Topological classification and enumeration of RNA structures by genus.

Area of Science:

• Computational Biology
• RNA Structure Analysis
• Topological Mathematics

Background:

• RNA pseudoknot structures possess an inherent topological surface associated with genus.
• Previous work by Harer-Zagier provides a foundation for enumerating combinatorial structures.

Purpose of the Study:

• To compute generating functions for RNA structures based on genus and minimum stack size.
• To classify RNA structures using their topological genus.
• To explore the relationship between RNA structure enumeration and Riemann's moduli space.

Main Methods:

• Calculation of the generating function Dg,σ(z) for RNA structures with fixed genus (g) and minimum stack size (σ).
• Analysis of the asymptotic behavior of the number of RNA structures (dg,σ(n)) using the derived generating function.
• Establishing a correspondence between RNA shapes and the Penner-Strebel decomposition of Riemann's moduli space.

Main Results:

• The generating function Dg,σ(z) for RNA structures is shown to be algebraic.
• The number of RNA structures with minimum stack size two (dg,2(n)) exhibits asymptotic behavior: dg,2(n) ∼ kg n^(3(g−1/2))γn^2, with γ2 ≈ 1.9685.
• The genus primarily influences the sub-exponential factor for stack sizes of two or more.
• A slow growth rate of RNA structures compared to the total number of RNA molecules implies the existence of neutral networks.

Conclusions:

• RNA structures can be topologically classified by genus, with implications for understanding their diversity.
• The algebraic nature of generating functions and the slow growth rate suggest extensive neutral networks in RNA sequence space.
• A direct link is established between RNA enumerative problems and the geometric study of Riemann's moduli space.