Spatial organization of bacterial transcription and translation

Significance

Protein synthesis is a mechanism of fundamental importance for life. The molecular components involved in this process are macromolecules called mRNAs and ribosomes. Here, we combine a theoretical and experimental analysis, and show that mRNAs congregate with ribosomes away from the nucleoid, namely at the cell poles. Overall, our model provides an understanding of the role of spatial localization in bacteria, providing potential insights on how such localization mechanisms may be useful to perform specific biological functions.

Abstract

In bacteria such as Escherichia coli, DNA is compacted into a nucleoid near the cell center, whereas ribosomes—molecular complexes that translate mRNAs into proteins—are mainly localized to the poles. We study the impact of this spatial organization using a minimal reaction–diffusion model for the cellular transcriptional–translational machinery. Although genome-wide mRNA-nucleoid segregation still lacks experimental validation, our model predicts that $\sim 90\text{\%}$ of mRNAs are segregated to the poles. In addition, our analysis reveals a “circulation” of ribosomes driven by the flux of mRNAs, from synthesis in the nucleoid to degradation at the poles. We show that our results are robust with respect to multiple, biologically relevant factors, such as mRNA degradation by RNase enzymes, different phases of the cell division cycle and growth rates, and the existence of nonspecific, transient interactions between ribosomes and mRNAs. Finally, we confirm that the observed nucleoid size stems from a balance between the forces that the chromosome and mRNAs exert on each other. This suggests a potential global feedback circuit in which gene expression feeds back on itself via nucleoid compaction.

The cytoplasm of many bacterial cells exhibits a striking spatial organization: rather than filling the entire cell volume, the DNA forms a condensed structure called a “nucleoid” that is generally localized near midcell (Fig. 1) (1, 2). Moreover, ribosomes—large molecular complexes that translate mRNAs into proteins—are observed to be antilocalized from the nucleoid (2). These observations raise two natural questions: (i) What physical processes are responsible for this subcellular organization? and (ii) How does this internal structure influence the basic processes of mRNA transcription and protein translation in the cell?

Fig. 1.

Schematic of the spatial organization of transcription and translation in E. coli. mRNAs (black solid curves) are transcribed in the nucleoid—the condensed DNA chromosome at the cell center (gray solid curve). Ribosomes (red circles) bind to mRNAs forming polysomes, i.e., mRNAs with multiple bound ribosomes. Polysomes diffuse preferentially out of the nucleoid due to excluded-volume effects. Eventually the mRNA molecules are degraded (black dashed lines). In our 1D model, the coordinate x runs along the long axis of the cell and, assuming symmetry, we model only the right half of the cell ($0\le x\le \ell )$.

In the model bacterium E. coli, $\sim 1.5\hspace{0.5em}\text{mm}$ of supercoiled DNA are compacted into an $\sim 1-\mu {\text{m}}^3$ nucleoid volume (3), thus forming a dense DNA mesh with average pore diameter $\sim 50\hspace{0.5em}\text{nm}$. As a result, free ribosomes, with diameter $\sim 20\hspace{0.5em}\text{nm}$, can readily diffuse into the nucleoid (4), whereas polysomes, molecular complexes composed of mRNAs with multiple bound ribosomes and having an effective diameter $\gtrsim 50\hspace{0.5em}\text{nm}$, antilocalize from the nucleoid due to excluded-volume effects. In vivo measurements of mRNA mobility suggest a typical diffusion coefficient of $D\sim 0.05\hspace{0.5em}{\text{μm}}^2/\text{s}$, implying that mRNAs formed in the nucleoid by transcription from DNA can diffuse out of the nucleoid to the ribosome-rich regions in a few seconds—a time significantly shorter than the typical mRNA lifetime of $\sim 5\hspace{0.5em}\text{min}$ (2, 5). These observations suggest that most mRNAs formed in the nucleoid diffuse out of the nucleoid to the ribosome-rich regions where ribosomes and mRNAs are colocalized, and where the bulk of translation occurs.

In recent years, the idea that mRNA localization may play a functional role in bacteria (6, 7) has inspired a variety of measurements of mRNA localization. These studies have provided evidence for multiple, mRNA-specific localization patterns, such as localization within the cell cytoplasm (8), to the cell membrane or to the cell poles (9), and at the nascent septum separating daughter cells (7). However, such experiments have proven to be challenging (10, 11) and limited to specific mRNAs (12, 13). As a result, genome-wide ribosome–mRNA colocalization still lacks experimental validation. In this study, we use known reaction–diffusion properties of mRNAs, ribosomes, and the nucleoid to predict the physical origin and the extent of overall, genome-wide mRNA localization.

Our approach also describes nonspecific, transient interactions between ribosomes and mRNAs. In this regard, recent studies in E. coli (2) and Caulobacter crescentus (14) observed an increase of the diffusion coefficient of nontranslating ribosomes under depletion of the pool of mRNAs, which has been interpreted as evidence for nonspecific, transient bindings between ribosomes and mRNA molecules (14). On the other hand, another study in E. coli reported that the diffusion coefficient of nontranslating ribosomes is not affected by mRNA depletion (4). Here, we show that if transient ribosome–mRNA interactions are significantly faster than other relevant timescales, the reaction–diffusion equations can be substantially simplified by treating such interactions as a local Poisson process. Finally, we consider the opposing forces exerted by the compressed nucleoid and by polarly localized mRNAs and confirm that the observed nucleoid size results from the balance of these dominant forces, and we present a simple analytical formula for the degree of nucleoid compaction under different physiological conditions, e.g., for different amounts of DNA and mRNA in the cell.

