Document Open Access Logo

Robust Predicate and Function Computation in Continuous Chemical Reaction Networks

Authors Kim Calabrese , David Doty , Mina Latifi

PDF

File

Thumbnail PDF
PDF
LIPIcs.DISC.2025.19.pdf
  • Filesize: 0.91 MB
  • 23 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
Keywords
  • chemical reaction networks
  • analog computation
  • mean-field limit

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We initiate the study of "rate-constant-independent" computation of Boolean predicates (decision problems) and numerical functions in the continuous model of chemical reaction networks (CRNs), which model the amount of a chemical species as a nonnegative, real-valued concentration, representing an average count per unit volume. Real-valued numerical functions have previously been studied [Chen et al., 2023], finding that exactly the continuous, piecewise rational linear (meaning linear with rational slopes) functions f: ℝ_{>0}^k → ℝ_{>0} can be computed stably (a.k.a., rate-independently), meaning roughly that the CRN gets the answer correct no matter the rate at which reactions occur. For example the reactions X₁ → Y and X₂+Y → ∅, starting with inputs X₁ ≥ X₂, converge to output Y having concentration equal to the initial difference of inputs X₁ - X₂, no matter the relative rate at which each reaction proceeds.
We first show that, contrary to the case of real-valued functions, continuous CRNs are severely limited in the Boolean predicates they can stably decide, reporting a yes/no answer based only on which inputs are 0 or positive, but not on the exact positive value of any input.
This limitation motivates a slightly relaxed notion of rate-independent computation in CRNs that we call robust computation. The standard mass-action rate model is used, in which each reaction (e.g., A+B →^k C) is assigned a rate (A ⋅ B ⋅ k in this example) equal to the product of its reactant concentrations and its rate constant k. We say the computation is correct in this model if it converges to the correct output for any positive choice of rate constants. This adversary is weaker than the adversary defining stable computation, the latter being able to run reactions at rates that are not those of mass-action for any choice of rate constants (e.g., the stable adversary may deactivate a reaction temporarily, even if all reactants are positive).
We show that CRNs can robustly decide every predicate that is a finite Boolean combination of threshold predicates, where a threshold predicate is defined by taking a rational weighted sum of the inputs x ∈ ℝ^k_{≥ 0} and comparing to a constant, answering the question "Is ∑_{i = 1}^k w_i ⋅ x(i) > h?", for rational weights w_i and real threshold h. Turning to function computation, we show that CRNs can robustly compute any piecewise affine function with rational coefficients, where threshold predicates determine which affine piece to evaluate for a given input x.

Cite As Get BibTex

Kim Calabrese, David Doty, and Mina Latifi. Robust Predicate and Function Computation in Continuous Chemical Reaction Networks. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 19:1-19:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.DISC.2025.19

Author Details

Kim Calabrese
  • University of California-Davis, CA, USA
David Doty
  • University of California-Davis, CA, USA
Mina Latifi
  • University of California-Davis, CA, USA

Funding

  • Calabrese, Kim: NSF awards 2211793, 1900931.
  • Doty, David: NSF awards 2211793, 1900931 and DoE award DE-SC0024467.
  • Latifi, Mina: NSF awards 2211793, 1900931 and DoE award DE-SC0024467.

