Stochastic Ratchet Mechanisms for Replacement of Proteins Bound to DNA

Experiments indicate that unbinding rates of proteins from DNA can depend on the concentration of proteins in nearby solution. Here we present a theory of multistep replacement of DNA-bound proteins by solution-phase proteins. For four different kinetic scenarios we calculate the dependence of protein unbinding and replacement rates on solution protein concentration. We find (1) strong effects of progressive “rezipping” of the solution-phase protein onto DNA sites liberated by “unzipping” of the originally bound protein, (2) that a model in which solution-phase proteins bind nonspecifically to DNA can describe experiments on exchanges between the nonspecific DNA-binding proteins Fis-Fis and Fis-HU, and (3) that a binding specific model describes experiments on the exchange of CueR proteins on specific binding sites.

Figure 1

The four proposed unbinding pathways. Brown squares show $N$ ($=3$ here) units of a protein bound to DNA (dark horizontal line). Circles show units of the invader proteins, with different colors corresponding to different proteins. Filled circles show units occupying the zipping site. The most likely replacement scenario at small concentration is shown with the blue invader protein. Parameters entering the rates are (i) the mean number $c$ of solution-phase proteins per binding site, in units of the elementary concentration $c_o=1/a^3$, of one particle per binding site, where $a$ is a length scale associated with the linear dimension of a binding site (for $a=1\mathrm{nm}$, $c_o=1M$ ) and (ii) the ratio of the unbinding and binding rates for one unit: $\rho =e^{-\epsilon }$, where $\epsilon$ is the binding energy in $k_{B}T$ units. Time is expressed in terms of the time scale $t_o$, equal to the self-diffusion time for one unit of the protein: $t_o=2\pi \eta a^3/k_{B}T\approx 1.6\times {10}^{-9}\mathrm{s}$, for $a\approx 1\mathrm{nm}$, $\eta =0.001\mathrm{Pa}\mathrm{s}$ and $k_{B}T=4\times {10}^{-21}\mathrm{J}$. In units of $1/t_0$ the zipping rate of a protein unit on a free binding site is equal to one.

Figure 2

Unbinding rates $r(c)$ (full lines) from Eq. (2) versus dimensionless concentration $c$ and for parameters $N=10$, $\epsilon =2$. The zero-concentration rate, $r(0)=1.8\times {10}^{-9}$ (units of $1/t_0$), is the same for all four models. The concentrations $c_R$ at which replacement starts to dominate over pure thermal unbinding, i.e., $r(c)$ starts to vary linearly with $c$, are indicated by the vertical dashed colored lines for the four models; $c_R=3\times {10}^{-2}$ for NZ, $2\times {10}^{-6}$ for Z-NS, ${2\times 10}^{-8}$ for Z-S-NSB, and $1.3\times {10}^{-7}$ for Z-S-SB (units of $c_0$). The offsets between the linear regimes of the rate curves [dotted lines, from Eq. (3)] and the $r=c$ (dashed black) line are $\log R$ (log of replacement rate, double arrow vertical lines); $R$ is approximatively equal to ${10}^{-1}$ for Z-S-NSB, ${10}^{-2}$ for Z-S-SB and ${10}^{-3}$ for Z-NS. The dotted lines are only visible when the linear approximation breaks down.

Figure 3

Fit of concentration-dependent unbinding rates of Fis bound to DNA in the presence of Fis (left) and HU (middle) proteins in solution [1], and for CueR dissociation rates as a function of CueR concentration in solution (right) [2], using $a=1\mathrm{nm}$ [9], and $\mathrm{N}=14$, $\epsilon =1.95$ for Fis-Fis, $\mathrm{N}=19$, $\epsilon =1.4$ for Fis-HU, and $\mathrm{N}=15$, $\epsilon =1.36$ for CueR-CueR. For replacement rates (slopes of unbinding rates versus concentration) we find $R^{\mathrm{Fis}\text{−}\mathrm{Fis}}=5\times {10}^4{M}^{-1}{\mathrm{s}}^{-1}$, $R^{\mathrm{Fis}\text{−}HU}=2.6\times {10}^3{M}^{-1}{\mathrm{s}}^{-1}$, and $R^{\text{CueR}\text{−}\text{CueR}}=2.9\times {10}^7{M}^{-1}{\mathrm{s}}^{-1}$, in agreement, considering the error bars, with experimental fits. Replacement concentrations found with the Z-NS model are $c_R=2\mathrm{n}M$ for Fis-Fis and $c_R=370\mathrm{n}M$ for Fis-HU, while we find $c_R=16\mathrm{n}M$ for CueR-CueR with the Z-S-NSB model.