The simplest 38 (or so) fold-classes of RNA (or DNA) by base-pairing. Knots and tangles from string sequences, and sequences for knots and tangles

Abstract: RNA is known to fold into 'pseudo knots' in a variety of viruses. Though true knots have not been detected, it is fair to wonder if they are likely in natural or synthetic biological systems. I will explore folding of a single string of RNA into double-helices, via Watson-Crick duplexing, allowing the possibility of strand junctions, found, for example, in transient Holliday junctions in vivo. Further, either parallel or antiparallel double-helices are admitted. Given some arrangement of complementary base pairs on a strand, an oriented rigid-vertex graph can be constructed which encodes the resulting fold. Conversely, given a fold, we can build a graph and deduce the the arrangement of complementary base pairs on the strand.

Simplest folds can be built by winding a single strand on oriented genus-g surfaces formed by tubifying graphs of cyclomatic number g, such that a duplexed pair of locally parallel strands wrap each tubule of the manifold. More complex folds have combinations of pairs of of tracks (2-track railways) and single tracks (1,2-track railways). 2-track railways give relaxed folds, so-named as they be embedded such that all base pairs are duplexed. 1,2-track railways necessarily have unduplexed hairpin-like strands within the fold. It turns out that for g<4, there are few folds among the 40 or so whose double-helices are all antiparallel - the preferred orientation for RNA and DNA duplexes.

A connection between folds and knots is possible via 2-track railways, whose consequences will be discussed if time permits.

computational geometryalgebraic topologycombinatoricsgeometric topologyprobability

Audience: advanced learners

Asia Pacific Seminar on Applied Topology and Geometry