New alphabet-dependent morphological transition in a random RNA alignment

O. V. Valba 1, 2, M. V. Tamm 3, S. K. Nechaev 1, 4

Physical Review Letters 109 (2012) 018102

We study the fraction $f$ of nucleotides involved in the formation of a cactus--like secondary structure of random heteropolymer RNA--like molecules. In the low--temperature limit we study this fraction as a function of the number $c$ of different nucleotide species. We show, that with changing $c$, the secondary structures of random RNAs undergo a morphological transition: $f(c)\to 1$ for $c \le c_{\rm cr}$ as the chain length $n$ goes to infinity, signaling the formation of a virtually 'perfect' gapless secondary structure; while $f(c)<1$ for $c>c_{\rm cr}$, what means that a non-perfect structure with gaps is formed. The strict upper and lower bounds $2 \le c_{\rm cr} \le 4$ are proven, and the numerical evidence for $c_{\rm cr}$ is presented. The relevance of the transition from the evolutional point of view is discussed.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Moscow Institute of Physics and Technology (MIPT),
Moscow Institute of Physics and Technology
• 3. Physics Department,
Moscow State University
• 4. P. N. Lebedev Physical Institute,
Russian Academy of Science