Phases of the secondary structures of RNA sequences

– Formation of RNA secondary structures is an example of the sequence-structure problem omnipresent in molecular biophysics. A basic theoretical issue concerns the phase behaviour of self-attracting random RNA sequences. By studying a simplified toy model of RNA folding, we show that there are two distinct possible phases for the random RNA below its melting transition —a molten phase in which an exponentially large number of allowed secondary structures have comparable free energies and coexist in thermal equilibrium, and a glass phase in which the equilibrium ensemble is dominated by one or a few structures with much lower free energies. Introduction. – RNA is an important biopolymer critical to all living systems [1]. Like the DNA, there are four types of nucleotides (or bases) A, C, G, and U which, when polymerized can form double helical structures consisting of stacks of stable Watson-Crick (A–U or G–C) pairs. However, unlike a long polymer of DNA, which is often accompanied by a complementary strand and forms otherwise featureless double-helical structures, a polymer of RNA is usually single stranded. It bends onto itself and forms elaborate spatial structures in order for bases located on different parts of the backbone to pair with each other. The structures encoded by the primary sequences often have important biological functions, much like the structures of proteins encoded by their amino acid sequences. Understanding the encoding of structure from the primary sequence has been an outstanding problem of theoretical biophysics. Most work in the past decade has been focused on the problem of protein folding, which is very difficult analytically and numerically [2]. Here, we study the problem of RNA folding, specifically the formation of RNA secondary structures which is more amenable to analytical and numerical approaches due to a separation of energy scales [3]. Efficient algorithms [4, 5] together with carefully measured free-energy parameters [6] describing the formation of various microscopic structures (e.g., stacks, loops, hairpins, etc.) allow the exact calculation of the ensemble of secondary structures formed by a given RNA molecule of up to a few thousand bases. In this work, we are not concerned with the structure formed by a specific sequence. Instead, we will study the statistics of secondary structures formed by the ensemble of long random RNA sequences. At high enough temperatures, the entropy gain of a freely fluctuating polymer backbone outweighs the binding energy to be gained by any base pairing and the RNA is in its denatured phase. As the temperature is lowered, base pairs are formed and compact structures appear. The transition between the denatured phase and compact phases