Document Open Access Logo

Scalable Enumeration of Pareto-Optimal Polymers for Computing Equilibrium Concentrations

Authors Archit Patil , Minki Hhan , David Soloveichik

PDF

File

Thumbnail PDF
PDF
LIPIcs.DNA.32.12.pdf
  • Filesize: 2.58 MB
  • 24 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Computer systems organization → Molecular computing
  • Theory of computation → Design and analysis of algorithms
  • Applied computing → Computational biology
Keywords
  • Molecular computation
  • Hilbert Basis
  • Thermodynamic Binding Network
  • covering design
  • equilibrium concentration

Metrics

Abstract

Predicting equilibrium concentrations of molecular complexes is essential for verifying the behavior of engineered DNA systems. However, a finite set of monomer types can in principle generate infinitely many complexes. We study this candidate-enumeration problem in a geometry-free, domain-level abstraction called a domain-monomer system, generalizing Thermodynamic Binding Networks (TBNs) to the unsaturated setting where not every possible bond need be formed. We define Pareto-suboptimal polymers as those that can be split into non-interacting parts, and show that restricting attention to Pareto-optimal polymers is thermodynamically justified: no Pareto-suboptimal polymer appears in any minimum free-energy configuration, and the total equilibrium concentration of such polymers is small. We prove that there are finitely many Pareto-optimal polymers and exactly characterize them via a Hilbert basis computation, extending prior work from the saturated TBN model. To scale this approach to large systems, we develop a framework that restricts the number of different monomer types that a single polymer contains, and uses combinatorial covering designs to reduce the number of Hilbert basis computations required. We benchmark the method on several families of DNA molecular programming systems, demonstrating order-of-magnitude speedups over direct computation while recovering nearly all equilibrium-relevant polymers.

Cite As Get BibTex

Archit Patil, Minki Hhan, and David Soloveichik. Scalable Enumeration of Pareto-Optimal Polymers for Computing Equilibrium Concentrations. In 32nd International Conference on DNA Computing and Molecular Programming (DNA 32). Leibniz International Proceedings in Informatics (LIPIcs), Volume 387, pp. 12:1-12:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DNA.32.12

Author Details

Archit Patil
  • University of Texas at Austin, TX, USA
Minki Hhan
  • Korea Advanced Institute of Science and Technology, Daejeon, South Korea
David Soloveichik
  • University of Texas at Austin, TX, USA

Funding

Schmidt Sciences Polymath Award to D.S., DOE grant DE-SC0024248, NSF SemiSynBio III: GOALI grant 2227578.

Acknowledgements

We thank And Kaan Ata Yilmaz for valuable discussions and insights.

