Method for the analysis of contribution of sliding and hopping to a facilitated diffusion of DNA-binding protein: Application to in vivo data

DNA-binding protein searches for its target, a specific site on DNA, by means of diffusion. The search process consists of many recurrent steps of one-dimensional diffusion (sliding) along the DNA chain and three-dimensional diffusion (hopping) after dissociation of a protein from the DNA chain. Here we propose a computational method that allows extracting the contribution of sliding and hopping to the search process in vivo from the measurements of the kinetics of the target search by the lac repressor in Escherichia coli [P. Hammar et al., Science 336, 1595 (2012)]. The method combines lattice Monte Carlo simulations with the Brownian excursion theory and includes explicitly steric constraints for hopping due to the helical structure of DNA. The simulation results including all experimental data reveal that the in vivo target search is dominated by sliding. The short-range hopping to the same base pair interrupts one-dimensional sliding while long-range hopping does not contribute significantly to the kinetics of the search of the target in vivo.

Figure 1

DNA-binding protein reassociation to the DNA chain depends on the helical structure of DNA. (a) The model of facilitated diffusion includes 3D translocations to nearby sites (hopping) interspersed with binding to nonspecific sites with diffusion along contiguous sites (sliding), i.e., along a helical path of DNA. The hopping leads to association only if the protein follows the DNA double helix (according to the angle $\varphi$ between rebinding events). When the protein reaches DNA we check this angle and allow the protein to associate only if the angle is within a given range of angles $\mathrm{\Delta }\varphi$ (this range of angles is a parameter in the model). (b) Normalized histograms of the TF rebinding angles as a function of the excursion (hopping) distance $r$. The rebinding to the DNA is focused within a narrow range of angles. (c) Histograms of the hop distances, along the axis of the DNA ($z$ direction), calculated for single excursions starting at $r_{\mathrm{start}}=5.51$ nm (black circles) and $r_{\mathrm{start}}=6.5$ nm (blue squares) and reaching the point of the DNA given by $r_{\min }=5.5$ nm. Closed data points correspond to the case with the whole DNA reactive surface while open data points result from incorporating the helical structure of DNA into the model with the reactive angle range $\mathrm{\Delta }\varphi ={36}^{\circ }$.

Figure 2

Outline of the simulation algorithm used in the interpretation of experiments performed for TF binding rates to two specific sites placed at a fixed distance on the DNA. The method requires as a prerequisite the calculation of the histograms of diffusion steps $n$ and rebinding angles $\varphi$ for the parameters $r_{\mathrm{start}}$ and $r_{\max }$ (the maximal distance of excursion after dissociation varied from 11 nm to 40 nm). The event-driven algorithm consists of the following steps: initialize, with $s^2, N$ (number of base pairs), and $M=0$; macroscopic association, in which a random number $\xi \in (1,N)$ is generated and the current position of the DNA binding protein $\mathcal{P}$ is $\mathcal{P}=\xi$; sliding, in which a random number $x$ is drawn from the uniform distribution in the unit interval and according to $x$ one of three events is chosen: (i) the position is changed to $\mathcal{P}+1$ [with probability $P_{+}=1/2P=s^2/(1+2s^2)]$, (ii) the position is changed to $\mathcal{P}-1$ [with probability $P_{-}=s^2/(1+2s^2)]$, or (iii) microscopic dissociation [with probability $P_d=1/(1+2s^2)]$ and hopping, in which the excursion distance $r$ (the excursion distance $r$ is generated from the distribution $dP\left(r\right)/dr=[ln(r_{\mathrm{start}}/r_{\min })]/\left\{r{[ln(r/r_{\min })]}^2\right\}$ by drawing a random number $x$ from the uniform distribution in the unit interval and taking $r=r_{\mathrm{start}}^{1/x}{r}_{\min }^{(1-x)/x}$) is chosen. If $r\ge r_{\max }$, then macroscopic dissociation will occur, $M=M+1$. If $r<r_{\max }$, the hopping distance $z$ is calculated along the DNA axis and the rebinding angle $\varphi$ is drawn from the histograms (see the text for details). If the pair $(z,\varphi )$ follows a helical pathway of the DNA a random number $x$ is drawn from the uniform distribution in the unit interval. If $x\le p$ ($p$ is the probability of microscopic association) the TF binds with the DNA; otherwise hopping is repeated. The final step is specific binding: The procedure is finished when $\mathcal{P}$ corresponds to the position of the operator or operators.

Figure 3

Application of the simulation algorithm to the in vivo kinetics data of lac repressor binding in E. coli [2]. (a)–(d) Results of the simulations for different sets of parameter value pairs $(\mathrm{\Delta }\varphi ,r_{\max })$ for $r_{\min }=5.5$ and $r_{\mathrm{start}}=5.51$ nm. (a) Changes in the number of steps $n_0$ between rebinding events as a function of the range of reactive angle (constant $r_{\max }$, black circles) and as a function of the macroscopic dissociation distance (constant $\mathrm{\Delta }\varphi$, red squares). (b) and (c) Residual error $\left({\chi }^2\right)$ of the model predictions and experimental data for lac repressor for the range of angles (b) $\mathrm{\Delta }\varphi ={36}^{\circ }$ and (c) $\mathrm{\Delta }\varphi =1^{\circ }$. The left data points correspond to the diffusion-limited reaction. (d) Comparison of experimental [2] and simulated ratios of association rates for $\mathrm{\Delta }\varphi ={36}^{\circ }$ and pairs of parameters $(s^2,r_{\max })$ marked by arrows in (b).