Results

We describe the coupled dynamics of ribosomes and mRNAs in an E. coli cell using a minimal, 1D reaction–diffusion model. We introduce a coordinate x running along the long axis of the cell and, given the approximate left–right symmetry of a typical E. coli cell, we consider only the right half, from $x=0$ at the cell center to $x=\ell$ at the right cell pole (Fig. 1). In vitro measurements of the assembly dynamics of the translation–initiation complex suggest that the in vivo binding rate of 30S ribosomal subunits to mRNAs is significantly larger than the unbinding rate, thus implying that the majority of mRNAs have a 30S subunit bound at the translation–initiation site (15). If so, translation is largely governed by the dynamics of 50S ribosomal subunits, and therefore we initially consider only the 50S units, which we refer to simply as “ribosomes.” We further assume that ribosomes may undergo transient, nonspecific binding to mRNAs; extended versions of our model including the two ribosomal subunits and disallowing nonspecific ribosome–mRNA interactions are presented later, and they confirm qualitatively the results obtained with the simple model discussed here; see SI Appendix, sections S10 and S14 for details. The 1D concentration of free (F) ribosomes, $c_{\text{F}}\left(x\right)$, denotes the number of F ribosomes per unit length in an infinitesimal slice of the cell perpendicular to the x axis. Similarly, ${\rho }_{m,n}\left(x\right)$ is the 1D concentration of mRNAs with m transiently bound (B) (14) ribosomes and n translating (T) ribosomes. As shown in SI Appendix, section S1, the average number of ribosomes per mRNA, $m+n\sim 12$, is well below the maximum total number of ribosomes, $m+n\sim 100$, that could be linearly packed onto a typical mRNA. Thus, we consider only mRNA species with $m\le m_{\text{max}}$, $n\le n_{\text{max}}$, where $m_{\text{max}}$ and $n_{\text{max}}$ are some maximal numbers of allowed ribosomes per mRNA, chosen large enough to account for all typical mRNA species present in the cell. Importantly, this choice reduces the number of mRNA species present in our model, thus making it computationally tractable.

The resulting reaction–diffusion equation for the F-ribosome concentration is

$$\begin{array}{r}\frac{\partial c_{\text{F}}\left(x,t\right)}{\partial t}=D_{\text{F}}\left[\frac{{\partial }^2{c}_{\text{F}}\left(x,t\right)}{\partial x^2}{v}_{\text{F}}\left(x\right)-c_{\text{F}}\left(x,t\right)\frac{d^2{v}_{\text{F}}\left(x\right)}{d{x}^2}\right]\\ -k_{\text{on}}^{\text{B}}{c}_{\text{F}}\left(x,t\right){\displaystyle \sum _m{\displaystyle \sum _n{\rho }_{m,n}\left(x,t\right)}}+k_{\text{off}}^{\text{B}}{\displaystyle \sum _m{\displaystyle \sum _{n}m\hspace{0.17em}{\rho }_{m,n}\left(x,t\right)}}\\ -k_{\text{on}}^{\text{T}}{c}_{\text{F}}\left(x,t\right){\displaystyle \sum _m{\displaystyle \sum _n{\rho }_{m,n}\left(x,t\right)}}+k_{\text{off}}^{\text{T}}{\displaystyle \sum _m{\displaystyle \sum _{n}n\hspace{0.17em}{\rho }_{m,n}\left(x,t\right)}}\\ +\beta {\displaystyle \sum _m{\displaystyle \sum _n\left(m+n\right)\hspace{0.17em}{\rho }_{m,n}\left(x,t\right),}}\end{array}$$ [1]

where only mRNA species with allowed values of $0\le m\le m_{\text{max}}$ and $0\le n\le n_{\text{max}}$ are considered. The first term on the right-hand side (RHS) represents diffusion including excluded-volume effects due to the condensed DNA (Fig. 1). In this term, $D_{\text{F}}$ is the diffusion coefficient for F ribosomes, which incorporates crowding effects due to ribosomes, mRNAs, and other macromolecules, whereas $v_{\text{F}}\left(x\right)$ is the fractional volume available to an F ribosome within the DNA mesh at position x (SI Appendix, Fig. S1); see SI Appendix, sections S2 and S4 for details. The first term in the second line represents F ribosomes binding to all possible mRNA species with m B ribosomes and n T ribosomes, and thus becoming B ribosomes. This term is proportional to the $\text{F}\to \text{B}$ transition rate $k_{\text{on}}^{\text{B}}$ and to the total density of mRNA. In principle, $k_{\text{on}}^{\text{B}}$ should decrease with the ribosome occupancy number $m+n$: however, here $m+n$ is much smaller than the maximum packing density, thus this effect is small (see above). The next term in the second line describes a B ribosome unbinding from an mRNA, where $k_{\text{off}}^{\text{B}}$ denotes the unbinding rate and the multiplicity factor m accounts for multiple B ribosomes on the mRNA. Similarly, the third line represents transitions between F and T ribosomes, where $k_{\text{on}}^{\text{T}}$ is assumed independent of m, n (see above). Because measurements suggest that the lifetime of the B state is significantly shorter than that of the T state (14), here the $\text{F}\to \text{B}\to \text{T}$ transition is incorporated into the $\text{F}\to \text{T}$ transition, with an effective rate $k_{\text{on}}^{\text{T}}$. Finally, the last line represents B and T ribosomes being freed from mRNA molecules as these are degraded at rate β (16).

