Facilitated Diffusion of DNA-Binding Proteins

The diffusion-controlled limit of reaction times for site-specific DNA-binding proteins is derived from first principles. We follow the generally accepted concept that a protein propagates via two competitive modes, a three-dimensional diffusion in space and a one-dimensional sliding along the DNA. However, our theoretical treatment of the problem is new. The accuracy of our analytical model is verified by numerical simulations. The results confirm that the unspecific binding of protein to DNA, combined with sliding, is capable to reduce the reaction times significantly.

Figure 1

Reaction time $\tau$ as a function of the sliding parameter $\xi$ [Eq. (9)] at a fixed cell radius $R=2$ and chain lengths $L=56$, 40, 24, 8 (top to bottom). The curves are plots of Eq. (8).

Figure 2

Reaction time $\tau$ as a function of the sliding parameter $\xi$ [Eq. (9)] at fixed chain length $L=56$ and with varying cell volumes ($8x$, $4x$, $2x$, and $1x$ the original volume $V_0=32\pi /3$, top to bottom). The curves are plots of Eq. (8).