2026-07-28T17:38:29Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:26781 2026-07-27T07:33:58Z ddc:004 open_access

Universal Spatial Distribution Approximation in Equilibrium Lattice Models with Local Pairwise Interactions Chalk, Cameron Winfree, Erik universality molecular programming neural networks self-assembly 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Cameron Chalk and Erik Winfree 2026 Is Part Of LIPIcs, Volume 387, 32nd International Conference on DNA Computing and Molecular Programming (DNA 32) (2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.DNA.32.11 urn:nbn:de:0030-drops-267812 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.32.11 eng https://creativecommons.org/licenses/by/4.0/legalcode