Facilitated diffusion with DNA coiling

Abstract

When DNA-binding proteins search for their specific binding site on a DNA molecule they alternate between linear 1-dimensional diffusion along the DNA molecule, mediated by nonspecific binding, and 3-dimensional volume excursion events between successive dissociation from and rebinding to DNA. If the DNA molecule is kept in a straight configuration, for instance, by optical tweezers, these 3-dimensional excursions may be divided into long volume excursions and short hops along the DNA. These short hops correspond to immediate rebindings after dissociation such that a rebinding event to the DNA occurs at a site that is close to the site of the preceding dissociation. When the DNA molecule is allowed to coil up, immediate rebinding may also lead to so-called intersegmental jumps, i.e., immediate rebindings to a DNA segment that is far away from the unbinding site when measured in the chemical distance along the DNA, but close by in the embedding 3-dimensional space. This effect is made possible by DNA looping. The significance of intersegmental jumps was recently demonstrated in a single DNA optical tweezers setup. Here we present a theoretical approach in which we explicitly take the effect of DNA coiling into account. By including the spatial correlations of the short hops we demonstrate how the facilitated diffusion model can be extended to account for intersegmental jumping at varying DNA densities. It is also shown that our approach provides a quantitative interpretation of the experimentally measured enhancement of the target location by DNA-binding proteins.

Fig. 1.

Schematic of the search mechanisms in Eq. 2 on a piece of stretched DNA. To locate the specific binding site (target) on the DNA molecule a DNA-binding protein diffuses one-dimensionally along the DNA chain with diffusivity D_1d, or it unbinds and performs a bulk excursion with diffusion constant D_3d. The virgin flux G denotes proteins that have not previously bound to DNA. Note the tendency to revisit sites when immediate rebindings and thus short hops are likely.

Fig. 2.

Search rate enhancement on coiled DNA: a protein that in the case of straight DNA would perform a short hop can now, because of DNA looping, be captured by a foreign (remote) segment of DNA. This leads to a more efficient search since local oversampling is reduced by these intersegmental jumps. The DNA configuration is assumed to be dynamic such that when the protein is released from the foreign segment the latter has moved further away from the original segment. Dashed lines, motion of the protein; dotted lines, DNA configurational changes.

Fig. 3.

Enhancement factor R of the search rate k_on in the maximally relaxed DNA configuration versus the search rate k_on for stretched DNA, as a function of NaCl concentration. The full line shows the values obtained from the experiment reported in ref. , and the dashed line represents the values from our theoretical approach. Interestingly the maximum occurs close to the physiological salt concentration of 100 mM.