Environment-specificity and universality of the microbial growth law | Panda Anku

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 χi. Here the index i represents one particular protein i. Mass spectrometry also allows us to measure the mass fractions ϕi of all proteins in the proteome12. The translation speed of ribosomes on the corresponding mRNAs is ki, which is the averaged mass of translated amino acids per unit time. Note that ki is averaged over the sequence of the corresponding mRNA so that each protein has one ki. We also assume that protein i degrades with a constant rate αi. 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 R0 is the number of inactive ribosomes. Our model is summarized in Fig. 1.

Fig. 1: A schematic of the model.
figure 1

Given a constant environment, cells allocate different fractions of active ribosomes (χi) to translate mRNAs corresponding to different proteins. In general, the translation speeds ki are heterogeneous among proteins. αi is the degradation rate of protein i. χi, ki and αi together determine the mass fraction of protein i. The ribosome allocation strategies reflect the adaptation of cells to different environments. In this schematic, we show three proteins for simplicity.

In this work, we focus on the effects of heterogeneous translation speeds ki and finite degradation rates αi. 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 ϕR as the growth rate decreases4.

We define the total protein mass M = ∑iMi, and the protein mass fraction ϕi = Mi/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 mR 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 = ϕR, k1 = kR, and α1 = αR. Here, kR and αR 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 (ki = k for all i) and protein degradations are negligible (αi = 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 αi = 0 for all proteins and only consider the effects of heterogeneous translation speeds ki. 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 ϕR(μ) curve. Interestingly, we find that akR − 〈kχ. If kR is smaller than 〈kχ, a is negative so that the second derivative of the ϕR(μ) curve is positive. In other words, the ϕR(μ) curve is upward bent in slow-growth conditions relative to a linear line.

kχ depends on both the elongation speeds ki and the ribosome allocations χi. To find its value, we further rewrite 〈kχ = 〈k〉(1 + Iχ,k). Here 〈k〉 is the arithmetic average of translation speeds over all non-ribosomal proteins. Iχ,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 Iχ,k is, the more ribosomes are allocated to mRNAs with higher translation speeds. Because the ribosomal allocations χi are generally different in different environments, we use Iχ,k to characterize an environment. Imagine that we grow cells in multiple environments with equal Iχ,k. We find that as long as Iχ,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 kR is much lower than 〈k9. Therefore, Eq. (3) predicts an upward bending of the ϕR(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 kR < 〈k〉. We randomly sample χi for each environment and compute the resulting growth rate μ and protein mass fractions ϕi. We show the results from environments with preselected Iχ,k, which agree well with the theoretical formula Eq. (3) (Fig. 2a).

Fig. 2: Numerical simulations of the growth law curves.
figure 2

a We simulate the case of heterogeneous translation speeds without protein degradations and compare our numerical simulations with model predictions (dashed lines). Each data point has its own randomly sampled χi, and we show the results with preselected Iχ,k values. The red dash line represents the predictions of the STM in which all proteins have the same translation speed 〈k〉. b Same analysis in which we simulate the case of finite protein degradation rates without heterogeneity in translation speeds.

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 Iϕ,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 Iϕ,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 Iϕ,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, c1, c2 and c3 are shown in Methods. We prove that given fixed Iχ,k and Iϕ,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, c2kR − 〈kχ, 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 ki and the allocation fractions χi. 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.

Fig. 3: Numerical simulations of the growth law curves with both heterogeneous translation speeds and protein degradation rates.
figure 3

a Numerical simulations with preselected Iχ,k and Iϕ,α. The red dashed line is the prediction of the STM, and other dashed lines represent our model predictions. b, e Two-dimensional Gaussian distribution of randomly sampled Iχ,k and Iϕ,α. The mean of Iχ,k is 0.5, and the mean of Iϕ,α is 0. The standard deviations σ are indicated in the legends. c, f The resulting growth law curve where each point has randomly sampled Iχ,k and Iϕ,α from (b) and (e). d, g The distribution of the fitting RMSE corresponding to randomly chosen points in (c) and (f).