The 1D reaction–diffusion equations for the mRNA densities are

$$\begin{array}{r}\frac{\partial {\rho }_{m,n}\left(x,t\right)}{\partial t}=\\ D\left[\frac{{\partial }^2{\rho }_{m,n}\left(x,t\right)}{\partial x^2}{v}_{m+n}\left(x\right)-{\rho }_{m,n}\left(x,t\right)\frac{d^2{v}_{m+n}\left(x\right)}{d{x}^2}\right]\\ -k_{\text{on}}^{\text{B}}{c}_{\text{F}}\left(x,t\right){\rho }_{m,n}\left(x,t\right)-k_{\text{off}}^{\text{B}}\hspace{0.17em}m\hspace{0.17em}{\rho }_{m,n}\left(x,t\right)\\ -k_{\text{on}}^{\text{T}}{c}_{\text{F}}\left(x,t\right){\rho }_{m,n}\left(x,t\right)-k_{\text{off}}^{\text{T}}\hspace{0.17em}n\hspace{0.17em}{\rho }_{m,n}\left(x,t\right)\\ +\hspace{0.17em}{k}_{\text{on}}^{\text{B}}{c}_{\text{F}}\left(x,t\right){\rho }_{m-1,n}\left(x,t\right)+k_{\text{off}}^{\text{B}}\left(m+1\right)\hspace{0.17em}{\rho }_{m+1,n}\left(x,t\right)\\ +\hspace{0.17em}{k}_{\text{on}}^{\text{T}}{c}_{\text{F}}\left(x,t\right){\rho }_{m,n-1}\left(x,t\right)+k_{\text{off}}^{\text{T}}\left(n+1\right)\hspace{0.17em}{\rho }_{m,n+1}\left(x,t\right)\\ +{\delta }_{m,0}\hspace{0.17em}{\delta }_{n,0}\hspace{0.17em}\alpha \left(x\right)-\beta \hspace{0.17em}{\rho }_{m,n}\left(x,t\right).\end{array}$$ [2]

Here D is the average mRNA diffusion coefficient in the cytoplasm, and $v_{m+n}\left(x\right)$ is the fractional available volume within the nucleoid for an mRNA with $m+n$ attached ribosomes (SI Appendix, Fig. S1); see SI Appendix, section S4 for details. The second and third lines represent binding and unbinding of B and T ribosomes from mRNAs of species m, n, whereas the fourth line represents B ribosomes binding to mRNAs of species $m-1$, n or unbinding from mRNAs of species $m+1$, n. Similarly, the terms in the fifth line represent T ribosomes binding and unbinding from mRNAs of species $m,n-1$ and $m,n+1$. Finally, the last line represents transcription of initially ribosome-free mRNAs according to the nucleoid profile $\alpha \left(x\right)$, and mRNA degradation at rate β.

As transcriptional and translational timescales ($\lesssim 1\hspace{0.5em}\min$) are fast compared with cell doubling times ($\gtrsim 20\hspace{0.5em}\text{min}$), we focus on steady-state conditions. At steady state, in Eq. 1 we enforce a constraint on the total number of ribosomes, $2{\displaystyle {\int }_0^{\ell }dx\left[c_{\text{F}}\left(x\right)+{\displaystyle {\sum }_{m=0}^{m_{\text{max}}}{\displaystyle {\sum }_{n=0}^{n_{\text{max}}}\left(m+n\right){\rho }_{m,n}\left(x\right)}}\right]}=N_{\text{tot}}$, and we set a no-flux boundary condition at the cell pole, ${\left[d{c}_{\text{F}}\left(x\right)/dx\hspace{0.17em}{v}_{\text{F}}\left(x\right)-c_{\text{F}}\left(x\right)d{v}_{\text{F}}\left(x\right)/dx\right]|}_{x=l}=0$ (SI Appendix, section S2). Similarly, in Eq. 2 we impose no-flux boundary conditions at the cell pole and at the cell center, ${\left[d{\rho }_{m,n}\left(x\right)/dx\hspace{0.17em}{v}_{m+n}\left(x\right)-{\rho }_{m,n}\left(x\right)d{v}_{m+n}\left(x\right)/dx\right]|}_{x=0,l}=0$, the latter reflecting the left–right symmetry of the cell. According to this symmetry, the flux of F ribosomes at midcell must also vanish, and this follows directly from the boundary conditions above (SI Appendix, section S5).

