Mechanical Properties of Transcription

The mechanical properties of transcription have recently been shown to play a central role in gene expression. However, a full physical characterization of this central biological process is lacking. In this Letter, we introduce a simple description of the basic physical elements of transcription where RNA elongation, RNA polymerase rotation, and DNA supercoiling are coupled. The resulting framework describes the relative amount of RNA polymerase rotation and DNA supercoiling that occurs during RNA elongation. Asymptotic behavior is derived and can be used to experimentally extract unknown mechanical parameters of transcription. Mechanical limits to transcription are incorporated through the addition of a DNA supercoiling-dependent RNA polymerase velocity. This addition can lead to transcriptional stalling and resulting implications for gene expression, chromatin structure and genome organization are discussed.

Figure 1

A cartoon depicting RNA elongation $x$ through shared RNAP $\theta$ and DNA $\varphi$ rotation. The DNA is attached to an optical bead for mechanical manipulation.

Figure 2

Supercoiling $\sigma$ and torque $\tau$ as a function of RNA elongation $x$ for $L=5$, 10, $100\mu \mathrm{m}$ (blue, orange, purple), with $\alpha =\frac{1}{2}$, $\eta v=1$ at a fixed force of $F=1/2\mathrm{pN}\mathrm{nm}$ ($S=582.0$, ${\tau }_0=9.5$, $P=189.6\mathrm{pN}\mathrm{nm}$).

Figure 3

Supercoiling $\sigma$ as a function of RNA elongation $x$ for $L=10\mu \mathrm{m}$ at a fixed force of $F=1\mathrm{pN}\mathrm{nm}$ ($S=622.6$, ${\tau }_0=14.5$, $P=189.6\mathrm{pN}\mathrm{nm}$). The solid and dashed curves are for $\alpha =\frac{1}{2}$ and $\alpha =\frac{1}{4}$, respectively, and the colors are such that ${\eta }_{\text{orange}}/{\eta }_{\text{purple}}=10$. The $y$ intercepts are determined by the drag coefficient $\eta$ and the slopes by the functional exponent $\alpha$. The plot illustrates how the asymptotic behavior of SC and torque would allow for direct measurements of the unknown mechanical parameters of the RNAC.

Figure 4

Supercoiling $\sigma$ and velocity $v(\sigma )$ as a function of RNA elongation $x$ for $L=5$, 10, $100\mu \mathrm{m}$ (blue, orange, purple), with $\alpha =\frac{1}{2}$, $\eta v=1$ at a fixed force of $F=1/2\mathrm{pN}\mathrm{nm}$ ($S=582.0$, ${\tau }_0=9.5$, $P=189.6\mathrm{pN}\mathrm{nm}$) for a system with a torque-dependent velocity. For each length, the torque cutoff was set at ${\tau }_c=12\mathrm{pN}\mathrm{nm}$.

Figure 5

Figure shows the number of transcripts $m_c$ that can be made for a gene of length $W$ against a barrier of length $L$ with no relaxation between transcription events. The mechanical values used are $\alpha =\frac{1}{2}$, $\eta v=1$ at a fixed force of $F=1/2\mathrm{pN}\mathrm{nm}$ ($S=582.0$, ${\tau }_0=9.5$, $P=189.6\mathrm{pN}\mathrm{nm}$) for a system with a torque-dependent velocity.