which includes RNA mass in their optimization procedure), the abundance of three core components of the tRNA cycle appear to be at the special point where the transition line plateau, that is set by total tRNA abundance, just crosses the EF-Tu-only optimum (blue line in Figure 3B). At this point, all three components are co-limiting.

Analogous to the case studies above, optimal concentrations for all core translation factors can be solved using the optimality condition (Equation 5) and their respective kinetics schemes (the case of translation initiation is solved in Appendix 4). The analytical forms of the optimal solutions are shown in Table 1. In the binding-limited regime, the ratios of growth-optimized tlF concentrations are independent of the growth rate (except for aaRS), and are dependent only on basic biophysical parameters, such as protein sizes and diffusion constants.

To obtain the numerical values of association rate constants needed for calculating the optimal tlF stoichiometry (Table 1), we used the measured ${\widehat{k}}_{on}^{TC}$ in vivo and estimated all other association rate constants using a biophysically motivated scaling ($\widehat{k}$ denotes the raw association rate constant in units µM⁻¹s⁻¹, which is different from the rescaled $k$, see section Conversion between concentration and proteome fraction). To our knowledge, the binding between TC and ribosomes, ${\displaystyle {\hat{k}}_{on}^{TC}=6.4}$ µM⁻¹s⁻¹ (Dai et al., 2016), is the only measured association rate constant for any tlFs in a physiological context. We estimate the association rate constants for other reactions by scaling ${\widehat{k}}_{on}^{TC}$ by the respective diffusion coefficients of the chemical species, that is for reaction involving species $A$ and ${\displaystyle B:{\hat{k}}_{on}^{AB}/{\hat{k}}_{on}^{TC}=(D_A+D_B)/(D_{TC}+D_{ribo})}$, where $D_i$ is the diffusion constant for the molecular species $i$ (see Appendix 5—table 2). Diffusion constants for several tlFs have been measured experimentally (Bakshi et al., 2012; Sanamrad et al., 2014; Plochowietz et al., 2017; Volkov et al., 2018), and uncharacterized ones can be estimated using the cubic-root scaling with number of codons per protein from the Stokes-Einstein relation (Nenninger et al., 2010) (see Appendix 5—table 1). For simplicity, this approach assumes that reactive radii and orientational constraints are similar for the different reactions (see 3 Discussion for additional assumptions). These strong assumptions are necessary given the lack of in vivo biochemical parameter measurements, and can be relaxed as refined empirical determination for more physiological association rates become available in the future. Nonetheless, we note that the square-root dependence on these parameters (Table 1) for our predictions makes the numerical values less sensitive to possible tlF-specific effects.

The estimated optimal tlF concentrations show concordance with the observed ones, both in terms of the absolute levels and the stoichiometry among tlFs (Figure 4 for fast growth, see Supplementary file 1 for data and Figure 4—figure supplement 1 for additional growth conditions). A hierarchy of expression levels emerges such that the factors involved in elongation are more abundant compared to initiation and termination factors. The separation of these two classes is driven by the scaling factor $\sqrt{⟨\mathrm{\ell }⟩}\approx 14$ in our analytical solutions, which reflects the fact that the flux for elongation factors is $⟨\mathrm{\ell }⟩\approx 200$ times higher than that for initiation and termination factors. Within each class, the finer hierarchy of expression levels can also be further explained by simple parameters. For example, EF-Tu is predicted to be more abundant than EF-G by a factor of $\sqrt{n_{aa}{\mathrm{\ell }}_{Tu}/{\mathrm{\ell }}_G}\approx 3.3$ (observed ${\varphi }_{Tu}/{\varphi }_G$: E. coli 3.9, B. subtilis 2.7, V. natriegens 3.3). A higher abundance is required for EF-Tu because it is bound to the different tRNAs, which effectively decreases the concentration by a factor of $n_{aa}\approx 20$ (see section Estimation of coarse-grained rates for derivation and discussion of why the factor is not equal to the number of different tRNAs). Taken together, our model offers straightforward explanations for the observed tlF stoichiometry.

