Undesignable motifs in structural RNAs and combinatorial consequences

RNA design aims at constructing RiboNucleic Acids (RNA) sequences that perform a predefined biological function, usually modeled by multiple constraints on the sequence and structure level. In its most popular setting, called the inverse folding problem, designed RNAs should adopt a predefined target secondary structure, preferentially to any alternative structure. It was previously observed that some secondary structures are undesignable, i.e. no RNA sequence can fold uniquely into the target structure while satisfying some criterion measuring how preferential this folding is compared to alternative conformations. We show that the proportion of designable secondary structures decreases exponentially with the size of the target secondary structure, for various popular combinations of energy models and design objectives. This exponential decay is, at least in part, due to the existence of undesignable motifs, which can be generically constructed, and jointly analyzed to yield asymptotic upper-bounds on the number of designable structures. Finally, we define a lower bound of the minimal ensemble defect of a secondary structure. We show that, across uniformly distributed secondary structures, such lower bound admits a normal limiting distribution whose two parameters, the expected value and the variance, both growing linearly with the size of secondary structure.