### Model of protein synthesis

Given a constant environment, we consider a population of cells with a constant growth rate, and the protein synthesis processes are in a steady state. Ribosome profiling allows us to quantify the fraction of ribosomes in the pool of total active ribosomes producing protein *i*, which we call ribosome allocation *χ*_{i}. Here the index *i* represents one particular protein *i*. Mass spectrometry also allows us to measure the mass fractions *ϕ*_{i} of all proteins in the proteome^{12}. The translation speed of ribosomes on the corresponding mRNAs is *k*_{i}, which is the averaged mass of translated amino acids per unit time. Note that *k*_{i} is averaged over the sequence of the corresponding mRNA so that each protein has one *k*_{i}. We also assume that protein *i* degrades with a constant rate *α*_{i}. The mass production rate of protein *i* becomes

$$frac{d{M}_{i}}{dt}={k}_{i}{chi }_{i}(R-{R}_{0})-{alpha }_{i}{M}_{i}.$$

(1)

Here *R* is the number of ribosomes, and *R*_{0} is the number of inactive ribosomes. Our model is summarized in Fig. 1.

In this work, we focus on the effects of heterogeneous translation speeds *k*_{i} and finite degradation rates *α*_{i}. Therefore, for simplicity, we assume them to be invariant of environments. We also mainly consider the effects of nutrient quality and do not consider the impact of antibiotics in this work, which can decrease the overall effective translation speed and increase *ϕ*_{R} as the growth rate decreases^{4}.

We define the total protein mass *M* = ∑_{i}*M*_{i}, and the protein mass fraction *ϕ*_{i} = *M*_{i}/*M*. Using Eq. (1), we find the fraction of ribosomal proteins in the proteome in the steady state, (see detailed derivations in Methods)

$${phi }_{R}=frac{{m}_{R}(mu +{sum }_{i}{alpha }_{i}{phi }_{i})}{{sum }_{i}{k}_{i}{chi }_{i}}+{phi }_{0}.$$

(2)

Here *μ* is the growth rate of the total protein mass (mu =dot{M}/M), and *m*_{R} is the total amino acid mass of a single ribosome. *ϕ*_{0} is the mass fraction of inactive ribosomes, which we assume to be constant for simplicity. In this work, *i* = 1 is reserved for ribosomal proteins so that *ϕ*_{1} = *ϕ*_{R}, *k*_{1} = *k*_{R}, and *α*_{1} = *α*_{R}. Here, *k*_{R} and *α*_{R} are the effective translation speed and degradation rate of the coarse-grained ribosomal protein averaged over all ribosomal proteins. They are approximately independent of environments due to the tight regulation of relative doses of different ribosomal proteins^{13} and their generally low degradation rates. It is easy to find that if all proteins have the same translation speed (*k*_{i} = *k* for all *i*) and protein degradations are negligible (*α*_{i} = 0), Eq. (2) is reduced to the STM.

### Universal and non-universal growth law curves

To disentangle the effects of heterogeneous translation speeds and protein degradations, we first simplify the model by taking *α*_{i} = 0 for all proteins and only consider the effects of heterogeneous translation speeds *k*_{i}. We rewrite ({sum }_{i}{k}_{i}{chi }_{i}={k}_{R}{chi }_{R}+(1-{chi }_{R})mathop{sum }nolimits_{i = 2}^{N}{k}_{i}{widetilde{chi }}_{i}) in Eq. (2). Here, *N* is the number of proteins and ({chi }_{i}=(1-{chi }_{R}){widetilde{chi }}_{i}) so that (mathop{sum }nolimits_{i = 2}^{N}{widetilde{chi }}_{i}=1). In the following, we define ({langle krangle }_{chi }=mathop{sum }nolimits_{i = 2}^{N}{k}_{i}{widetilde{chi }}_{i}) as the *χ*-weighted average translation speed over all non-ribosomal proteins. As we derive in Methods, the fraction of ribosomal proteins can be written exactly as a Hill function of the growth rate:

$${phi }_{R}=frac{mu }{amu +b}+{phi }_{0},$$

(3)

where the expressions of *a*, *b* are shown in Methods. We are particularly interested in the sign of *a* because it determines the shape of the *ϕ*_{R}(*μ*) curve. Interestingly, we find that *a* ∝ *k*_{R} − 〈*k*〉_{χ}. If *k*_{R} is smaller than 〈*k*〉_{χ}, *a* is negative so that the second derivative of the *ϕ*_{R}(*μ*) curve is positive. In other words, the *ϕ*_{R}(*μ*) curve is upward bent in slow-growth conditions relative to a linear line.

