Minimum length RNA folding trajectories

The Kinfold and KFOLD programs for RNA folding kinetics implement the Gillespie algorithm to generate stochastic folding trajectories from an initial structure s to a target structure t, in which each intermediate secondary structure is obtained from its predecessor by the addition, removal or shift of a single base pair. Define MS2 distance between secondary structures s and t to be the minimum path length to refold s to t, where a move from MS2 is applied in each step. We describe algorithms to compute the shortest MS2 folding trajectory between any two given RNA secondary structures. These algorithms include an optimal integer programming (IP) algorithm, an accurate and efficient near-optimal algorithm, a greedy algorithm, a branch-and-bound algorithm, and an optimal algorithm if one allows intermediate structures to contain pseudoknots. Our optimal IP [resp. near-optimal IP] algorithm maximizes [resp. approximately maximizes] the number of shifts and minimizes [resp. approximately minimizes] the number of base pair additions and removals by applying integer programming to (essentially) solve the minimum feedback vertex set (FVS) problem for the RNA conflict digraph, then applies topological sort to tether subtrajectories into the final optimal folding trajectory. We prove NP-hardness of the problem to determine the minimum barrier energy over all possible MS2 folding pathways, and conjecture that computing the MS2 distance between arbitrary secondary structures is NP-hard. Since our optimal IP algorithm relies on the FVS, known to be NP-complete for arbitrary digraphs, we compare the family of RNA conflict digraphs with the following classes of digraphs (planar, reducible flow graph, Eulerian, and tournament) for which FVS is known to be either polynomial time computable or NP-hard. Source code available at http://bioinformatics.bc.edu/clotelab/MS2distance/.