Bridging ribosomal synthesis to cell growth through the lens of kinetics

Luan Quang Le; Kaicheng Zhu; Haibin Su

doi:10.1016/j.bpj.2022.12.028

Biophys J. 2023 Feb 7; 122(3): 544–553.

Published online 2022 Dec 22. doi: 10.1016/j.bpj.2022.12.028

PMCID: PMC9941725

PMID: 36564946

Bridging ribosomal synthesis to cell growth through the lens of kinetics

Luan Quang Le,^1,² Kaicheng Zhu,² and Haibin Su^2,^3,^∗

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Associated Data

Supplementary Materials: Document S1. Figures S1–S13 and Tables S1–S6
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Document S2. Article plus supporting material
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Abstract

Understanding prokaryotic cell growth requires a multiscale modeling framework from the kinetics perspective. The detailed kinetics pathway of ribosomes exhibits features beyond the scope of the classical Hopfield kinetics model. The complexity of the molecular responses to various nutrient conditions poses additional challenge to elucidate the cell growth. Herein, a kinetics framework is developed to bridge ribosomal synthesis to cell growth. For the ribosomal synthesis kinetics, the competitive binding between cognate and near-cognate tRNAs for ribosomes can be modulated by $[{Mg}^{2 +}]$ . This results in distinct patterns of the speed – accuracy relation comprising “trade-off” and “competition” regimes. Furthermore, the cell growth rate is optimized by varying the characteristics of ribosomal synthesis through cellular responses to different nutrient conditions. In this scenario, cellular responses to nutrient conditions manifest by two quadratic scaling relations: one for nutrient flux versus cell mass, the other for ribosomal number versus growth rate. Both are in quantitative agreement with experimental measurements.

Significance

Ribosomes play essential roles in protein synthesis which underpins cell growth. $[{Mg}^{2 +}]$ -dependent ribosomal synthesis performance in terms of speed – accuracy relation is illustrated by the generalized Hopfield model. Integrating the characteristics of ribosomal synthesis into cell growth model yields adaptive cell growth kinetics through $[{Mg}^{2 +}]$ -dependent responses to different nutrient conditions.

Introduction

Protein synthesis is one of the most resource-intensive processes that occurs during bacterial growth. Since ribosomes are responsible for polypeptide synthesis, studying ribosomal kinetics across multiscale is important because the molecular performances will ultimately affect the cellular behaviors. At the molecular level, protein synthesis has the elongation speed of 10∼22 amino acids per second (1,2,3) and the accuracy of one mistake per $3 \times 10^{3}$ codons (4,5,6), enabling the rapid growth to prevail over rivals for high fitness (7). Studying the ribosomal kinetics in terms of speed – accuracy relation is thus necessary to understand the performance of polypeptide synthesis (8,9,10). At the cellular level, cells adopt several different strategies for optimal growth in response to nutrient conditions (11). Cells can increase protein synthesis speed for rapid growth, manifesting in the Michaelis-Menten relation between protein synthesis speed and growth rate (12). The variation of protein synthesis accuracy was observed to adjust cell growth (13,14), suggesting the responses on cellular activities from the ribosomal speed and accuracy relation. (15,16).

The kinetics pathway of ribosomes comprises two stages: initial selection and proofreading (17). At initial selection, ribosomes select cognates over near-cognates with significantly faster forward rate constants owing to the correct codon-anticodon interactions. In case near-cognates pass the initial selection, ribosomes can discard near-cognates at proofreading energized from the irreversible guanosine-5′-triphosphate (GTP) hydrolysis (18,19,20). The cognates are clearly favored during the peptide bond formation within the ribosomes which invoke the delicate stepwise “proton shuttle” process. (21). The in vitro discrimination factor of selecting cognates over near-cognates is 30∼60 for initial selection (4,19,22) and 20∼100 for proofreading (4,19,20,23). The product of the discrimination factor of the two stages agree with the in vivo accuracy of $3 \times 10^{3}$ (4,5).

Ribosomes exhibit the trade-off relation between speed and accuracy during peptide synthesis. The mutation in ribosomes contains an asparagine instead of lysine at position 42 of protein S12, enabling ribosomes to achieve better accuracy in protein production by reducing the overall elongation speed for cognates and near-cognates (24). Alternatively, increasing $[{Mg}^{2 +}]$ enables ribosomes to produce proteins more rapidly but with less accuracy because the speed of near-cognates accelerates at a faster rate with respect to $[{Mg}^{2 +}]$ than that of the cognates (15,16). The acceleration of near-cognates is attributed to less dissociation of the complexes between ribosomes and near-cognates, thus stabilizing near-cognates’ binding to ribosomes and causing more competition for cognates selection (25). Hopfield model can explain the trade-off relation by suggesting that synthesis speed increases and accuracy reduces when rejection rate constants decrease (18). Recently, ribosomes have been shown to discriminate near-cognates against cognates with sluggish forward speed (e.g., the net speed of initial selection is 1.6 and 100 $μ M^{- 1} s^{- 1}$ for near-cognates and cognates, respectively) (19,20). Moreover, less rejection can lead to enhanced competition with more ribosomes occupying near-cognates, especially at high $[{Mg}^{2 +}]$ due to diminishing contribution of proofreading to total accuracy (16,26). Consequently, near-cognates can slow down the translation speed once they bind to ribosomes. Insides cells, ribosomes frequently encounter near-cognates because of their much higher number compared with cognates (20,27). However, it has remained unclear how the kinetics difference between cognates and near-cognates influences ribosomal partition between the corresponding two complexes and the peptide synthesis speed – accuracy relation.

