Abstract
As the nutrient quality changes, the fractions of ribosomal proteins in the proteome are usually positively correlated with the growth rates due to the autocatalytic nature of ribosomes. While this growth law is observed across multiple organisms, the relation between the ribosome fraction and growth rate is often more complex than linear, beyond models assuming a constant translation speed. Here, we propose a general framework of protein synthesis considering heterogeneous translation speeds and protein degradations. We demonstrate that the growth law curves are generally environmentspecific, e.g., depending on the correlation between the translation speeds and ribosome allocations among proteins. Our predictions of ribosome fractions agree quantitatively with data of Saccharomyces cerevisiae. Interestingly, we find that the growth law curve of Escherichia coli nevertheless appears universal, which we prove must exhibit an upward bending in slowgrowth conditions, in agreement with experiments. Our work provides insights on the connection between the heterogeneity among genes and the environmentspecificity of cell behaviors.
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Introduction
Cells can adapt to different environments and alter the expression levels of multiple genes. The genomewide gene expression profile can change dramatically as cells switch between different environments. However, proliferating cells, including bacteria and unicellular eukaryotes, exhibit a growth law as the nutrient quality changes: the fraction of ribosomal proteins in the proteome (ϕ_{R}) and the growth rate (μ) are positively correlated. The growth law curve (ϕ_{R} vs. μ) is often fit by a linear relation, ϕ_{R} = μ/κ + ϕ_{0}^{1,2,3,4,5,6}, which can be rationalized by a simple translation model (STM): ribosomes are engaged in translation with a constant translation speed that is proportional to κ^{2,4}. ϕ_{0} represents the fraction of inactive ribosomes that are not producing proteins, independent of environments in the STM. While the STM is simple and intuitive, it does not always provide a good empirical fitting to the experimental growth law curves, e.g., it apparently breaks down in slowgrowth conditions of Escherichia coli (doubling time longer than 60 min at 37 °C) in which more ribosomes are produced than the prediction of the STM^{7}. A similar breakdown was also observed for other bacteria^{1}.
We remark that two important biological features beyond the STM are crucial to interpreting the growth law curve, as we show in this work. The first is the heterogeneous translation speeds of ribosomes producing different proteins. Recent studies showed that the translation speeds are highly heterogeneous among different proteins due to multiple mechanisms, including codon usages^{8} and amino acid compositions^{9}. In particular, the translation speeds of ribosomal proteins are much slower than the average translation speed over nonribosomal proteins due to the abundance of positively charged amino acids in ribosomal proteins^{9}. Nowadays, the ribosome profiling technique allows us to quantify the allocation of ribosomes toward the production of different proteins. These experimental techniques enable us to rethink the growth law in the presence of heterogeneity in translation speeds^{9}.
The second feature is finite protein degradation rates. The STM neglects protein degradation and predicts that at zero growth rate, ϕ_{R} = ϕ_{0} so that all ribosomes are inactive. However, this contradicts experiments of nongrowing bacteria in which translation activities are observed^{10}. Protein degradation must be considered at zero growth rate to balance protein production to ensure a constant protein mass. Therefore, protein degradation must be important to the growth law, at least in slowgrowth conditions.
In this work, we show that the heterogeneous translation speeds and protein degradations significantly influence the growth law by introducing a general theoretical framework of protein synthesis. We find that the fraction of ribosomal proteins ϕ_{R} depends not only on the growth rate but also on the statistical properties of environments. Besides the growth rate, ϕ_{R} depends on two correlation coefficients among proteins. One is between the translation speeds and ribosome allocations towards the production of different proteins. The other is between the degradation rates and mass fractions of proteins. Both correlation coefficients are environmentspecific. We compute the above correlation coefficients using proteomics and ribosomal profiling datasets of S. cerevisiae^{11}. Interestingly, we find that the correlation between the translation speed and ribosome allocations becomes stronger when the growth rate decreases; cells tend to produce more proteins with higher translation speeds in poor nutrients. In contrast, the correlation between the protein degradation rates and mass fractions is almost independent of growth rates.
