Modelling Protein Target-Search in Human Chromosomes

Several processes in the cell, such as gene regulation, start when key proteins recognise and bind to short DNA sequences. However, as these sequences can be hundreds of million times shorter than the genome, they are hard to find by simple diffusion: diffusion-limited association rates may underestimate $in~vitro$ measurements up to several orders of magnitude. Moreover, the rates increase if the DNA is coiled rather than straight. Here we model how this works $in~vivo$ in mammalian cells. We use chromatin-chromatin contact data from state-of-the-art Hi-C experiments to map the protein target-search onto a network problem. The nodes represent a DNA segment and the weight of the links is proportional to measured contact probabilities. We then put forward a master equation for the density of searching protein that allows us to calculate the association rates across the genome analytically. For segments where the rates are high, we find that they are enriched with active genes and have high RNA expression levels. This paper suggests that the DNA's 3D conformation is important for protein search times $in~vivo$ and offers a method to interpret protein-binding profiles in eukaryotes that cannot be explained by the DNA sequence itself.