An efficient algorithm to compute the minimum free energy of interacting nucleic acid strands

The information-encoding molecules RNA and DNA form a combinatorially large set of secondary structures through nucleic acid base pairing. Thermodynamic prediction algorithms predict favoured, or minimum free energy (MFE), secondary structures, and can assign an equilibrium probability to any structure via the partition function: a Boltzman-weighted sum over the set of secondary structures. MFE is NP-hard in the presence pseudoknots, base pairings that violate a restricted planarity condition. However, unpseudoknotted structures are amenable to dynamic programming: for a single DNA/RNA strand there are polynomial time algorithms for MFE and partition function. For multiple strands, the problem is more complicated due to entropic penalties. Dirks et al [SICOMP Review; 2007] showed that for O(1) strands, with N bases, there is a polynomial time in N partition function algorithm, however their technique did not generalise to MFE which they left open. We give the first polynomial time (O(N^4)) algorithm for unpseudoknotted multiple (O(1)) strand MFE, answering the open problem from Dirks et al. The challenge lies in considering rotational symmetry of secondary structures, a feature not immediately amenable to dynamic programming algorithms. Our proof has two main technical contributions: First, a polynomial upper bound on the number of symmetric secondary structures to be considered when computing rotational symmetry penalties. Second, that bound is leveraged by a backtracking algorithm to find the MFE in an exponential space of contenders. Our MFE algorithm has the same asymptotic run time as Dirks et al's partition function algorithm, suggesting efficient handling of rotational symmetry, although higher space complexity. It also seems reasonably tight in the number of strands since Codon, Hajiaghayi & Thachuk [DNA27, 2021] have shown that unpseudoknotted MFE is NP-hard for O(N) strands.