Cellular adaptation to nutrient fluxes manifests in a linear relation between ribosomal mass fraction and growth rate when bacteria reach exponential, steady-state growth at different nutrient quality (28,29,30,31). The linear relation emerges from the optimal proteomic arrangement for the fastest growth (32). The effects of protein synthesis accuracy on cell growth were extensively investigated. The growth rate of E. coli cells decreased by 26% due to a twofold decrease in the accuracy caused by wrong isoleucine incorporation into the poly-phenylalanine chain (33). The decrease in accuracy is attributed to mutations at ribosomal proteins S4 and S5 disabling termination of protein synthesis at stop codons (i.e., readthrough of stop codon) (33). Readthrough of stop codon is also induced by mutations tufA8 and tufB103 in elongation factor Tu (34). The mutations caused a fourfold increase in misincorporation of leucine into the poly-phenylalanine chain, consequently reducing the growth rate of S. typhimurium cells by nearly 30% (34). Besides, cells have signaling pathways to maintain intracellular $[{Mg}^{2 +}]$ in response to perturbations amid the cell growth (35,36,37). If $M g^{2 +}$ import is enhanced due to overexpression of the MgtE gene (35,36), intracellular $[{Mg}^{2 +}]$ increases and causes both accuracy and cell growth to reduce. Increasing $[{Mg}^{2 +}]$ from 4.5 to 14 mM caused a nearly tenfold increase of leucine misincorporation into the poly-phenylalanine chain (38), and further raising $[{Mg}^{2 +}]$ to 1.25 M reduced growth rate of E. coli cells sixfold (39).

In this article, we address the speed – accuracy relation of ribosomes and its connection to cell growth. Hopfield model is extended into generalized Hopfield model (GHM) to account for the in vivo conditions where the competition between cognates and near-cognates requires ribosomes to distinguish the two species with different kinetics. The GHM shows that $[{Mg}^{2 +}]$ modulates the ribosomal speed – accuracy relation from trade-off to positive correlation. We then study the effects of ribosomal speed – accuracy relation on cell growth via a kinetics coupled bacterial growth model (BGM). Cellular response toward nutrient conditions can be regulated by $[{Mg}^{2 +}]$ which enables cells to optimize growth rate.

Materials and methods

The kinetics scheme of ribosomes for peptide bond formation is elaborated. The free energy landscape analysis is invoked for the study of $[{Mg}^{2 +}]$ -dependent rate constants, thereby capturing the overall trend of ribosomal speed – accuracy as a function of $[{Mg}^{2 +}]$ . The results of GHM are integrated into the BGM, as illustrated in Fig. S1, to describe how cells control $[{Mg}^{2 +}]$ to modulate accuracy and speed for the optimal growth in response to nutrient conditions. The codes are available at https://github.com/haibinsu/Ribosome_cell_growth.

Protein synthesis from the GHM

The GHM includes three key states of ribosome: free ribosomes $(R_{f r e e})$ , GTP-activated state of initial selection $(R_{A c t})$ , and hydrolyzed ternary complexes involved proofreading $(R_{PR})$ , as shown in Fig. 1 a. Initially, $R_{f r e e}$ waits for the binding of ternary complexes $T_{3}$ to form $R_{A c t}$ with bimolecular rate constants $k_{f}^{c / n c}$ . The superscripts $c, n c$ denote cognates and near-cognates, respectively. The $k_{f}$ corresponds to the overall measured speed (denoted as $k_{c a t} / K_{M}$ ) of initial selection consisting of binding between ternary complexes and ribosomes, codon recognition, and GTPase activation (Fig. S2 a; Eq. S19). Once ribosomes reach $R_{A c t}$ , they quickly enter $R_{PR}$ due to rapid hydrolysis of GTP with rate constant $k_{h y d}$ . At proofreading, ribosomes can either form a peptide bond or reject hydrolyzed ternary complexes with rate constants $k_{p e p f}^{c / n c}$ and $k_{r e j}^{c / n c}$ , respectively. If peptide formation occurs, ribosomes translocate to the next codon of messenger RNA, become $R_{f r e e}$ , and start the next round of peptide synthesis. If rejection happens, which is most likely for near-cognates, ribosomes become $R_{f r e e}$ and repeat the selection process until a peptide bond is formed. The two scenarios indicate that ribosomes eventually return to $R_{f r e e}$ to wait for $T_{3}$ . The three states form a cycle where ribosomes continuously synthesize protein by selecting cognates, which are outnumbered by the pool of near-cognates. The selection of cognates under the competition of near-cognates is represented as the two reaction pathways with different kinetics as shown in Fig. 1 a.

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Figure 1

The generalized Hopfield model (GHM) of ribosomal protein synthesis. (a) Three states are: free ribosomes ( $R_{f r e e}$ ), GTPase activation at initial selection $(R_{A c t})$ , and proofreading $(R_{PR})$ comprising peptide formation and rejection. Arrow thickness reflects the magnitude of rate constants depending on cognates or near-cognates. (b) The partition of ribosomes in three states at various $[{Mg}^{2 +}]$ is dominated by $R_{f r e e}$ (white), $R_{P R^{c}}$ for cognates (green), and $R_{P R^{n c}}$ for near-cognates (brown). (c) The relation among $k_{s y n}^{e f f}$ , accuracy, and $[{Mg}^{2 +}]$ . The magenta and blue dots indicate the maximum value of $k_{s y n}^{e f f}$ . To see this figure in color, go online.

