  Chromatin and associated proteins constitute the highly folded structure of
chromosomes. We consider a self-avoiding polymer model of the chromatin,
segments of which may get cross-linked via protein binders that repel each
other. The binders cluster together via the polymer mediated attraction, in
turn, folding the polymer. Using molecular dynamics simulations, and a mean
field description, we explicitly demonstrate the continuous nature of the
folding transition, characterized by unimodal distributions of the polymer size
across the transition. At the transition point the chromatin size and
cross-linker clusters display large fluctuations, and a maximum in their
negative cross-correlation, apart from a critical slowing down. Along the
transition, we distinguish the local chain morphologies in terms of topological
loops, inter-loop gaps, and zippering. The topologies are dominated by simply
connected loops at the criticality, and by zippering in the folded phase.
