The Repressor-Lattice: Feed-Back, Commensurability, and Dynamical Frustration

A repressilator consists of a loop made up of three repressively interacting genes. We construct a hexagonal lattice with repressilators on each triangle, and use this as a model system for multiple interacting feedback loops. Using symmetry arguments and stability analysis we argue that the repressor-lattice can be in a non-frustrated oscillating state with only three distinct phases. If the system size is not commensurate with three, oscillating solutions of several different phases are possible. As the strength of the interactions between the nodes increases, the system undergoes many transitions, breaking several symmetries. Eventually dynamical frustrated states appear, where the temporal evolution is chaotic, even though there are no built-in frustrations. Applications of the repressor-lattice to real biological systems, such as tissues or biofilms, are discussed.