Complexity reduction by symmetry: Uncovering the minimal regulatory network for logical computation in bacteria.

Symmetry principles play an important role in geometry, and physics, allowing for the reduction of complicated systems to simpler, more comprehensible models that preserve the system's features of interest. Biological systems are often highly complex and may consist of a large number of interacting parts. Using symmetry fibrations, the relevant symmetries for biological "message-passing" networks, we introduce a scheme, called Complexity Reduction by Symmetry or CoReSym, to reduce the gene regulatory networks of Escherichia coli and Bacillus subtilis bacteria to core networks in a way that preserves the dynamics and uncovers the computational capabilities of the network. Gene nodes in the original network that share isomorphic input trees are collapsed by the fibration into equivalence classes called fibers, whereby nodes that receive signals with the same "history" belong to one fiber and synchronize. Then we reduce the networks to its minimal computational core via k-core decomposition. This computational core consists of a few strongly connected components or "signal vortices," in which signals can cycle through. While between them, these "signal vortices" transmit signals in a feedforward manner. These connected components perform signal processing and decision making in the bacterial cell by employing a series of genetic toggle-switch circuits that store memory, plus oscillator circuits. These circuits act as the central computation device of the network, whose output signals then spread to the rest of the network. Our reduction method opens the door to narrow the vast complexity of biological systems to their minimal parts in a systematic way by using fundamental theoretical principles of symmetry.