Abstract
Protein synthesis is the most expensive process in fastgrowing bacteria. Experimentally observed growth rate dependencies of the translation machinery form the basis of powerful phenomenological growth laws; however, a quantitative theory on the basis of biochemical and biophysical constraints is lacking. Here, we show that the growth ratedependence of the concentrations of ribosomes, tRNAs, mRNA, and elongation factors observed in Escherichia coli can be predicted accurately from a minimization of cellular costs in a mechanistic model of protein translation. The model is constrained only by the physicochemical properties of the molecules and has no adjustable parameters. The costs of individual components (made of protein and RNA parts) can be approximated through molecular masses, which correlate strongly with alternative cost measures such as the molecules’ carbon content or the requirement of energy or enzymes for their biosynthesis. Analogous cost minimization approaches may facilitate similar quantitative insights also for other cellular subsystems.
Introduction
Protein translation is central to the selfreplication of biological cells. While the workings of its individual components are well understood, the translation apparatus is a complex machine with many degrees of freedom, where the same rate of protein production could be achieved with very different relative abundances of its components. Although a large body of quantitative experimental data on these abundances in E. coli across different growth conditions is available, it is still unclear according to which organizing principle(s)—if any—they are set by the cell. Given that translation is the energetically most expensive process in fast growing E. coli cells, accounting for up to 50% of the proteome^{1} and 2/3 of cellular ATP consumption^{2}, it is likely that natural selection acted to optimize the efficiency of translation. But what exactly is efficiency in the evolutionary context?
In the late 1950s, it was hypothesized that ribosomes operate at a constant, maximal rate^{3,4}, consistent with the observed linear dependence of ribosome concentration on growth rate^{3,5,6,7}. This hypothesis was later proven untenable, as the activity of ribosomes was observed to increase with growth rate^{8}. Klumpp et al.^{9} suggested that optimal translational efficiency corresponds to the parsimonious usage of translationassociated proteins, most notably ribosomal proteins, elongation factor Tu, and tRNA synthetases. While these authors were able to fit a coarsegrained phenomenological model to the data, their suggested evolutionary objective could also not explain the observed growth rate dependencies quantitatively (see Supplementary Note 1 for a discussion of ref. ^{9} and of the phenomenological model of bacterial growth it is based on^{5,10}). Thus, it is currently unclear to what extent translation has indeed been optimized by natural selection, and—if such optimization indeed occurred—whether its action can be expressed in terms of a simple objective function.
In principle, these questions would best be addressed in the context of a wholecell model of balanced growth that combines mechanistic descriptions of metabolism and protein production. However, while such models have been described conceptually^{11,12}, kinetic parameterizations are unavailable for a majority of the relevant enzymatic reactions^{13}, preventing a truly mechanistic description that combines metabolism and protein translation. Thus, we here focus on protein production alone, taking the experimentally observed output of translation (proteome production rate and composition in a given growth condition), the corresponding input (charged tRNAs), and the kinetics of individual translation reactions as given. We then use this mechanistic description of translation to find the combination of the concentrations of mRNA, ribosomes, elongation factors, and tRNAs that results in minimal cellular costs in the given condition. Thus, our estimate of the optimal efficiency of the translation machinery is not based on the maximization of ribosome activity, but on the minimization of the combined cost of the complete translation machinery at an observed protein production output.
We base our cost definition on the experimental observation that cellular dry mass per cell volume is approximately constant across environments and growth rates in E. coli^{14}, as is the total mass concentration in the cytosol^{15}. If the cell allocates more of this limited mass concentration budget to one particular process, less is available to other processes. The upper bound for the cytosolic mass concentration, beyond which diffusion becomes inefficient, is a fundamental constraint on cellular growth^{16,17,18}, and we thus use the cytosolic mass concentration of a particular molecule type as an approximation to its cost. Theoretical models of cellular growth that account for all major biochemical and biophysical constraints indicate that the limit on cellular dry mass indeed represents a dominant constraint on bacterial growth rates^{12}.
We hypothesize that to maximize the E. coli growth rate in a given environment, natural selection minimizes the total mass concentration of translation components utilized to achieve the required protein production rate. A corresponding optimality principle has been used to understand the relationship between the concentrations of enzymes and their substrates^{19}.
We find that a theoretical minimization of the combined cellular costs of the translation machinery components indeed leads to accurate predictions for their abundances, the resulting elongation rate, and the RNA/protein ratio. In addition to molecular masses, we also examine four alternative cost measures for cellular components that have been explored in the literature: (i) their protein content^{9,20}; (ii) their carbon content^{21}; and (iii) the energy^{22,23} or (iv) the amount of catalysts^{5,24} required for their production. We find that these alternative cost measures are strongly correlated for the studied components of the translation machinery and lead to very similar predictions for their abundances; the only cost measure that leads to substantially different predictions is the protein content, which does not assign any cost to tRNA and mRNA molecules.
Results
Cost minimization in a mechanistic model of translation
To test our hypothesis, we constructed a translation model consisting of 274 biochemical reactions, including 119 reactions with nonlinear kinetics. Figure 1 shows the modeled reactions for a subset of the 61 codons and the 40 species of charged tRNAs; for details see Methods, Supplementary Table 1, and Supplementary Data 1. This mechanistic model accounts for the concentrations of mRNA, the ribosome, the different charged tRNAs, and the elongation factors Ts (EFTs) and Tu (EFTu). We fully parameterized the model with molecular masses and kinetic constants measured experimentally^{25,26,27}; the only exceptions are the translation initiation parameters, which were previously estimated from gene expression data^{25}, and the ribosomal Michaelis constant for the ternary complexes, which was previously estimated based on the diffusion limit^{9}. The model is based purely on biochemical and biophysical considerations; it contains no adjustable parameters, nor does it include any explicit growthrate dependencies.