We derive the general form of growth law involving the above correlations and demonstrate that for environments with similar correlation coefficients, the growth law curve is universal and has the following form, ϕ_{R} = (μ + c_{1})/(c_{2}μ + c_{3}) where c_{1}, c_{2}, and c_{3} are constants depending on the above correlation coefficients. In particular, c_{2}, which sets the nonlinearity of the growth law curve, is finite due to the slow translation speed of ribosomal proteins. We prove that a universal growth law curve must be monotonically increasing and convex. Surprisingly, we find that a universal growth law applies to E. coli and our theories justify the upward bending of the growth law curve of E. coli in slowgrowth conditions relative to a linear line^{7}. However, if the experiments are implemented in multiple environments with dramatically different correlation coefficients, the growth law curve is nonuniversal and environmentspecific. Our analysis of experimental data suggests that this scenario may apply to S. cerevisiae. We fit the experimentally measured growth law curves by our model predictions, from which we can estimate the translation speed of ribosomal and nonribosomal proteins. Consistent with direct experimental measurements^{9}, the estimated translation speed of ribosomal proteins is indeed much slower than nonribosomal proteins.
Results
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
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)
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 coarsegrained 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 nonuniversal 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 \({\langle k\rangle }_{\chi }=\mathop{\sum }\nolimits_{i = 2}^{N}{k}_{i}{\widetilde{\chi }}_{i}\) as the χweighted average translation speed over all nonribosomal 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 slowgrowth 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 nonribosomal 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 nonribosomal 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 slowgrowth 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 nonribosomal 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, 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 slowgrowth 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 environmentspecific, 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 nonuniversal 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 environmentspecific.
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 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 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 ϕaveraged 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 nonuniversal 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 nondegradable 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}Da^{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 nonuniversal.
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 environmentspecificity 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 nonproliferating 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 slowgrowth 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 slowgrowth 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 genomewide 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 genomewide 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 environmentspecific parameters: k_{R}, 〈k〉, α_{R}, and 〈α〉.
Methods
Derivations of Equation (2)
All variables are summarized in Table S3.
Because the total protein mass M = ∑_{i}M_{i}, we sum over all proteins on both sides of Eq. (1) and obtain
We then divide both sides by M and obtain the expression of the growth rate
from which Eq. (2) is obtained.
We can also find the changing rate of ϕ_{i} = M_{i}/M using Eq. (1),
which leads to the expression of ϕ_{i} in the steady state as
Since all proteins grow in the same rate in the steadystate, the growth rates of protein i defined as
must be equal to μ, which can be easily verified using Eq. (10). Using the normalization condition ∑_{i}ϕ_{i} = 1, we can write ϕ_{i} using Eq. (10) as
We can also write the normalization condition as
Details of the simplified model without protein degradation
In deriving Eq. (3), we neglect protein degradation and rewrite Eq. (2) as
Meanwhile, we compute the growth rate using the autocatalytic nature of ribosomal proteins,
The above equation allows us to replace χ_{R} by μ in Eq. (14), from which we obtain Eq. (3) where
Details of the simplified model with finite protein degradation rates
We now discuss the effects of finite protein degradation rates and assume that the translation speeds are homogeneous and equal to k for all proteins. We rewrite the ∑_{i}α_{i}ϕ_{i} term in Eq. (2) such that \({\sum }_{i}{\alpha }_{i}{\phi }_{i}={\alpha }_{R}{\phi }_{R}+(1{\phi }_{R})\mathop{\sum }\nolimits_{i = 2}^{N}{\alpha }_{i}{\widetilde{\phi }}_{i}\). Here, \({\phi }_{i}=(1{\phi }_{R}){\widetilde{\phi }}_{i}\) so that \(\mathop{\sum }\nolimits_{i = 2}^{N}{\widetilde{\phi }}_{i}=1\). We define the ϕ − averaged degradation rates over all nonribosomal proteins as \({\langle \alpha \rangle }_{\phi }=\mathop{\sum }\nolimits_{i = 2}^{N}{\alpha }_{i}{\widetilde{\phi }}_{i}\). Therefore, Eq. (2) can be written as
where
To find the sign of d, we further rewrite 〈α〉_{ϕ} as 〈α〉_{ϕ} = 〈α〉(1 + I_{ϕ,α}) where 〈α〉 is the arithmetic average of degradation rates over all nonribosomal proteins.
Imagine that we grow cells in multiple environments with equal I_{ϕ,α}. We assume that the degradation rate of ribosomal protein α_{R} is slower than the average of nonribosomal proteins 〈α〉, which is biologically reasonable since ribosomal proteins are generally nondegraded. Therefore, as long as I_{ϕ,α} is not too close to −1, which we confirm using experimental data, d is positive since α_{R} is always smaller than 〈α〉_{ϕ}. Therefore, our model predicts that the growth law curve is linear given a constant I_{ϕ,α} and finite protein degradation decreases the slope relative to the STM. The intercept at μ = 0 is also larger than ϕ_{0}. Therefore, a finite fraction of ribosomes are still translating at zero growth rate. We verify the above theoretical predictions by numerical simulations and randomly sample the protein degradation rates that are fixed for all environments, with α_{R} < 〈α〉 satisfied.
