Kinetic Accessibility of Buried DNA Sites in Nucleosomes

Using a theoretical model for spontaneous partial DNA unwrapping from histones, we study the transient exposure of protein-binding DNA sites within nucleosomes. We focus on the functional dependence of the rates for site exposure and reburial on the site position, which is measurable experimentally and pertinent to gene regulation. We find the dependence to be roughly described by a random walker model. Close inspection reveals a surprising physical effect of flexibility-assisted barrier crossing, which we characterize within a toy model, the “semiflexible Brownian rotor.”

Figure 1

(a) Illustration of our nucleosome model. The DNA-histone interaction is localized at contact points attracting the red (dark) beads. The DNA is shown in the ground state as well as a conformation where the first contact is open. (b) Illustration of the semiflexible Brownian rotor (SBR) model. In this toy model, the tradeoff between bending energy and DNA-histone interaction in the nucleosome is mimicked by an angular potential $V(\phi )$, exerting a torque on the attachment angle $\phi$ of a semiflexible polymer at the origin.

Figure 2

(a) Equilibrium distribution of the DNA angle $\phi$ defined in Fig. 1a. The two peaks at $\phi =0$ and $\phi \approx 45\mathrm{deg}$ correspond to the fully wrapped state and the state with contact 1 open, respectively. (b) Kinetics of DNA site exposure within our nucleosome model. The dwell time in the inaccessible state (squares) increases roughly exponentially with the number of contacts that must open to render a DNA site accessible. The dashed line is a fit to Eq. (3). The circles show the average time the $n$th contact point remains open.

Figure 3

(a) The dependence of the dwell time ${\tau }_a(n=1)$ on the overhanging DNA length (diamonds) is compatible with ${\tau }_a(n)$ when $n$ is converted to contour length (gray open circles). The dashed line indicates the diffusion limit (see main text for details). (b) The average barrier crossing time ${\tau }_w$ (open circles) for the SBR model of Fig. 1b. At small lengths, the barrier crossing time follows that of a stiff rod (indicated by the dotted line). Beyond a crossover length ${\ell }_c\ll {\ell }_p$, barrier crossing is much faster than for a stiff rod. For large lengths, ${\tau }_w$ approaches the diffusion limit, i.e., ${\tau }_w$ of the free SBR (open squares). With $L<{\ell }_p$, free diffusion of the SBR is virtually indistinguishable from free diffusion of a rigid rod (dashed line). The crossover from the rodlike regime to the intermediate regime is well described by the theoretical analysis (solid line), see main text.

Figure 4

Dynamics at the barrier. (a) The unstable eigenmode for three different lengths. Polymers shorter than ${\ell }_c$ rotate without significant deformation, while long polymers form a bulge of size $\sim {\ell }_c$ at the origin. (b) The prefactor of the Kramers time ${\tilde{\tau }}_w=1/{\lambda }_{-}$ as a function of the length. The prefactor increases as $L^3$ if $L\ll {\ell }_c$ and is constant if $L\gg {\ell }_c$.