Diffusion, dimensionality, and noise in transcriptional regulation

The precision of biochemical signaling is limited by randomness in the diffusive arrival of molecules at their targets. For proteins binding to specific sites on DNA and regulating transcription, the ability of the proteins to diffuse in one dimension by sliding along the length of the DNA, in addition to their diffusion in bulk solution, would seem to generate a larger target for DNA binding, consequently reducing the noise in the occupancy of the regulatory site. Here we show that this effect is largely canceled by the enhanced temporal correlations in one-dimensional diffusion. With realistic parameters, sliding along DNA has surprisingly little effect on the physical limits to the precision of transcriptional regulation.

Figure 1

The transcription factors can either be free in solution at concentration $c$, or they can enter a region on the DNA where they diffuse by sliding. The 1D concentration is denoted by $\xi \left(x,t\right)$. The specific binding site on the DNA is at location $x_0=0$; $k_{+}\xi \left(x_0\right)$ and $k_{-}$ are the on and off rates for transitions between the 1D sliding state and the bound state on the specific site; ${\kappa }_{+}c$ is the rate per unit length for transition onto the DNA from free solution, and ${\kappa }_{-}={\tau }_{\text{r}}^{-1}$ is the rate for dissociating from 1D solution into the 3D solution. The effective radius of the DNA molecule is $R$ and the “sliding length,” or the average distance along the contour covered in a single 1D random walk before dissociation, is $b$.

Figure 2

(Color online) Effective concentration noise level, compared to the pure 3D diffusion model, as a function of parameters $\alpha$ and $\beta$. Parameter $\alpha ={\kappa }_{+}/4\pi D_3$ is the per unit length on rate for binding nonspecifically to the DNA, normalized by the Smoluchowski diffusion limit. Parameter $\beta ={\left(R/\pi b\right)}^2$ is approximately the square of the ratio between the radius of the DNA cylinder and the typical sliding length during a 1D diffusion excursion along the DNA contour length; in this plot it covers the relevant range if typical 1D diffusion length is as expected from search time optimality arguments (order hundred base pairs). Three values for $\alpha$ are shown, spanning 2 orders of magnitude ($\alpha =1$ solid, $\alpha =0.1$ dotted, $\alpha =0.01$ dashed). For each value of $\alpha$, two curves are plotted: the thin curve with ${\Lambda }_x/\Lambda =R/a\to \infty$, and the thick curve with ${\Lambda }_x/\Lambda =1$ (thin curves at given $\alpha$ have larger effective noise). The latter case in $\beta \to \infty$ (short sliding length) limit converges to the pure 3D diffusion result, as explained in Eq. (74). Importantly, note that while in both cases $\beta$ extends over 4 orders of magnitude, the relative decrease in noise is only of order 1.