Stochastic dynamics and ribosome-RNAP interactions in transcription-translation coupling

Abstract

Under certain cellular conditions, transcription and mRNA translation in prokaryotes appear to be "coupled," in which the formation of mRNA transcript and production of its associated protein are temporally correlated. Such transcription-translation coupling (TTC) has been evoked as a mechanism that speeds up the overall process, provides protection against premature termination, and/or regulates the timing of transcript and protein formation. What molecular mechanisms underlie ribosome-RNAP coupling and how they can perform these functions have not been explicitly modeled. We develop and analyze a continuous-time stochastic model that incorporates ribosome and RNAP elongation rates, initiation and termination rates, RNAP pausing, and direct ribosome and RNAP interactions (exclusion and binding). Our model predicts how distributions of delay times depend on these molecular features of transcription and translation. We also propose additional measures for TTC: a direct ribosome-RNAP binding probability and the fraction of time the translation-transcription process is "protected" from attack by transcription-terminating proteins. These metrics quantify different aspects of TTC and differentially depend on parameters of known molecular processes. We use our metrics to reveal how and when our model can exhibit either acceleration or deceleration of transcription, as well as protection from termination. Our detailed mechanistic model provides a basis for designing new experimental assays that can better elucidate the mechanisms of TTC.

Figure 1

Schematic of translation of a nascent mRNA transcript (polypeptide not shown). Ribosomes translocate at rate p while RNAPs elongate at rate q. (A) The transcript associated with the RNAP at position n along the gene is shown with a leading ribosome at position m along the mRNA. Ribosomes attach to open initiation sites at rate α. (B) A nearly complete transcript is shown. If the leading ribosome has caught up to the RNAP and m is close to n, the two may bind with rate $k_{\text{a}}$ to form a “coupled expressome.” Ribosome-RNAP complexes can spontaneously dissociate with rate $k_{\text{d}}$. We assume that the leading ribosome “terminates” upon reaching the stop codon (not shown). Protein-mediated expressomes (not shown) form larger complexes that can accommodate longer mRNA segments within it.

Figure 2

(A) State space of the stochastic model defined in terms of the leading ribosome and RNAP positions $\left(m\text{,}n\right)$. The initial time $t=0$ is defined as the time RNAP first produces a ribosome initiation site, starting the system in $\left(m=0,n=1\right)$. For $t>0$, as the RNAP is elongating, the first ribosome binds at rate α. Here, a ribosome binds after the RNAP first reaches position $n=n_0$. Red and blue trajectories indicate scenarios in which the RNAP is relatively fast and slow, respectively. Within each position $\left(m\text{,}n\right)$ exist internal molecular microstates. (B) In the “interior” states $n-m>\ell$ ($\ell =2$ in this example), the ribosome and RNAP are too distant to be bound, and only stalled and processing RNAP states arise, with transition rates $k_{\pm }$ between them. (C) When $d=n-m=0$, the ribosome and RNAP are adjacent without any intervening mRNA, allowing them to associate with rate $k_{\text{a}}$. The RNAP can be in either stalled or processive states. In the stalled state, whether associated or not, the adjacent volume-excluding ribosome entropically “pushes” the stalled RNAP, catalyzing its transition to a processive state so that $k_{+}^{\text{∗}}>k_{+}$. (D) When $0<n-m<\ell$, the ribosome and the RNAP are close enough to bind with rate $k_{\text{a}}$. Here, the intervening mRNA dissipates the entropic pushing (so that the stalled RNAP $\to$ processing RNAP transition rate is $k_{+}$) and also allows an RNAP in the processive state to elongate with rate q, regardless of whether it is bound to the ribosome. (E) Only when the ribosome and the RNAP are separated by $d=\ell$ is a bound RNAP prevented from processing as this would reel in more mRNA than can be fit inside the complex, either a collided or NusG-mediated expressome. Molecular binding prevents complexed ribosome and RNAP to be separated by more than $\ell$ mRNA codons.

