« Coherent collective behaviour of coupled genetic circuits »
Darka Labavić, Jacobs University, Bremen
We study the versatile performance of networks of coupled circuits. Each of these circuits is composed of a positive and a negative feedback loop in a motif that is frequently found in genetic and neural networks. When two of these circuits are coupled with mutual repression, the system can function as a toggle switch. The variety of its states can be controlled by two parameters as we demonstrate by a detailed bifurcation analysis. In the bistable regimes, switches between the coexisting attractors can be induced by noise. When we couple larger sets of these units, we numerically observe collective coherent modes of individual fixed-point and limit-cycle behavior. It is there that the monotonic change of a single bifurcation parameter allows one to control the onset and arrest of the synchronized oscillations. This mechanism may play a role in biological applications, in particular, in connection with the segmentation clock. While tuning the bifurcation parameter a variety of transient patterns emerges upon approaching the stationary states, in particular, a self-organized pacemaker in a completely uniformly equipped ensemble, so that the symmetry breaking happens dynamically. A change of a single bifurcation parameter also triggers the mutual conversion of regimes of collective fixed-point behavior and collective synchronized oscillations. Of particular interest is the arrest of oscillations. We identify the criterion that determines the seeds of arrest and the propagation of arrest fronts in terms of the vicinity to the future attractor. Due to a high degree of multistability we observe versatile patterns of phase locked motion in the oscillatory regime. Quenching the system into the regime, where oscillatory states are metastable, we observe qualitatively distinct approaches of the fixed-point attractor, depending on the initial seeds. If the seeds of arrest are isolated single sites of the lattice, the arrest propagates via bubble formation, visually similar to nucleation processes ; if the seed is extended along a line of lowest amplitudes, the freezing follows the spatial patterns of phaselocked motion with long interfaces between arrested and oscillating units. For spiral patterns of oscillator phases these interfaces are arranged along the arms of the spirals.