Minimum length RNA folding trajectories

Background: Existent programs for RNA folding kinetics, such as Kinefold, Kinfold and KFOLD,implement the Gillespie algorithm to generate stochastic folding trajectories from an initial structure sto a target structure t, in which each intermediate secondary structure is obtained from its predecessorby the application of a move from a given move set. The Kinfold move set MS1 [resp. MS2] allows theaddition or removal [resp. addition, removal or shift] of a single base pair. Define the MS1 [resp. MS2]distance between secondary structures s and t to be the minimum path length to refold s to t, wherea move from MS1 [resp. MS2] is applied in each step. The MS1 distance between s and t is triviallyequal to the cardinality of the symmetric difference of s and t, i.e the number of base pairs belonging toone structure but not the other; in contrast, the computation of MS2 distance is highly non-trivial.Results: We describe algorithms to compute the shortest MS2 folding trajectory between any two givenRNA 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. A 10-fold slowerversion of our IP algorithm appeared in WABI 2017; the current version exploits special treatment ofclosed 2-cycles.Our optimal IP [resp. near-optimal IP] algorithm maximizes [resp. approximately maximizes] thenumber of shifts and minimizes [resp. approximately minimizes] the number of base pair additions andremovals 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 thefinal optimal folding trajectory.We prove NP-hardness of the problem to determine the minimum barrier energy over all possibleMS2 folding pathways, and conjecture that computing the MS2 distance between arbitrary secondarystructures is NP-hard. Since our optimal IP algorithm relies on the FVS, known to be NP-completefor arbitrary digraphs, we compare the family of RNA conflict digraphs with the following classes ofdigraphs – planar, reducible flow graph, Eulerian, and tournament – for which FVS is known to be eitherpolynomial time computable or NP-hard.Conclusion: This paper describes a number of optimal and near-optimal algorithms to compute theshortest MS2 folding trajectory between any two secondary structures. Source code for our algorithmsis available at http://bioinformatics.bc.edu/clotelab/MS2distance/.