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Strongly nonlinear degrade-and-fire (DF) oscillations may emerge in genetic circuits having a delayed negative feedback loop as their core element. Here we study the synchronization of DF oscillators coupled through a common repressor field. For weak coupling, initially distinct oscillators remain desynchronized. For stronger coupling, oscillators can be forced to wait in the repressed state until the global repressor field is sufficiently degraded, and then they fire simultaneously forming a synchronized cluster. Our analytical theory provides necessary and sufficient conditions for clustering and specifies the maximum number of clusters that can be formed in the asymptotic regime. We find that in the thermodynamic limit a phase transition occurs at a certain coupling strength from the weakly clustered regime with only microscopic clusters to a strongly clustered regime where at least one giant cluster has to be present. [ABSTRACT FROM AUTHOR]

The development of synthetic gene oscillators has not only demonstrated our ability toforward engineer reliable circuits in living cells, but it has also proven to be an excellent testing ground for the statistical behavior of coupled noisy oscillators. Previous experimental studies demonstrated that a shared positive feedback can reliably synchronize such oscillators, though the theoretical mechanism was not studied in detail. In the present work, we examine an experimentally motivated stochastic model for coupled degrade-and-fire gene oscillators, where a core delayed negative feedback establishes oscillations within each cell, and a shared delayed positive feedback couples all cells. We use analytic and numerical techniques to investigate conditions for one cluster and multicluster synchrony. A nonzero delay in the shared positive feedback, as expected for the experimental systems, is found to be important for synchrony to occur. [ABSTRACT FROM AUTHOR]

Quantitative predictions for the spread of mutations in bacterial populations are essential to interpret evolution experiments and to improve the stability of synthetic gene circuits. We derive analytical expressions for the suppression factor for beneficial mutations in populations that undergo periodic dilutions, covering arbitrary population sizes, dilution factors, and growth advantages in a single stochastic model. We find that the suppression factor grows with the dilution factor and depends nontrivially on the growth advantage, resulting in the preferential elimination of mutations with certain growth advantages. We confirm our results by extensive numerical simulations. [ABSTRACT FROM AUTHOR]