Based on Fig. 1 a, the kinetics of ribosomes is as follows:

\frac{d [R_{f r e e}]}{d t} = (k_{r e j}^{c} + k_{p e p f}^{c}) [R_{P R^{c}}] + (k_{r e j}^{n c} + k_{p e p f}^{n c}) [R_{P R^{n c}}] - (k_{f}^{c} f^{c} + k_{f}^{n c} f^{n c}) [T_{3}] [R_{f r e e}] + k_{b}^{c} [R_{A c t^{c}}] + k_{b}^{n c} [R_{A c t^{n c}}],

(1a)

\frac{d [R_{A c t^{c}}]}{d t} = k_{f}^{c} f^{c} [T_{3}] [R_{f r e e}] - (k_{b}^{c} + k_{h y d}) [R_{A c t^{c}}],

(1b)

\frac{d [R_{A c t^{n c}}]}{d t} = k_{f}^{n c} f^{n c} [T_{3}] [R_{f r e e}] - (k_{b}^{n c} + k_{h y d}) [R_{A c t^{n c}}],

(1c)

\frac{d [R_{P R^{c}}]}{d t} = k_{h y d} [R_{A c t^{c}}] - (k_{r e j}^{c} + k_{p e p f}^{c}) [R_{P R^{c}}],

(1d)

and

\frac{d [R_{P R^{n c}}]}{d t} = k_{h y d} [R_{A c t^{n c}}] - (k_{r e j}^{n c} + k_{p e p f}^{n c}) [R_{P R^{n c}}] .

(1e)

The total concentration of ribosomes $[R_{t}]$ is 10 $μ$ M, and it is assumed constant because of enzyme conservation (i.e., the sum of (1a), (1b), (1c), (1d), (1e) is zero). $[R_{t}]$ is partitioned into five chemical states during amino acid incorporation (Fig. 1 a):

[R_{f r e e}] + [R_{A c t^{c}}] + [R_{A c t^{n c}}] + [R_{P R^{c}}] + [R_{P R^{n c}}] = [R_{t}] .

(1f)

$[T_{3}] = 100 μ$ M, while the number fraction of cognate and near-cognate species are $f^{c} = 0.02$ and $f^{n c} = 0.15$ , respectively. All values correspond to in vivo conditions (20,27,40). The values of rate constants $k_{f}^{c}$ and $k_{f}^{n c}$ are from the in vitro data measured at different $[{Mg}^{2 +}]$ (16,41). The backward rate constants $k_{b}^{c / n c}$ are neglected because they are much smaller than the forward rate constants $k_{f}^{c / n c}$ . The remaining rate constants are given in Table S1. Eqs. 1a–1e are solved at steady state where there are no changes in concentrations with time. Eq. 1f acts as a constraint since there are only four independent equations among Eqs. 1a–1e.

The influence of $[{Mg}^{2 +}]$ on the protein synthesis speed – accuracy relation is studied by using the GHM in conjunction with the experimental data (16,41). In particular, the bimolecular rate constant $k_{f}$ corresponds to the $k_{c a t} / K_{M}$ of initial selection (Eq. S19), and the net rate constant from $R_{f r e e}$ to $R_{PR}$ is the $k_{c a t} / K_{M}$ of overall peptide formation (Table S1; supporting material section 2.2). Expressions of $k_{f}^{c}$ and $k_{f}^{n c}$ are based on the free energy landscape of ribosomes at initial selection (supporting material section 2.3). The expressions show good agreement with the experimental data as depicted in Fig. S2 b, thereby capturing the rate constants’ dependence on $[{Mg}^{2 +}]$ .

Cell growth from the BGM

Based on Fig. 2 a, the kinetics of the BGM is expressed as follows:

M = m_{R} R + m_{P} P + \tilde{a} = ρ V,

(2a)

\frac{d R}{d t} = \frac{u_{R}}{m_{R}} R \cdot m (A) \frac{k_{s y n m a x}^{c e l l} (A) \tilde{a}}{K_{M}^{\tilde{a}} (A) N_{A} V + \tilde{a}} - k_{d} R,

(2b)

\frac{d P}{d t} = \frac{1 - u_{R}}{m_{P}} R \cdot m (A) \frac{k_{s y n m a x}^{c e l l} (A) \tilde{a}}{K_{M}^{\tilde{a}} (A) N_{A} V + \tilde{a}} - k_{d} P,

(2c)

and

\frac{d \tilde{a}}{d t} = \frac{P k_{m e t}^{e f f}}{1 + {(\frac{\tilde{a}}{K_{e} N_{A} V})}^{2}} - R \cdot m (A) \frac{k_{s y n m a x}^{c e l l} (A) \tilde{a}}{K_{M}^{\tilde{a}} (A) N_{A} V + \tilde{a}} - J_{m t n},

(2d)

where

R, P, and $\tilde{a}$ denote the number of ribosomes, metabolic proteins, and amino acids, respectively. $m_{R}$ and $m_{P}$ are the masses of ribosome and metabolic protein, respectively, with all masses expressed in terms of the number of amino acids (Table S2). The three components approximately contribute to total cell mass M (Table S2). The approximation follows the experimental results showing that protein and RNA dry mass can take up to 70% of cell dry mass (44). Protein mass density is approximately constant across different growth rates (45,46) as shown in Fig. S6 b. The Avogadro constant $N_{A}$ is used to express concentrations in term of molecular numbers. Protein synthesis flux depends on accuracy A via accuracy dependent $K_{M}^{\tilde{a}} (A)$ , $k_{s y n m a x}^{c e l l} (A)$ , and $m (A)$ . The $K_{M}^{\tilde{a}}$ of amino acid is presumably proportional to the $K_{M}^{e f f}$ of the ternary complex (Eqs. S47 and S48; Table S2). The maximal protein synthesis speed $k_{s y n m a x}^{c e l l}$ also couples with accuracy in the framework of GHM. The modulation factor $m$ can capture the effect of accuracy in the cells. The dependence of $m$ on accuracy across various nutrient conditions is illustrated in Fig. S7 a. Ribosomes synthesize themselves with allocation $u_{R}$ , suggesting that the self-catalytic activity of ribosomes is the origin of exponential growth (29,32). The remaining allocation of protein synthesis flux is for metabolic proteins that supply amino acids (29,32). Cells limit the number of amino acids to avoid overproduction by negative feedback via inhibition constant $K_{e}$ (32,47). For moderate-fast growth, the maintenance flux $J_{m t n}$ can be neglected because most of the nutrient intake flux is allocated for growth (48). We choose the values of all variables and expressions so that the output parameters, such as growth rates and the number and mass fraction of ribosomes, are within the range of experimental data. The physical meanings and values of remaining parameters are further provided in Tables S2, S5, and S6.

