  The genetic instructions stored in the genome require an additional layer of
information to robustly determine cell fate. This additional regulation is
provided by the interplay between chromosome-patterning biochemical
("epigenetic") marks and three-dimensional genome folding. Yet, the physical
principles underlying the dynamical coupling between three-dimensional genomic
organisation and one-dimensional epigenetic patterns remain elusive. To shed
light on this issue, here we study by mean field theory and Brownian dynamics
simulations a magnetic polymer model for chromosomes, where each monomer
carries a dynamic epigenetic mark. At the single chromosome level, we show that
a first order transition describes the unlimited spreading of epigenetic marks,
a phenomenon that is often observed in vivo. At the level of the whole nucleus,
experiments suggest chromosomes form micro-phase separated compartments with
distinct epigenetic marks. We here discover that for a melt of magnetic
polymers such a morphology is thermodynamically unstable, but can be stabilised
by a non- equilibrium and ATP-mediated epigenetic switch between different
monomer states.