We fix the model parameters from experimental data as follows. We considered E. coli cells in glucose minimal media with an $\sim 1/\text{h}\hspace{0.5em}$ growth rate (SI Appendix, section S3), and focused on the midphase of the division cycle by selecting cells with length within $5\mathrm{\%}$ of a typical, medium length of $2\hspace{0.17em}\ell =3\hspace{0.17em}\mu \text{m}$; compare Fig. 2 (Left Inset). We then rescaled the resulting DNA fluorescence profiles to a cell length of $2\hspace{0.17em}\ell$, and we estimated the nucleoid profile along the long cell axis by averaging over multiple cells; see the main panel in Fig. 2 and SI Appendix, section S4. The F-ribosome diffusion coefficient was taken to be $D_{\text{F}}=0.4\hspace{0.5em}{\text{μm}}^2/\text{s}$, whereas the diffusion coefficient of mRNAs was set at the average diffusion coefficient of polysomes, $D=0.05\hspace{0.5em}{\text{μm}}^2/\text{s}$ (2, 4). These diffusion coefficients were obtained from measurements of mean-square displacements of free and mRNA-bound ribosomal subunits in living E. coli cells (2), and thus include the effects of cytoplasmic crowding. The F-ribosome available volume $v_{\text{F}}\left(x\right)$ was estimated from the concentration profiles of free ribosomal subunits (4), and similarly for the mRNA available-volume profiles $v_{m+n}\left(x\right)$ (SI Appendix, section S4). The mRNA degradation rate was taken to be $\beta =3\times {10}^{-3}/\text{s}$, corresponding to a mean mRNA lifetime of $\sim 5\hspace{0.5em}\text{min}$ (5). The total mRNA production rate ${\alpha }_{\text{tot}}=2{\displaystyle {\int }_0^{\ell }dx\hspace{0.17em}\alpha \left(x\right)}$ was obtained from the total number of mRNAs per cell via the relation ${\alpha }_{\text{tot}}/\beta =\hspace{0.17em}{N}_{\text{mRNA}}=5\times {10}^3$ (17), whereas the profile of mRNA production $\alpha \left(x\right)$ was chosen to be proportional to the DNA density $\mathit{\phi }\left(x\right)$; see SI Appendix, section S4 for details. The average time for a ribosome to complete protein translation is estimated to be ${\tau }_{\text{T}}=40\hspace{0.17em}\text{s}$ (18), so we took the $\text{T}\to \text{F}$ transition rate to be $k_{\text{off}}^{\text{T}}=1/{\tau }_{\text{T}}=2.5\times {10}^{-2}/\text{s}$. To set the other transition rates we used the observation that $\sim 80\mathrm{\%}$ of ribosomes are T ribosomes (2), with non-T ribosomes estimated to spend $\sim 90\mathrm{\%}$ of their time as B ribosomes and $\sim 10\mathrm{\%}$ of their time as F ribosomes, where the $90\mathrm{\%}:10\mathrm{\%}$ division of non-T ribosomes between B and F ribosomes is inferred from ribosome diffusion rates in C. crescentus (14). Global equilibrium between T and F ribosomes at steady state requires the $\text{F}\to \text{T}$ transition rate $k_{\text{on}}^{\text{T}}$ to satisfy $N_{\text{F}}\hspace{0.17em}\overline{\rho }\hspace{0.17em}{k}_{\text{on}}^{\text{T}}=N_{\text{T}}\hspace{0.17em}{k}_{\text{off}}^{\text{T}}$, where $N_{\text{F}}$ and $N_{\text{T}}$ are the total number of F and T ribosomes, respectively, and $\overline{\rho }\simeq 1.7\times {10}^3/\text{μm}$ is the average total mRNA axial density (17). Also, in the equilibrium condition above we neglected the rate $\beta N_{\text{T}}$ at which mRNA-bound T ribosomes are freed by mRNAs that are being degraded, because this rate is much smaller than $N_{\text{T}}{k}_{\text{off}}^{\text{T}}$. According to the estimate above for the average number of F, T, and B ribosomes, we have $N_{\text{T}}/N_{\text{F}}=80\mathrm{\%}/\left(20\mathrm{\%}\times 10\mathrm{\%}\right)$, which yields $k_{\text{on}}^{\text{T}}=6\times {10}^{-4}\text{μm}/\text{s}$. The total number of ribosome in the cell was taken to be $N_{\text{tot}}=6\times {10}^4$ (2). As for the binding–unbinding rates $k_{\text{on}}^{\text{B}}$, $k_{\text{off}}^{\text{B}}$ of the $\text{F}\leftrightarrow \text{B}$ transition, the equilibrium condition reads $N_{\text{F}}\hspace{0.17em}\overline{\rho }\hspace{0.17em}{k}_{\text{on}}^{\text{B}}=N_{\text{B}}\hspace{0.17em}{k}_{\text{off}}^{\text{B}}$ where, because the $\text{F}\leftrightarrow \text{B}$ transition occurs on timescales not longer than $\sim 1\hspace{0.5em}\text{s}$ (14), we neglected the mRNA-decay term $\beta N_{\text{B}}\ll N_{\text{B}}\hspace{0.17em}{k}_{\text{off}}^{\text{B}}$. Together with the estimation above for the average number of F, T, and B ribosomes $N_{\text{F}}/N_{\text{B}}=2\mathrm{\%}/18\mathrm{\%}$, this equilibrium condition provides an estimate for the ratio $k_{\text{on}}^{\text{B}}/k_{\text{off}}^{\text{B}}\simeq 5.4\times {10}^{-3}\hspace{0.5em}\text{μm}$, but it does not specify the individual values of $k_{\text{on}}^{\text{B}}$, $k_{\text{off}}^{\text{B}}$. However, because the timescale of the $\text{F}\leftrightarrow \text{B}$ transition is significantly shorter than other relevant timescales (14), these processes can be treated in the limit where they are at rapid equilibrium. In this limit, the problem can be significantly simplified, and the set of $1+\left(m_{\text{max}}+1\right)\left(n_{\text{max}}+1\right)$ reaction–diffusion equations 1 and 2 reduces to a set of $n_{\text{max}}+2$ equations which completely characterize the solution $c_{\text{F}}\left(x\right)$ and ${\rho }_{m,n}\left(x\right)$ for any m and $0\le n\le n_{\text{max}}$. Importantly, these rapid-equilibrium equations depend on $k_{\text{on}}^{\text{B}}$, $k_{\text{off}}^{\text{B}}$ only through their ratio; see SI Appendix, section S6 for details.

Fig. 2.