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For a few tlFs, the observed concentrations are two- to fivefold higher than the predicted optimal levels (e.g. EF-Ts, RF4, and IF1 in Figure 4). A potential explanation is that the corresponding reactions may not be binding or diffusion-limited, which would lead to a non-negligible fraction of tlFs sequestered at the catalytic step and thereby require higher total concentrations. Indeed, recent detailed modeling of the EF-Ts (Hu et al., 2020) cycle estimated only a small fraction (6% to 48%) of its abundance was in the free form in the cell, consistent with the large deviation we observe for this factor from our diffusion only prediction. Our optimization model can also be solved analytically in the non-binding-limited regime (Table 1), with the finite catalytic rate leading to an additional contribution of the form $\propto \mathrm{\ell }{\lambda }^{*}/k_{cat}$. However, the numerical values for these solutions are in general difficult to obtain because the estimates for catalytic rates are sparse and often inconsistent with estimates of kinetics in live cells. As an example, median estimated aaRS catalytic rates (Jeske et al., 2019) measured in vitro is ≈3 s⁻¹, well below the minimal value of 15 s⁻¹, required to sustain translation flux at the measured value (Appendix 5), suggesting substantial deviation between in vitro and in vivo kinetics. While technically demanding, the fraction of free vs. bound factors can in principle be determined through live cell microscopy of tagged factors by partitioning the diffusive states of the tagged enzyme. Using that approach, Volkov et al., 2018 estimated that EF-Tu was in its bound state <10% of the time (consistent with our diffusion-limited prediction closed to the observed value for this factor).

Another potential explanation for the observed deviations from our predictions is that the selective pressure for these tlFs may be lower compared to the more highly expressed tlFs. This explanation is unlikely both because their stoichiometry are observed to be conserved (Figure 1B, Figure 4—figure supplement 2) and given that the expression of other lowly expressed tlFs (e.g. RF1, RF2, and individual aaRSs) has been shown to acutely affect cell growth (Lalanne et al., 2021; Parker et al., 2020). Nevertheless, the deviations from the predicted optimal levels suggest that a more refined model may be required than our first-principles derivation.

We calculate the average number of codons per protein, weighted by expression, as

where ${\mathrm{\ell }}_i$ is the number of codon for the protein product of gene $i$, and e_i is the protein synthesis rate (as estimated from ribosome profiling [Li et al., 2014; Lalanne et al., 2018]) for gene $i$. For a stable proteome (in fast growing bacteria, the cell doubling time is shorter than the active degradation of most proteins [Larrabee et al., 1980]), the protein synthesis rate equals to the proteome mass fraction (Li et al., 2014). Changes in the expression of genes across growth conditions do not lead to substantial changes in $⟨\mathrm{\ell }⟩$. In E. coli, across growth conditions spanning ≈20 min doubling time to ≈120 min, $⟨\mathrm{\ell }⟩$ changes by about 20%. Specifically, we find $⟨\mathrm{\ell }⟩=$ 196, 210, and 240 in respectively MOPS complete (≈20 min doubling time [Li et al., 2014]), MOPS minimal (≈56 min doubling time [Li et al., 2014]), and NQ1390 forced glucose limitation (≈120 min doubling time [Mori et al., 2021]), based on ribosome profiling data. Here for simplicity, we take $⟨\mathrm{\ell }⟩\approx 200$ throughout.

Throughout, we use both units of concentration (molar), denoted as for example, $[A]$ for protein $A$, and proteome fraction, denoted by ${\varphi }_A$ (Scott et al., 2010). The correspondence between the two is ${\varphi }_A=[A]{\mathrm{\ell }}_A/P$, where ${\mathrm{\ell }}_A$ is the number of amino acid in protein $A$, and $P$ is the in-protein amino acid concentration in the cell. ${\displaystyle P\approx 2.6\times {10}^6}$ µM, and has a value approximately independent of growth rate (Klumpp et al., 2013; Bremer and Dennis, 2008). This change in units also relates to how association constants are defined in units of proteome fraction: ${\widehat{k}}_{on}[A]:=k_{on}{\varphi }_A$, where the hat $\widehat{\cdot }$ refers to the association constant in usual units of µM⁻¹ s⁻¹ (used to connect to empirical data). Hence, $k_{on}:={\widehat{k}}_{on}P{\mathrm{\ell }}^{-1}$ is the rescaled association rate in units of proteome fraction.

