Epigenetic Transitions and Knotted Solitons in Stretched Chromatin

Abstract

The spreading and regulation of epigenetic marks on chromosomes is crucial to establish and maintain cellular identity. Nonetheless, the dynamical mechanism leading to the establishment and maintenance of a given, cell-line specific, epigenetic pattern is still poorly understood. In this work we propose, and investigate in silico, a possible experimental strategy to illuminate the interplay between 3D chromatin structure and epigenetic dynamics. We consider a set-up where a reconstituted chromatin fibre is stretched at its two ends (e.g., by laser tweezers), while epigenetic enzymes (writers) and chromatin-binding proteins (readers) are flooded into the system. We show that, by tuning the stretching force and the binding affinity of the readers for chromatin, the fibre undergoes a sharp transition between a stretched, epigenetically disordered, state and a crumpled, epigenetically coherent, one. We further investigate the case in which a knot is tied along the chromatin fibre, and find that the knotted segment enhances local epigenetic order, giving rise to "epigenetic solitons" which travel and diffuse along chromatin. Our results point to an intriguing coupling between 3D chromatin topology and epigenetic dynamics, which may be investigated via single molecule experiments.

The Chern–Simons current in time series of knots and links in proteins

“…We modify the Khovanov cohomology [14] to define a new mathematical object in the framework of modified Grothendieck cohomology [15,16] for the definition of time series [17] of knots and links in proteins. The source of folding diversity from the primary structure of protein is considered to be a source of knot hidden super state of link [18] between interaction of all species. This happens in the space of superstates over principal fiber bundle of secondary protein with curvature of docking.…”