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 Iχ,k and Iϕ,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 Iχ,k and Iϕ,α (Fig. 3f). To quantify the effects of heterogeneous Iχ,k and Iϕ,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 Iχ,k and Iϕ,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 Iχ,k, and the correlation coefficients between the protein degradation rates and protein mass fractions Iϕ,α. Given the values of μ, Iχ,k, and Iϕ,α, we predicted the fraction of ribosomal proteins ϕR 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 ϕR (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).

Fig. 4: Experimental analysis and tests of theoretical predictions.
figure 4

a Experimental measured ϕR of S. cerevisiae along with the predictions (dashed lines) of our model. b The growth rate dependences of the correlation coefficients Iχ,k and Iϕ,α. c The normalized enrichment scores (NES) of GSEA of enriched gene sets. A positive NES of ki-ordered GSEA means that the genes in the corresponding gene set are enriched in the regime of higher ki. A positive NES of ({log }_{2})FC-ordered GSEA means that the genes in the corresponding gene set are enriched in the regime of increasing χi when the nutrient changes from glucose to glycerol. d Summary of the variables and parameters in the analysis of experimental data. The effective mass of ribosomal proteins mR is calculated based on molecular weights of ribosomal proteins detected in the proteome (Methods). SC synthetic complete medium, Glu glucose, Gly glycerol.

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 Iϕ,α ≈ −0.33 for all the conditions we computed. However, Iχ,k are negatively correlated with the growth rates, suggesting cells tend to allocate more ribosomes to translate mRNAs with higher ki 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 ki and the GSEA where genes are ordered by ({log }_{2}) fold change (({log }_{2})FC) of χi when the nutrient changes (Fig. S3a).

We find that five gene sets related to stress response are enriched in the regime of higher ki and increasing χi 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 ki and decreasing χi (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 αR and the mass of ribosomal proteins mR as given. We approximate the ϕ-averaged degradation rate 〈αϕ by 〈α〉(1 + Iϕ,α) where Iϕ,α = −0.33, justified by the observations that Iϕ,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, kR and 〈kχ 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 Iχ,k and Iϕ,α exist among environments (Fig. 3d).

Fig. 5: Fitting of the full model to different datasets.
figure 5

a The non-linear fitting to data of S. cerevisiae from Ref. 6. The shadow represents the 95% prediction interval. b The non-linear fitting to data of E. coli from Ref. 7. c Detailed fitting results of (a) and (b). Note that the reference value of 〈kχ of (a) is approximated by 〈k〉(1 + Iχ,k) where the minimal Iχ,k = 0.608 and the maximal Iχ,k = 2.415 (Fig. 4d).

We also apply our theories to E. coli7 (Fig. 5b). Because most proteins are non-degradable in bacteria20,21, we set αR and 〈αϕ as 0, and the mass of ribosomal protein mR = 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 kR, and 〈kχ are consistent with previous studies22,23,24 (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 Iχ,k and Iϕ,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 Iχ,k and Iϕ,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 (ϕR) 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 (Iχ,k); the other is between the protein mass fractions and protein degradation rates (Iϕ,α). By analyzing the dataset from14, we found that Iϕ,α is independent of growth rate, while Iχ,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 ϕR 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 Iχ,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 Iχ,k and Iϕ,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 Iχ,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, ϕi must equal the ribosome allocation χi. Indeed, these two datasets are often highly correlated among proteins in E. coli12,28. However, in a more realistic scenario, ϕi also depends on the translation speed and protein degradation rate. Given the same χi, 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 ϕi,pre based on the ribosome allocations χi14, the translation speeds ki9, and the protein degradation rates αi11 do not correlate strong enough with the measured ϕi. We note that these datasets are from different references, and the deviation is likely due to the noise in the measurements of ki (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 Iχ,k and Iϕ,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: kR, 〈k〉, αR, and 〈α〉.

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