〈*k*〉_{χ} depends on both the elongation speeds *k*_{i} and the ribosome allocations *χ*_{i}. To find its value, we further rewrite 〈*k*〉_{χ} = 〈*k*〉(1 + *I*_{χ,k}). Here 〈*k*〉 is the arithmetic average of translation speeds over all non-ribosomal proteins. *I*_{χ,k} is a metric we use to quantify the correlation between the ribosome allocations and the translation speeds:

$${I}_{chi ,k}=frac{langle {widetilde{chi }}_{i}{k}_{i}rangle -langle {widetilde{chi }}_{i}rangle langle krangle }{langle {widetilde{chi }}_{i}rangle langle krangle }.$$

(4)

Here, the bracket represents an average over all non-ribosomal proteins. Biologically, the higher *I*_{χ,k} is, the more ribosomes are allocated to mRNAs with higher translation speeds. Because the ribosomal allocations *χ*_{i} are generally different in different environments, we use *I*_{χ,k} to characterize an environment. Imagine that we grow cells in multiple environments with equal *I*_{χ,k}. We find that as long as *I*_{χ,k} is not too close to −1, which we confirm later using experimental data, *a* is always negative since the translation speed of ribosomal proteins *k*_{R} is much lower than 〈*k*〉^{9}. Therefore, Eq. (3) predicts an upward bending of the *ϕ*_{R}(*μ*) curve in slow-growth conditions.

We verify the above theoretical predictions by numerically simulating the model of protein synthesis (Methods). The translation speeds are randomly sampled among proteins and fixed for all environments, with *k*_{R} < 〈*k*〉. We randomly sample *χ*_{i} for each environment and compute the resulting growth rate *μ* and protein mass fractions *ϕ*_{i}. We show the results from environments with preselected *I*_{χ,k}, which agree well with the theoretical formula Eq. (3) (Fig. 2a).

We also consider another simplified model in which the translation speeds are homogeneous, but protein degradation rates are finite and heterogeneous. We find that in this model, the growth law curve is linear with a reduced slope and increased intercept compared with the STM (see details in Methods). The actual shape of the growth law curve depends on the parameter *I*_{ϕ,α}, which is a metric to characterize an environment by quantifying the correlation between the protein mass fractions and degradation rates:

$${I}_{phi ,alpha }=frac{langle {widetilde{phi }}_{i}{alpha }_{i}rangle -langle {widetilde{phi }}_{i}rangle langle alpha rangle }{langle {widetilde{phi }}_{i}rangle langle alpha rangle }.$$

(5)

Here, the bracket represents an average over all non-ribosomal proteins and ({widetilde{phi }}_{i}=(1-{phi }_{R}){phi }_{i}). Biologically, a high *I*_{ϕ,α} value means that the proteome are enriched with proteins with high degradation rates. We verify the above theoretical predictions by numerical simulations and randomly sample the protein degradation rates that are fixed for all environments. We show the results from environments with preselected *I*_{ϕ,α} and our theoretical predictions Eq. (18) are nicely confirmed (Fig. 2b).