For E. coli growing under different experimental conditions, we used measured growth rates and protein concentrations^{28} to determine the required translation rate and the proportions of the different amino acids incorporated into the elongating proteins. At this required protein production rate, we minimized the combined cost of the translation machinery in our model, treating the concentrations of all components as free variables; the values of individual reaction fluxes result deterministically from these concentrations according to the respective rate laws (Methods). As the modeled kinetic rate laws are nonlinear, all optimizations were performed numerically. In repeated optimization runs with two different solvers, we never found alternative optima, indicating that the optimization problem may be convex. The results shown in the main text and figures are based on the assumption that costs are proportional to molecular masses; results based on other cost functions are shown in Supplementary Figs. 5 and 7.
Predicted concentrations agree with observations
We first compared our predictions to experimental data for exponentially growing E. coli in different conditions^{8,28,29,30,31}. Figure 2 shows the results for growth in a glucoselimited chemostat at growth rate μ = 0.35 h^{−1}; for other conditions, see Supplementary Fig. 1. The mechanistic model accurately predicts the absolute concentrations of ribosomes, EFTu, EFTs, mRNA, and total tRNA in each condition. Predictions for individual tRNA concentrations are less accurate but are still mostly within a 2fold error (Fig. 2, Supplementary Fig. 1); the discrepancies may be due to the simplifying assumption of the same ribosomal Michaelis constant K_{m} for all tRNA species^{9}.
We next tested if this systemslevel view on the total cost of translation explains the observed growth ratedependencies of the expression of translation machinery components^{8,28,29,31,32}, of the elongation rate^{32}, and of the RNA/protein ratio^{5,32}, considering experimental data across 20 diverse conditions (14 minimal media, including 3 stress conditions; 4 chemostats; and 2 rich media)^{28}. The predicted concentrations of ribosomes, EFTu, and EFTs increase with growth rate in line with experimental observations (Fig. 3).
Predicted absolute abundances of EFTu (Fig. 3a), EFTs (Fig. 3b), and mRNA (Supplementary Fig. 2a) account quantitatively for the experimental data^{8,28,29,30,31}, with average deviations (geometric mean folderror) GMFE ≤ 27% for the elongation factors and GMFE = 6% for mRNA. At low growth rates, experimentally observed concentrations of EFTu (Fig. 3a) and tRNA (Supplementary Fig. 2b) are higher than predicted. The model only includes charged (aminoacyl) tRNA concentrations, and it is likely that the unknown fraction of uncharged tRNA explains at least part of this deviation. Overall, the largest deviations between observed concentrations and predictions are seen in the two nonminimal conditions, which also exhibit the fastest growth (µ > 1 h^{−1}). A recent analytical study of balanced cellular growth indicates that these deviations may result from the influence of an increased growthrelated dilution of cofactors and other intermediate metabolites, a phenomenon not included in our simulations^{12}.
Active and deactivated ribosome fractions
At low growth rates (µ < 0.3 h^{−1}; Fig. 3c), observed ribosome concentrations exceed those predicted from cost minimization, a deviation consistent with a substantial reserve of deactivated ribosomes at low growth rates^{32}. Such deactivated ribosomes may provide fitness benefits in changing environments^{33,34}, but cannot be maximally efficient in a constant environment and thus cannot be predicted by our optimization strategy. To allow a meaningful comparison between predictions and experiment, we thus estimated the experimental concentration of ribosomes actively involved in elongation (Methods). Cost minimization predicts these experimental estimates with high accuracy across the full range of assayed growth rates; observed values deviate from predictions on average by GMFE = 14% (Fig. 3d).
The remaining, nonactive ribosome fraction comprises two parts: the deactivated ribosome reserve currently unavailable for translation^{32}, and free, potentially active ribosomes not currently bound to mRNA (see Supplementary Note 2 for the nomenclature on ribosome states). As our model quantifies the abundance of both active and free ribosomes, their subtraction from observed total ribosome concentrations provides an estimate of the deactivated ribosome reserve as a function of the growth rate (Fig. 4). While this reserve accounts for less than 20% of total ribosomes at fast to moderate growth, it reaches almost 50% at the lowest growth rate assayed in ref. ^{28}.
RNA/protein ratio and elongation rate
A linear correlation between the RNA/protein ratio and growth rate was discovered in the 1950s^{3,6,7,35} and forms the basis of phenomenological bacterial growth laws^{5,9,32}. Relating the predicted total RNA (ribosomal RNA + tRNA + mRNA) with measured protein concentrations^{28} indeed results in a nearlinear relationship, accurately matching observed values at high to intermediate growth rates (µ > 0.3 h^{−1}; Fig. 5a). At lower growth rates, model predictions are slightly too low, likely because of the deactivated ribosome reserve^{32} (Fig. 4). At low growth rates (µ = 0.12 h^{−1}), predictions of RNA and proteins allocated to an optimally efficient translation machinery (including deactivated ribosomes) account for 13% of total dry mass, rising almost linearly to 49% at high growth rates (µ = 1.9 h^{−1}; Supplementary Fig. 3).
The concentrations of the individual components of the translation machinery determine the average translation elongation rate (ribosomal activity), defined as the total cellular translation rate divided by the total active ribosome content^{34}. The predicted elongation rates closely match the experimental data^{32} over a broad range of growth rates (Fig. 5b).
Cost minimization predicts response to antibiotics
The expression of E. coli’s translation machinery reacts strongly to the exposure to antibiotics that inhibit the ribosome, such as chloramphenicol^{5,32,36}. The details of these changes can also be understood from our hypothesis of cost minimization. The concentrations of ribosomes and EFTu, the RNA/protein ratio, and the elongation rate of active ribosomes increase under chloramphenicol stress (Supplementary Fig. 4); these changes partially compensate for the reduced fraction of active ribosomes. The concentration of EFTs instead decreases with increasing chloramphenicol concentration (Supplementary Fig. 4c). EFTs contributes to translation by converting EFTu·GDP to EFTu·GTP, which then forms a ternary complex with charged tRNA. Under chloramphenicol stress, fewer ternary complexes are turned over, and hence less EFTs is needed.
Alternative cost measures lead to similar results
The biological fitness of E. coli cells depends on many factors, and hence any simple assignment of fitness costs to molecules can only be approximate. The results presented so far are based on the assumption that costs are proportional to molecular masses. To test if alternative cost measures lead to consistent results, we repeated our calculations using four distinct costs that have been employed in the literature.