In steady-state exponential growth, the ribosome flux in and out of each intermediate state is equal to the total flux. This results from the fact that no ribosome can accumulate in any intermediate state. Since the flux out of state $i$ is given by ${\varphi }_{ribo}^i/{\mathrm{\tau }}_i$, we must have:

As a consequence, the proportion of ribosome in each state is equal to the proportion of time spent at that given step, for example for translation initiation:

In order to write the mass action kinetic scheme for more complex models, it is useful to recast our framework in terms of the protein number production flux $J$, defined as the number of full length proteins produced per cell volume per unit time. The production of each protein requires a ribosome to go through the full synthesis cycle, and as such $J$ provides a convenient quantity in mass action schemes formulated in molar units.

In steady-state of exponential growth (Monod, 1949; Scott et al., 2010; Dai et al., 2016), there is a direct relationship between the growth rate λ (defined through $\text{d}N/\text{d}t=\lambda N$, where $N$ is the number of cells per unit volume of culture) and the protein production flux $J$. Explicitly, the protein mass accumulation rate is $\lambda M$, where $M$ is the total protein mass per unit volume of culture. If $V$ is the mean cell volume, then $\lambda M/V=N{m}_{aa}⟨\mathrm{\ell }⟩J$, where $m_{aa}$ is the mean amino acid mass. Defining $P:=M/(m_{aa}NV)$, the in-protein amino acid concentration per cell (Materials and methods, section Conversion between concentration and proteome fraction), the connection between protein production flux $J$ and growth rate λ is then $J=\frac{P\lambda }{⟨\mathrm{\ell }⟩}$. This relationship will be used to convert between molar and proteome fraction in some equations below.

Solutions for the factor predicted optimal abundances as a function of effective biochemical parameters and the growth rate at the optimum, are presented in Table 1. The table breaks down terms in each solution by categories: direct diffusion term (arising from diffusive search time), catalytic sequestration, and delay incurred by the diffusion of other proteins in part of the cycle of the factor of interest. Solutions are listed in terms of on-rate ${\widehat{k}}_{on}$ (units of µM⁻¹s⁻¹). The aaRS solution follows a different form:

Our coarse-grained model of ribosome transitions between categories of initiation, elongation, and termination need to be distinguished from the individual molecular times of the respective steps in one important regard: ribosome traffic on mRNAs can lead to effective delays arising from transient queuing. For example, if translation termination is slow and ribosomes start to pile up and form queues upstream of stop codons on mRNAs, the molecular time of termination (time between ribosome arrival to the stop codon and its recycling to the free ribosome pool) will not be a correct reflection of the actual termination time of a ribosome, because of the additional wait time in the queue. A similar argument can be made for transient queuing forming in the body of genes for elongating ribosomes.

We connect these two (molecular and coarse-grained) levels of description by noting that our mass action schemes relating the translation factor abundance to the times of the specific steps can be used as input parameters in traffic models of ribosome movement along mRNAs taking into account possible many-body interactions (e.g. totally asymmetric exclusion processes [Shaw et al., 2003; Kavčič et al., 2020]). Solving these traffic models can then be used to obtain transition times in our coarse-grained translation cycle model. As we show below, corrections arising from transient queuing are small (for endogenous translation factor abundances) based on current estimates the absolute rates of initiation, elongation, and termination, on individual mRNAs, such that stochastic queuing does not play a dominant role in determining optimal translation factor expression levels.

As a first example, we relate the on-stop codon molecular termination time ${\mathrm{\tau }}_{ter}$, which we obtain from solving our mass action scheme (see Equation 6), to the termination time in presence of queuing: ${\mathrm{\tau }}_{ter}^{full}$. The difference between the two, as described above, being related to possible queues upstream of stop codons leading to further delays in the process of translation termination, and thus to a longer termination time than that of the molecular on-stop codon termination. The delay factor will be denoted $𝒬\left({\mathrm{\tau }}_{ter}\right)$, defined through:

