Facilitated Diffusion of Proteins on Chromatin

We present a theoretical model of facilitated diffusion of proteins in the cell nucleus. This model, which takes into account the successive binding and unbinding events of proteins to DNA, relies on a fractal description of the chromatin which has been recently evidenced experimentally. Facilitated diffusion is shown quantitatively to be favorable for a fast localization of a target locus by a transcription factor and even to enable the minimization of the search time by tuning the affinity of the transcription factor with DNA. This study shows the robustness of the facilitated diffusion mechanism, invoked so far only for linear conformations of DNA.

Figure 1

Facilitated diffusion on chromatin: A chromatin binding protein (blue) searches for a target locus (yellow) on chromatin (red). The binding (unbinding) rate is denoted by ${\lambda }_2$ (${\lambda }_1$).

Figure 2

Distribution of the GFPT for diffusion in fractal medium in the compact case. Numerical simulations for different system sizes $N$ (symbols) are plotted against the theoretical prediction of Eq. (3) (plain lines). (a) Sierpinski gasket (target at the apex) and (b) critical bond percolation cluster embedded in a 3D cubic lattice (average taken over random targets and cluster realizations).

Figure 3

Mean search time for facilitated diffusion on critical bond percolation clusters embedded in a 3D cubic lattice with a randomly located target. Numerical simulations (symbols) and theoretical prediction [Eq. (7), plain lines] of the mean GFPT, rescaled by its value for ${\lambda }_1=0$, are plotted as a function of $x_0=\sqrt{A⟨{\tau }_T⟩{\lambda }_1}$ for various ${\lambda }_2$ and system size $N$.