  The 3D folding of a mammalian gene can be studied by a polymer model, where
the chromatin fibre is represented by a semiflexible polymer which interacts
with multivalent proteins, representing complexes of DNA-binding transcription
factors and RNA polymerases. This physical model leads to the natural emergence
of clusters of proteins and binding sites, accompanied by the folding of
chromatin into a set of topologies, each associated with a different network of
loops. Here we combine numerics and analytics to first classify these networks
and then find their relative importance or statistical weight, when the
properties of the underlying polymer are those relevant to chromatin. Unlike
polymer networks previously studied, our chromatin networks have finite average
distances between successive binding sites, and this leads to giant differences
between the weights of topologies with the same number of edges and nodes but
different wiring. These weights strongly favour rosette-like structures with a
local cloud of loops with respect to more complicated non-local topologies. Our
results suggest that genes should overwhelmingly fold into a small fraction of
all possible 3D topologies, which can be robustly characterised by the
framework we propose here.
