CANA v1.0.0 and schematodes: efficient quantification of symmetry in Boolean automata

The biomolecular networks underpinning cell function exhibit canalization, or the buffering of fluctuations required to function in a noisy environment. One understudied putative mechanism for canalization is the functional equivalence of a biomolecular entity's regulators (e.g., among the transcription factors for a gene). In these discrete dynamical systems, activation and inhibition of biomolecular entities (e.g., transcription of genes) are modeled as the activity of coupled 2-state automata, and thus the equivalence of regulators can be studied using the theory of symmetry in discrete functions. To this end, we present a new exact algorithm for finding maximal symmetry groups among the inputs to discrete functions. We implement this algorithm in Rust as a Python package, schematodes. We include schematodes in the new CANA v1.0.0 release, an open source Python library for analyzing canalization in Boolean networks, which we also present here. We compare our exact method implemented in schematodes to the previously published inexact method used in earlier releases of CANA and find that schematodes significantly outperforms the prior method both in speed and accuracy. We also apply CANA v1.0.0 to study the symmetry properties of regulatory function from an ensemble of experimentally-supported Boolean networks from the Cell Collective. Using CANA v1.0.0, we find that the distribution of a previously reported symmetry parameter, $k_s/k$, is statistically significantly different in the Cell Collective than in random automata with the same in-degree and activation bias (Kolmogorov-Smirnov test, $p<0.001$). In particular, its spread is much wider than in our null model (IQR 0.31 vs IQR 0.20 with equal medians), demonstrating that the Cell Collective is enriched in functions with extreme symmetry or asymmetry.