Across most conditions, we obtained very similar predictions for the concentrations of translation machinery components when our model assigned molecular costs based on the carbon content of the molecules^{21}, on the amount of energy^{22,23} spent on their production (ATP cost), or on the total investment into macromolecular catalysts^{24} required for their production (synthesis cost) (Supplementary Figs. 5 and 6; we estimated ATP costs based on refs. ^{37,38}, and calculated synthesis costs using flux balance analysis with molecular crowding^{39,40,41}, see Methods). All components whose concentrations we predict consist of protein, RNA, or both, and all costs examined are approximately proportional to the lengths of RNA and protein molecules. Thus, the relative costs of all components are essentially a function of the RNA/protein cost ratio r, i.e., the cost of RNA per nucleotide divided by the cost of protein per amino acid. We assume that the cost of RNA per nucleotide is identical for tRNA and rRNA; the corresponding cost ratio is broadly similar between molecular masses (r = 3.0), carbon content (r = 2.0), ATP cost (r = 1.6), and synthesis costs (r = 1.7–2.1) across minimal growth conditions (µ < 1 h^{−1}; Supplementary Fig. 7). In contrast, assuming that costs are proportional to only the protein content of the molecular assemblies^{9,20} results in an RNA/protein cost ratio of zero. Predictions based on protein costs hence overestimate mRNA and tRNA concentrations (which cost nothing), resulting in corresponding underestimates of EFTs and especially EFTu concentrations (Supplementary Figs. 5, 6).
We note that in rich medium (µ = 1.9 h^{−1}), the RNA/protein cost ratio for synthesis costs is much lower than across minimal media, falling to r = 0.21 for tRNA and rRNA (Supplementary Fig. 7). This results in an overprediction of the observed tRNA concentration^{8,29,31} by a factor of almost 2 (Supplementary Fig. 6). Moreover, while the predicted tRNA concentration is also almost twice the predicted EFTu concentration, experimental estimates for tRNA and EFTu are very similar^{8,28,29,31,42}. We conclude that if the translation machinery has been optimized for efficiency at high growth rates by natural selection, the synthesis cost of its components is unlikely to have been central to this optimization.
In sum, cost minimization in a mechanistic bottomup model of optimal translation efficiency, fully parameterized with known kinetic constants and molecular masses, accounts quantitatively for the concentrations of all molecule species involved without any adjustable parameters. The optimal concentrations of different components change differentially with growth rate, explaining the observed scaling of E. coli’s translation machinery composition, RNA composition, and elongation rate. At least for the translation machinery part of the cellular economy, whose components consist largely of protein and RNA, approximate cost measures appear to be sufficient: several alternative cost measures provided predictions very similar to those based on molecular masses, emphasizing the generality of our findings.
We conclude that E. coli’s translation machinery works close to optimal cost efficiency. Accordingly, our findings are consistent with the hypothesis that natural selection has minimized a cost function similar to those examined here. Our results further support the idea that phenomenological growth laws of proteome composition^{5,9,32,36} may have their root in the costs associated with the nonprotein molecules involved in particular processes, and that their explicit inclusion in systems biology models of cellular growth^{9,25,43,44} may eventually allow these models to abandon any reliance on phenomenological parameters.
Methods
Experimental concentrations of ribosomes, EFTu, and EFTs
We used molar concentrations (µM) in the model; thus, all experimental data were converted to molar concentrations. We first calculated the total protein density during exponential growth on a glucose minimal medium at growth rate μ = 0.58 h^{−1}. In this condition, the total protein mass per cell is 280 fg (Supplementary Note 3 in ref. ^{28}), and cell volume is 1.90 fL (the cell volume 2.84 fL modified by a factor of 0.67 according to Supplementary Note 3 in ref. ^{28}). Accordingly, the protein mass density on glucose is ρ_{p,glc} = 147.15 g L^{−1}.
We then fitted a secondorder polynomial function ϕ(μ) to the fraction of total protein in dry mass provided in ref. ^{1} across different growth rates µ. With ϕ(μ), ρ_{p,glc} = 147.15 g L^{−1} at μ = 0.58 h^{−1}, and the observed constant dry mass density of E. coli across growth conditions^{14,15}, we obtained the conditionspecific total protein concentration, ρ_{p}, for all other growth conditions based on the respective observed growth rates.
For a given growth rate, μ′, the total protein concentration ρ_{p, μ = μ′} is given by
With the measured fraction of each protein (f_{p,i}) in the proteome^{28}, the molar concentration of each protein was then calculated as
where MW_{p,i} is the molecular weight of protein i in g·mole^{−1}. The ribosome concentration was calculated as the arithmetic mean of the molar concentrations of all ribosomal proteins.
Experimental concentration of active ribosome
Active ribosomes are defined here as ribosomes engaged in peptide elongation. Dai et al. estimated the fraction of active ribosomes, f_{active}, in E. coli at different growth rates^{32}. We fitted a MichaelisMenten type equation to their data, resulting in f_{active} = μ/(0.124 + μ). For each total ribosome concentration c_{ribosome} in ref. ^{28}, we then estimated the corresponding active ribosome concentration as c_{activeribosome} = f_{active}·c_{ribosome}.
Experimental concentrations of GTP and GDP
GTP and GDP concentrations are from ref. ^{45}. We chose the data for growth on glucose for all simulations (See Supplementary Note 3 for details).
Experimental concentration of mRNA
mRNA concentration was calculated from the data given in ref. ^{30} as the ratio of mRNA copy number per cell and cell volume. To estimate the mRNA concentrations at the growth rates shown in Fig. 2 and Supplementary Fig. 1, we fitted a second order polynomial to the mRNA concentration as a function of growth rate; we then read off the values at the required growth rates. mRNA concentrations were assayed only at growth rates between 0.11 h^{−1} and 0.49 h^{−1}, and we did not attempt to extrapolate values beyond this range.
