Fig. 5

Non-trivial circuits leading to synchronization: Multi-layer composite fiber. a Circuit consists of two layers of fibers: add, dsbG, gor, grxA, hemH, oxyS, trxC classified with \(|n = 0, \ell = 1 \rangle\) and rbsR, oxyR classified with \(|n = 1, \ell = 1 \rangle\), therefore forming a multi-layer composite fiber \(|n = 0, \ell = 1 \rangle \oplus |n = 1, \ell = 1 \rangle\). Fibration \(\psi\) of this circuit “collapses” both fibers and leaves the regulator untouched. b Genes in the red fiber receive one input from the gene in the green fiber, which in turn receives an input from itself and the regulator. Therefore, input trees of genes in the red fiber resemble the sum of an input tree of \(|n = 0, \ell = 1 \rangle\), followed by the input tree of \(|n = 1, \ell = 1 \rangle\). Input trees of the green fiber are those of the FFF. Regulator node has no inputs. Thus, multi-layer composite has two non-trivial fibers that can synchronize their activity. Note, gene add is separated from the rest of the red fiber by two steps, therefore allowing for a long range synchronization in the network