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An Efficient Algorithm to Compute the Minimum Free Energy of Interacting Nucleic Acid Strands

Authors Ahmed Shalaby , Damien Woods

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LIPIcs.ICALP.2025.130.pdf
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ACM Subject Classification
  • Theory of computation → Algorithm design techniques
  • Theory of computation → Dynamic programming
Keywords
  • Minimum free energy
  • MFE
  • partition function
  • nucleic acid
  • DNA
  • RNA
  • secondary structure
  • computational complexity
  • algorithm analysis and design
  • dynamic programming

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Abstract

The information-encoding molecules RNA and DNA bind via base pairing to form an exponentially large set of secondary structures. Practitioners need algorithms to predict the most favoured structures, called minimum free energy (MFE) structures, or to compute a partition function that allows assigning a probability to any structure. MFE prediction is NP-hard in the presence pseudoknots - base pairings that violate a restricted planarity condition. However, for single-stranded unpseudoknotted structures, there are polynomial time dynamic programming algorithms. For multiple strands, the problem is significantly more complicated: Codon, Hajiaghayi and Thachuk [DNA27, 2021] proved it NP-hard for N bases and 𝒪(N) strands. Dirks, Bois, Schaeffer, Winfree and Pierce [SIAM Review, 2007] gave a polynomial time partition function algorithm for multiple (𝒪(1)) strands, now widely-used, however their technique did not generalise to MFE which they left open. 
We give an 𝒪(N⁴) time algorithm for unpseudoknotted multiple (𝒪(1)) strand MFE prediction, answering the open problem from Dirks et al. The challenge lies in considering the rotational symmetry of secondary structures, a global feature not immediately amenable to local subproblem decomposition used in dynamic programming. Our proof has two main technical contributions: First, a characterisation of symmetric secondary structures implying only quadratically many need to be considered when computing the rotational symmetry penalty. Second, that bound is leveraged by a backtracking algorithm to efficiently find the MFE in an exponential space of contenders.

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Ahmed Shalaby and Damien Woods. An Efficient Algorithm to Compute the Minimum Free Energy of Interacting Nucleic Acid Strands. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 130:1-130:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.130

Author Details

Ahmed Shalaby
  • Hamilton Institute and Department of Computer Science, Maynooth University, Ireland
Damien Woods
  • Hamilton Institute and Department of Computer Science, Maynooth University, Ireland

Funding

Supported by Science Foundation Ireland (SFI) under grant number 20/FFP-P/8843, European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 772766, Active-DNA project), and Funded by the European Union - European Innovation Council (EIC) and SMEs Executive Agency (EISMEA). Grant number 101115422, DISCO project. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union, ERC, EIC, SMEs Executive Agency (EISMEA) or SFI. Neither the European Union nor the granting authority can be held responsible for them.

Acknowledgements

We thank Constantine Evans for his helpful comments on the origin of the MFE rotational symmetry penalty from statistical mechanics, Mark Fornace, Niles Pierce, Dave Doty, Erik Winfree and Anne Condon for helpful comments.

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