  The mechanism through which cells determine their fate is intimately related
to the spreading of certain biochemical (so-called epigenetic) marks along
their genome. The mechanisms behind mark spreading and maintenance are not yet
fully understood, and current models often assume a long-range infection-like
process for the dynamics of marks, due to the polymeric nature of the chromatin
fibre which allows looping between distant sites. While these existing models
typically consider antagonising marks, here we propose a qualitatively
different scenario which analyses the spreading of a single mark. We define a
1D stochastic model in which mark spreading/infection occurs as a long-range
process whereas mark erasure/recovery is a local process, with an enhanced rate
at boundaries of infected domains. In the limiting case where our model
exhibits absorbing states, we find a first-order-like transition separating the
marked/infected phase from the unmarked/recovered phase. This suggests that our
model, in this limit, belongs to the long-range compact directed percolation
universality class. The abrupt nature of the transition is retained in a more
biophysically realistic situation when a basal infection/recovery rate is
introduced (thereby removing absorbing states). Close to the transition there
is a range of bistability where both the marked/infected and unmarked/recovered
states are metastable and long lived, which provides a possible avenue for
controlling fate decisions in cells. Increasing the basal infection/recovery
rate, we find a second transition between a coherent (marked or unmarked)
phase, and a mixed, or random, one.
