Classification and predictions of RNA pseudoknots based on topological invariants

We propose a new topological characterization of ribonucleic acid (RNA) secondary structures with pseudoknots based on two topological invariants. Starting from the classic arc representation of RNA secondary structures, we consider a model that couples both (i) the topological genus of the graph and (ii) the number of crossing arcs of the corresponding primitive graph. We add a term proportional to these topological invariants to the standard free energy of the RNA molecule, thus obtaining a novel free-energy parametrization that takes into account the abundance of topologies of RNA pseudoknots observed in RNA databases.

Figure 1

In the arc-diagram representation of an RNA, crossing arcs indicate the presence of a pseudoknot.

Figure 2

The arc-diagam representation of an “ABAB” H-pseudoknot, its double-line representation, and its embedding on a torus on which the arc diagram can be drawn without crossings. This corresponds to a topological genus $g=1$.

Figure 3

There are four type of primitive pseudoknots with genus $g=1$.

Figure 4

There are eight primitive pseudoknots with genus $g=2$ and crossing number $N_c=3$.

Figure 5

A reducible diagram can be decomposed in two disconnected components by cutting the backbone once (dotted line).

Figure 6

Arc renormalization: Any sheaf of parallel arcs can be mapped into a single arc.