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Documents authored by Shalaby, Ahmed


Document
Algorithmic Hardness of the Partition Function for Nucleic Acid Strands

Authors: Gwendal Ducloz, Ahmed Shalaby, and Damien Woods

Published in: LIPIcs, Volume 347, 31st International Conference on DNA Computing and Molecular Programming (DNA 31) (2025)


Abstract
To understand and engineer biological and artificial nucleic acid systems, algorithms are employed for prediction of secondary structures at thermodynamic equilibrium. Dynamic programming algorithms are used to compute the most favoured, or Minimum Free Energy (MFE), structure, and the Partition Function (PF) - a tool for assigning a probability to any structure. However, in some situations, such as when there are large numbers of strands, or pseudoknotted systems, NP-hardness results show that such algorithms are unlikely, but only for MFE. Curiously, algorithmic hardness results were not shown for PF, leaving two open questions on the complexity of PF for multiple strands and single strands with pseudoknots. The challenge is that while the MFE problem cares only about one, or a few structures, PF is a summation over the entire secondary structure space, giving theorists the vibe that computing PF should not only be as hard as MFE, but should be even harder. We answer both questions. First, we show that computing PF is #P-hard for systems with an unbounded number of strands, answering a question of Condon Hajiaghayi, and Thachuk [DNA27]. Second, for even a single strand, but allowing pseudoknots, we find that PF is #P-hard. Our proof relies on a novel magnification trick that leads to a tightly-woven set of reductions between five key thermodynamic problems: MFE, PF, their decision versions, and #SSEL that counts structures of a given energy. Our reductions show these five problems are fundamentally related for any energy model amenable to magnification. That general classification clarifies the mathematical landscape of nucleic acid energy models and yields several open questions.

Cite as

Gwendal Ducloz, Ahmed Shalaby, and Damien Woods. Algorithmic Hardness of the Partition Function for Nucleic Acid Strands. In 31st International Conference on DNA Computing and Molecular Programming (DNA 31). Leibniz International Proceedings in Informatics (LIPIcs), Volume 347, pp. 1:1-1:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)


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@InProceedings{ducloz_et_al:LIPIcs.DNA.31.1,
  author =	{Ducloz, Gwendal and Shalaby, Ahmed and Woods, Damien},
  title =	{{Algorithmic Hardness of the Partition Function for Nucleic Acid Strands}},
  booktitle =	{31st International Conference on DNA Computing and Molecular Programming (DNA 31)},
  pages =	{1:1--1:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-399-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{347},
  editor =	{Schaeffer, Josie and Zhang, Fei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.31.1},
  URN =		{urn:nbn:de:0030-drops-238504},
  doi =		{10.4230/LIPIcs.DNA.31.1},
  annote =	{Keywords: Partition function, minimum free energy, nucleic acid, DNA, RNA, secondary structure, computational complexity, #P-hardness}
}
Document
Track A: Algorithms, Complexity and Games
An Efficient Algorithm to Compute the Minimum Free Energy of Interacting Nucleic Acid Strands

Authors: Ahmed Shalaby and Damien Woods

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)


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.

Cite as

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)


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@InProceedings{shalaby_et_al:LIPIcs.ICALP.2025.130,
  author =	{Shalaby, Ahmed and Woods, Damien},
  title =	{{An Efficient Algorithm to Compute the Minimum Free Energy of Interacting Nucleic Acid Strands}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{130:1--130:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.130},
  URN =		{urn:nbn:de:0030-drops-235071},
  doi =		{10.4230/LIPIcs.ICALP.2025.130},
  annote =	{Keywords: Minimum free energy, MFE, partition function, nucleic acid, DNA, RNA, secondary structure, computational complexity, algorithm analysis and design, dynamic programming}
}
Document
Domain-Based Nucleic-Acid Minimum Free Energy: Algorithmic Hardness and Parameterized Bounds

Authors: Erik D. Demaine, Timothy Gomez, Elise Grizzell, Markus Hecher, Jayson Lynch, Robert Schweller, Ahmed Shalaby, and Damien Woods

Published in: LIPIcs, Volume 314, 30th International Conference on DNA Computing and Molecular Programming (DNA 30) (2024)


Abstract
Molecular programmers and nanostructure engineers use domain-level design to abstract away messy DNA/RNA sequence, chemical and geometric details. Such domain-level abstractions are enforced by sequence design principles and provide a key principle that allows scaling up of complex multistranded DNA/RNA programs and structures. Determining the most favoured secondary structure, or Minimum Free Energy (MFE), of a set of strands, is typically studied at the sequence level but has seen limited domain-level work. We analyse the computational complexity of MFE for multistranded systems in a simple setting were we allow only 1 or 2 domains per strand. On the one hand, with 2-domain strands, we find that the MFE decision problem is NP-complete, even without pseudoknots, and requires exponential time algorithms assuming SAT does. On the other hand, in the simplest case of 1-domain strands there are efficient MFE algorithms for various binding modes. However, even in this single-domain case, MFE is P-hard for promiscuous binding, where one domain may bind to multiple as experimentally used by Nikitin [Nat Chem., 2023], which in turn implies that strands consisting of a single domain efficiently implement arbitrary Boolean circuits.