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Figure 2

Cellular characteristics by the bacterial growth model (BGM). (a) The schematic illustration of BGM (b) The characteristics of ribosomal synthesis in terms of protein synthesis speed $k_{s y n e f f}^{c e l l}$ -accuracy relation (at $[T_{3}] = 100 μ M$ ). Experimental data are from refs. 16 and 41. (c) $[{Mg}^{2 +}]$ regulated optimal cell growth rate under different nutrient conditions. The optimal growth rates $λ^{*}$ as large, solid dots. (d) The quadratic scaling between number of ribosomes per cell $R^{*}$ and optimal growth rate $λ^{*}$ with the experimental data of E. coli (1,42). The inset shows a linear relation between $λ^{*}$ and ribosomal mass fraction $φ_{R}^{*}$ with the experimental data of E. coli (3,29,31). (e) The quadratic scaling between intake flux $J_{M}^{*}$ and cell mass $M^{*}$ with the experimental data of E. coli (1,30,42). The right inset shows metabolic fluxes of different bacteria species (43). The linear $M^{*} - λ^{*}$ relation is shown in the left inset. To see this figure in color, go online.

The important parameters in the BGM are amino acid-production rate constant $k_{m e t}^{e f f}$ , cell mass M, and accuracy A, as well as accuracy dependent maximal speed $k_{s y n m a x}^{c e l l}$ , Michaelis constant $K_{M}^{\tilde{a}}$ , and modulation factor $m$ . The values of $k_{m e t}^{e f f}$ , listed in Table S5, fall within the experimental range for various substrates summarized in Table S3. The dependence of $k_{m e t}^{e f f}$ on nutrient conditions is captured through its increasing values indicating better nutrient quality (32). The values of cell mass M, shown in Table S5, are physically consistent with the experimental values in Table S4. Besides, M presumably follows growth rate in a linear fashion. The presumption is based on the experimental results shown in Fig. S6 a (30) and the left inset of Fig. 2 e (1). For bacterial growth, the maximal value of $k_{s y n m a x}^{c e l l}$ is 22 $s^{- 1}$ in order to match in vivo condition by changing $k_{p e p f}^{c}$ to 22 $s^{- 1}$ . The values of ribosomal rate constants, shown in Fig. S9 and Table S6, follow the experimental values summarized in Table S1. The values of accuracy, $k_{s y n m a x}^{c e l l}$ , and $K_{M}^{\tilde{a}}$ are calculated based on Tables S2 and S6 and Eqs. 3b, 4b, and 4c, respectively.

Results and discussion

The GHM extends Hopfield model by considering the kinetics difference between ternary complexes and their competition for ribosomes, thereby establishing that the coupling between protein synthesis speed and accuracy is the basis for the speed – accuracy relation. The GHM also establishes quantitatively the modulation of $[{Mg}^{2 +}]$ on ribosomal speed – accuracy relation. Overall, the speed – accuracy relation subject to $[{Mg}^{2 +}]$ regulation is incorporated in the BGM to demonstrate that nutrient quality dictates the cellular regulation of $[{Mg}^{2 +}]$ intake flux to optimize both accuracy and ribosomal content for optimal growth.

The protein synthesis

The GHM is an extension of Hopfield model to address $[{Mg}^{2 +}]$ effect on protein synthesis speed – accuracy relation. Like Hopfield model (18,49,50), the protein synthesis network in the GHM has three chemical states: an available ribosomal state waiting for ternary complexes and two other states where ribosomes are occupied with tRNA (49,50) (e.g., $R_{A c t}$ and $R_{PR}$ in the GHM). The transition between two tRNA-occupied states is driven by the irreversible hydrolysis enabling ribosomes to reject near-cognates at proofreading. The rate constants $k_{f}, k_{h y d}$ , and $k_{p e p f}$ are independent of ternary complexes in Hopfield model as depicted in Fig. S3 (18). In the GHM, these rate constants are based on biochemical measurements, for example, $k_{f}^{c} ≃ 100 μ M^{- 1} s^{- 1}$ and $k_{f}^{n c} ≃ 1.6 μ M^{- 1} s^{- 1}$ (19,20). Peptide formation rate constant of cognates is also faster than that of near-cognates (i.e., $k_{p e p f}^{c}$ is 7∼22 $s^{- 1}$ (10), whereas $k_{p e p f}^{n c}$ is 0.3 $s^{- 1}$ (19)). Moreover, near-cognates face higher rejection due to much larger $k_{r e j}^{n c}$ than $k_{r e j}^{c}$ of cognates, which is less than 0.1 $s^{- 1}$ (19). The kinetics difference between cognates and near-cognates is also considered in previous work (49,50). The $k_{f}$ in those models corresponds to the codon recognition step (24); the values of $k_{f}$ are first-order rate constants. In contrast, the $k_{f}$ in the GHM is the bimolecular rate constant, which corresponds to the net rate of ribosomes transitioning from the free to the GTPase-activated state (Eq. S19). The competition between cognates and near-cognates for ribosomes depends on not only the bimolecular rate constants $k_{f}^{c}$ and $k_{f}^{n c}$ but also the concentrations of the two ternary complexes. The GHM considers the concentration difference via the number fraction of cognates $f^{c}$ and near-cognates $f^{n c}$ with respect to total $[T_{3}]$ . The values of $f^{c}$ and $f^{n c}$ are in good agreement with in vivo measurements (20,27,40).