Figure 3

Comparison of different TTC indices. Common parameters for all these plots are $\alpha =1$/s, $E_{+}=2$, $E_{\text{a}}=3$, $k_{\text{d}}=k_{\text{a}}{e}^{-E_{\text{a}}}$, $\ell =4$, $L=335$, and $k_{\text{a}}=100$/s, unless stated otherwise. (A) The delay-time density $\rho \left(\text{Δ}T\right)$ plotted for $p=12$ codons/s and $q=30$ codons/s. Densities for $k_{+}=0.4$/s, $k_{-}=0.3$/s (rarely pausing RNAPs, green curves), and $k_{+}=4.0$/s, $k_{-}=3.0$/s (frequently pausing RNAPs, red curves) are shown. Within these cases, strong-binding ($a\sim 1$, $k_{\text{a}}=100$/s, $k_{\text{d}}=k_{\text{a}}{e}^{-3}$) and no-binding ($a=0$, $k_{\text{a}}=0$) subcases are indicated by solid and dashed curves, respectively. (B) Mean delay $\mathbb{E}\left[\text{Δ}T\right]$ as a function of p and q. (C) The direct coupling coefficient C as a function of the relative velocity $p/\overline{q}$. Each point represents C evaluated at specific values of (p,q), each chosen from all integers between 3 and 27 codons/s. The dashed curve represents the analytic approximation given by Eq. 15. (D) Heatmap of $C\left(p\text{,}q\right)$. (E) Values of $\mathbb{E}\left[F_T\right]$, each derived from 1000 kinetic Monte-Carlo (kMC) trajectories, plotted against $p/\overline{q}$. The analytic approximation given by Eq. 16 is shown by the dashed curve. (F) The heatmap of $\mathbb{E}\left[F_T\left(p\text{,}q\right)\right]$.

Figure 4

Slowdown induced by molecular coupling. The common parameters used are the same as those in Fig. 3. (A) Effective velocities as a function activation energy reduction $E_{+}=\text{log}\left(k_{+}^{\text{∗}}/k_{+}\right)$ in ribosome-induced RNAP unstalling, $p=20$ codons/s and $q=30$ codons/s. (B) Effective velocity of RNAP as a function of the free ribosome translation rate p. (C) The trade-off between translation efficiency and mean fraction of time protected $\mathbb{E}\left[F_T\right]$. The efficiency $\eta \equiv {\overline{V}}_{\text{rib}}/p$ is defined by the ratio of the mean ribosome speed to the translation rate of an isolated ribosome. In (B) and (C), $q=30$ codons/s. The variances (not shown) for the plotted quantities are large, typically overlapping the mean-value curves in (A and B). The dashed curve in (A) and solid curves in (B) are the analytical predictions of the effective velocities using Eq. 17. To see this figure in color, go online.

Figure 5

Effects of molecular coupling. For those parameters not varied, we use the same values used to generate Figs. 3 and 4. (A) For $\ell =4$, the effective transcription velocity ${\overline{V}}_{\text{RNAP}}$ as a function of binding-energy depth between the ribosome and RNAP $E_{\text{a}}$. (B) Mean protected-time fraction $\mathbb{E}\left[F_T\right]$ as a function of binding energy depth $E_{\text{a}}$ ($\ell =4$). (C) The trade-off between efficiency and protection for $\ell =4$. (D) Rescaled heatmap of the delay-time distribution $\rho \left(\text{Δ}T\right)$ as a function of ribosome translocation rate p. The brightness indicates the relative probability, and the inset shows the probability distribution at $p=12$ codons/s indicated by the dashed white line. Here, the binding energy $E_{\text{a}}=3$ and $\ell =4$. For $p\approx 9$ codons/s, $\rho \left(\text{Δ}T\right)$ is bimodal in $\text{Δ}T$. (E) Delay-time distribution in the absence of ribosome-RNAP binding ($k_{\text{a}}=0$). Here, the $\ell$ dependence disappears and $\rho \left(\text{Δ}T\right)$ is mono-modal. (F) Delay-time distribution for $\ell =40$ and $E_{\text{a}}=3$. Bimodality arises in more than one regime of p. To see this figure in color, go online.