Finally, we turn to the full model with both the heterogeneities in the translation speeds and protein degradation rates. We find that the growth law curve has the following general form,

$${phi }_{R}=frac{mu +{c}_{1}}{{c}_{2}mu +{c}_{3}},$$

(6)

where the expressions of the constants, *c*_{1}, *c*_{2} and *c*_{3} are shown in Methods. We prove that given fixed *I*_{χ,k} and *I*_{ϕ,α} (as long as they are not too close to −1), the growth law curve must be monotonically increasing and convex, which suggests an upward bending in slow-growth conditions (Methods). In particular, *c*_{2} ∝ *k*_{R} − 〈*k*〉_{χ}, which means that it is the slower translation speed of ribosomal proteins than other proteins that generates the nonlinear shape of the growth law curve. The simulation results match the theoretical predictions (Fig. 3a). We note that the uncertainness of real environments often leads to random production of proteins and random allocation of ribosomes. To address this question, we also simulate models in which noises exist in the translation speeds *k*_{i} and the allocation fractions *χ*_{i}. We find that both noises do not affect our conclusions qualitatively (Fig. S1). Note that adding noises to the translation speeds and allocation fractions only makes the resulting growth law curves even noisier and therefore does not affect our main conclusion that the growth law curve is generally environment-specific, as we show later.

In real situations, we remark that the actual growth curve shape depends on the particular environments. To verify this, we compute the resulting growth law curve with multiple environments, and the *I*_{χ,k} and *I*_{ϕ,α} of each environment are randomly sampled from Gaussian distributions (Fig. 3b, e) (Methods). We find that when the Gaussian distributions have large standard deviations, the growth law curve is non-universal and depends on the particular chosen environments (Fig. 3c). This means that if we randomly pick some environments from Fig. 3c, the resulting growth law curves are generally different. In contrast, when the Gaussian distributions have small standard deviations, the growth law curve is well captured by our theoretical predictions Eq. (6) because the environments share similar *I*_{χ,k} and *I*_{ϕ,α} (Fig. 3f). To quantify the effects of heterogeneous *I*_{χ,k} and *I*_{ϕ,α} across environments, we repeatedly sample 20 random points from Fig. 3c, f and fit them using Eq. (6) (Methods), imitating the sampling processes in real experiments. We find that when the chosen environments have significantly different *I*_{χ,k} and *I*_{ϕ,α}, the median root mean squared error RMSE = 1.69 × 10^{−2} (Fig. 3d). In contrast, in the case of similar environments, RMSE = 4.44 × 10^{−3} (Fig. 3g). The above results suggest that we can use the fitting error as a criterion of the universality of the growth law curve, which we apply to the experimental data later.

### Experimental tests of theories

In this section, we test our model using published datasets of *S. cerevisiae*^{14} (Methods). For each strain and nutrient quality, we computed the correlation coefficients between the translation speeds and ribosome allocations *I*_{χ,k}, and the correlation coefficients between the protein degradation rates and protein mass fractions *I*_{ϕ,α}. Given the values of *μ*, *I*_{χ,k}, and *I*_{ϕ,α}, we predicted the fraction of ribosomal proteins *ϕ*_{R} using Eq. (6) (Fig. 4a, d). We note that one parameter *ϕ*_{0} is not known experimentally. By choosing a common *ϕ*_{0} = 0.048, our model predictions nicely match the experimentally measured values of *ϕ*_{R} (with one data point slightly above the theoretical prediction). We find that regardless of the data processing procedures, the relative relationships between the predicted curves always agree with those of the experimental values (Methods and Fig. S2).

Our model is simplified as we assume that the translation speeds and protein degradation rates do not depend on environments. Remarkably, our model predictions still quantitatively match the experimental observations, suggesting that our assumptions may be reasonable for most situations. While our model cannot predict the growth rate dependence of *ϕ*_{0}, our results show that a constant *ϕ*_{0} is consistent with three of four data points in Fig. 4a, and the outlier may have a higher *ϕ*_{0} in that particular environment. Our analysis cannot exclude the possibility that *ϕ*_{0} is also environment-specific.