Derivations of the full model
In this section we derive the full model considering both the heterogeneities in the translation speeds and protein degradation rates. We rewrite Eq. (2) in the main text as
Meanwhile, the growth rate is
Combining Eq. (21) and Eq. (22) allows us to solve ϕ_{R} as a function of μ and we obtain Eq. (6)
where
It is straightforward to find that the condition for Eq. (6) to be monotonically increasing is that c_{3} > c_{1}c_{2}. Using the above expressions, we find that
We find that the first two terms are always positive, and the last term is positive as long as I_{α,ϕ} is not too close to −1. Therefore, the ϕ_{R}(μ) curve must increase monotonically. It is straightforward to find that the second derivative of the ϕ_{R}(μ) curve is proportional to (c_{1}c_{2} − c_{3})c_{2}, which is always positive as long as I_{χ,k} is not too close to −1.
Details of the numerical simulations
We summarize the parameters we use in the numerical simulations in Table S1. We consider a cell with 4000 genes. We set the elongation speed k_{i} and the degradation rates α_{i} of nonribosomal genes to follow lognormal distributions. We set k_{R} = 2.07 × 10^{4} Da/min, 〈k〉 = 4.80 × 10^{4} Da/min, α_{R} = 4.83 × 10^{−4}min^{−1}, and 〈α〉 = 1.10 × 10^{−3}min^{−1} as the experimentally measured values of S. cerevisiae^{9,11}. The coefficients of variation (CV) of the lognormal distributions can be found in Table S1. In all simulations, we set ϕ_{0} = 0.08. We note that in Fig. 2a, we set α_{i} = 0 for all proteins and in Fig. 2b, we set k_{i} = 〈k〉 for all proteins.
To simulate a random environment, we generate a random χ_{R}. Meanwhile, a lognormal distribution of χ_{i} of nonribosomal genes is also randomly generated. The CV of the lognormal distribution is included in Table S1. We then search for the ϕ_{R} and μ that simultaneously satisfy Eq. (22) and Eq. (13). ϕ_{i}, I_{χ,k} and I_{ϕ,α} are then calculated using Eq. (10), Eq. (4) and Eq. (5), respectively. For a chosen pair of I_{χ,k} and I_{ϕ,α}, the predicted ϕ_{R}(μ) curve is obtained using Eq. (6). To obtain Fig. 3d, g, we randomly sample 20 points from Fig. 3c, f respectively, fit them using Eq. (6), and calculate the resulting RMSE. We repeat the above process 5000 times.
Details of the experimental data analysis
For the ribosome profiling data^{14}, we first trim the adapter with Cutadapt (version 3.4)^{29}. Then we use Bowtie2 (version 2.4.2)^{30} to eliminate ribosomal RNAs (rRNA) as mentioned in^{31}. The cleaned reads are then mapped to S. cerevisiae genome R64.1.1 with HISAT2 (version 2.2.1)^{32}. Read counts are then generated with featureCount (version 2.0.1)^{33}. We then manually eliminate the noncoding RNAs. The ribosome allocation χ_{i} is calculated based on the mean count fraction of all samples (Supplementary Data 3).
For the proteomics data^{14}, we perform the absolute quantification (or the insample relative quantification) of proteins based on the intensities of peptides using xTop (version 1.2)^{12}. The intensity ratio of 2 proteins in the same sample of proteomics data does not directly represent the real abundance (either the mass or the copy number) ratio, so the abundance fraction can not be replaced with the intensity fraction^{12,34}. xTop is a software that accurately calculates the insample relative protein copy number with the maximum a posteriori probability (MAP) algorithm^{12}. We then calculate all proteins’ mass fraction ϕ_{i} with the xTop results and the protein molecular mass (Supplementary Data 2). In^{12}, the authors further calibrated ϕ_{i} with ribosome profiling data assuming homogeneous k_{i}. In this work, we alternatively calibrate ϕ_{i} with L^{−0.57} where L is the protein length, as mentioned in^{12}. Calibration with L^{−0.57} is independent of ribosome profiling data, although it reduces the distance between χ_{i} and calibrated ϕ_{i}^{12}. We also show the result with calibration of L^{−1} or without calibration in Fig. S2b, c. To compute ϕ_{R}, we sum up the ϕ_{i} of all proteins annotated as the cytoplasmic ribosomal protein in the Saccharomyces Genome Database (SGD).
