How does a protein search for the specific site on DNA: The role of disorder

Proteins can locate their specific targets on DNA up to two orders of magnitude faster than the Smoluchowski three-dimensional diffusion rate. This happens due to nonspecific adsorption of proteins to DNA and subsequent one-dimensional sliding along DNA. We call such a one-dimensional route towards the target an “antenna.” We studied the role of the dispersion of nonspecific binding energies within the antenna due to a quasirandom sequence of natural DNA. A random energy profile for sliding proteins slows the searching rate for the target. We show that this slowdown is different for macroscopic and mesoscopic antennas.

Figure 1

(Color online) Distribution of the nonspecific adsorption energy $ϵ$ and chemical potential $\mu \left(x\right)$ along a DNA molecule. The target site is located at $x=0$; $\lambda$ is the antenna length.

Figure 2

(Color online) The phase diagram of the scaling regimes for $\mid w\mid >\sigma >kT$. Each line marks a smooth crossover between scaling regimes. The red line $\mid w\mid =3{\sigma }^2∕2kT$ marks border 1 between macroscopic regimes $(A,B)$ and mesoscopic regimes $(C,D)$. The blue line $\mid w\mid =kT\ln (v∕L{b}^2)-{\sigma }^2∕2kT$ marks border 2 between weak and strong adsorption regimes. They intersect at ${\sigma }_0=kT{\left[(1∕2)\ln (v∕L{b}^2)\right]}^{1∕2}$, $\mid w\mid =kT(3∕4)\ln (v∕L{b}^2)$.

Figure 3

(Color online) Dependence of the antenna length $\lambda$ on the disorder strength $\sigma$. Dashed lines represent the asymptotic limits.

Figure 4

(Color online) Schematic plot of the dependences of the rate enhancement $J∕J_s$ on $\mid w\mid$ at $\sigma ={\sigma }_1$ (upper solid curve) and $\sigma ={\sigma }_2$ (lower solid curve). Letters $A$, $B$, $C$, and $D$ represent the domains of Fig. 2 they go through. The dashed line shows the limit case of the flat energy profile with $\sigma =0$.