  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.