Interestingly, we found that *I*_{ϕ,α} ≈ −0.33 for all the conditions we computed. However, *I*_{χ,k} are negatively correlated with the growth rates, suggesting cells tend to allocate more ribosomes to translate mRNAs with higher *k*_{i} in poor nutrient conditions (Fig. 4b). To find out what genes acquire more resources when the environment is shifted, we perform Gene Set Enrichment Analysis (GSEA)^{15,16} for wild type cells (Methods) and find that eight gene sets from the Gene Ontology (GO)^{17,18} database are enriched in both the GSEA where genes are ordered by *k*_{i} and the GSEA where genes are ordered by ({log }_{2}) fold change (({log }_{2})FC) of *χ*_{i} when the nutrient changes (Fig. S3a).

We find that five gene sets related to stress response are enriched in the regime of higher *k*_{i} and increasing *χ*_{i} when the environment is changed from 2% glucose to 2% glycerol (Fig. 4c). This is consistent with the environmental stress response (ESR) of *S. cerevisiae* as an adaptation to the shifts of environments^{19}. We propose that higher translation speeds of stress response proteins enable cells to respond rapidly to environmental changes, which is evolutionarily advantageous. We also find two gene sets related to the rRNA process enriched in the regime of lower *k*_{i} and decreasing *χ*_{i} (Fig. 4c). We also perform GSEA for *natA*Δ cells and get similar results (Fig. S3b, c).

### Applications of theories to experimental growth law curves

An important application of our theories is that one can estimate the translation speeds by fitting the experimental growth law curve to our model prediction Eq. (6) (Methods). Because there are 6 unknown parameters in the definition of *c*_{1}, *c*_{2}, and *c*_{3} (Eqs. (23)–(25)), we can estimate three of the parameters given the values of the other three. For the *S. cerevisiae* data from Ref. ^{6}, we use the experimentally measured degradation rate of ribosomal proteins *α*_{R} and the mass of ribosomal proteins *m*_{R} as given. We approximate the *ϕ*-averaged degradation rate 〈*α*〉_{ϕ} by 〈*α*〉(1 + *I*_{ϕ,α}) where *I*_{ϕ,α} = −0.33, justified by the observations that *I*_{ϕ,α} is independent of environments (Fig. 4b). We find that the fitted parameters *c*_{1}, *c*_{2} and *c*_{3} having a wide range of 95% confidence intervals (Fig. 5a, c) with RMSE = 1.35 × 10^{−2}. The inferred ranges of *ϕ*_{0}, *k*_{R} and 〈*k*〉_{χ} are also unreasonably large (Fig. 5c). All these results suggest that the growth law curves of *S. cerevisiae* are non-universal and large variations of *I*_{χ,k} and *I*_{ϕ,α} exist among environments (Fig. 3d).

We also apply our theories to *E. coli*^{7} (Fig. 5b). Because most proteins are non-degradable in bacteria^{20,21}, we set *α*_{R} and 〈*α*〉_{ϕ} as 0, and the mass of ribosomal protein *m*_{R} = 8.07 × 10^{5}*D**a*^{12}. In this case, the fitted parameters have much smaller range of 95% confidence intervals with RMSE = 3.60 × 10^{−3}. The estimated *k*_{R}, and 〈*k*〉_{χ} are consistent with previous studies^{22,23,24} (Fig. 5c). Our analysis of experimental data demonstrates that the translation speed of ribosomal proteins is indeed smaller than the *χ* − averaged translation speed, in agreement with experimental observations^{9}. Our results suggest that E. coli has similar values of *I*_{χ,k} and *I*_{ϕ,α} in the chosen environments of Ref. ^{7} so that it has a universal growth law curve. In contrast, *S. cerevisiae* appears to have significantly different *I*_{χ,k} and *I*_{ϕ,α} across different environments of Ref. ^{6} so that the growth law curve depends on the chosen environments and therefore non-universal.

