Ultrametricity of Energy Minimum Configurations of RNA Secondary Structures in the Nussinov Model

We present a numerical study of the ultrametric properties of the set of RNA secondary structures with the maximum number of base pairs (energetically degenerate minima) within the maximum matching model (Nussinov algorithm). Using 18 natural small nuclear RNAs as examples, we show that the degree of nontrivial ultrametricity varies widely. We consider the optimization problem for the degree of nontrivial ultrametricity of RNA secondary structures with the maximum number of base pairs under a fixed nucleotide composition. It is found that permuting the nucleotide sequence can strongly change the degree of ultrametricity, indicating the key role of nucleotide order in shaping the hierarchical properties of RNA secondary structures.