Random words in free groups, non-crossing matchings and RNA secondary structures

Gadgil, S and Krishnapur, M (2022) Random words in free groups, non-crossing matchings and RNA secondary structures. In: Indian Journal of Pure and Applied Mathematics .

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Abstract

Consider a random word Xn= (X1, … , Xn) in an alphabet consisting of 4 letters, with the letters viewed either as A, U, G and C (i.e., nucleotides in an RNA sequence) or α, α¯ , β and β¯ (i.e., generators of the free group ⟨ α, β⟩ and their inverses). We show that the expected fraction ρ(n) of unpaired bases in an optimal RNA secondary structure (with only Watson-Crick bonds and no pseudo-knots) converges to a constant λ2 with 0 < λ2< 1 as n→ ∞. Thus, a positive proportion of the bases of a random RNA string do not form hydrogen bonds. We do not know the exact value of λ2, but we derive upper and lower bounds for it. In terms of free groups, ρ(n) is the ratio of the length of the shortest word representing X in the generating set consisting of conjugates of generators and their inverses to the word length of X with respect to the standard generators and their inverses. Thus for a typical word the word length in the (infinite) generating set consisting of the conjugates of standard generators grows linearly with the word length in the standard generators. In fact, we show that a similar result holds for all non-abelian finitely generated free groups ⟨ α1, ⋯ , αk⟩ , k≥ 2.

Item Type:	Journal Article
Publication:	Indian Journal of Pure and Applied Mathematics
Publisher:	Indian National Science Academy
Additional Information:	The copyright for this article belongs to the Indian National Science Academy, Springer.
Keywords:	Conjuacy-invariant norms; Non-abelian free groups; Noncrossing matchings; RNA secondary structure
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	04 Jul 2022 06:21
Last Modified:	04 Jul 2022 06:21
URI:	https://eprints.iisc.ac.in/id/eprint/74232

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