Diffusion in fractal globules

Abstract: Recent experiments suggest that the human genome (all of our DNA) is organised as a so-called fractal globule. The fractal globule is a knot--free dense polymer that easily folds and unfolds any genomic locus, for example a group of nearby genes. Proteins often need to locate specific target sites on the DNA, for instance to activate a gene. To understand how proteins move through the DNA polymer, we simulate diffusion of particles through a fractal globule. The fractal globule was generated on a cubic lattice as spheres connected by cylinders. With the structure in place, we simulate particle diffusion and measure how their mean squared displacement ($\langle R^2(t)\rangle$) grows as function of time $t$ for different particle radii. This quantity allows us to better understand how the three dimensional structure of DNA affects the protein's motion. From our simulations we found that $\langle R^2(t)/t\rangle$ is a decaying function when the particle is sufficiently large. This means that the particles diffuse slower than if they were free. Assuming that $\langle R^2(t) \rangle \propto t^\alpha$ for long times, we calculated the growth exponent $\alpha$ as a function of particle radius $r_p$. When $r_p$ is small compared to the average distance between two polymer segments $d$, we find that $\alpha \approx 1$. This means the polymer network does not affect the particle's motion. However, in the opposite limit $r_p\sim d$ we find that $\alpha<1$ which means that the polymer strongly slows down the particle's motion. This behaviour is indicative of sub-diffusive dynamics and has potentially far reaching consequences for target finding processes and biochemical reactions in the cell.