[Paper Review] Asymptotic connectivity for the network of RNA secondary structures

Peter Clote|arXiv (Cornell University)|Aug 16, 2015

RNA and protein synthesis mechanisms27 references3 citations

TL;DR

This paper derives the asymptotic expected network connectivity of RNA secondary structures in the homopolymer model using algebraic combinatorics and the Flajolet-Odlyzko theorem. It proves that the average number of neighboring structures (base pair distance 1) for a length-n RNA sequence is asymptotically 0.473418⋯n, revealing a key topological property of the RNA secondary structure network.

ABSTRACT

Given an RNA sequence a, consider the network G = (V;E), where the set V of nodes consists of all secondary structures of a, and whose edge set E consists of all edges connecting two secondary structures whose base pair distance is 1. Define the network connectivity, or expected network degree, as the average number of edges incident to vertices of G. Using algebraic combinatorial methods, we prove that the asymptotic connectivity of length n homopolymer sequences is 0:473418 ? n. This raises the question of what other network properties are characteristic of the network of RNA secondary structures. Programs in Python, C and Mathematica are available at the web site http://bioinformatics.bc.edu/clotelab/ RNAexpNumNbors.

Motivation & Objective

• To characterize the network topology of RNA secondary structures by computing the expected number of neighboring structures.
• To determine the asymptotic behavior of network connectivity in the homopolymer model where any base can pair with any other.
• To apply advanced analytic combinatorics to derive precise asymptotic expressions for network properties of RNA structures.
• To extend the analysis beyond the homopolymer model to arbitrary stickiness values using minor modifications to algorithms.

Proposed method

• Uses generating functions and recurrence relations to compute Qn (total neighbor count) and Zn (total number of secondary structures) for subsequences of length n.
• Applies the Flajolet-Odlyzko theorem to extract asymptotic coefficients from generating functions of Qn and Zn.
• Models the network as a graph G = (V, E), where vertices are secondary structures and edges connect structures differing by one base pair.
• Derives asymptotic expansions for Qn and Zn using singularity analysis of generating functions, focusing on the dominant singularity at ρ ≈ 0.436911.
• Establishes that [x^n]q2 ~ 0.0644456 ⋅ 2.288795^n / n^{1/2} and [x^n]z1 ~ 0.1361285 ⋅ 2.288795^n / n^{3/2}.
• Computes the ratio [x^n]q2 / [x^n]z1 to obtain the asymptotic expected connectivity as 0.4734176431521986 ⋅ n.

Experimental results

Research questions

• RQ1 What is the asymptotic expected number of neighboring RNA secondary structures for a homopolymer sequence of length n?
• RQ2 How does the network connectivity of RNA secondary structures scale with sequence length n?
• RQ3 What algebraic and analytic techniques can be used to derive asymptotic expressions for network properties of RNA structures?
• RQ4 How does the network topology of RNA secondary structures compare to small-world or other complex network models?

Key findings

• The asymptotic expected network connectivity for length-n homopolymer RNA sequences is 0.4734176431521986 ⋅ n.
• The dominant singularity of the generating function occurs at ρ ≈ 0.436911, governing the exponential growth rate of secondary structures.
• The asymptotic number of secondary structures grows as ~0.1361285 ⋅ 2.288795^n / n^{3/2}.
• The total number of neighbor relationships across all structures scales as ~0.0644456 ⋅ 2.288795^n / n^{1/2}.
• The ratio of neighbor counts to total structures yields a linear scaling of connectivity with sequence length n.
• The result confirms that RNA secondary structure networks exhibit a well-defined, predictable average connectivity in the asymptotic regime.