References

  1. Dan Alistarh, James Aspnes, David Eisenstat, Rati Gelashvili, and Ronald L Rivest. Time-space trade-offs in population protocols. In Proceedings of the twenty-eighth annual ACM-SIAM symposium on discrete algorithms, pages 2560-2579. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.169.
  2. Dan Alistarh, James Aspnes, and Rati Gelashvili. Space-optimal majority in population protocols. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2221-2239. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.144.
  3. Dan Alistarh, Rati Gelashvili, and Milan Vojnović. Fast and exact majority in population protocols. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pages 47-56, 2015. URL: https://doi.org/10.1145/2767386.2767429.
  4. Dana Angluin, James Aspnes, Zoë Diamadi, Michael J Fischer, and René Peralta. Computation in networks of passively mobile finite-state sensors. In Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing, pages 290-299, 2004. URL: https://doi.org/10.1145/1011767.1011810.
  5. Dana Angluin, James Aspnes, Zoë Diamadi, Michael J. Fischer, and René Peralta. Computation in networks of passively mobile finite-state sensors. Distributed Computing, pages 235-253, March 2006. URL: https://doi.org/10.1007/S00446-005-0138-3.
  6. Dana Angluin, James Aspnes, and David Eisenstat. Stably computable predicates are semilinear. In Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing, pages 292-299, 2006. URL: https://doi.org/10.1145/1146381.1146425.
  7. Dana Angluin, James Aspnes, and David Eisenstat. Fast computation by population protocols with a leader. Distributed Computing, 21:183-199, 2008. URL: https://doi.org/10.1007/S00446-008-0067-Z.
  8. Dana Angluin, James Aspnes, and David Eisenstat. A simple population protocol for fast robust approximate majority. Distributed Computing, 21(2):87-102, 2008. URL: https://doi.org/10.1007/S00446-008-0059-Z.
  9. Dana Angluin, James Aspnes, David Eisenstat, and Eric Ruppert. The computational power of population protocols. Distributed Computing, 20(4):279-304, 2007. URL: https://doi.org/10.1007/S00446-007-0040-2.
  10. Guillaume Aupy and Olivier Bournez. On the number of binary-minded individuals required to compute sqrt(1/2). Theoretical Computer Science, 412(22):2262-2267, 2011. URL: https://doi.org/10.1016/j.tcs.2011.01.003.
  11. Stav Ben-Nun, Tsvi Kopelowitz, Matan Kraus, and Ely Porat. An o (log3/2 n) parallel time population protocol for majority with o (log n) states. In Proceedings of the 39th Symposium on Principles of Distributed Computing, pages 191-199, 2020. URL: https://doi.org/10.1145/3382734.3405747.
  12. Petra Berenbrink, Robert Elsässer, Tom Friedetzky, Dominik Kaaser, Peter Kling, and Tomasz Radzik. A population protocol for exact majority with o (log5/3 n) stabilization time and θ (log n) states. In 32nd International Symposium on Distributed Computing, DISC 2018, page 10. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl Publishing, 2018. URL: https://doi.org/10.4230/LIPIcs.DISC.2018.10.
  13. Petra Berenbrink, Robert Elsässer, Tom Friedetzky, Dominik Kaaser, Peter Kling, and Tomasz Radzik. Time-space trade-offs in population protocols for the majority problem. Distributed Computing, 34(2):91-111, 2021. URL: https://doi.org/10.1007/S00446-020-00385-0.
  14. Andreas Bilke, Colin Cooper, Robert Elsässer, and Tomasz Radzik. Brief announcement: Population protocols for leader election and exact majority with o (log2 n) states and o (log2 n) convergence time. In Proceedings of the ACM Symposium on Principles of Distributed Computing, pages 451-453, 2017. URL: https://doi.org/10.1145/3087801.3087858.
  15. Olivier Bournez, Philippe Chassaing, Johanne Cohen, Lucas Gerin, and Xavier Koegler. On the convergence of population protocols when population goes to infinity. Applied Mathematics and Computation, 215(4):1340-1350, 2009. URL: https://doi.org/10.1016/j.amc.2009.04.056.
  16. Olivier Bournez, Pierre Fraigniaud, and Xavier Koegler. Computing with large populations using interactions. In Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Bratislava, Slovakia, August 27-31, 2012. Proceedings, volume 7464, pages 234-246, 2012. URL: https://doi.org/10.1007/978-3-642-32589-2_23.
  17. Olivier Bournez, Daniel S Graça, and Amaury Pouly. Polynomial time corresponds to solutions of polynomial ordinary differential equations of polynomial length. Journal of the ACM (JACM), 64(6):1-76, 2017. URL: https://doi.org/10.1145/3127496.
  18. Kim Calabrese, David Doty, and Mina Latifi. Robust predicate and function computation in continuous chemical reaction networks, 2025. URL: https://doi.org/10.48550/arXiv.2506.06590.
  19. Cameron Chalk, Niels Kornerup, Wyatt Reeves, and David Soloveichik. Composable rate-independent computation in continuous chemical reaction networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 18(1):250-260, 2019. Google Scholar