Experimental concentration of tRNA
We collected three independent datasets of tRNA concentrations. Dataset 1^{29} contains tRNA concentration for each individual tRNA, whereas both dataset 2^{8} and dataset 3^{31} contain only total tRNA concentrations. In each of these experiments, tRNA abundance was measured as the ratio of tRNA to ribosomal RNA (rRNA). We scaled these values to absolute tRNA concentrations assuming that the rRNA concentration corresponds to the ribosome concentration estimated from the proteomics data (see the subsection “Experimental concentrations of ribosomes, EFTu, and EFTs”). To estimate the tRNA concentrations at the growth rates shown in Fig. 2 and Supplementary Fig. 1, we used the same fitting procedure as for mRNA.
Concentrations of individual tRNAs and relationship to model
Our model differentiates tRNAs by their anticodons (see below for details). Thus, 40 tRNAs were used to represent all elongator tRNAs. The tRNAs modeled in this work are listed in Supplementary Table 1 together with their common names used in dataset 1^{29} and their gene IDs.
In the experiments by Dong et al. (dataset 1)^{29}, tRNAs were classified into 41 distinct sets based on twodimensional polyacrylamide gel electrophoresis. We combined two tRNA sets corresponding to different tRNA weights if they have the same anticodon (i.e., the pairs of Val2A + Val2B, Thr1 + Thr3, and Tyr1 + Tyr2). The experimenters could not distinguish between the tRNAs Gly1 and Gly2, as these have very similar molecular weights and isoelectric point; the same was true for Ile1 and Ile2. We estimated the individual concentrations of these four tRNAs based on the ratios 3:2 between Gly1 and Gly2 and 20:1 between Ile1 and Ile2 observed by Ikemura et al.^{46}.
To estimate the tRNA concentrations at the growth rates shown in Fig. 2 and Supplementary Fig. 1, we fitted the concentration of each tRNA in dataset1 (measured for growth rates ranging from 0.28 h^{−1} to 1.73 h^{−1}) to a second order polynomial of growth rate and extended this function to the required range, 0.12 h^{−1} to 1.9 h^{−1}.
Combination of tRNAs that are predicted to be nonexpressed
The predicted concentrations of 6 tRNAs are 0 μM; these are highlighted in red in Supplementary Table 2. This result is a straightforward consequence of the model structure. The relationship between codons and tRNAs is not onetoone in E. coli. Consider a given codon (codon1) that has more than one cognate tRNA, say, tRNA1 and tRNA2. If tRNA2 is also the cognate tRNA of another codon (codon2), the predicted concentration of tRNA1 will be zero: for the same “price” (the same contribution to the limited total mass concentration), tRNA2 can service two codons, while tRNA1 can service only one. For example, codon GGG has two cognate tRNAs, gly1 and gly2; gly2 is also the cognate tRNA of codon GGA. Thus, both gly1 and gly2 can translate GGG, but gly2 can translate GGA, too, and is thus more valuable to the cell if we assume that both tRNAs are processed equally efficiently, as done in the model. Therefore, the predicted optimal concentration of gly1 will be zero (note that this might not occur in models that consider different ribosomal k_{cat} or K_{m} values for the two tRNAs). To compare our predictions to the experimental data^{29}, we combined tRNAs with predicted zero concentration with their cofunctioning tRNAs in both the predictions and the experimental data. The resulting six combined tRNA pairs are: GLy1 + Gly2; Leu1 + Leu3; Leu4 + Leu5; Pro1 + Pro3; Ser2 + Ser1; Thr2 + Thr4. In all reported figures, the total number of tRNAs shown is thus 34.
Concentrations of amino acids and total protein
We first calculated a typical molecular protein mass for each condition. MW_{p} is the weighted average over the molecular masses of all proteins assayed by Schmidt et al., where the weights are the corresponding proteome fractions in that condition.The molar concentration of protein, c_{protein}, is given by
where ρ_{p} is the mass concentration of total protein at the given condition (Eq. 1).
The concentration of amino acids encoded by codoni is given by:
where L_{protein} is the abundanceweighted average protein length at the given condition; f_{codoni} is the frequency of codon i in the genome, where each gene is weighted by its relative abundance in the proteome.
Note that for an amino acid AA_{j} encoded by multiple synonymous codons, c_{codoni} is not the total concentration of AA_{j} in cellular proteins, but only of the fraction encoded by codoni; the total concentration of AA_{j} is obtained by summing the c_{codoni} values for all synonymous codons for AA_{j}.
For simulations under chloramphenicol stress, c_{protein} and c_{codoni} are not available. We approximated their values by the corresponding concentrations for growth on glucose in the absence of the antibiotic.
Mass fraction of translation machinery in total dry weight
The dry mass fraction of the translation machinery shown in Supplementary Fig. 3 includes ribosome, mRNA, charged tRNAs, EFTu, and EFTs; it does not include GDP, GTP, free tRNA, tRNAsynthetases, and elongation factor G (FusA). We converted from protein fractions to mass fractions of total dry weight using the relationship between total protein mass and dry mass discussed in the subsection “Experimental concentrations of ribosomes, EFTu, and EFTs”.
Experimental estimate: The mass fraction of the translation machinery in total dry weight is the sum of two parts: (1) protein and (2) RNA.
(1) We calculated the mass fraction of translational proteins (including ribosomal protein, EFTu, and EFTs) in dry weight from the proteomics data in ref. ^{28}.
(2) We fitted the reported total RNA/protein ratio in refs. ^{5,32} to a second order polynomial of growth rate. We then used this fitted function to calculate the RNA/protein ratio at the growth rates assayed by Schmidt et al.^{28}, and multiplied this ratio with the dry mass fraction of protein.
Theoretical prediction: The predicted dry mass fraction of the translation machinery was the ratio of the total mass concentration of the translation machinery (including free ribosome, active ribosome, EFTu, EFTs, charged tRNA, and mRNA) to the total dry mass density. The total dry mass density was estimated as ρ_{p,glc}/ϕ(0.58 h^{−1}) = 147.15 g L^{−1}/0.631 = 233.30 g L^{−1}. For the prediction including deactivated ribosome concentrations (dashed line in Supplementary Fig. 3), we added estimates of deactivated ribosome concentrations according to Fig. 4 of the main text.
Molecular weights
Molecular weights of ribosome, charged tRNAs (aatRNAs), EFTu, EFTs, and the ternary complexes (TC, EFTu·GTP·aatRNA) were calculated from their sequences. The stoichiometry of ribosomal proteins and RNAs in the ribosome was obtained from the EcoCyc database^{47}; the stoichiometry of all components is 1 except for RplL, for which it is 4.
We used an average mRNA to represent the total mRNA. The molecular weight of an average mRNA (MW_{mRNA}) is the sum of two parts: (1) the molecular weight of the coding sequence (CDS) of mRNA (MW_{mRNACDS}), which was calculated from proteinexpressionweighted mRNA length and nucleotide composition of E. coli proteincoding sequences in each growth condition^{28}; (2) the weight of the untranslated region (UTR) of mRNA (MW_{mRNAUTR}), which was calculated from the average nucleotide composition of the genome and the typical length of UTR. The length of UTR was assumed to be 85 nt, a typical length of the untranslated region in E. coli^{48}. Thus,
In the simulations of translation under antibiotic stress, the molecular weight of chloramphenicol was set to 0.
Alternative costs of translation machinery components
The hypothesis underlying our analysis is that the components of the translation machinery are expressed to minimize the total cost of translation at a given protein production rate. For the calculations underlying the figures of the main text, we assumed that molecular costs are proportional to molecular mass concentrations. To test alternative cost measures proposed in the literature, we estimated the costs of each translation machinery component in terms of (1) its carbon content^{21} (carbon cost); (2) the total number of highenergy phosphate bonds required for its production (ATP cost); (3) the total enzyme mass required for its production (synthesis cost)^{24}; and (4) its protein content^{9,20} (protein content). Some of these cost measures are conditiondependent. The estimated costs are provided in Supplementary Data 2; the RNA/protein cost ratios are compared in Supplementary Fig. 7 and listed in Supplementary Table 3. All costs were estimated per component (i.e., per molecule or per macromolecular complex). Note that all components whose costs are considered in the model consist of protein, RNA, or both.
Carbon cost as an alternative cost measure
The carbon cost of a component is its total number of carbon atoms.
ATP cost as an alternative cost measure
The ATP cost of a component is the number of highenergy phosphate bonds (denoted ~P) that were invested into its production. The ATP costs includes (1) the ATP invested into the synthesis of the precursors (nucleoside triphosphates or amino acids) and (2) the ATP cost of polymerization during RNA transcription or protein translation.
The ATP cost of amino acid synthesis: For cells growing on minimal carbon media, the ATP cost of amino acid production was obtained from ref. ^{37}. Since the ATP costs of a given amino acid are very similar across minimal media with different carbon sources^{37}, we used the ATP costs for amino acid synthesis on glucose for all minimal media considered. For E. coli growing on glycerol + amino acids^{28}, the ATP production cost for amino acids was assumed to be zero.
The polymerization cost of protein: The polymerization cost of protein is 4 ATP per amino acid: two ATP for tRNA charging, 1 ATP for EFTu in elongation, and 1 ATP for elongation factor G (EFG) in elongation.
The ATP cost of NTP synthesis: For cells growing on minimal carbon media, the de novo synthesis cost of NTP was obtained from ref. ^{38}. The glycerol + amino acids medium used in the proteomics study^{28} also contains adenine and uracil; here, we assumed that the synthesis of ATP and GTP starts from adenine and that the synthesis of UTP and CTP starts from uracil. PRPP (5phosphoαDribose 1diphosphate), whose production consumes 29 ATP^{37}, was considered to be the donor of ribose to the synthesis of NTPs. Finally, we estimated the total energy (~P) costs in the glycerol + amino acids medium for ATP, UTP, CTP, and GTP as 31, 31, 29, and 32, respectively. We did not attempt to estimate the ATP cost of NTP synthesis in the LB condition, as it is not clear to what extent NTPs are taken up from the medium.
The polymerization cost of RNA: The polymerization cost of RNA is 0 ATP per base, as no highenergy phosphate bonds beyond those of the polymerized NTPs are required.
The degradation cost of mRNA: The degradation rates of tRNA and rRNA are much lower than their production rates, and hence we did not account for their degradation. In contrast, mRNA is degraded much more quickly than tRNA and rRNA, and we thus considered the influence of degradation on the mRNA polymerization cost. At steady state, all degraded mRNA (in the form of nucleoside monophosphates, NMPs) is assumed to be recycled to retranscribe mRNA. These recycled NMPs require two ~P to form NTPs, so the mRNA recycling cost (cost_{NTPdeg}) is 2 ~P per NTP. At a given mRNA concentration c_{mRNANTP} (in units of NTPs built into mRNA), the production of mRNA must offset the combination of mRNA degradation and mRNA dilution by cellular growth at rate µ. The rate of ATP consumption for this production is thus given by
where cost_{NTP} is the synthesis cost of NTPs (estimated above, “The ATP cost of NTP synthesis”) and k_{deg} is the mRNA degradation rate constant. k_{deg} is calculated from mRNA halflife (t_{half}), k_{deg} = ln(2)/t_{half}, with t_{half} = 5 min for all growth conditions^{49}. To obtain the ATP cost per NTP in mRNA, we must divide this rate by μ· c_{mRNANTP}:
Thus, mRNA degradation dominates the ATP cost of mRNA at very low growth rates, but becomes insignificant at growth rates higher than the mRNA degradation rate.
Synthesis cost as an alternative cost measure
The synthesis cost is the total macromolecular dry mass, which includes transporters, enzymes, RNA polymerase, and ribosome, that is needed to synthesize each component of the translation machinery. To estimate the synthesis cost of each component, we first estimated the macromolecular dry mass that is needed to synthesize one millimole of amino acid (cost_{AA}) and one millimole of nucleotide (cost_{nucl}). Based on these estimates, the synthesis cost of a protein is cost_{protein} = L_{protein}·cost_{AA}, where L_{protein} is the protein length in amino acids, and the synthesis cost of an RNA molecule is cost_{RNA} = L_{RNA}·cost_{nucl}, where L_{RNA} is the length of the RNA molecule in nucleotides.
