Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology

Abstract

Grid diagrams with their relatively simple mathematical formalism provide a convenient way to generate and model projections of various knots. It has been an open question whether these 2D diagrams can be used to model a complex 3D process such as the topoisomerasemediated preferential unknotting of DNA molecules. We model here topoisomerase-mediated passages of double-stranded DNA segments through each other using the formalism of grid diagrams. We show that this grid diagram-based modelling approach captures the essence of the preferential unknotting mechanism, based on topoisomerase selectivity of hooked DNA juxtapositions as the sites of intersegmental passages. We show that grid diagram-based approach provide an important, new and computationally convenient framework for investigating entanglement in biopolymers. arXiv:1909.05937v1 [q-bio.BM]

GridPyM: A Python module to handle grid diagrams

“…, O n ], where the i-th component denotes the position (from the left, and starting from 0) of the marking on the i-th row, where rows are enumerated from the bottom up. For example, the grid in the left of Figure 1 is described by the pair of markings ‫ޘ‬ = [7, 2, 4, 5, 6, 1, 0, 3] and ‫ޏ‬ = [5,6,7,0,3,4,1,2]. Note that the combinatorial requirement on the markings implies that ‫ޘ‬ and ‫ޏ‬ are a pair of collision-free permutations of {0, .…”

“…As an example, they have been used to quantify the intensity of crossing change-mediated fluxes between different knot types, and to demonstrate the dependency of those fluxes on the local geometry of knotted configurations. In turn, this provided strong evidence that the simplification action of specific DNA enzymes is driven by a geometric selection of sites [4]. One further potential use of grid diagrams is to help with the search of band changes and the determination of Gordian distance between knot types [6; 7; 18; 26].…”

Molecular Dynamics of Topological Barriers on the Crystallization Behavior of Ring Polyethylene Melts with Trefoil Knots

“…We believe that expanding such a cooperation between disciplines will deepen our understanding of polymer science. Finally, we expect that our developed mathematical tool will also be applicable for DNA trefoil knots − and synthesized trefoil knots. − Notably, DNA trefoil knots are interesting due to the action of topoisomerase in interconverting the trivial and trefoil knot topologies. We propose that the developed mathematical tool could be immediately applied when three-dimensional observation data are available for the DNA trefoil knot structures.…”