The steady-state peptide production fluxes $J_{t}$ and accuracy A are defined as follows:

J_{t} = J_{c} + J_{n c} = [R_{P R^{c}}] k_{p e p f}^{c} + [R_{P R^{n c}}] k_{p e p f}^{n c}

(3a)

and

A = \frac{J_{c}}{J_{n c}} = \frac{[R_{P R^{c}}] k_{p e p f}^{c}}{[R_{P R^{n c}}] k_{p e p f}^{n c}} .

(3b)

The accuracy A, defined as the peptide flux ratio between cognates and near-cognates in Eq. 3b, comprises the contribution from initial selection and proofreading (Eq. S10), as shown by in vivo measurements (18,40). The definition also takes into account the concentration difference between cognates and near-cognates. The protein synthesis speed $k_{s y n}^{e f f}$ can be further expressed in the form of Michaelis-Menten kinetics (supporting material section 2.2):

k_{s y n}^{e f f} = \frac{J_{t}}{[R_{t}]} = \frac{k_{s y n}^{\max} [T_{3}]}{K_{M}^{e f f} + [T_{3}]},

(4a)

where $k_{s y n}^{\max}$ and $K_{M}^{e f f}$ are approximated as follows (Eqs. S8 and S9):

k_{s y n}^{\max} \approx f^{c} k_{f}^{c} {(\frac{f^{c} k_{f}^{c}}{k_{p e p f}^{c}} + \frac{f^{n c} k_{f}^{n c}}{k_{r e j}^{n c}})}^{- 1}

(4b)

and

K_{M}^{e f f} \approx {(\frac{f^{c} k_{f}^{c}}{k_{p e p f}^{c}} + \frac{f^{n c} k_{f}^{n c}}{k_{r e j}^{n c}})}^{- 1} .

(4c)

$k_{s y n}^{\max}$ and $K_{M}^{e f f}$ represent the maximum protein synthesis speed and Michaelis constant (i.e., indicating ternary complex concentration $[T_{3}]$ that gives half of $k_{s y n}^{\max}$ ), respectively. Both $k_{s y n}^{\max}$ and $K_{M}^{e f f}$ depend on the competition between cognates and near-cognates through their number fraction $f^{c}$ and $f^{n c}$ . Besides, $k_{s y n}^{\max}$ and $K_{M}^{e f f}$ can vary because some rate constants (e.g., $k_{f}^{c}$ and $k_{f}^{n c}$ ) depend on $[{Mg}^{2 +}]$ . More importantly, changes in the rate constants of some ribosomal states affect $k_{s y n}^{\max}$ , $K_{M}^{e f f}$ , and accuracy A simultaneously (Eqs. 3b and S10), suggesting the intimate interplay among the three quantities. The dependence of the speed – accuracy relation on $[{Mg}^{2 +}]$ is elaborated in following subsection.

[ ${Mg}^{2 +}$ ] effect on protein synthesis

The partition of ribosomes at different $[{Mg}^{2 +}]$ relates to their $[{Mg}^{2 +}]$ -dependent rate constants. The ribosomal partition provides molecular insight on how the allocation of ribosomes in the network effectively manifests into $k_{s y n}^{e f f}$ and accuracy. The three dominant states are $R_{f r e e}$ , $R_{P R^{n c}}$ , and $R_{P R^{c}}$ , which are approximated as follows (for $[{Mg}^{2 +}]$ within 1∼8 mM):

[R_{f r e e}] \approx [R_{t}] r_{0} \exp (- a_{0} [{Mg}^{2 +}]),

(5a)

[R_{P R^{n c}}] \approx [R_{t}] r_{pr}^{n c} \exp (a_{pr}^{n c} [{Mg}^{2 +}]),

(5b)

and

[R_{P R^{c}}] \approx [R_{t}] [1 - r_{0} \exp (- a_{0} [{Mg}^{2 +}]) - r_{pr}^{n c} \exp (a_{pr}^{n c} [{Mg}^{2 +}])],

(5c)

where $r_{0} = 0.082, a_{0} = 0.3 m M^{- 1}, r_{pr}^{n c} = 0.002, a n d a_{pr}^{n c} = 0.62 m M^{- 1}$ (Eqs. S40a–S40c). The approximation captures the trend of the data calculated based on the experimental rate constants and the GHM (Figs. S2 d and S4 a). Fig. 1 b shows the ribosomal percentage distribution of the three states at different $[{Mg}^{2 +}]$ . The total percentage of $R_{P R^{c}}$ and $R_{P R^{n c}}$ rose and reached plateau, while $R_{f r e e}$ plummeted as $[{Mg}^{2 +}]$ increased. At low $[{Mg}^{2 +}]$ , $R_{P R^{c}}$ dominated and occupied almost 95% at 3 mM because ribosomes prefer cognates to near-cognates. Beyond 3 mM, $R_{P R^{n c}}$ accumulated, while the concentrations of $R_{P R^{c}}$ and $R_{f r e e}$ reduced. The dependence of ribosomal partition on $[{Mg}^{2 +}]$ is closely related to the modulation of $[{Mg}^{2 +}]$ on $k_{s y n}^{e f f}$ and accuracy. The relation between $k_{s y n}^{e f f}$ , accuracy, and $[{Mg}^{2 +}]$ was depicted as the three-dimensional curve (3D curve) shown in Fig. 1 c. When the 3D curve was projected onto the $k_{s y n}^{e f f}$ - $[{Mg}^{2 +}]$ plane, $k_{s y n}^{e f f}$ fell into “trade-off” and “competition” regimes at low (1∼3 mM) and high $[{Mg}^{2 +}]$ (3∼8 mM), respectively.