We calculated cost_{AA} and cost_{nucl} using ccFBA^{40,41}, which is an implementation of the MOMENT^{39} algorithm for flux balance analysis with molecular crowding, featuring an improved treatment of cofunctional enzymes. Briefly, ccFBA assigns each enzyme a constant catalytic rate (k_{cat}) and molecular weight, and then finds the flux distribution that maximizes biomass production while not exceeding a threshold on the total enzyme mass.
Synthesis cost of protein: We first added a protein synthesis reaction to the iML1515 model implemented in ccFBA. The stoichiometric coefficient of each amino acid consumed in this reaction was set to its proportion in the biomass reaction of the iML1515 model. In E. coli, there are approx. 9 TCs per ribosome at high growth rates^{29,42}; we thus designated the ribosome plus 9 ternary complexes (TCs) as the “enzyme” of the protein synthesis reaction. We parameterized this “enzyme” with k_{cat} = 22 s^{−1} and molecular weight = 2933.241 kD^{1}. We then set the objective function to the rate of protein production, v_{AA}, instead of the biomass production rate v_{bio}. We simulated different growth conditions by only allowing the model to import nutrients available in the respective medium, setting the lower bound of the corresponding exchange reactions to −1000 (mmol·g_{DW}^{−1}·h^{−1}). For cells growing on LB medium, the lower bound of all exchange reactions was set to 1000, i.e., all metabolites for which there is a transporter in the model can be taken up. We maximized the protein synthesis rate v_{AA}, given a limit on the dry mass fraction of macromolecules involved in metabolism (a “budget”) of C = 0.27. As the lower bound chosen for the exchange rates was very high, the optimizations were constrained by C, i.e., the optimal v_{AA} value is the maximal rate of protein production with this macromolecular budget. Accordingly, the synthesis cost of amino acids is cost_{AA} = 0.27/v_{AA}, expressed as the dry mass fraction required to produce 1 mmol·g_{DW}^{−1}·h^{−1} of amino acids. The computed protein synthesis costs for all 20 conditions considered are shown in Supplementary Table 4.
Synthesis cost of stable RNA: To estimate the cost per nucleotide of stable RNA production, cost_{nucl}, we implemented an analogous algorithm. We first added an RNA synthesis reaction to the iML1515 model^{50} and set this reaction as the objective function. The stoichiometry of NTPs consumed in this reaction was set to the corresponding fractions of NTPs in the biomass reaction of the iML1515 model. We designated the RNA polymerase (molecular weight = 389.11 kD) as the enzyme catalyzing this reaction. For stable RNA (tRNA and rRNA) synthesis, the turnover rate of the RNA polymerase (RNAP)^{1} is k_{RNAPsRNA} = 85 s^{−1}. We maximized the flux of the RNA synthesis reaction, v_{sRNA}, constrained by the macromolecular budget C = 0.27. The synthesis cost of stable RNA per nucleotide was then calculated as cost_{nuclsRNA} = 0.27/v_{sRNA}, expressed as the dry mass fraction required to produce 1 mmol·g_{DW}^{−1}·h^{−1} of NTP in stable RNA. The computed stable RNA synthesis costs for all 20 conditions considered are listed in Supplementary Table 4.
Synthesis cost of mRNA: In E. coli, mRNA transcription is slower than stable RNA transcription. The turnover rate of RNAP for mRNA is k_{RNAPmRNA} = 66 s^{−1} (i.e., the fastest rate of mRNA transcription has been observed to be about three times the maximal translation rate)^{51}. As for the calculation of the ATP cost of mRNA, we need to account for mRNA degradation at rate k_{deg} (which is much faster than the degradation of stable RNA). We again assumed that nucleotides from degraded mRNA are reused by RNAP to synthesize mRNA. As before, we note that at a given mRNA concentration c_{mRNA}, the production of mRNA must offset the combination of mRNA degradation and mRNA dilution by cellular growth at rate µ. The rate of mRNA production by the RNAP is thus
To obtain the concentration of RNAP necessary to catalyze this rate, we need to divide this expression by k_{RNAPmRNA}:
Thus, the concentration of RNAP required for mRNA production is larger by a factor (1 + k_{deg}/μ) when accounting for mRNA degradation than it would be otherwise. To account for mRNA degradation, we thus set the effective turnover number of RNAP to \(k_{{\mathrm{eff}}  {\mathrm{mRNA}}} = k_{{\mathrm{RNAP}}  {\mathrm{mRNA}}}/\left( {1 + k_{{\mathrm{deg}}}/\mu } \right)\). We maximized the flux of the mRNA synthesis reaction, v_{mRNA}, constrained by the macromolecular budget C = 0.27. The synthesis cost of mRNA per nucleotide was then calculated as cost_{nuclmRNA} = 0.27/v_{mRNA}, expressed as the dry mass fraction required to produce 1 mmol g_{DW}^{−1} h^{−1} of NTP in mRNA. The computed costs for all 20 conditions considered are provided in Supplementary Table 4.
Protein content as an alternative cost measure
The protein content was considered as the only relevant cost of the translation machinery in two previous models^{9,20}. These models accurately predicted the ribosomal protein fraction in total protein, whereas they substantially underestimated the EFTu proteome fraction compared to measurements data^{9,20}.
Following these earlier works, we calculated the protein cost of a component as the number of amino acid residues it incorporates. Since mRNA and tRNA have zero protein content, this definition assigns no costs to their expression, potentially leading to a prediction of infinite concentrations. To avoid such pathological predictions, we set the protein content costs of mRNA and charged tRNAs to small, arbitrary values in the model (mRNA: 10 amino acid residues; tRNA: 5 amino acid residues).
Model overview
The mechanistic translation model encompasses the processes of translation initiation, elongation, termination, nucleotide exchange in EFTu, and ternary complex (TC) formation. Figure 1 of the main text illustrates the modeled reaction; for better readability, the figure shows only a subset of codon/charged tRNA combinations. In total, the model includes 274 reactions.