DNA fluorescence along the long cell axis for $3-\text{μm}$-long E. coli cells grown in glucose minimal media. Cells were stained with SYTOX Orange and imaged at exponential phase. Fluorescence for a few representative cells (gray), and resulting average fluorescence over 35 cells with SEM (red), both symmetrized and normalized to unit area. (Left Inset) Nucleoid vs. cell length, where the gray area includes cells with length within $5\mathrm{\%}$ of $3\hspace{0.5em}\text{μm}$. (Right Inset) Ribosomal protein S2-YFP (green) and nucleoid (red). (Scale bar: $1\hspace{0.5em}\text{μm}$.)

We numerically solved Eqs. 1 and 2 at steady state in the rapid-equilibrium limit for B ribosomes, by fixing the maximal number of allowed T ribosomes per mRNA at $n_{\text{max}}=24$; see SI Appendix, section S6 for details. The resulting mRNA profiles and ribosome concentrations are shown in Figs. 3 and 4, respectively. As shown in Fig. 3, the total mRNA profile ${\rho }_{\text{tot}}\left(x\right)={\displaystyle {\sum }_{m=0}^{\infty }{\displaystyle {\sum }_{n=0}^{n_{\text{max}}}{\rho }_{m,n}\left(x\right)}}$ is markedly localized outside the nucleoid region—most of the mRNAs are segregated from the nucleoid because of excluded volume. Indeed, the density profiles ${\rho }_{m,n}\left(x\right)$ show that the more ribosomes an mRNA has bound, the more segregated the mRNA is from the nucleoid; see SI Appendix, Fig. S2 for details. Because mRNAs are generated at the nucleoid, the strong segregation of mRNAs away from the nucleoid at steady state must result from the majority of mRNAs being loaded with ribosomes so that excluded volume biases their diffusion away from the nucleoid. This conclusion is confirmed in Fig. 3 (Inset), which shows a heat map of the total number $N_{m,n}={\displaystyle {\int }_0^{\ell }dx{\rho }_{m,n}\left(x\right)}$ of mRNAs of species $m,n$. Most mRNAs are loaded with $\sim 10$ T ribosomes and $\sim 2$ B ribosomes. These average loading numbers are in agreement with the experimental estimates above for the ribosome numbers (2, 17): The number of T ribosomes per mRNA is $N_{\text{T}}/N_{\text{mRNA}}\sim \left(80\text{\%}\hspace{0.17em}{N}_{\text{tot}}\right)/N_{\text{mRNA}}\sim 10$, and a similar estimate yields $\sim 2$ B ribosomes per mRNA. In addition, Fig. 3 (Inset) shows that the chosen value $n_{\text{max}}=24$ is large enough to encompass all typical mRNA species that are present.

Fig. 3.

Steady-state mRNA and polysome distributions. Total mRNA density ${\rho }_{\text{tot}}\left(x\right)$ (red) and density ${\rho }_{m,n}\left(x\right)$ of mRNAs with m transiently bound (B) ribosomes and n translating (T) ribosomes for $0\le m\le m_{\text{max}}=8$ and $0\le n\le n_{\text{max}}=24$ (gray). The density ${\rho }_{\mathrm{0,0}}\left(x\right)$ of ribosome-free mRNAs (green) and the density ${\rho }_{\mathrm{8,24}}\left(x\right)$ of mRNAs with the largest number of T and B ribosomes considered (blue) are also shown. The profiles ${\rho }_{m,n}\left(x\right)$, ${\rho }_{\mathrm{0,0}}\left(x\right)$, and ${\rho }_{\mathrm{8,24}}\left(x\right)$ are normalized to unit area. (Inset) Distribution of mRNA species, shown as a heat map of the number $N_{m,n}$ of mRNAs with m B ribosomes and n T ribosomes in the right half of the cell. The maximal number of T ribosomes per mRNA used in our model, $n_{\text{max}}=24$, is indicated.

Fig. 4.

Steady-state ribosome concentrations and ribosome fluxes. (Top) Concentrations $c_{\text{T}}\left(x\right)$, $c_{\text{B}}\left(x\right)$, and $c_{\text{F}}\left(x\right)$ of translating (T), transiently bound (B), and free (F) ribosomes in the right half of the cell (compare Fig. 1). (Bottom) Fluxes of T, B, and F ribosomes along the cell’s long axis depicted in the top panel. The arrow length is proportional to local ribosome flux, and the arrows in the legends correspond to a flux of $20/\text{s}$.

Because each ribosome has a linear size of $\sim 20\hspace{0.17em}\text{nm}$ (19), the effective size of an mRNA molecule with $\sim 10$ bound ribosomes is significantly larger than the pore size of the DNA mesh in the nucleoid, which we estimate to be $\sim 50\hspace{0.17em}\text{nm}$; see SI Appendix, section S4 for details. Thus, the majority of mRNAs experience strong excluded-volume effects which push them out of the nucleoid region.