The “trade-off” regime exhibits the trade-off with increasing $k_{s y n}^{e f f}$ and reducing accuracy. When $[{Mg}^{2 +}]$ increased from 1 to 3 mM, $k_{s y n}^{e f f}$ increased and reached its maximum value. The increase of $k_{s y n}^{e f f}$ is attributed to $K_{M}^{e f f}$ , which showed a nearly threefold decrease (Fig. S4 b), while $k_{s y n}^{\max}$ remained almost constant within 1∼3 mM. The increase also relates to the available 6% of $R_{f r e e}$ (Fig. 1 b) serving as an extra surplus to associate with cognates and near-cognates. $R_{f r e e}$ can bind more frequently to both cognates and near-cognates because $k_{f}^{c}$ and $k_{f}^{n c}$ increase, but the majority of $R_{f r e e}$ associates with cognates and converts into $R_{P R^{c}}$ due to much faster $k_{f}^{c}$ . However, the accuracy experiences an almost sixfold reduction due to more ribosomes associating with near-cognates. The enhanced association of near-cognates closely relates to $k_{f}^{n c}$ that increases more than $k_{f}^{c}$ with increasing $[{Mg}^{2 +}]$ (Fig. S2 b; Table S1), leading to the trade-off curve as shown in the $k_{s y n}^{e f f}$ -accuracy plane.

The “competition” regime shows the positive correlation between $k_{s y n}^{e f f}$ and accuracy. $k_{s y n}^{e f f}$ reduced quickly because of less formation at $R_{P R^{c}}$ and more accumulation at $R_{P R^{n c}}$ . The reduction indicates the enhanced association of near-cognates for limited $R_{f r e e}$ . Near-cognates with increasing $k_{f}^{n c}$ can rapidly associate with free ribosomes to form $R_{P R^{n c}}$ with less rejection due to decreasing $k_{r e j}^{n c}$ (Table S1; Fig. S2, b and c). Consequently, the concentration of $R_{P R^{n c}}$ increases (Fig. 1 b) and accuracy reduces (Fig. 1 c). At the proofreading stage, sluggish peptide formation and little rejection increase the number of ribosomes binding to hydrolyzed near-cognates. The near-cognates-bound ribosomes are prohibited from being released to bind to cognates, causing drastic reduction of $k_{s y n}^{\max}$ and a modest decrease of $K_{M}^{e f f}$ (Fig. S4 b). The overall consequence is the positive correlation between $k_{s y n}^{e f f}$ and accuracy when $[{Mg}^{2 +}]$ increased from 3 to 8 mM as indicated in the $k_{s y n}^{e f f}$ -accuracy plane. The “competition” regime was also studied by single ribosome stochastic modelling (49). The optimal $k_{s y n}^{e f f}$ is located at the transition from “trade-off” to “competition” at $[{Mg}^{2 +}]$ of nearly 3 mM (Figs. 1 c and S4 a), which is within the measured range of $[{Mg}^{2 +}]$ inside E. coli (1∼5 mM) (16,36,37). Besides, the maximal speed can be maintained for $[{Mg}^{2 +}]$ within 1∼3 mM (supporting material section 2.3). Coincidentally, the in vitro globin synthesis by ribosomes in mice also showed a similar transition with our result (51) (supporting material section 3.5).

Two aspects of ribosomal kinetics can be further incorporated in GHM. Firstly, the in vivo protein synthesis shows that most of the errors fall into the error hotspots of some types of codons (e.g., mismatch of codons GGA and GAU to ${tRNA}_{U U C}^{G l u}$ with error frequencies of $3 \sim 5 \times 10^{- 4}$ (52)). The error frequencies are two orders of magnitude higher than other codons (52). The effects of codons on peptide bond formation kinetics and accuracy need to be taken into account in GHM for studying the variation of error frequency. Secondly, considering that high $[{Mg}^{2 +}]$ drastically slowed down the mRNA translocation rate (53), the translocation step can be incorporated into GHM by partitioning the total protein synthesis time into amino acid incorporation and translocation for better characterization of ribosomal protein synthesis kinetics.

The bacterial growth

Cell growth depends on cellular regulation of $[{Mg}^{2 +}]$ in response to the environment. Cellular regulation of $[{Mg}^{2 +}]$ has been observed under two conditions: regular (2 mM) and low (0.1 $μ M$ ∼1 mM) concentration of magnesium ions in the media. Under regular $[{Mg}^{2 +}]$ condition, B. subtilis cells modulated $[{Mg}^{2 +}]$ intake flux to cope with the antibiotic spectinomycin (54) and osmotic stress (55). Exposure to a sublethal amount of spectinomycin (2 mg/L) reduced protein synthesis speed and cell growth (54). Cells increased their growth rate by increasing $[{Mg}^{2 +}]$ intake flux so that ${Mg}^{2 +}$ can stabilize ribosomal structure and restore protein synthesis speed (54). Under osmotic stress induced by 1.2 M NaCl, cells restored growth rate by increasing $[{Mg}^{2 +}]$ intake to compensate for the loss of ${Mg}^{2 +}$ ions (supporting material section 3.5) (55). Under low $[{Mg}^{2 +}]$ , both S. typhimurium and E. coli cells slowed down protein synthesis and used regulatory pathways to import ${Mg}^{2 +}$ ions (supporting material section 3.5) (56,57). Cell growth can be enhanced by adding $[{Mg}^{2 +}]$ . For example, S. typhimurium wild-type cells had larger colonies on agar when $[{Mg}^{2 +}]$ in the agar increased from 40 to 500 $μ M$ (58). E. coli cells exhibited faster growth rate when extracellular $[{Mg}^{2 +}]$ increased within the range of 0.1 $μ M$ to 1 mM (59). In this work, we integrate $[{Mg}^{2 +}]$ -dependent ribosomal speed – accuracy relation into the cell growth process analyzed by BGM.