For the elongation speed k_{i}, we first calculate v_{i} as mentioned in^{9}. k_{i} is then calculated using the relationship k_{i} = v_{i}a_{i} where \(v_i\) is the number of translated amino acids per unit time, and \(a_i\) is the averaged mass of amino acids over the sequence of protein \(i\) (Supplementary Data 1). For the degradation rate α_{i}, data is obtained from^{11}. We calculate the experimental I_{χ,k}, I_{ϕ,α}, 〈k〉 and 〈α〉 for nonribosomal genes that exist in all data sets of χ_{i}, ϕ_{i}, k_{i} and α_{i}. We also calculate the χaveraged k of ribosomal proteins as k_{R} and ϕaveraged α of ribosomal proteins as α_{R}.
For the molecular mass of the ribosome, we calculate the effective m_{R}. Considering the efficiency of the mass spectrometry (MS), not all proteins can be detected. Therefore, we define the effective m_{R} as the molecular weights of ribosomal proteins detected in the proteome. Because most of the ribosomal proteins can be expressed by two paralogous genes in S. cerevisiae, we count the average molecular mass when both proteins of the paralogs are detected in the proteome. We also show our predictions of ϕ_{R} using the actual ribosome mass (m_{R} = 1.40 × 10^{6} Da) in Fig. S2a.
For the growth rate μ (Supplementary Data 4), it is obtained from the growth curve, OD_{600} versus time from^{14} with the method mentioned in^{35}. Briefly, the slopes of \(\ln ({{{\mbox{OD}}}}_{600})\) versus time in 5point windows are calculated. Then windows with slopes at least 95% of the maximum slope are extracted. The slope of points within these windows is calculated as the growth rate. With these results, we predict the corresponding ϕ_{R}(μ) curves and compare them with the experimental data points.
We further calculate the predicted mass fraction ϕ_{i,pre} of nonribosomal proteins with Eq. (12). Pearson correlation coefficients ρ between ϕ_{i,pre} and ϕ_{i} are calculated. We also compute ρ under the assumptions that α_{i} = 0 or k_{i} = 〈k〉 (Table S2).
For GSEA analysis, we first perform the differential expression analysis on the ribosome profiling data of WT or natAΔ cells using the package DEseq2 (version 1.24.0)^{36} in R (version 3.6.1). The \({\log }_{2}\) fold changes in counts when cells changed from SC+2% glucose to SC+2% glycerol, the pvalue of the twosided Wald test, and the FDR qvalues are calculated. Ribosomal genes and genes with FDR qvalue > 0.05 are eliminated. We then pick out genes that also exist in the data sets of k_{i}. GSEA on these genes is performed twice using the R package clusterProfiler (version 3.12.0)^{37} and org.Sc.sgd.db (version 3.8.2)^{38}. In the first GSEA, genes are ordered by the \({\log }_{2}\) fold change (denoted as \({\log }_{2}\)FCordered GSEA). In the second GSEA, genes are ordered by k_{i} (denoted as k_{i}ordered GSEA). We then find the common gene sets from GO database^{17,18} enriched in these two GSEA. The cutoff criteria are set as the pvalue < 0.05 (singlesided permutation test) and the FDR qvalue < 0.25. The number of permutations used in the analysis is 10^{5}.
Details of fitting in Fig. 5
Nonlinear fitting is performed with MATLAB (version R2020b). We obtain the fitting parameters c_{1}, c_{2} and c_{3} with their 95% confidence intervals, and then compute ϕ_{0}, k_{R} and 〈k〉_{χ} using Eqs. (23), (24), (25). To compute the ranges of these values, we numerically find the maximum and the minimum value of the multivariate functions ϕ_{0}(c_{1}, c_{2}, c_{3}), k_{R}(c_{1}, c_{2}, c_{3}) and 〈k〉_{χ}(c_{1}, c_{2}, c_{3}) as their upper and lower bounds, where the ranges of c_{1}, c_{2} and c_{3} are their 95% confidence intervals.
Statistics and reproducibility
We use the twosided Wald test in the differential expression analysis of the ribosome profiling data. In GSEA, we use a singlesided permutation test. As for reproducibility, no biological experiments are performed in our work, and all data are acquired from public repositories (see Data Availability).
