Enumeration of RNA Structures by Matrix Models

We enumerate the number of RNA contact structures according to their genus, i.e., the topological character of their pseudoknots. By using a recently proposed matrix model formulation for the RNA folding problem, we obtain exact results for the simple case of an RNA molecule with an infinitely flexible backbone, in which any arbitrary pair of bases is allowed. We analyze the distribution of the genus of pseudoknots as a function of the total number of nucleotides along the phosphate-sugar backbone.

Figure 1

All possible arc diagrams with $L=4$. Diagrams with $i$ arcs are associated with the power $v^i$, and $g$ is the genus.

Figure 2

Left: The normalized number of diagrams $a_{L,g}/\mathcal{N}$ at fixed $g$ as a function of $L$. Right: The same quantity at fixed $L$ as a function of $g$. In both cases we put $v=1$.