Accuracy effect on cell growth

Cellular modulation of $[{Mg}^{2 +}]$ is represented by the speed – accuracy relation and modulation factor $m$ . The effective protein synthesis speed $k_{s y n e f f}^{c e l l}$ couples with accuracy because $k_{s y n e f f}^{c e l l}$ is expressed in the Michaelis-Menten kinetics form consisting of $k_{s y n m a x}^{c e l l}$ and $K_{M}^{e f f}$ , as depicted in Fig. 2b. The coupling is also controlled by $[{Mg}^{2 +}]$ with each $[{Mg}^{2 +}]$ corresponding to a particular set of values of $k_{s y n e f f}^{c e l l}$ and accuracy. As $[{Mg}^{2 +}]$ increased, $k_{s y n e f f}^{c e l l}$ rose and fell while accuracy kept reducing as shown in Fig. 2 b. The $[{Mg}^{2 +}]$ -dependent protein synthesis speed – accuracy relation thereby suggests that cells can regulate $[{Mg}^{2 +}]$ to tune accuracy for the fastest speed and the maximum growth rate. The sigmoid function of accuracy with the maximum value of one is used in the accuracy function $m$ (Fig. S7 a) that reflects cellular regulatory network with the accuracy threshold below which cells cannot tolerate. Ribosomes with low accuracy produce nonfunctional proteins (6,60) and slow down growth rate (34,61). The functionality of some proteins is compromised by wrong amino acid incorporation (40). The probability of producing N proteins with at least T error-free proteins is a cumulative binomial distribution that behaves similarly to sigmoid function (62). The BGM also encodes the nutrient dependence into $m$ to reflect that better nutrient qualities lead to the enhancement of accuracy and protein quality (Fig. S7 a).

For a given nutrient quality, cells optimize growth rate by adjusting proteomic content and accuracy (14,32). In the BGM, this optimization is achieved by the following two steps (supporting material section 3.2). Firstly, the growth rate is maximized locally at a given accuracy by tuning ribosomal mass fraction (32) (Fig. S8). Secondly, the maximum values of locally optimized growth rates will be taken as the optimal growth rates. All the maximal growth rates are depicted in Fig. 2 c as open dots, and the optimal among them at a given nutrient condition as solid dots represented as $λ^{*}$ . As nutrient quality improves (e.g., increasing glucose concentration), cells can regulate $[{Mg}^{2 +}]$ intake fluxes for the optimal growth rates to occur within the typical intracellular $[{Mg}^{2 +}]$ of 1∼5 mM as observed in bacteria (16,36,37). Within this range, growth rate exhibits trade-off with accuracy as depicted in Fig. 2 c. Cells grow slowly with high accuracy under poor nutrient quality, while a rich nutrient condition enables cells to sacrifice accuracy for fast growth (14). Food quality thereby can trigger cells to regulate the $[{Mg}^{2 +}]$ intake flux to maximize growth rate in a similar manner to the aforementioned behaviors under stresses (e.g., antibiotics, osmotic difference, $[{Mg}^{2 +}]$ shortage).

Improvement in nutrient quality enables cells to achieve fast growth rate at the expense of accuracy, as shown in Fig. 2 c. Our analysis is consistent with (14), which also suggests that cells need to ensure the majority of proteins are functional to provide amino acids by maintaining slow and accurate protein synthesis when growth is hampered. When there is food abundance, cells divert their efforts to protein synthesis for rapid growth. By producing more ribosomes, cells need to ensure that all ribosomes work near their maximal speed $k_{s y n m a x}^{c e l l}$ by reducing the effect of proofreading to lower accuracy for fast growth (14). Besides, the BGM developed here can explain the positive correlation between growth rate and accuracy. Accuracy is considered at the cellular level comprising not only the performance of ribosomes but also cellular regulatory networks to maintain functional proteins. Therefore, the accuracy function depends on cellular responses toward external conditions. These conditions can perturb the relation between growth rate and accuracy, causing growth rate to positively correlate with accuracy. For example, mutations in elongation factor Tu caused a nearly fourfold decrease of accuracy and almost one-third reduction of growth rate (34). Raising the antibiotic streptomycin concentration from 2 to 10 $μ M$ increased the misincorporation of two amino acids nearly tenfold and decreased cell density almost threefold (63). Increasing $[{Mg}^{2 +}]$ from 4.5 to 14 mM reduced accuracy almost tenfold and decreased the growth rate sixfold when further raising $[{Mg}^{2 +}]$ to 1.25 M (38,39).

Cellular properties in the high accuracy regime

Under the normal growth conditions, the average accuracy of protein synthesis is around $3 \times 10^{3}$ (4,5), where growth rate and accuracy exhibit a trade-off scenario, as shown in Fig. 2 c. The majority of proteins are functional with minimal cellular control as modulated by the accuracy function. The values of $k_{s y n m a x}^{c e l l}$ are approximately constant across different optimal accuracy values (Fig. 2, b and c), as shown by the linear relation between ribosomal mass fraction and growth rate (inset of Fig. 2 d; Eq. S69).

