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Universal Spatial Distribution Approximation in Equilibrium Lattice Models with Local Pairwise Interactions

Authors Cameron Chalk , Erik Winfree

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LIPIcs.DNA.32.11.pdf
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Subject Classification

ACM Subject Classification
  • Hardware → Neural systems
  • Applied computing → Physics
  • Applied computing → Systems biology
Keywords
  • universality
  • molecular programming
  • neural networks
  • self-assembly

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Abstract

Universality is one of the defining hallmarks of a useful neural network model: it means that the model is expressive enough, in principle, to learn arbitrary target behavior rather than being limited to a narrow class. Inspired by rigorous formal analogies between stochastic neural networks and lattice models in which molecules occupy discrete sites, contribute species-dependent interaction energies with neighbors, and rearrange according to the Boltzmann distribution, we ask whether interactions can be programmed to yield arbitrary spatial distributions. At the level of fine-grained spatial arrangements of species, universality fails: arbitrary spatial microstate distributions cannot be represented, as shown by a counting argument and explicit construction of unrepresentable distributions. Yet, for a natural observable - spatial patterns of molecular labels - universality can be achieved. We show that a grand-canonical lattice model of anisotropic molecules is universal for arbitrary label distributions, and a similar model with isotropic molecules is also universal given a simple symmetry-breaking boundary condition. Our results suggest that equilibrium systems of multivalent molecules undergoing stochastic rearrangement governed by local pairwise interactions can act similarly to probabilistic neural networks: with sufficient microstate detail in hidden dimensions, they can be universal in a meaningful macrostate observable.

Cite As Get BibTex

Cameron Chalk and Erik Winfree. Universal Spatial Distribution Approximation in Equilibrium Lattice Models with Local Pairwise Interactions. In 32nd International Conference on DNA Computing and Molecular Programming (DNA 32). Leibniz International Proceedings in Informatics (LIPIcs), Volume 387, pp. 11:1-11:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DNA.32.11

Author Details

Cameron Chalk
  • California Institute of Technology, Pasadena, CA, USA
Erik Winfree
  • California Institute of Technology, Pasadena, CA, USA

Funding

  • Chalk, Cameron: NSF CCF/FET 2212546
  • Winfree, Erik: NSF CCF/FET 2212546

Acknowledgements

The authors would like to thank David Doty and Damien Woods for tangential yet inspirational discussions.

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