Cite as

Erik D. Demaine, Timothy Gomez, Elise Grizzell, Markus Hecher, Jayson Lynch, Robert Schweller, Ahmed Shalaby, and Damien Woods. Domain-Based Nucleic-Acid Minimum Free Energy: Algorithmic Hardness and Parameterized Bounds. In 30th International Conference on DNA Computing and Molecular Programming (DNA 30). Leibniz International Proceedings in Informatics (LIPIcs), Volume 314, pp. 2:1-2:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{demaine_et_al:LIPIcs.DNA.30.2,
  author =	{Demaine, Erik D. and Gomez, Timothy and Grizzell, Elise and Hecher, Markus and Lynch, Jayson and Schweller, Robert and Shalaby, Ahmed and Woods, Damien},
  title =	{{Domain-Based Nucleic-Acid Minimum Free Energy: Algorithmic Hardness and Parameterized Bounds}},
  booktitle =	{30th International Conference on DNA Computing and Molecular Programming (DNA 30)},
  pages =	{2:1--2:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-344-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{314},
  editor =	{Seki, Shinnosuke and Stewart, Jaimie Marie},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.30.2},
  URN =		{urn:nbn:de:0030-drops-209304},
  doi =		{10.4230/LIPIcs.DNA.30.2},
  annote =	{Keywords: Domain-based DNA designs, minimum free energy, efficient algorithms, NP-hard, P-hard, NC, fixed-parameter tractable}
}
Document
Minimum Free Energy, Partition Function and Kinetics Simulation Algorithms for a Multistranded Scaffolded DNA Computer

Authors: Ahmed Shalaby, Chris Thachuk, and Damien Woods

Published in: LIPIcs, Volume 276, 29th International Conference on DNA Computing and Molecular Programming (DNA 29) (2023)


Abstract
Polynomial time dynamic programming algorithms play a crucial role in the design, analysis and engineering of nucleic acid systems including DNA computers and DNA/RNA nanostructures. However, in complex multistranded or pseudoknotted systems, computing the minimum free energy (MFE), and partition function of nucleic acid systems is NP-hard. Despite this, multistranded and/or pseudoknotted systems represent some of the most utilised and successful systems in the field. This leaves open the tempting possibility that many of the kinds of multistranded and/or pseudoknotted systems we wish to engineer actually fall into restricted classes, that do in fact have polynomial time algorithms, but we've just not found them yet. Here, we give polynomial time algorithms for MFE and partition function calculation for a restricted kind of multistranded system called the 1D scaffolded DNA computer. This model of computation thermodynamically favours correct outputs over erroneous states, simulates finite state machines in 1D and Boolean circuits in 2D, and is amenable to DNA storage applications. In an effort to begin to ask the question of whether we can naturally compare the expressivity of nucleic acid systems based on the computational complexity of prediction of their preferred energetic states, we show our MFE problem is in logspace (the complexity class L), making it perhaps one of the simplest known, natural, nucleic acid MFE problems. Finally, we provide a stochastic kinetic simulator for the 1D scaffolded DNA computer and evaluate strategies for efficiently speeding up this thermodynamically favourable system in a constant-temperature kinetic regime.

Cite as

Ahmed Shalaby, Chris Thachuk, and Damien Woods. Minimum Free Energy, Partition Function and Kinetics Simulation Algorithms for a Multistranded Scaffolded DNA Computer. In 29th International Conference on DNA Computing and Molecular Programming (DNA 29). Leibniz International Proceedings in Informatics (LIPIcs), Volume 276, pp. 1:1-1:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{shalaby_et_al:LIPIcs.DNA.29.1,
  author =	{Shalaby, Ahmed and Thachuk, Chris and Woods, Damien},
  title =	{{Minimum Free Energy, Partition Function and Kinetics Simulation Algorithms for a Multistranded Scaffolded DNA Computer}},
  booktitle =	{29th International Conference on DNA Computing and Molecular Programming (DNA 29)},
  pages =	{1:1--1:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-297-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{276},
  editor =	{Chen, Ho-Lin and Evans, Constantine G.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.29.1},
  URN =		{urn:nbn:de:0030-drops-187840},
  doi =		{10.4230/LIPIcs.DNA.29.1},
  annote =	{Keywords: thermodynamic computation, model of computation, molecular computing, minimum free energy, partition function, DNA computing, DNA self-assembly, DNA strand displacement, kinetics simulation}
}
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