We study the fraction $f$ of nucleotides involved in the formation of a cactuslike 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 $c\le c_{\mathrm{cr}}$ as the chain length $n$ goes to infinity, signaling the formation of a virtually perfect gapless secondary structure; while $f(c)<1$ for $c>c_{\mathrm{cr}}$, which means that a nonperfect structure with gaps is formed. The strict upper and lower bounds $2\le c_{\mathrm{cr}}\le 4$ are proven, and the numerical evidence for $c_{\mathrm{cr}}$ is presented. The relevance of the transition from the evolutional point of view is discussed.

• Received 24 December 2011

DOI:https://doi.org/10.1103/PhysRevLett.109.018102