Translation initiation
During initiation, mRNA converts free ribosomes to active ribosomes:
Translation initiation consists of multiple elementary reactions^{52}. However, a recent study found that at steady state, the kinetics of initiation effectively follow Michaelis–Menten kinetics, with mRNA in the enzyme position (with concentration c_{mRNA}) and free (unbound) ribosomes in the substrate position (with concentration c_{ribofree})^{53},
with k_{cat–mRNA} = 1.33 s^{−1} and K_{Mribo} = 8.5 µM from ref. ^{25}.
We assume that the turnover number of mRNA for ribosome binding (k_{catmRNA}) is growth rateindependent, and hence that the observed growth ratedependent activity is due to changes in the concentration of free ribosomes available for initiation^{54}. Thus, we used the maximal reported mRNA activity as an estimate of k_{catmRNA}.
Ternary complex formation and EFTu nucleotide exchange
Ternary complex formation and nucleotide exchange in EFTu are the processes by which the translation machinery recycles its substrates, the ternary complexes, for elongation.
TC formation: The binding of EFTu·GTP to chargedtRNA (aatRNA) forms the ternary complex (ternary complex formation in Fig. 1), which is the substrate of translation elongation.
Nucleotide exchange in EFTu: EFTu·GDP is released after the formation of a new peptide bond. Elongation factor Ts (EFTs) binds to EFTu·GDP and induces the exchange of GDP for GTP (right panel in Fig. 1).
The individual steps of these two processes are modeled with mass action kinetics^{26,27}.
In the implementation of the model, we divide each reversible reaction into two irreversible reactions. The TC formation reaction is a set of reactions that include the binding of 40 aatRNAs to Tu·GTP, and its rate constants depend on which amino acid is involved^{27} (for the parameter values see Supplementary Data 1).
Elongation
Elongation is a very complex process^{55,56}. For simplicity, we model elongation as a single reaction with an active ribosome as the enzyme and TC as the substrate, which was proposed by Klumpp et al.^{9}. In this single reaction model, TC is discriminated by the anticodon and all TCs are treated with the same activity. Michaelis–Menten kinetics are used to describe the reaction rate^{9}. There are 40 anticodons in total for all elongator tRNAs in E. coli; accordingly, our model uses 40 tRNAs to represent all tRNAs.
At steady state, the total translation rate (per cytosolic volume) of each codon remains constant. We do not model the translation of a whole protein. Instead, we decompose protein synthesis into the translation of 61 codons (see also “Modeling” below).
Active ribosome state: We distinguish active ribosomes according to their binding codons (codon in ribosome A site). Thus, there are 61 types of active ribosome in the model, distinguished in the model by subscripts indicating the codon currently presented by the ribosome:
here, codoni is one of the 61 codons and Ribosome_{codoni} is the active ribosome that binds codoni. This reaction is constrained by mass balance, but is considered to be instantaneous. We assume that when an active ribosome binds with a specific codon, it only translates the codon’s cognate tRNA.
In the model, there are 61 codons and 40 tRNAs, and the relation between tRNA and codon is not onetoone. Based on the number of cognate tRNAs, we partition the 61 codons into 2 classes: class 1 codons have one cognate tRNA, whereas class 2 codons have two cognate tRNAs. The lists of class 1 and class 2 codons are provided in Supplementary Table 5.
For class 1 codons (n = 51), the elongation reaction is:
where TC_{codoni} is the cognate TC of codoni, tRNA_{codoni} is the released free tRNA, and aa_{codoni} symbolizes the amino acid that was just appended to the growing peptide. The tRNA_{codoni} and aa_{codoni} are included here for completeness, but are not included explicitly in the optimized model, as they do not influence the results once appropriate exchange reactions have been added. Simultaneously, the Ribosome_{codoni} is converted to Ribosome_{active}, which is ready to participate in the next round of elongation,
For simplicity, we combine r3 and r4 into the new reaction r5, with Ribosome_{codoni} as a substrate and Ribosome_{active} as a product (the same for r6 and r7),
The translation rate of codoni is described by Michaelis–Menten kinetics,
where c_{ribocodoni} is the concentration of ribosomes that present codoni (Ribosome_{codoni}), c_{TCcodoni} is the concentration of cognate TC of codoni (TC_{codoni}), k_{catribo} = 22 s^{−1}, and K_{MTC} = 3 μM (parameters from ref. ^{9}).
For class 2 codons (n = 10), the active ribosome can translate two TCs and thus there are two reactions:
and
The translation rate of codoni is the sum of these two reactions:
with
and
Termination
In termination, an active ribosome is converted to a free ribosome:
The termination rate is equal to the protein synthesis rate at steady state:
where c_{protein} (Eq. 3) is the absolute concentration of protein measured experimentally.
Modeling exchange reactions
Besides the reactions mentioned above, we also add exchange reactions that allow the influx of free ribosome, charged tRNA, mRNA, EFTu, EFTs, and GTP into the system. We also add exchange reactions that allow efflux of GDP out of the system.
Modeling antibiotic stress
To model chloramphenicol (cm) stress, we add the exchange reaction for chloramphenicol. All forms of ribosome can be inhibited by chloramphenicol:
where Ribosome includes both free and active ribosomes. The reaction rate is described by mass action kinetics with k_{on} = 0.00057 µM s^{−1} (0.034 µM min^{−1}) and k_{off} = 0.0014 s^{−1} (0.084 min^{−1})^{57}.
For simplicity, we make the following assumptions:

(1)
chloramphenicol diffuses freely across cell membranes, such that the intracellular concentration of free chloramphenicol is the same as that in the medium;

(2)
chloramphenicol bound to an active ribosome (ribosome·cm) causes the ribosome·cm complex to dissociate quickly from the mRNA, thus not affecting further translation of the mRNA.
We assume that the reaction of ribosome and chloramphenicol binding is at steady state (dynamic equilibrium), and thus the active ribosome concentration will be constant. However, not all active ribosomes will be able to successfully finish translation, as the active ribosome can be inhibited by chloramphenicol during translation. Thus, to estimate the production rate of functional proteins, we need to estimate the probability that the active ribosome can finish translation without chloramphenicol inhibition.
