Counting RNA Loop Interaction Networks of Homology Group Rank Zero

An enumerative study of RNA secondary structures according to various characteristics is a topic of key importance in computational biology. RNA secondary structure pairs have also been studied in various contexts. Recently, the homology groups of the simplicial complices induced by pairs of secondary structures have been studied by Bura, He, and Reidys, providing a new way for characterizing these structure pairs. In particular, the homology group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mrow></mml:math> corresponding to any pair has been shown to be a free group. In this article, we provide enumerative results, both exactly and asymptotically, for those pairs giving a free group of rank zero. The asymptotic number of these structure pairs of length <i>n</i> is shown to be (0.2774624151…)(4.8105752536…)^<i>n</i><i>n</i>^-3/2. We also prove that the distribution of the number of base pairs in those pairs of secondary structures is asymptotically normal.