Because mRNAs are created by transcription in the nucleoid but end up segregated away from the nucleoid, there must be a flux of mRNAs toward the cell poles. Given that new mRNAs are rapidly loaded with T ribosomes at a rate $k_{\text{on}}^{\text{T}}{N}_{\text{F}}/\left(2\hspace{0.17em}\ell \right)=6\times {10}^{-4}\text{μm}/\text{s}\times \mathrm{1,200}/3\hspace{0.5em}\text{μm}\simeq 0.24/\text{s}$, implying full occupation by $\sim 10$ T ribosomes in $\sim 3\hspace{0.5em}\text{s}$, the poleward flux of mRNAs carries with it a poleward flux of ribosomes. Because ribosomes are conserved in our model, reflecting the long half-life of ribosomal components (20), there must be a compensating flux of F ribosomes from the poles toward the nucleoid. In Fig. 4 we show the steady-state concentration of F ribosomes $c_{\text{F}}\left(x\right)$, the concentrations of T and B ribosomes, $c_{\text{T}}\left(x\right)={\displaystyle {\sum }_{m=0}^{\infty }{\displaystyle {\sum }_{n=1}^{n_{\text{max}}}n\hspace{0.17em}{\rho }_{m,n}\left(x\right)}}$, $c_{\text{B}}\left(x\right)={\displaystyle {\sum }_{m=1}^{\infty }{\displaystyle {\sum }_{n=0}^{n_{\text{max}}}m\hspace{0.17em}{\rho }_{m,n}\left(x\right)}}$, the flux $J_{\text{F}}=-D_{\text{F}}\left[d{c}_{\text{F}}\left(x\right)/dx\hspace{0.17em}{v}_{\text{F}}\left(x\right)-c_{\text{F}}\left(x\right)d{v}_{\text{F}}\left(x\right)/dx\right]$ of F ribosomes, and the fluxes $J_{\text{T}}$ and $J_{\text{B}}$ of T and B ribosomes; compare SI Appendix, Eqs. S45 and S46. As expected from the observed segregation of mRNAs, the T and B ribosomes are markedly excluded from the nucleoid, and there is a net poleward flux of T and B ribosomes. Notably, the effect of excluded volume in the nucleoid is so strong that mRNAs and their associated ribosomes flow from a low- to a high-concentration region. By contrast, F ribosomes are small enough to freely penetrate the nucleoid, and a flux of F ribosomes is driven by the gradient of these ribosomes from the poles to the nucleoid. Overall, these results illustrate and quantify a “circular” flux for the ensemble of T, B, and F ribosomes; compare Fig. 1: First, multiple F ribosomes bind to mRNAs made in the cell nucleoid. Each mRNA is thus loaded with $\sim 10$ T ribosomes, and $\sim 2$ B ribosomes to become a polysome. Second, the effects of excluded volume in the nucleoid result in a net flow of these polysomes to the cell poles. Once polysomes reach the poles, they ultimately decay and free their ribosomes. This “pumping” of T and B ribosomes from the nucleoid to the poles results in an excess of F ribosomes at the poles, and thus in a diffusive return flux of F ribosomes to the nucleoid.

The existence of a steady ribosome circulation implies that there must be an external source of energy driving these circular fluxes. There are two possible candidates within our model: Process (A) is the nonequilibrium creation and degradation of mRNAs, and process (B) is mRNA and F-ribosome binding in the nucleoid and subsequent expulsion from the nucleoid by excluded-volume effects. Process (A) should be strictly dependent on new mRNA production, whereas process (B) should persist even in the limit where the mRNA production and degradation rates are both low, with the total number of mRNAs fixed and equal to $N_{\text{mRNA}}$. Therefore, we varied the mRNA production and degradation rates together, keeping the total mRNA number constant: The circulation vanished as the mRNA rates slowed (SI Appendix, Fig. S3), thus identifying process (A), the flux of new mRNAs from nucleoid to pole, as the driver of ribosome circulation. This conclusion is confirmed by an analytical estimate for the poleward flux of T and B ribosomes, which is shown to be proportional to the mRNA production rate; see SI Appendix, section S8 for details.

Before discussing other implications of our results, it is worth considering that mRNA transcription takes a finite amount of time. For the average mRNA length of $\sim 3\times {10}^3\hspace{0.17em}\text{nt}$ discussed in SI Appendix, section S1 and an average mRNA elongation speed of $\sim 50\hspace{0.5em}\text{nt}/\text{s}$ (21), we obtain a typical transcription time of $\sim 1\hspace{0.5em}\text{min}$, during which nascent mRNAs are bound to DNA while being elongated. We therefore extended our model to include these nucleoid-bound mRNAs, whose axial densities we denote by ${\rho }_{m,n}^{\text{∗}}\left(x\right)$: These mRNAs turn into free mRNAs at a rate $\gamma =1/\left(1\hspace{0.5em}\text{min}\right)$; see SI Appendix, section S9 for details. Besides confirming the picture in the model with one mRNA species, this extended model gives insights into the mechanism of cotranscriptional translation, namely the observation that ribosomes translate nascent mRNAs while these are being transcribed in the nucleoid (22). Whereas it has been previously hypothesized that most of the ribosomes in the dense nucleoid region translate cotranscriptionally (2), our analysis shows that only $\sim 34\mathrm{\%}$ of these ribosomes carry out cotranscriptional translation, whereas a comparable fraction of $\sim 37\mathrm{\%}$ translates posttranscriptionally, mostly on polysomes loaded with a relatively small number of ribosomes. The extended model also allows us to address the effects of cotranscriptional translation on the protein-synthesis rate: Introducing the efficiency $\epsilon =2\hspace{0.17em}{k}_{\text{off}}^{\text{T}}{\displaystyle {\sum }_{m=0}^{\infty }{\displaystyle {\sum }_{n=1}^{n_{\text{max}}}n{\displaystyle {\int }_0^{\ell }dx{\rho }_{m,n}\left(x\right)/N_{\text{tot}}}}}$, i.e., the average number of proteins translated per unit time per ribosome, we find that cotranscriptional translation implies an $\sim 3\mathrm{\%}$ increase in ribosome efficiency, under the assumption of B-ribosome binding to all transcripts; see SI Appendix, section S9 for details.