To calculate this probability, we use the method proposed in ref. ^{32}. The probability of chloramphenicol binding to a ribosome in a given time unit is:
where k_{on} is the binding constant. The probability that the ribosome is bound n times in the time interval t follows the Poisson distribution
where t_{tl} is the experimental measured translation time of the translated gene. The probability that the ribosome can finish translation without inhibition by chloramphenicol is P(0):
Here, t_{tl} is the time for LacZ and t_{tl} = 72 s (from ref. ^{32}). For simplicity, we assume that all codonpresenting active ribosomes (61 forms of active ribosome) have the same P_{tl}, and thus the effective concentration of the ribosome presenting codoni is
Under inhibition by chloramphenicol, this effective ribosome concentration of codoni (c_{riboeffi}) replaces the ribosome concentration of (c_{riboi}) in the model.
Model optimization
We assume translation at steady state and use a constraintbased optimization model. The constraints are given by the above equations and by the requirement of a given total cellular rate of protein synthesis (estimated as the product of growth rate and experimental proteome composition at this growth rate). For simplicity, we decomposed protein synthesis into the translation of 61 codons, and so the constraint on protein synthesis rate is implemented as 61 individual equations, each representing the translation rate of one codon. At steady state, the codon translation rate equals the dilution rate of amino acids incorporated at protein positions encoded by codoni, i.e., the growth rate μ multiplied by the concentration of amino acids coded by codoni in proteome data:
note that if multiple codons encode the same amino acids, then the total concentration of that amino acid in cellular proteins is the sum over the c_{codoni} values for the individual codons.
Given these constraints, we minimize the total mass concentration of the modeled translation apparatus, consisting of ribosome, EFTu, EFTs, mRNA, GTP, GDP, and charged tRNAs (aa⋅tRNA),
where the molecule types together form the set C, MW_{m} is the molecular weight, and c_{m} is the concentration of molecule type m. We do not minimize the contributions of GTP and GDP, who participate in multiple other cellular processes^{58} and whose concentrations are thus unlikely to be dominated by translation; their concentrations are consequently fixed to experimentally observed values in our simulations (see Supplementary Note 3).
Let R be the set of reactions that together comprise translation. We also consider the dilution of the molecules involved in these reactions due to cellular volume growth at rate µ:
where S is the stoichiometric matrix for the reactions in R, c is a vector of the concentrations c_{m}, and v(c) is the corresponding vector of reaction rates v_{i}, with the concentrationdependent kinetics described above.
Thus, we solve the nonlinear constrained optimization problem:
subject to:
\({\mathbf{S}}\cdot {\mathbf{v}}\left( {\mathbf{c}} \right)  \mu \cdot {\mathbf{c}} = 0\)
\(v_{tl  codon  i} = \mu \cdot c_{codon  i}\) for i=1, …, 61
\(v_{{\mathrm{term}}} = \mu \cdot c_{{\mathrm{protein}}}\)
We formulated the optimization problem in GAMS and used the BARON global solver^{59} on NEOS Server^{60} with 8 h as the time limit to solve this problem. Because the problem is nonlinear, it is not clear a priori if it is convex, in which case only a single optimum would exist. The problem is conceptionally similar to the one studied by Noor et al.^{24}, and thus convexity is conceivable. A first optimum was returned by the solver within at most a few minutes for all optimizations performed. However, to guard against the existence of overlooked alternative optima, we allowed the search to continue for a total of 8 h in each case. In addition, we repeated all simulation with the global nonlinear solver in Lingo 13 (LINDO Systems, Inc., https://www.lindo.com/index.php/products/lingoandoptimizationmodeling). No alternative optima were ever found; we thus have no evidence of nonconvexity.
To assess the alternative costs, the molecular weight of each component was replaced by the corresponding cost measure (see the section “Alternative costs of translation machinery components”). The costs for each component are shown in Supplementary Data 2, and the corresponding predictions are shown in Supplementary Data 3.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Data availability
The experimental data that support the findings of this study are available from the original publications^{5,8,28,29,30,31,32,36}. Source data are provided with this paper.
Code availability
Model: The optimization problem, including the model and its parameterization, is provided as an SBML file (Supplementary Data 4) and as a GAMS input file (Supplementary Data 5, with protein production requirements set to those for growth on minimal glucose medium). In addition, the model has been submitted to Biomodels (MODEL2006210001). GAMS (General Algebraic Modeling System, https://www.gams.com/) is a modeling platform for mathematical optimization, it provides dozens of solvers for a variety of optimization problems. The model is written in GAMS (version 25.1.2). BARON (BranchAndReduce Optimization Navigator) is a computational system for solving nonconvex optimization problems to global optimality. NEOS Server is a free online server that provides stateoftheart solvers for numerical optimization. We used the BARON solver (version 19.12.7) on NEOS Server (Version 5.0) for our optimizations. sybilccFBA is a package in R that provides an improved implementation of MOMENT (MetabOlic Modeling with ENzyme kineTics). We used version 3.0.1. Lingo (https://www.lindo.com/index.php/products/lingoandoptimizationmodeling) is a commercial optimization software. Lingo version 13 was used as the alternative solver to double check the results from BARON.
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Acknowledgements
We thank Uri Alon, Matteo Mori, and Terence Hwa for helpful discussions and Abdelmoneim Desouki for providing the sybilccFBA R package. This work was supported by the Volkswagen Foundation under the “Life?” initiative, and by the German Research Foundation (DFG) through grants IRTG 1525, CRC 680, CRC 1310, and, under Germany’s Excellence Strategy, through grant EXC 2048/1 (Project ID: 390686111).
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H.D. conceived of the study. X.P.H. developed, implemented, and parameterized the model and performed the analyses. P.S. prepared Fig. 1 and the S.B.M.L. model description. M.J.L. supervised the study. X.P.H. and M.J.L. interpreted the results and wrote the manuscript.
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Hu, XP., Dourado, H., Schubert, P. et al. The protein translation machinery is expressed for maximal efficiency in Escherichia coli. Nat Commun 11, 5260 (2020). https://doi.org/10.1038/s4146702018948x
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