When lowering temperature, the in vivo circadian clock in cyanobacteria follows and surpasses the in vitro protein clock trough the Hopf bifurcation

Abstract: The in vivo circadian clock in single cyanobacteria is studied here by time-lapse fluorescence microscopy when the temperature is lowered below 25{\deg}C . We first disentangle the circadian clock behavior from the bacterial cold shock response by identifying a sequence of "death steps" based on cellular indicators. By analyzing only "alive" tracks, we show that the dynamic response of individual oscillatory tracks to a step-down temperature signal is described by a simple Stuart-Landau oscillator model. The same dynamical analysis applied to in vitro data (KaiC phosphorylation level following a temperature step-down) allows for extracting and comparing both clock's responses to a temperature step down. It appears, therefore, that both oscillators go through a similar supercritical Hopf bifurcation. Finally, to quantitatively describe the temperature dependence of the resulting in vivo and in vitro Stuart-Landau parameters $\mu(T)$ and $\omega_c(T)$, we propose two simplified analytical models: temperature-dependent positive feedback or time-delayed negative feedback that is temperature compensated. Our results provide strong constraints for future models and emphasize the importance of studying transitory regimes along temperature effects in circadian systems.