Fibration symmetry uncovers minimal regulatory networks for logical computation in bacteria

Symmetry principles have proven important in physics, deep learning and geometry, allowing for the reduction of complicated systems to simpler, more comprehensible models that preserve the system’s features of interest. Biological systems often show a high level of complexity and consist of a high number of interacting parts. Using symmetry fibrations, the relevant symmetries for biological ’message-passing’ networks, we reduced the gene regulatory networks of E. coli and B. subtilis bacteria in a way that preserves information flow and highlights the computational capabilities of the network. Nodes that share isomorphic input trees are grouped into equivalence classes called fibers, whereby genes that receive signals with the same ’history’ belong to one fiber and synchronize. We further reduce the networks to its computational core by removing "dangling ends" via k-core decomposition. The computational core of the network consists of a few strongly connected components in which signals can cycle while signals are transmitted between these "information vortices" in a linear feed-forward manner. These components are in charge of decision making in the bacterial cell by employing a series of genetic toggle-switch circuits that store memory, and oscillator circuits. These circuits act as the central computation machine of the network, whose output signals then spread to the rest of the network.