  Epigenetics is a driving force of important and ubiquitous phenomena in
nature such as cell differentiation or even metamorphosis. Oppositely to its
widespread role, understanding the biophysical principles that allow
epigenetics to control and rewire gene regulatory networks remains an open
challenge. In this work we study the effects of epigenetic modifications on the
spatial folding of chromosomes -- and hence on the expression of the underlying
genes -- by mapping the problem to a class of models known as magnetic
polymers. In this work we show that a first-order phase transition underlies
the simultaneous spreading of certain epigenetic marks and the conformational
collapse of a chromosome. Further, we describe Brownian Dynamics simulations of
the model in which the topology of the polymer and thermal fluctuations are
fully taken into account and that confirms our mean-field predictions.
Extending our models to allow for non-equilibrium terms yields new stable
phases which qualitatively agrees with observations in vivo. Our results show
that statistical mechanics techniques applied to models of magnetic polymers
can be successfully exploited to rationalize the outcomes of experiments
designed to probe the interplay between a dynamic epigenetic landscape and
chromatin organization.
