Searching Fast for a Target on DNA without Falling to Traps

Genomic expression depends critically on both the ability of regulatory proteins to locate specific target sites on DNA within seconds and on the formation of long-lived (many minutes) complexes between these proteins and the DNA. Equilibrium experiments show that indeed regulatory proteins bind tightly to their target site. However, they also find strong binding to other nonspecific sites which act as traps that can dramatically increase the time needed to locate the target. This gives rise to a conflict between the speed and stability requirements. Here we suggest a simple mechanism which can resolve this long-standing paradox.

Figure 1

An illustration of the model. (a) A time sequence of a protein sliding in the $s$ mode [light gray (green) circle], diffusing off the DNA [dark gray (blue) circle], and entering the target site in the $r$ mode (red oval). (b) A protein finding the target after entering the $r$ state. (c) An illustration of the rates and the energy landscape which governs them at each location, $i=1,\dots ,N$, along the DNA. Here ${\lambda }_r^i\propto e^{-(E_b^i-E_s^i)/k_{B}T}$, ${\lambda }_s^i\propto e^{-(E_b^i-E_r^i)/k_{B}T}$, and ${\lambda }_u\propto e^{-E_s^i/k_{B}T}$, while ${\lambda }_b$ depends on details of the three-dimensional diffusion process.

Figure 2

A plot of $\mathcal{R}(t)$ for $N={10}^6$ (empty circles) and $N={10}^8$ (filled squares). Lines correspond to Eq. (4), with ${\tau }_1$, ${\tau }_2$, and $q$ derived analytically. Here ${\lambda }_u={10}^{-2}{\lambda }_0$, ${\lambda }_b=0.1{\lambda }_0$, ${\lambda }_r={10}^{-7}{\lambda }_0$, and ${\lambda }_s={10}^{-9}{\lambda }_0$, in agreement with 24. These correspond to energies, measured relative to the energy of the unbound state, of $E_s=-4.6k_{B}T$, $E_b=11.5k_{B}T$, and $E_r=-9.2k_{B}T$. Experiments suggest ${\lambda }_0\simeq {10}^6{\mathrm{s}}^{-1}$ for the Lac repressor [5].

Figure 3

A plot of ${\mathcal{R}}_{n_p}(t)$ for $n_p=1$ (empty circles) and $n_p=10$ (filled squares). Here $N={10}^6$, ${\lambda }_u={10}^{-4}{\lambda }_0$, ${\lambda }_b=0.1{\lambda }_0$, ${\lambda }_r={10}^{-7}{\lambda }_0$, and ${\lambda }_s={10}^{-9}{\lambda }_0$ (see 24). These correspond to energies, measured relative to the unbound state, of $E_s=-9.2k_{B}T$, $E_b=6.9k_{B}T$, and $E_r=-13.8k_{B}T$. Lines correspond to Eq. (4) with calculated values of ${\tau }_1$, ${\tau }_2$, and $q$. Note that here ${\lambda }_u$ is different from Fig. 2.

Figure 4

Disordered case for $n_p=1$ (empty circles) and $n_p=10$ (filled squares), where $N={10}^6$, ${\lambda }_u={10}^{-2}{\lambda }_0$ ($E_s=-4.6k_{B}T$), ${\lambda }_b=0.1{\lambda }_0$, and $E_0=25.4k_{B}T$. (a) Plot of ${\mathcal{R}}_{n_p}(t)$ for $\sigma =5.3k_{B}T$. The lines were obtained by fitting the form $1-[q{e}^{-t/{\tau }_1}+(1-q){]}^{n_p}$ to the numerical simulations with $q=0.2817$, ${\lambda }_0{\tau }_1=1.7\times {10}^7$, and ${\tau }_2=\infty$. These are close to the mean-field prediction $q=0.2827$, ${\lambda }_0{\tau }_1=1.1\times {10}^7$. The average height of the barrier at the target site is then $6.25k_{B}T$, which corresponds to a transition rate of $2\times {10}^3{\mathrm{s}}^{-1}$. (b) $p_{\mathrm{cat}}$ as a function of $\sigma$ for $n_p=1$ and $n_p=10$. (c) $t^{\mathrm{typ}}$ for $n_p=10$ and ${\tau }_1$ are plotted as a function of $\sigma$. Using ${\lambda }_0={10}^6{\mathrm{s}}^{-1}$ [5] for $n_p=10$ at the minimal $p_{\mathrm{cat}}$ we find $t^{\mathrm{typ}}\simeq 10\mathrm{s}$. Note that by moderate changes in $E_0$ similar results can be obtained for longer DNA sequences.