We then extended our model to take account of both ribosomal subunits. During translation initiation, a 30S subunit binds to the mRNA initiation site first, and then a 50S subunit binds to the 30S subunit forming a translating 30S–50S (70S) pair (23). To model this process and the spatiotemporal dynamics of the two ribosomal subunits, we introduce mRNA species with l B 30S subunits, m B 50S subunits, n 70S pairs, and no 30S subunit at the initiation site, and denote their densities by ${\rho }_{l,m,n}\left(x,t\right)$. Similarly, we denote by ${\rho \prime }_{l,m,n}\left(x,t\right)$ the density of mRNAs with l B 30S subunits, m B 50S subunits, n 70S pairs, and a 30S subunit at the initiation site. Assuming rapid equilibrium for B 30S and 50S subunits, we solved the reaction–diffusion equations for the mRNA densities ${\rho }_{l,m,n}\left(x\right)$ and ${\rho \prime }_{l,m,n}\left(x\right)$, and for the concentrations $c_{30\text{S}\hspace{0.17em}\text{F}}\left(x,t\right)$ and $c_{50\text{S}\hspace{0.17em}\text{F}}\left(x,t\right)$ of free 30S and 50S subunits; see SI Appendix, section S10 for details. The results shown in Fig. 5 and SI Appendix, Figs. S7 and S18 confirm the picture obtained with the simpler one-subunit model. In particular, the mRNAs, both with and without a 30S subunit on the translation–initiation site, are strongly segregated from the nucleoid, and the larger l, m, n, the stronger the segregation. In Fig. 5 we show the concentrations and fluxes of the ribosomal subunits: the 70S subunits, the B 30S and 50S subunits, and the 30S subunits bound to translation–initiation sites are all strongly segregated from the nucleoid (compare Fig. 4), whereas F 30S and 50S subunits are only slightly excluded from the nucleoid. Note that the concentrations of F and B 30S subunits are lower than the corresponding concentrations of F and B 50S subunits because for our choice of parameters the majority of nontranslating 30S subunits are bound to translation–initiation sites, thus reducing the number of F and B 30S subunits. The fluxes of 70S, B 30S, and 50S subunits, and of 30S subunits bound to translation–initiation sites are directed toward the cell poles, whereas the compensating fluxes of F 30S and 50S subunits are directed toward the cell center.

Fig. 5.

Steady-state ribosomal-subunit concentrations and fluxes for the model including 30S and 50S ribosomal subunits. (Top) Concentrations $c_{70\text{S}}\left(x\right)$, $c_{30\text{S}\hspace{0.17em}\text{B}}\left(x\right)$, $c_{30\text{S}\hspace{0.17em}\text{F}}\left(x\right)$, and ${c\prime }_{30\text{S}}\left(x\right)$ of 70S, transiently bound (B) 30S, free (F) 30S subunits, and of 30S subunits bound to the translation–initiation site, respectively. We also show the concentrations $c_{50\text{S}\hspace{0.17em}\text{B}}\left(x\right)$ and $c_{50\text{S}\hspace{0.17em}\text{F}}\left(x\right)$ of B and F 50S subunits, respectively. (Bottom) Fluxes $J_{70\text{S}}\left(x\right)$, $J_{30\text{S}\hspace{0.17em}\text{B}}\left(x\right)$, and $J_{30\text{S}\hspace{0.17em}\text{F}}\left(x\right)$ of 70S, B and F 30S subunits, and flux ${J\prime }_{30\text{S}}\left(x\right)$ of 30S subunits bound to the translation–initiation site. The fluxes $J_{50\text{S}\hspace{0.17em}\text{B}}\left(x\right)$ and $J_{50\text{S}\hspace{0.17em}\text{F}}\left(x\right)$ of B and F 50S subunits are also shown. Fluxes are represented along the cell’s long axis depicted in the top panel, the arrow length is proportional to local ribosome flux, and the arrows in the legends correspond to a flux of $30/\text{s}$.

The two-subunit model was then extended to include additional, biologically relevant features: In SI Appendix, section S11 we incorporated in the model the mechanism of mRNA degradation by RNase enzymes (16, 24) (SI Appendix, Figs. S9–S12), and in SI Appendix, section S12 we extended the model to cells in the late phase of their division cycle (SI Appendix, Figs. S14–S16). We then extended the analysis above for glucose minimal media to different growth rates: We imaged E. coli cells in glycerol minimal and defined rich media, with growth rates $\sim 0.5/\text{h}\hspace{0.5em}$ and $\sim 2/\text{h}\hspace{0.5em}$, respectively, extracted the nucleoid profiles, and present the resulting model predictions in SI Appendix, section S13; compare SI Appendix, Figs. S17–S20. Finally, as the existence of B ribosomes is an open question (2, 4, 14), in SI Appendix, section S14 we considered a version of the model with two ribosomal subunits and no B subunits. Overall, these results confirm all of the qualitative features of the simpler, two-subunit model discussed above.

So far, our analysis has shown that the excluded volume due to DNA localization at midcell segregates the majority of mRNAs to the cell poles. In what follows, we will extend this analysis and show that the converse is also true, i.e., mRNA segregation to the poles causes nucleoid compaction at midcell. Specifically, in SI Appendix, section S15 we show that mRNAs, like particles in a gas, exert an entropic force (pressure) on the nucleoid directed toward the cell center, and that this force can be computed directly from the reaction–diffusion equations 1 and 2. On the other hand, the natural tendency of the compressed DNA polymer to increase its configurational entropy results in an effective “spring” force pushing outward on the mRNAs. Exploiting the condition that these two forces must balance at mechanical equilibrium, we self-consistently determined the mRNA and F ribosome profiles, as well as the DNA density profile $\phi \left(x\right)$, and the results are shown in SI Appendix, Figs. S23 and S24. In particular, denoting by $2x_0$ the width of the nucleoid region centered at midcell, the resulting value of $2x_0\approx 1.43\hspace{0.5em}\text{μm}$ matches well the experimentally observed nucleoid size in Fig. 2. Finally, a simplified version of our analysis provides a straightforward prediction for the nucleoid width under different physiological conditions: We assume that mRNAs are uniformly distributed outside the nucleoid, which they cannot penetrate, that the nucleoid is confined in a region of width $2x_0$ centered at midcell, and we neglect the small force exerted by F ribosomes on the nucleoid (see SI Appendix, section S15 for details). As a result, the nucleoid size $x_0$ can be determined by solving the following force–balance equation:

$$\frac{N_{\text{mRNA}}}{2\left(\ell -x_0\right)}=\frac{{\pi }^2}{6}\frac{{\xi }^2{N}^{2\nu }}{{\left(2\pi R^2\right)}^{2/3}{x}_0^{5/3}},$$ [3]

where $\xi =200\hspace{0.5em}\text{nm}$ is twice the estimated persistence length of a segment of supercoiled DNA, $N=7.5\times {10}^3$ is the total number of such segments in the chromosome (25), and $R=0.5\hspace{0.5em}\text{μm}$ is the radius of a circular cell slice. In SI Appendix, Fig. S25 we show the predicted nucleoid size for different values of the total number of mRNAs—the larger the number of mRNAs, the more the nucleoid shrinks toward midcell due to the entropic force exerted by the polysomes.

Discussion

The study of intracellular mRNA localization has attracted growing interest in recent years (10–13). In eukaryotic systems, mRNA localization is a well-established mechanism for achieving a variety of functions, such as the targeted expression of proteins to specific regions of the cell, the control of intracellular signaling, or the partition of mRNAs into daughter cells for cell-fate differentiation (26, 27). Some functional, mRNA-specific localization patterns have also been observed in bacteria (6): Two examples are the membrane localization of mRNAs which code for proteins transporting lactose into the cell (9) and mRNA localization at the cell poles, which has been shown to play a functional role in controlling sugar-utilization genes (9, 28).

In this study, we analyze the extent of bacterial genome-wide mRNA localization by means of a minimal reaction–diffusion model for the transcriptional and translational machinery in E. coli. The experimental observation that ribosomes in E. coli are markedly localized to the cell poles, and thus segregated from the nucleoid located at the cell center (2), suggests that the majority of mRNAs are also likely to be segregated from the nucleoid. Whereas experiments on mRNA localization in E. coli have proven to be challenging (10, 11), and are limited to specific mRNAs (12, 13), our model makes a prediction for strong, genome-wide mRNA localization away from the nucleoid, indicating that $\sim 90\mathrm{\%}$ of mRNAs are typically located outside the nucleoid, and demonstrating that the total mRNA profile resembles that of translating ribosomes (compare Figs. 3 and 4). A specific prediction of our model is that mRNA segregation is due to excluded-volume effects resulting from the condensed nucleoid DNA. Overall, this result provides insights into the mechanisms of mRNA segregation: Whereas other studies for specific mRNAs raised the possibility that mRNA segregation may be associated with dynamical rearrangements of the nucleoid (29), our analysis indicates that genome-wide mRNA segregation can arise entirely from excluded-volume effects. Also, our result that segregation primarily affects mRNAs loaded with multiple ribosomes is in line with the recent experimental observation that mRNAs with multiple bound fluorescent proteins localize to the cell poles in live E. coli cells (10).

Our model also reveals a “circulation” of ribosomes within the cell driven by the flux of newly synthesized mRNAs from the nucleoid to the poles: mRNA-bound ribosomes flow from the nucleoid to the cell poles, where they unbind from mRNAs and then diffuse back to the nucleoid to bind newly synthesized mRNA molecules. Using our model, we also analyze the extent of cotranscriptional translation, namely the observation that ribosomes translate mRNAs that are being transcribed in the nucleoid (22). Although it has been recently hypothesized that most of the ribosomes in the DNA-rich region translate cotranscriptionally (2), we find that only about one-third of these ribosomes carry out cotranscriptional translation, whereas a slightly higher fraction translates posttranscriptionally in polysomes with a relatively small loading number.

We incorporated in our model the mechanism of mRNA degradation by RNase enzymes, different phases of the cell division cycle, different growth rates, and the effect of nonspecific, transient interactions between ribosomes and mRNAs (2, 4, 14), showing that our results are stable with respect to such variety of conditions.

Finally, we extended our analysis to study the consequences of mRNA localization on nucleoid compaction. Using our calculated ribosome and mRNA densities, we confirmed that mRNA segregation to the poles quantitatively accounts for nucleoid compaction. Physically, the observed nucleoid size reflects the balance of two competing entropic forces—the compressive force that mRNAs (polysomes) at the poles exert on the DNA, and the expansive force exerted by the DNA on these mRNAs. Our detailed analysis supports a simplified analytical formula (3) that predicts nucleoid size for different physiological conditions; compare SI Appendix, Fig. S25. Biologically, the compaction of the nucleoid by mRNAs creates a potential global feedback circuit: Gene expression drives mRNA levels, which, by compacting the nucleoid, impact transcription-factor access and hence gene expression (30).

To summarize, whereas localization of the transcriptional–translational machinery is a well-known, functional mechanism in eukaryotes, the function of such localization in bacteria is not yet well established. In this regard, our model provides insights into the mechanisms governing the spatial structure of transcription and translation in bacteria, and can help guide the molecular manipulation of these functions, with potentially broad applications in molecular synthetic biology and biotechnology.