Enumeration of RNA complexes via random matrix theory

Area of Science:

• * Mathematical Physics
• * Computational Biology
• * Random Matrix Theory

Background:

• * RNA complex formation involves intricate topological arrangements.
• * Quantifying these topological structures is a significant challenge.
• * Existing methods may not fully capture the complexity of RNA topologies.

Purpose of the Study:

• * To derive a method for calculating the numbers of RNA complexes with arbitrary topology.
• * To establish a connection between RNA complex numbers and a specific Hermitian matrix model.
• * To explore the mathematical significance of these numbers in combinatorics and geometry.

Main Methods:

• * Utilizing a Hermitian matrix model with a specific potential function V(x)=x^2/2-stx/(1-tx).
• * Employing the topological recursion formalism, a technique from random matrix theory.
• * Relating the free energy of the matrix model to the counts of RNA complexes.

Main Results:

• * The free energy of the specified matrix model directly encodes the numbers of RNA complexes.
• * Generating parameters 's' and 't' correspond to the number of RNA molecules and hydrogen bonds.
• * The derived numbers also count mathematical objects like chord diagrams and cells in moduli spaces.

Conclusions:

• * The Hermitian matrix model and topological recursion provide a powerful framework for enumerating RNA complex topologies.
• * This work bridges concepts in physics, biology, and mathematics.
• * The findings offer new perspectives on understanding RNA structure and its mathematical underpinnings.