  20. Ho-Lin Chen, David Doty, Wyatt Reeves, and David Soloveichik. Rate-independent computation in continuous chemical reaction networks. Journal of the ACM, 70(3), May 2023. URL: https://doi.org/10.1145/3590776.
  21. Ho-Lin Chen, David Doty, and David Soloveichik. Deterministic function computation with chemical reaction networks. Natural Computing, 13(4):517-534, 2014. Special issue of invited papers from DNA 2012. URL: https://doi.org/10.1007/s11047-013-9393-6.
  22. Yuan-Jyue Chen, Neil Dalchau, Niranjan Srinivas, Andrew Phillips, Luca Cardelli, David Soloveichik, and Georg Seelig. Programmable chemical controllers made from DNA. Nature nanotechnology, 8(10):755-762, 2013. Google Scholar
  23. Rachel Cummings, David Doty, and David Soloveichik. Probability 1 computation with chemical reaction networks. Natural Computing, 15(2):245-261, 2016. Special issue of invited papers from DNA 2014. URL: https://doi.org/10.1007/s11047-015-9501-x.
  24. David Doty, Mahsa Eftekhari, Leszek Gąsieniec, Eric Severson, Przemyslaw Uznański, and Grzegorz Stachowiak. A time and space optimal stable population protocol solving exact majority. In 2021 IEEE 62nd annual symposium on foundations of computer science (FOCS), pages 1044-1055. IEEE, 2022. Google Scholar
  25. David Doty and Monir Hajiaghayi. Leaderless deterministic chemical reaction networks. Natural Computing, 14(2):213-223, 2015. Preliminary version appeared in DNA 2013. URL: https://doi.org/10.1007/S11047-014-9435-8.
  26. Moez Draief and Milan Vojnović. Convergence speed of binary interval consensus. SIAM Journal on control and Optimization, 50(3):1087-1109, 2012. URL: https://doi.org/10.1137/110823018.
  27. François Fages, Guillaume Le Guludec, Olivier Bournez, and Amaury Pouly. Strong Turing completeness of continuous chemical reaction networks and compilation of mixed analog-digital programs. In Computational Methods in Systems Biology: 15th International Conference, CMSB 2017, Darmstadt, Germany, September 27-29, 2017, Proceedings 15, pages 108-127. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-67471-1_7.
  28. Daniel T Gillespie. Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry, 81(25):2340-2361, 1977. Google Scholar
  29. Cato M. Guldberg and Peter Waage. Studies concerning affinity. Forhandlinger: Videnskabs-Selskabet i Christinia. Norwegian Academy of Science and Letters, 35, 1864. English translation in [Waage and Gulberg, 1986]. Google Scholar
  30. Hooman Hashemi, Ben Chugg, and Anne Condon. Composable Computation in Leaderless, Discrete Chemical Reaction Networks. In DNA 26: 26th International Conference on DNA Computing and Molecular Programming, volume 174, pages 3:1-3:18, 2020. URL: https://doi.org/10.4230/LIPIcs.DNA.2020.3.
  31. George B Mertzios, Sotiris E Nikoletseas, Christoforos L Raptopoulos, and Paul G Spirakis. Determining majority in networks with local interactions and very small local memory. Distributed Computing, 30(1):1-16, 2017. URL: https://doi.org/10.1007/S00446-016-0277-8.
  32. Yves Mocquard, Emmanuelle Anceaume, James Aspnes, Yann Busnel, and Bruno Sericola. Counting with population protocols. In 2015 IEEE 14th International Symposium on Network Computing and Applications, pages 35-42. IEEE, 2015. URL: https://doi.org/10.1109/NCA.2015.35.
  33. Yves Mocquard, Emmanuelle Anceaume, and Bruno Sericola. Optimal proportion computation with population protocols. In 2016 IEEE 15th International Symposium on Network Computing and Applications (NCA), pages 216-223. IEEE, 2016. URL: https://doi.org/10.1109/NCA.2016.7778621.
  34. Eric E Severson, David Haley, and David Doty. Composable computation in discrete chemical reaction networks. Distributed Computing, 34(6):437-461, 2021. special issue of invited papers from PODC 2019. URL: https://doi.org/10.1007/S00446-020-00378-Z.
  35. David Soloveichik, Matthew Cook, Erik Winfree, and Jehoshua Bruck. Computation with finite stochastic chemical reaction networks. natural computing, 7:615-633, 2008. URL: https://doi.org/10.1007/S11047-008-9067-Y.
  36. David Soloveichik, Georg Seelig, and Erik Winfree. DNA as a universal substrate for chemical kinetics. Proceedings of the National Academy of Sciences, 107(12):5393-5398, 2010. Google Scholar
  37. PG Spirakis, L Gasieniec, Russell Martin, D Hamilton, and G Stachowiac. Deterministic population protocols for exact majority and plurality. LIPIcs, 70:14-1, 2017. URL: https://doi.org/10.4230/LIPIcs.OPODIS.2016.14.
  38. Niranjan Srinivas, James Parkin, Georg Seelig, Erik Winfree, and David Soloveichik. Enzyme-free nucleic acid dynamical systems. Science, 358(6369):eaal2052, 2017. Google Scholar
  39. Peter Waage and Cato Maximilian Gulberg. Studies concerning affinity. Journal of chemical education, 63(12):1044, 1986. English translation; original paper is [Cato M. Guldberg and Peter Waage, 1864]. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact