Asymptotic Number of Hairpins of Saturated RNA Secondary Structures

In the absence of chaperone molecules, RNA folding is believed to depend on the distribution of kinetic traps in the energy landscape of all secondary structures. Kinetic traps in the Nussinov energy model are precisely those secondary structures that are saturated, meaning that no base pair can be added without introducing either a pseudoknot or base triple. In this paper, we compute the asymptotic expected number of hairpins in saturated structures. For instance, if every hairpin is required to contain at least θ=3 unpaired bases and the probability that any two positions can base-pair is p=3/8, then the asymptotic number of saturated structures is 1.34685?n ^?3/2?1.62178 ⁿ , and the asymptotic expected number of hairpins follows a normal distribution with mean $0.06695640 \cdot n + 0.01909350 \cdot\sqrt{n} \cdot\mathcal{N}$ . Similar results are given for values θ=1,3, and p=1,1/2,3/8; for instance, when θ=1 and p=1, the asymptotic expected number of hairpins in saturated secondary structures is 0.123194?n, a value greater than the asymptotic expected number 0.105573?n of hairpins over all secondary structures. Since RNA binding targets are often found in hairpin regions, it follows that saturated structures present potentially more binding targets than nonsaturated structures, on average. Next, we describe a novel algorithm to compute the hairpin profile of a given RNA sequence: given RNA sequence a ₁,…,a _n , for each integer k, we compute that secondary structure S _k having minimum energy in the Nussinov energy model, taken over all secondary structures having k hairpins. We expect that an extension of our algorithm to the Turner energy model may provide more accurate structure prediction for particular RNAs, such as tRNAs and purine riboswitches, known to have a particular number of hairpins. Mathematica^? computations, C and Python source code, and additional supplementary information are available at the website http://bioinformatics.bc.edu/clotelab/RNAhairpinProfile/.