  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.
