Combinatorial properties of RNA secondary structures

Area of Science:

• Bioinformatics
• Computational Biology
• RNA Structure Analysis

Background:

• RNA secondary structure is crucial for molecular interactions and stability.
• Classifying RNA structures by order has numerous applications.
• Quantifying RNA secondary structures of a specific order (p) has been a long-standing challenge since Waterman's 1978 framework.

Purpose of the Study:

• To derive a precise asymptotic equivalent for a(n,p), the number of RNA secondary structures of size n and order p.
• To extend enumeration to include the number of unpaired bases as a parameter.
• To compute average order and moments of RNA secondary structures and enumerate substructures.

Main Methods:

• Utilizing generating functions for binary tree structures with Horton-Strahler numbers.
• Deriving generating functions for RNA secondary structures of order p based on Viennot et al.'s observation.
• Applying these functions to compute asymptotic equivalents and related enumeration problems.

Main Results:

• A precise asymptotic equivalent for a(n,p) has been computed.
• The number of structures can be determined with unpaired bases as an additional parameter.
• The average order, r-th moments, and counts of substructures (hairpins, bulges) are computable.

Conclusions:

• The developed method provides a robust framework for enumerating RNA secondary structures based on their order.
• This approach offers new insights into RNA structural complexity and the distribution of substructures.
• The findings have implications for understanding RNA function and interactions in biological systems.