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Data availability
The Ribosome profiling data from^{14} was deposited at GEO (GSE140255) (link https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE140255). The Proteomics data from^{14} was deposited at ProteomeXchange (PXD015217) (link http://proteomecentral.proteomexchange.org/cgi/GetDataset?ID=PXD015217). The growth rate data was acquired from the figures of OD600versustime curves in^{14}. The Ribosomal protein list was acquired from Saccharomyces Genome Database (SGD) (link https://yeastgenome.org). The data needed calculating elongation speeds was acquired from the supplementary materials of^{9}. The protein degradation rate data was acquired from the supplementary materials of^{11}. The ϕ_{R}μ data of E. coli was acquired from the supplementary materials of^{7}. The ϕ_{R}μ data of S. cerevisiae was acquired from the figure in^{6}. Calculated data of \(k_i\), \(\phi_i\), \(\chi_i\), and \(\mu\) have been provided in Supplementary Data 14.
Code availability
All codes for mathematical simulations and data analysis are available in the following link (https://github.com/QirunWang/CodesforEnvironmentspecificityanduniversalityofthegrowthlaw).
References
Neidhardt, F. C. & Magasanik, B. Studies on the role of ribonucleic acid in the growth of bacteria. Bioch. Biophys. Acta 42, 99–116 (1960).
Maaløe, O. Regulation of the proteinsynthesizing machinery—ribosomes, trna, factors, and so on. In Biological regulation and development, 487–542 (Springer, 1979).
Bremer, H. & Dennis, P. P. Modulation of chemical composition and other parameters of the cell at different exponential growth rates. EcoSal Plus 3 (2008).
Scott, M., Gunderson, C. W., Mateescu, E. M., Zhang, Z. & Hwa, T. Interdependence of cell growth and gene expression: origins and consequences. Science 330, 1099 (2010).
Hui, S. et al. Quantitative proteomic analysis reveals a simple strategy of global resource allocation in bacteria. Mol. Syst. Biol. 11, 784 (2015).
MetzlRaz, E. et al. Principles of cellular resource allocation revealed by conditiondependent proteome profiling. Elife 6, e28034 (2017).
Dai, X. et al. Reduction of translating ribosomes enables escherichia coli to maintain elongation rates during slow growth. Nat. Microbiol. 2, 1–9 (2016).
Klumpp, S., Dong, J. & Hwa, T. On ribosome load, codon bias and protein abundance. PLoS ONE 7, e48542 (2012).
Riba, A. et al. Protein synthesis rates and ribosome occupancies reveal determinants of translation elongation rates. Proc. Natl Acad. Sci. 116, 15023–15032 (2019).
Gefen, O., Fridman, O., Ronin, I. & Balaban, N. Q. Direct observation of single stationaryphase bacteria reveals a surprisingly long period of constant protein production activity. Proc. Natl Acad. Sci. 111, 556–561 (2014).
Lahtvee, P.J. et al. Absolute quantification of protein and mrna abundances demonstrate variability in genespecific translation efficiency in yeast. Cell Syst. 4, 495–504.e5 (2017).
Mori, M. et al. From coarse to fine: the absolute escherichia coli proteome under diverse growth conditions. Mol. Syst. Biol. 17, e9536 (2021).
Li, G.W., Burkhardt, D., Gross, C. & Weissman, J. S. Quantifying absolute protein synthesis rates reveals principles underlying allocation of cellular resources. Cell 157, 624–635 (2014).
Friedrich, U. A. et al. Nαterminal acetylation of proteins by nata and natb serves distinct physiological roles in saccharomyces cerevisiae. Cell Rep. 34, 108711 (2021).
Subramanian, A. et al. Gene set enrichment analysis: a knowledgebased approach for interpreting genomewide expression profiles. Proc. Natl Acad. Sci. 102, 15545–15550 (2005).
Mootha, V. K. et al. Pgc1alpharesponsive genes involved in oxidative phosphorylation are coordinately downregulated in human diabetes. Nat. Genet. 34, 267–273 (2003).
Ashburner, M. et al. Gene ontology: tool for the unification of biology. Nat. Genet. 25, 25–29 (2000).
Consortium, T. G. O. The Gene Ontology resource: enriching a Gold mine. Nucleic Acids Res. 49, D325–D334 (2020).