### Discussion

It has been known since the 1950s that the chemical compositions and cell size of bacteria are functions of growth rate and seem to be independent of the medium used to achieve the growth rate^{25}. This view has been broadly accepted in the study of bacteria physiologies in the past decades. The growth law acquired its name because of the independence of the environment. However, recent findings hint at an unforeseen complexity in the growth law. For example, bacterial cell sizes have been shown to depend on the presence of antibiotics, and overexpression of useless proteins^{26}, and dramatically different cell sizes can exist at the same growth rate^{27}. Our study focuses on the growth law regarding the fraction of ribosomal proteins in the proteome and further uncovers the importance of environment-specificity to microbial physiologies. We go beyond the simple translation model and take account of the heterogeneous translation speeds and finite protein degradations. Given the translation speeds and protein degradation rates, our model is completely general and virtually applies to any cells, including both proliferating cells (*μ* > 0) and non-proliferating cells (*μ* = 0). In this work, we mainly consider the scenario in which the growth rate changes due to the nutrient quality and the fraction of ribosomal proteins (*ϕ*_{R}) increases monotonically as the growth rate increases.

We demonstrate that the growth law curve generally has the form of Eq. (6). The actual shape of the growth law curve depends on two correlation coefficients: one is between the ribosome allocations and the translation speeds (*I*_{χ,k}); the other is between the protein mass fractions and protein degradation rates (*I*_{ϕ,α}). By analyzing the dataset from^{14}, we found that *I*_{ϕ,α} is independent of growth rate, while *I*_{χ,k} appears to be negatively correlated with the growth rate. This means that cells tend to produce proteins with faster translation speeds in slow-growth conditions, which can be an economic strategy under evolutionary selection. Remarkably, our theoretical predictions of *ϕ*_{R} reasonably match the experimentally measured values^{14}. We note that the upward bending of the growth law curves of bacteria compared with a linear relation appears to hint at an increasing fraction of inactive ribosomes in slow-growth conditions. While this mechanism appears plausible, it has not been confirmed experimentally as we realize. In our work, we demonstrate that the apparent upward bending can be merely a consequence of heterogeneous translation speeds among proteins and therefore raises caution on the biological interpretation of the shape of the growth law curve.

We apply our model predictions to the growth law curves of *S. cerevisiae*^{6} and *E. coli*^{7}. In the former case, data fitting to our model prediction is subject to very large uncertainty. This observation agrees with the computed *I*_{χ,k} that are variable across conditions using the ribosome profiling and mass spectrometry data from^{14} (Fig. 4b). In contrast, the fitting of *E. coli* data exhibits a much smaller uncertainty, suggesting that common *I*_{χ,k} and *I*_{ϕ,α} may apply to all the nutrient qualities used in the experiments of Ref. ^{7}. We expect that this idea can be tested when genome-wide measurements, such as the translation speeds of *E. coli* are available in the future so that critical parameters such as *I*_{χ,k} can be calculated for *E. coli*.

We remark that in the absence of heterogeneous translation speeds and protein degradation, the mass fraction of protein *i*, *ϕ*_{i} must equal the ribosome allocation *χ*_{i}. Indeed, these two datasets are often highly correlated among proteins in *E. coli*^{12,28}. However, in a more realistic scenario, *ϕ*_{i} also depends on the translation speed and protein degradation rate. Given the same *χ*_{i}, proteins with higher translation speeds or lower degradation rates should have higher mass fractions (Methods). We note that using the current genome-wide datasets of *S. cerevisiae*, the predicted protein mass fractions *ϕ*_{i,pre} based on the ribosome allocations *χ*_{i}^{14}, the translation speeds *k*_{i}^{9}, and the protein degradation rates *α*_{i}^{11} do not correlate strong enough with the measured *ϕ*_{i}. We note that these datasets are from different references, and the deviation is likely due to the noise in the measurements of *k*_{i} (Table S2). We expect our theories to be further verified when more accurate measurements of translation speeds are available.

For simplicity, in this work, we assume that the translation speeds and protein degradation rates are invariant as the nutrient quality changes. Therefore, we can use the two correlation coefficients *I*_{χ,k} and *I*_{ϕ,α} to characterize a particular environment. We remark that our model can be generalized to more complex scenarios in which the translation speeds or protein degradation rates depend on the growth rate^{7}. In this case, one just needs to include four additional environment-specific parameters: *k*_{R}, 〈*k*〉, *α*_{R}, and 〈*α*〉.