Two quadratic relations emerge from the optimal growth rates $λ^{*}$ and the corresponding cellular properties. The first relation is the number of ribosomes per cell $R^{*}$ proportional to the square of growth rates $λ^{*}$ , i.e., $R^{*} \propto {(λ^{*})}^{2}$ . Fig. 2 d shows that the quadratic relation captures well the monotonous increasing trend of growth rate as a function of the number of ribosomes. The second relation is the nutrient intake flux following a quadratic relationship with cell mass, i.e., $J_{M}^{*} \propto {(M^{*})}^{2}$ . Both the theoretical and experimental fluxes were in good agreement with each other (1,30,42), as depicted in Fig. 2 e. The experimental protein synthesis fluxes are the product of the number of ribosomes per cell and protein synthesis speed (Table S4; supporting material section 3.1). Moreover, the theoretical values of $J_{M}^{*} - M^{*}$ relation had a slope of two in log-log scale, and the slope is similar to the slope of 1.96 in (43) showing the scaling between metabolic flux and cell mass across bacteria species in the right inset of Fig. 2 e. The nearly quadratic scaling was presumably related to the scaling between genome size and cell mass with the power of 0.35 (43). Our analysis suggests another reason can be the scaling between ribosomes and cell mass across bacteria species. The quadratic scaling $J_{M}^{*} \propto {(M^{*})}^{2}$ can also indicate the efficiency of utilizing nutrient intake flux for cell growth (supporting material section 3.4.2). The biomass conversion figure of merit Z, defined as $λ^{*} / J_{M}^{*}$ , reduced with growth rate in the region of moderate to fast growth (Fig. S7 b), while it increased with growth rate in slow growth region (Fig. S11 b).

The coefficient involved in these two quadratic scaling relations can be written as: $R^{*} = c_{F} {(λ^{*})}^{2} / k_{s y n m a x}^{c e l l}$ and $J_{M}^{*} = {(M^{*})}^{2} / c_{F}$ . (Eqs. S53 and S54, respectively). The first quadratic relation between ribosomes number and growth rate originates from the two linear relations: cell mass versus growth rate (Fig. S6 a and the left inset of Fig. 2 e) and ribosomal mass fraction versus growth rate (the inset of Fig. 2 d). Since nutrient intake flux equals the product of the protein synthesis speed and the number of ribosomes, this results in the second quadratic relation of nutrient intake flux versus cell mass. The quadratic scaling equations captured the experimental data well due to the common factor $c_{F}$ of $2.15 \times 10^{9}$ $(a a \cdot h)$ , as shown in Fig. S12. The common factor $c_{F}$ indicates the importance of ribosomes and connects various cellular properties: the number of ribosomes, growth rate, cell mass, protein synthesis speed and metabolic flux (1,30,42).

Conclusion

In summary, we investigated the kinetics of prokaryotic protein synthesis at the ribosomal level and coupled it to cell growth. Translation accuracy couples with the protein synthesis speed at different $[{Mg}^{2 +}]$ . The speed – accuracy relation of peptide bond formation sheds light on the mechanism of growth rate maximization via tuning translational accuracy and partitioning proteomic content. During the adaptation to the improvement in nutrient quality, cells can tune accuracy to optimize growth rate by regulating $[{Mg}^{2 +}]$ . Our results thus provide a theoretical framework to understand the mechanism of adaptive response of bacterial cell growth for optimal fitness.

Author contributions

H.S. conceived and supervised the work. L.Q.L. and K.Z. performed research. All authors wrote the manuscript.

Acknowledgments

The authors are grateful to Magnus Johansson, Måns Ehrenberg, Rudy Marcus, Daniela Rhodes, Ada Yonath, and Venki Ramakrishnan for enlightening discussions. H.B.S. is grateful to Weicheng Su and Kerson Huang for enlightening advice at the early stage of this work. We thank Zhihao Liu, Zhouyi He, Mengqi Jia, Chiming Kan, and Shanqi Yap for technique assistance. The authors would like to express their gratitude to the editor Jianhua Xing and anonymous reviewers for their careful reading of our manuscript as well as their insightful comments and suggestions. This work is supported in part by the Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) (SMSEGL20SC01) and HKUST grant (R9418).

Declaration of interests

The authors declare no competing financial interests.

Notes

Editor: Jianhua Xing.

Footnotes

Luan Quang Le and Kaicheng Zhu contributed equally to this work.

Supporting material can be found online at https://doi.org/10.1016/j.bpj.2022.12.028.

Supporting citations

References (64,65,66,67,68,69,70,71,72,73,74,75,76,77,78) appear in the supporting material.

Supporting material

Document S1. Figures S1–S13 and Tables S1–S6:

Click here to view.^{(1.4M, pdf)}

Document S2. Article plus supporting material:

Click here to view.^{(2.3M, pdf)}

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Articles from Biophysical Journal are provided here courtesy of The Biophysical Society

Bridging ribosomal synthesis to cell growth through the lens of kinetics

Luan Quang Le

Kaicheng Zhu

Haibin Su

Associated Data

Abstract

Significance

Introduction

Materials and methods

Protein synthesis from the GHM

Cell growth from the BGM

Results and discussion

The protein synthesis

[Mg2+] effect on protein synthesis

The bacterial growth

Accuracy effect on cell growth

Cellular properties in the high accuracy regime

Conclusion

Author contributions

Acknowledgments

Declaration of interests

Notes

Footnotes

Supporting citations

Supporting material

References

[ ${Mg}^{2 +}$ ] effect on protein synthesis