To derive the expression for the $𝒬$ factor, note that in steady-state, ribosome numbers in a given state is directly proportional to the time to transition out of that state. Let m_i be the mRNA concentration for gene $i$ in the cell, $n_{ter}({\alpha }_i,{\mathrm{\tau }}_{ter})$ the number of terminating ribosomes (including queues if present) on a transcript with per mRNA translation initiation rate (i.e. translation efficiency [Li, 2015]) ${\alpha }_i$, then:

whereas

with ${\displaystyle n_{ter}^{\mathrm{\varnothing }\mathcal{𝒬}}({\alpha }_i,{\tau }_{ter})}$ the average number of terminating ribosomes on a transcript with translation efficiency ${\alpha }_i$, assuming no queue upstream of the stop codon. Note that ${\displaystyle n_{ter}({\alpha }_i,{\tau }_{ter})\ge n_{ter}^{\mathrm{\varnothing }\mathcal{𝒬}}({\alpha }_i,{\tau }_{ter})}$ (the differences being queued ribosomes). Hence, the queuing factor $𝒬$ is:

Formally, $n_{ter}$ can be obtained by solving a TASEP model (Shaw et al., 2003), but a simplified queue model (Bergmann and Lodish, 1979; Lalanne et al., 2021) disregarding spatial information recapitulates the statistics of queue formation (as verified by full stochastic simulations, data not shown). The state space of the queue model is the number of ribosomes $N$ in the queue. Ribosomes arrive at a rate α (initiation rate on the transcript), and leave at the molecular termination rate ${\mathrm{\tau }}_{ter}^{-1}$. The ribosome arrival rate at the queue is rigorously correct in steady-state, unless the queue becomes large enough to affect the initiation process (fully jammed transcript), or RNA degradation. The stochastic process (away from the jammed state) is then described by: $N\to N+1$ at rate α, and $N\to N-1$ at rate ${\mathrm{\tau }}_{ter}^{-1}$ for $N>0$. The probability for the queue to have $N$ ribosomes, $P(N)$, can be obtained as the steady-state from the resulting master equation, leading to a geometric series: $P(N)={\left(\alpha {\mathrm{\tau }}_{ter}\right)}^N\left(1-\alpha {\mathrm{\tau }}_{ter}\right)$. Hence, the prevalence of higher order queues scales as the ratio of the initiation to termination rate on the transcript. The average queue size, corresponding to $n_{ter}({\alpha }_i,{\mathrm{\tau }}_{ter})$, is:

Above, the solution of the simple model is truncated at the value where the transcript becomes fully jammed with ${\mathrm{\ell }}_i/{\mathrm{\ell }}_{footprint}$ ribosomes (${\mathrm{\ell }}_i$ and ${\mathrm{\ell }}_{footprint}$ being the size of gene $i$ and the size occupied by a ribosome respectively). The no queue ribosome number is simply equal to a model where queues with $N>1$ do not arise, hence ${\displaystyle n_{ter}^{\mathrm{\varnothing }\mathcal{𝒬}}({\alpha }_i,{\tau }_{ter})={\alpha }_i{\tau }_{ter}}$. Therefore, the queuing factor, under the stated assumptions (and assuming no transcript is in the jammed state), is

Expanding for fast termination gives $𝒬-1=\frac{{\mathrm{\tau }}_{ter}⟨{\alpha }^2⟩}{⟨\alpha ⟩}$ as the leading order correction, where the averages are weighted by mRNA levels. The above was derived assuming exponentially distributed initiation and termination times, but could be modified to account for more complex dynamics of the initiation and initiation steps.

The queuing factor can be estimated based on absolute measurements of the initiation and termination rates in cells. Kennell and Riezman, 1977 estimate 3.2 s between initiation events on the lacZ mRNA (at 48 min per cell doubling). Bremer and Dennis, 2008 estimate 1 s per ribosome initiation events at 20 min doubling time. Recent calibrated high-throughput measurements report a genome-wide median of 5.6 s per initiation events (Gorochowski et al., 2019). To our knowledge, estimation of absolute in vivo termination rates have not been performed, but we can estimate bounds. Indirect assessment based on steady-state protein production measurements place the fraction of actively elongating ribosome at about 95% (Dai et al., 2016). Assuming (upper bound) that the 5% of non elongating ribosomes are in the process of termination would give a termination time of ${\displaystyle 5\mathrm{\%}\times 11.1s\approx 0.6s}$ (fraction of ribosomes in a given state equal to the ratio of transition times), where we have used that the elongation time of an average protein is about 11.1 s (${\displaystyle 200/18s^{-1}}$) at fast growth (Dai et al., 2016). This upper bound is still much smaller than the reported median initiation time, suggesting that the queuing factor for termination is small. As additional support to the view that translation is far from being termination limited, small that queues at stop codons are only globally observed in ribosome profiling upon severe perturbations (Kavčič et al., 2020; Baggett et al., 2017; Mangano et al., 2020; Saito et al., 2020; Lalanne et al., 2021).