Supplementary Materials

References

  1. 4ti2 Team. 4ti2 - a software package for algebraic, geometric and combinatorial problems on linear spaces. URL: https://4ti2.github.io.
  2. Hamidreza Akef, Minki Hhan, and David Soloveichik. Computing and bounding equilibrium concentrations in athermic chemical systems. In 31st International Conference on DNA Computing and Molecular Programming (DNA 31), pages 10-1. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.DNA.31.10.
  3. Keenan Breik, Cameron Chalk, David Doty, David Haley, and David Soloveichik. Programming substrate-independent kinetic barriers with thermodynamic binding networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 18(1):283-295, 2021. URL: https://doi.org/10.1109/TCBB.2019.2959310.
  4. Keenan Breik, Chris Thachuk, Marijn Heule, and David Soloveichik. Computing properties of stable configurations of thermodynamic binding networks. Theoretical Computer Science, 785:17-29, 2019. URL: https://doi.org/10.1016/J.TCS.2018.10.027.
  5. W. Bruns, B. Ichim, C. Söger, and U. von der Ohe. Normaliz: Algorithms for rational cones and affine monoids. https://www.normaliz.uni-osnabrueck.de. Google Scholar
  6. Winfried Bruns and Bogdan Ichim. Normaliz: Algorithms for affine monoids and rational cones. Journal of Algebra, 324(5):1098-1113, 2010. Google Scholar
  7. Kevin M Cherry and Lulu Qian. Supervised learning in DNA neural networks. Nature, 645(8081):639-647, 2025. Google Scholar
  8. Evelyne Contejean and Hervé Devie. An efficient incremental algorithm for solving systems of linear diophantine equations. Information and Computation, 113(1):143-172, 1994. URL: https://doi.org/10.1006/INCO.1994.1067.
  9. Pilu Crescenzi, Federico Montecalvo, and Gianluca Rossi. Optimal covering designs: Complexity results and new bounds. Discrete Applied Mathematics, 144(3):281-290, 2004. URL: https://doi.org/10.1016/J.DAM.2003.11.006.
  10. Robert M Dirks, Justin S Bois, Joseph M Schaeffer, Erik Winfree, and Niles A Pierce. Thermodynamic analysis of interacting nucleic acid strands. SIAM Review, 49(1):65-88, 2007. URL: https://doi.org/10.1137/060651100.
  11. David Doty, Trent A Rogers, David Soloveichik, Chris Thachuk, and Damien Woods. Thermodynamic binding networks. In International Conference on DNA-Based Computers, pages 249-266. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-66799-7_16.
  12. Mark E Fornace, Jining Huang, Cody T Newman, Avinash Nanjundiah, Nicholas J Porubsky, Marshall B Pierce, and Niles A Pierce. NUPACK: Analysis and design of nucleic acid structures, devices, and systems. ChemRxiv, 2025. Google Scholar
  13. Daniel Gordon. La jolla coverings repository, April 2026. URL: https://doi.org/10.5281/zenodo.19735294.
  14. Daniel M Gordon, Oren Patashnik, and Greg Kuperberg. New constructions for covering designs. Journal of Combinatorial Designs, 3(4):269-284, 1995. Google Scholar
  15. David Haley and David Doty. Computing properties of thermodynamic binding networks: An integer programming approach. In Matthew R. Lakin and Petr Šulc, editors, 27th International Conference on DNA Computing and Molecular Programming (DNA 27), volume 205 of Leibniz International Proceedings in Informatics (LIPIcs), pages 2:1-2:16, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.DNA.27.2.
  16. Raymond Hemmecke. On the computation of hilbert bases of cones. In Mathematical Software: Proceedings of the First International Congress of Mathematical Software, page 307. World Scientific, 2002. Google Scholar
  17. Miki Hermann, Laurent Juban, and Phokion G Kolaitis. On the complexity of counting the hilbert basis of a linear diophantine system. In International Conference on Logic for Programming Artificial Intelligence and Reasoning, pages 13-32. Springer, 1999. URL: https://doi.org/10.1007/3-540-48242-3_2.
  18. Loic Pottier. The euclidean algorithm in dimension n. In Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, pages 40-42, 1996. URL: https://doi.org/10.1145/236869.236894.
  19. Johanan Schonheim. On coverings. Pacific Journal of Mathematics, 14(4):1405-1411, 1964. Google Scholar
  20. Tianqi Song and Lulu Qian. Heat-rechargeable computation in DNA logic circuits and neural networks. Nature, 646(8084):315-322, 2025. Google Scholar
  21. Tristan Stérin, Abeer Eshra, Janet Adio, Constantine Glen Evans, and Damien Woods. A thermodynamically favoured molecular computer: Robust, fast, renewable, scalable. bioRxiv, 2025. Google Scholar
  22. Boya Wang, Cameron Chalk, David Doty, and David Soloveichik. Molecular computation at equilibrium via programmable entropy. Science Advances, 12:eadx3969, 2026. URL: https://doi.org/10.1126/sciadv.adx3969.
  23. And Kaan Ata Yilmaz and David Soloveichik. Modularity of signal propagation networks in the thermodynamic binding network model. In 32nd International Conference on DNA Computing and Molecular Programming (DNA32), 2026. Google Scholar
  24. Fu-Yao Yu, Sahil Shah, Yash Mittal, Paul Bessler, Aamir Mohsin, Jeffrey Geng, Arnav Vats, and David Soloveichik. COFFEE: A high-performance approach to convex optimization for thermodynamic equilibrium computations. In 2025 Systems and Information Engineering Design Symposium (SIEDS), pages 161-166. IEEE, 2025. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact