Environment-specificity and universality of the microbial growth law

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 proteome¹². 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

Here R is the number of ribosomes, and R₀ 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 decreases⁴.

We define the total protein mass M = ∑_iM_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)

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. ϕ₀ 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 ϕ₁ = ϕ_R, k₁ = k_R, and α₁ = 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 proteins¹³ 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:

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:

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:

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,

where the expressions of the constants, c₁, c₂ and c₃ 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, c₂ ∝ 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. cerevisiae¹⁴ (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 ϕ₀ is not known experimentally. By choosing a common ϕ₀ = 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 ϕ₀, our results show that a constant ϕ₀ is consistent with three of four data points in Fig. 4a, and the outlier may have a higher ϕ₀ in that particular environment. Our analysis cannot exclude the possibility that ϕ₀ 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 environments¹⁹. 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 natAΔ 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 c₁, c₂, and c₃ (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 c₁, c₂ and c₃ having a wide range of 95% confidence intervals (Fig. 5a, c) with RMSE = 1.35 × 10⁻². The inferred ranges of ϕ₀, 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. coli⁷ (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 × 10⁵Da¹². 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 observations⁹. 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 rate²⁵. 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 proteins²⁶, and dramatically different cell sizes can exist at the same growth rate²⁷. 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 from¹⁴, 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 values¹⁴. 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. cerevisiae⁶ and E. coli⁷. 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 from¹⁴ (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_i¹⁴, the translation speeds k_i⁹, and the protein degradation rates α_i¹¹ 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 rate⁷. In this case, one just needs to include four additional environment-specific parameters: k_R, 〈k〉, α_R, and 〈α〉.