Gutin, J., Sadeh, A., Rahat, A., Aharoni, A. & Friedman, N. Conditionspecific genetic interaction maps reveal crosstalk between the camp/pka and the hog mapk pathways in the activation of the general stress response. Mol. Syst. Biol. 11, 829 (2015).
Erickson, D. W. et al. A global resource allocation strategy governs growth transition kinetics of escherichia coli. Nature. 551, 119–123 (2017).
Goldberg, A. L. & John, A. C. S. Intracellular protein degradation in mammalian and bacterial cells: Part 2. Annu. Rev. Biochem. 45, 747–804 (1976).
Arkin, A., Ross, J. & McAdams, H. H. Stochastic kinetic analysis of developmental pathway bifurcation in phage lambdainfected escherichia coli cells. Genetics. 149, 1633–1648 (1998).
Gouy, M. & Grantham, R. Polypeptide elongation and trna cycling in escherichia coli: a dynamic approach. FEBS Lett. 115, 151–155 (1980).
Karpinets, T. V., Greenwood, D. J., Sams, C. E. & Ammons, J. T. RNA: protein ratio of the unicellular organism as a characteristic of phosphorous and nitrogen stoichiometry and of the cellular requirement of ribosomes for protein synthesis. BMC Biol. 4, 1–10 (2006).
Schaechter, M., MaalØe, O. & Kjeldgaard, N. O. Dependency on medium and temperature of cell size and chemical composition during balanced growth of salmonella typhimurium. Microbiology 19, 592–606 (1958).
Basan, M. et al. Inflating bacterial cells by increased protein synthesis. Mol. Syst. Biol. 11, 836 (2015).
Vadia, S. & Levin, P. A. Growth rate and cell size: a reexamination of the growth law. Curr. Opin. Microbiol. 24, 96–103 (2015).
Liu, T.Y. et al. Timeresolved proteomics extends ribosome profilingbased measurements of protein synthesis dynamics. Cell Systems 4, 636–644.e9 (2017).
Martin, M. Cutadapt removes adapter sequences from highthroughput sequencing reads. EMBnet. Journal 17, 10–12 (2011).
Langmead, B. & Salzberg, S. L. Fast gappedread alignment with bowtie 2. Nat. Methods 9, 357–359 (2012).
Galmozzi, C. V., Merker, D., Friedrich, U. A., Döring, K. & Kramer, G. Selective ribosome profiling to study interactions of translating ribosomes in yeast. Nat. Protocols 14, 2279–2317 (2019).
Kim, D., Paggi, J. M., Park, C., Bennett, C. & Salzberg, S. L. Graphbased genome alignment and genotyping with hisat2 and hisatgenotype. Nat. Biotechnol. 37, 907–915 (2019).
Liao, Y., Smyth, G. K. & Shi, W. featureCounts: an efficient general purpose program for assigning sequence reads to genomic features. Bioinformatics 30, 923–930 (2013).
CalderónCelis, F., Encinar, J. R. & SanzMedel, A. Standardization approaches in absolute quantitative proteomics with mass spectrometry. Mass Spectrometry Rev. 37, 715–737 (2018).
Hall, B. G., Acar, H., Nandipati, A. & Barlow, M. Growth rates made easy. Mol. Biol. Evol. 31, 232–238 (2013).
Love, M. I., Huber, W. & Anders, S. Moderated estimation of fold change and dispersion for rnaseq data with deseq2. Genome Biol. 15, 550 (2014).
Yu, G., Wang, L.G., Han, Y. & He, Q.Y. clusterProfiler: an R package for comparing biological themes among gene clusters. OMICS: J. Integrative Biol. 16, 284–287 (2012).
Carlson, M. org.Sc.sgd.db: genome wide annotation for yeast. https://bioconductor.org/packages/release/data/annotation/html/org.Sc.sgd.db.html (2019).
Acknowledgements
We thank Poyi Ho for useful discussions related to this work. The research was funded by National Key R&D Program of China (2021YFF1200500) and supported by grants from PekingTsinghua Center for Life Sciences.
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Q.W. and J.L. conceived, designed, and carried out the theoretical and numerical part of this work. Q.W. performed the analysis of experimental data. All the authors contributed to the preparation of the manuscript.
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Wang, Q., Lin, J. Environmentspecificity and universality of the microbial growth law. Commun Biol 5, 891 (2022). https://doi.org/10.1038/s4200302203815w
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DOI: https://doi.org/10.1038/s4200302203815w
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