(A) Schematic of the framework. In the light portion of the day, the oscillator runs along the ‘light’ limit cycle (orange) and accumulates phase at constant rate ; in the dark, the oscillator runs along the ‘dark’ limit cycle (blue) with frequency . At dawn (solid blue circle) and dusk (solid orange circle), the oscillator transitions between the light and dark limit cycles (blue and orange arrows) and incurs instantaneous phase shifts given by at dawn and at dusk. and can be approximated by linear functions of phase, such that and . Colors along the ‘light’ and ‘dark’ limit cycles indicate the magnitude of phase shifts due to light-to-dark () and dark-to-light () responses throughout the cycle for a single set of linearized and , shown in (B), generated by bootstrapping the measurements in Figure 3E–F. refers to the minimum of the KaiC phosphorylation rhythm. (B) One set of linearized step response functions and for the KaiABC oscillator determined by SDS-PAGE analysis of KaiC phosphorylation rhythms (see Figure 3E–F). The functions plotted represent one pair of and generated by bootstrapping (see Computational Methods). Colors denote phase shift magnitude as in (A). Solid circular markers correspond to dawn and dusk transitions in (A). (C) (left) Schematic of phase response analysis in the phase oscillator framework. The phase shift is a linear function of the dark pulse time (tDP) and duration (δ) if and the are linear functions. Slopes of linear dependencies on tDP and duration δ can be computed based on clock frequencies in light and dark (, ) and slopes of and functions. (right) Simulation of a phase response experiment for a phase oscillator with experimentally determined parameters (, , l, and d from the same bootstrapped parameter set as in (B)). A 9 hr dark pulse applied 4 hr after the beginning of the simulation to an oscillator (green) results in a ≈8 hr phase delay relative to a control (black) that remains in light throughout the simulation. (D) (top) Schematic of analysis of seasonal entrainment in the phase oscillator framework. Entrained phase is a linear function of day length (τ) and driving period (T) if and the are linear functions. Slopes of linear dependencies on τ and T can be computed based on four parameters (, , l, and d). (bottom) Simulation of entrainment to a LD 12:12 cycle for a phase oscillator with experimentally determined parameters (, , l, and d from the same bootstrapped parameter set as in (B)).