Metabolic network visualization using constraint planar graph drawing algorithm

ABSTRACTWe develop tools to explore and catalogue the topologies of knotted or pseudoknotted circular folds due to secondary and tertiary interactions within a closed loop of RNA which generate multiple double-helices due (for example) to strand complementarity. The fold topology is captured by a ‘contracted fold’ which merges helices separated by bulges and removes hairpin loops. Contracted folds are either trivial or pseudoknotted. Strand folding is characterised by a rigid-vertex ‘polarised strand graph’, whose vertices correspond to double-helices and edges correspond to strands joining those helices. Each vertex has a plumbline whose polarisation direction defines the helical axis. That polarised graph has a corresponding circular ribbon diagram and canonical alphanumeric fold label. Key features of the ‘fully-flagged’ fold are the arrangement of complementary domains along the strand, described by a numerical bare fold label, and a pair of binary ‘flags’: a parity flag that specifies the twist in each helix (even or odd half-twists), and an orientation flag that characterises each double-helix as parallel or antiparallel. A simple algorithm is presented to translate an arbitrary fold label into a polarised strand graph. Any embedding of the graph in 3-space is an admissible fold geometry; the simplest embeddings minimise the number of edge-crossings in a planar graph drawing. If that number is zero, the fold lies in one of two classes: (a)-type ‘relaxed’ folds, which contain conventional junctions and (b)-type folds whose junctions are described as meso-junctions in H. Wang and N.C. Seeman, Biochem, vol. 34, pp920-929. (c)-type folds induce polarised strand graphs with edge-crossings, regardless of the planar graph drawing. Canonical fold labelling allows us to sort and enumerate all ‘semi-flagged’ folds with up to six contracted double-helices as windings around the edges of a graph-like fold skeleton, whose cyclomatic number - the ‘fold genus’ - ranges from 0 – 3, resulting in a pair of duplexed strands along each skeletal edge. Those semi-flagged folds admit both even and odd double-helical twists. Appending specific parity flags to those semi-flagged folds gives fully-flagged (a)-type folds, which are also enumerated up to genus-3 cases. We focus on all-antiparallel folds, characteristic of conventional ssRNA and enumerate all distinct (a), (b) and (c)-type folds with up to five double-helices. Those circular folds lead to pseudoknotted folds for linear ssRNA strands. We describe all linear folds derived from (a) or (b)-type circular folds with up to four contracted double-helices, whose simplest cases correspond to so-called H, K and L pseudoknotted folds, detected in ssRNA. Fold knotting is explored in detail, via constructions of so-called antifolds and isomorphic folds. We also tabulate fold knottings for (a) and (b)-type folds whose embeddings minimise the number of edge-crossings and outline the procedure for (c)-type folds. The inverse construction - from a specific knot to a suitable nucleotide sequence - results in a hierarchy of knots. A number of specific alternating knots with up to 10 crossings emerge as favoured fold designs for ssRNA, since they are readily constructed as (a)-type all-antiparallel folds.

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