With regard to translation elongation, transient queuing in the body of gene can also lead to a difference between molecular and coarse-grained transition times in our model. However, the fraction of ribosomes transiently stalled due to this queuing scales as $\alpha {\mathrm{\tau }}_{aa}$ in the low-density phase (defined by requirements $\alpha {\mathrm{\tau }}_{ter}<1$ and $\alpha {\mathrm{\tau }}_{aa}<{(1+\sqrt{{\mathrm{\ell }}_{footprint}})}^{-1}\approx 0.25)$ of the TASEP model (Shaw et al., 2003). Since measured estimates place $\alpha {\mathrm{\tau }}_{aa}\sim 0.01$ (Dai et al., 2016; Gorochowski et al., 2019), we do not consider the queuing effect for elongating ribosomes within our optimization framework for elongation factor abundances.

If translation termination is not diffusion limited, terms corresponding to the finite catalytic times must be included in addition to the diffusive contributions in the termination time (Equation 6). Under our simplified scheme (Figure 2A) and with a single stop codons (grouping RF1 and RF2), the molecular termination time is then sum of the four separate times corresponding to distinct events:

The two novelties compared to the diffusion-limited regime (Equation 6) are: (1) addition of the catalytic times $k_{cat}^{-1}$ for the two steps, and importantly (2) the mass action diffusion terms now involve the free concentration of release factors. Generally, the free concentration of the tlFs can be obtained by solving the steady-state solutions of kinetic schemes under constraints imposed by conservation equations. The examples in e.g., sections B.3, C.3, and D.1 below provide the mathematical details associated with the procedure.

Here, the difference between the total and free concentration of release factor arises from the finite catalytic turnover of the enzymes, and corresponds to the concentration of ribosome bound release factors. Given the flux $J$ through the system in steady-state of growth, the concentration of ribosome bound release factor (e.g. for RF4) is $J/k_{cat}^{RF4}$, which becomes $\frac{{\mathrm{\ell }}_{RF4}\lambda }{⟨\mathrm{\ell }⟩k_{cat}^{RF4}}$ upon converting to proteome fraction. This quantity sets the absolute minimum for the release factor abundance necessary to sustain growth λ for a given $k_{cat}$. The free concentrations for the release factors are then:

Hence, the final solution for the steady-state termination time as a function of the total abundance of the release factors and growth rate is:

The relationship above, between termination time, total tlF abundance, and growth rate λ closes the solution of the kinetic scheme. Substituting the above in the optimality condition (Equation 5) leads to the solution:

The additional terms $\propto {\lambda }^{*}$ correspond to the contribution to the optimal abundance arising from the finite catalytic rates, no present in the diffusion limited regime (Equation 7).

The full model with three different stop codons (UAA, UGA, UAG) and RF1/RF2 with different specificities (RF1: UAA, UAG; RF2: UAA, UGA) can also be solved exactly, leading to a small correction on the summed optimal abundance for RF1 and RF2 of $\sqrt{1+2\sqrt{f_{UAG}{f}_{UGA}}}<1.05$ (fast growing species considered, where $f_{UAG}$ and $f_{UGA}$ are the fractional fluxes through the RF1 and RF2 stop codons, respectively) compared to the single stop codon optimum derived above (${\varphi }_{RFI}^{*}$, Equation 20). We provide details below. With three stop codons, the coarse-grained reaction scheme is shown in Appendix 2—figure 1. The relevant chemical species and parameters are listed in Appendix 2—table 1.

Appendix 2—figure 1

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