Nature of the glassy phase of RNA secondary structure

– We characterize the low temperature phase of a simple model for RNA secondary structures by determining the typical energy scale E(l) of excitations involving l bases. At zero temperature, we find a scaling law E(l) ∼ l θ with θ ≈ 0.23, and this same scaling holds at low enough temperatures. Above a critical temperature, there is a different phase characterized by a relatively flat free energy landscape resembling that of a homopolymer with a scaling exponent θ = 1. These results strengthen the evidence in favour of the existence of a glass phase at low temperatures. Introduction. – The folding of RNA or single stranded DNA as described by its secondary structure is both a relevant problem in molecular biology and a challenging task for the statistical mechanics of disordered systems. Several authors [1–6] have recently addressed the topic and put forward some evidence for the existence of a glassy phase at low temperatures. Numerical studies of the specific heat demonstrate the existence of a higher order phase transition [2]. The nature and properties of the low temperature phase are less clear because of large finite size corrections. While the overlap distribution is certainly broad for systems of up to 1000 bases [1, 2], indicating a kind of glassy phase, its asymptotic behaviour for long sequences cannot be deduced reliably from the present simulations [1–4]. Bundschuh and Hwa [5, 6] have recently argued in favour of the existence of a glass phase. They showed analytically that in the disordered case the asymptotic pre-exponential scaling of the partition function cannot be the same as for homogeneous RNA at low temperatures. They also showed that the system of two attractively coupled replicas of the same disordered sequence exhibits a phase transition from a strongly coupled low temperature phase to a phase at high temperatures where the replicas are essentially independent. Both results favour the existence of a glass transition at finite temperature, but a rigorous proof is still missing. Numerically, the same authors characterize the RNA conformation via the free energy cost of an imposed pairing (pinching) of two bases. Concentrating on the largest possible pinching excitations in the groundstate of a given RNA sequence they argue in favour of an excitation energy scale that grows logarithmically with the number N of bases in the sequence, a weak power law not being ruled out. However, it is not clear in …