Topological phase transition in a RNA model in the de Gennes regime

We study a simplified model of the RNA molecule proposed by G. Vernizzi, H. Orland and A. Zee in the regime of strong concentration of positive ions in solution. The model considers a flexible chain of equal bases that can pairwise interact with any other one along the chain, while preserving the property of saturation of the interactions. In the regime considered, we observe the emergence of a critical temperature T_c separating two phases that can be characterized by the topology of the predominant configurations: in the large temperature regime, the dominant configurations of the molecule have very large genera (of the order of the size of the molecule), corresponding to a complex topology, whereas in the opposite regime of low temperatures, the dominant configurations are simple and have the topology of a sphere. We determine that this topological phase transition is of first order and provide an analytic expression for T_c. The regime studied for this model exhibits analogies with that for the dense polymer systems studied by de Gennes

💡 Research Summary

The paper investigates a highly simplified statistical‑mechanical model of an RNA molecule under conditions of strong positive‑ion concentration, a regime often referred to as the de Gennes regime. The model consists of a flexible polymer chain of N identical bases. Any base can pair with any other base along the chain, but the “saturation” constraint forbids a base from forming more than one pair simultaneously. This restriction translates, in the language of matrix models, into a topological expansion in terms of the genus g of the associated Feynman‑type diagrams (or “fat graphs”).

The authors formulate the partition function Z(N,T) as a matrix integral and perform a 1/N expansion. The free energy per monomer can be written as a sum over genera, \