Interacting RNA polymerase motors on a DNA track: Effects of traffic congestion and intrinsic noise on RNA synthesis

RNA polymerase (RNAP) is an enzyme that synthesizes a messenger RNA (mRNA) strand which is complementary to a single-stranded DNA template. From the perspective of physicists, an RNAP is a molecular motor that utilizes chemical energy input to move along the track formed by DNA. In many circumstances, which are described in this paper, a large number of RNAPs move simultaneously along the same track; we refer to such collective movements of the RNAPs as RNAP traffic. Here we develop a theoretical model for RNAP traffic by incorporating the steric interactions between RNAPs as well as the mechanochemical cycle of individual RNAPs during the elongation of the mRNA. By a combination of analytical and numerical techniques, we calculate the rates of mRNA synthesis and the average density profile of the RNAPs on the DNA track. We also introduce, and compute, two different measures of fluctuations in the synthesis of RNA. Analyzing these fluctuations, we show how the level of intrinsic noise in mRNA synthesis depends on the concentrations of the RNAPs as well as on those of some of the reactants and the products of the enzymatic reactions catalyzed by RNAP. We suggest appropriate experimental systems and techniques for testing our theoretical predictions.

Figure 1

(a) A schematic representation of RNAP traffic where the three dashed squares represent three TECs. The solid lines connecting filled circles represent the two strands of the double-stranded DNA while the string of open circles denotes the elongating RNA molecule. The dashed lines connecting the circles denote the unbroken noncovalent bonds between the complementary subunits on the DNA and RNA strands. Each of the gray ovals represents the catalytic site on the corresponding RNAP. (b) A simplified version of (a). The DNA track for the RNAP motors is assumed to be, effectively, a one-dimensional lattice. Each TEC has been replaced by a rectangular black box that can cover $r$ lattice sites simultaneously ($r=6$ in this figure). The RNAP in each TEC can exist in either of the two chemical states.

Figure 2

A schematic representation of the mechanochemical cycle of each RNAP in our model in the elongation stage. No ${\text{PP}}_i$ is bound to the RNAP in the state 1 whereas the ${\text{PP}}_i$-bound state of the RNAP is labeled by the index 2.

Figure 3

The steady-state flux of the RNAPs, under periodic boundary conditions, plotted as a function of the coverage density ${\rho }_{cov}$ for (a) three different values of [NTP] at $\left[{\text{PP}}_i\right]=1\mu \text{M}$, and (b) three different values of $\left[{\text{PP}}_i\right]$ at $\left[\text{NTP}\right]=1\text{mM}$. The lines correspond to our mean-field theoretic predictions whereas the discrete data points have been obtained from computer simulations.

Figure 4

The steady-state flux of the RNAPs, under open boundary conditions, plotted as a function of (a) ${\omega }_{\alpha }$, for three sets of values of the pair of parameters [NTP], $\left[{\text{PP}}_i\right]$; (b) ${\omega }_{21}^f$ for three values of the parameter ${\omega }_{\alpha }$. The lines correspond to our mean-field theoretic predictions whereas the discrete data points have been obtained from computer simulations. The insets show the average density profiles for (a) three different values of ${\omega }_{\alpha }$ and (b) three different values of ${\omega }_{21}^f$.

Figure 5

(Color online) The distributions of the run times and time headways in our model, under open boundary conditions, plotted in (a) and (b), respectively, for two different values of ${\omega }_{21}^f$. The discrete data points have been obtained from computer simulations. The curves fitted to these data points are drawn with the lines. The variation of the standard deviations of the distributions of run times and time headways with the increase of the parameter ${\omega }_{21}^f$ are shown in the insets; the discrete points have been obtained from the simulation data and the best fit curve through these points has been drawn by a line.

Figure 6

(Color online) The distributions of the run times and time headways in our model, under open boundary conditions, plotted in (a) and (b), respectively, for two different values of ${\omega }_{21}$. The discrete data points have been obtained from computer simulations. The curves fitted to these data points are drawn with the lines. The variation of the standard deviation of the distributions of run times with the increase of the parameter ${\omega }_{21}$ are shown in the insets; the discrete points have been obtained from the simulation data and the best fit curve through these points has been drawn by a line.