article: Circadian rhythms are biological oscillations that schedule daily changes in physiology. Outside the laboratory, circadian clocks do not generally free-run but are driven by daily cues whose timing varies with the seasons. The principles that determine how circadian clocks align to these external cycles are not well understood. Here, we report experimental platforms for driving the cyanobacterial circadian clock both in vivo and in vitro. We find that the phase of the circadian rhythm follows a simple scaling law in light-dark cycles, tracking midday across conditions with variable day length. The core biochemical oscillator comprised of the Kai proteins behaves similarly when driven by metabolic pulses in vitro, indicating that such dynamics are intrinsic to these proteins. We develop a general mathematical framework based on instantaneous transformation of the clock cycle by external cues, which successfully predicts clock behavior under many cycling environments.
  Circadian clocks generate biological rhythms that temporally organize physiology to match the 24 hr diurnal cycle. Although these clocks continue to oscillate in constant laboratory conditions, in natural environments they are driven by the external cycle of day and night. Because of latitude-dependent changes in the duration of day and night throughout the year, clock architectures must incorporate mechanisms that respond appropriately to environmental signals, such as dawn and dusk, whose schedule changes throughout the year. Data from several species indicate that circadian clocks adapt to different day lengths by modulating their phases relative to the light-dark cycle (de Montaigu et al. , 2015; Rémi et al. , 2010). In this way, circadian oscillators are able to coordinate physiological events relative to specific times of day and, in multicellular organisms, specialized mechanisms exist that allow oscillators in different cells to follow distinct times of day (Herzog, 2007; Daan et al. , 2003). Because the biochemical circuits governing circadian rhythms and light-dark sensing in such organisms are complex, it has been difficult to identify which features of circadian systems are responsible for seasonal adaptation. More generally, it remains unclear what features oscillators must have to respond appropriately to varying light-dark cycles and how those features are implemented molecularly in specific systems. Cyanobacteria present a unique opportunity to elucidate molecular mechanisms in circadian biology because the core oscillator responsible for driving genome-wide transcriptional rhythms can be reconstituted using purified KaiABC proteins, and these proteins have been extensively studied biochemically in constant conditions (Nishiwaki et al. , 2004; Rust et al. , 2007). Cyanobacteria must contend with large seasonal variations in day length because their natural aquatic environments span a wide range of latitudes (Flombaum et al. , 2013). Yet, comparatively little is known about how the cyanobacterial clock functions when driven by light-dark cycles that mimic days in different seasons. To study how the circadian clock is affected by light-dark cycles with different day lengths, we developed multiplexed LED illumination devices to grow cyanobacteria in a wide range of light-dark conditions (Figure 1A). We used square-wave illumination patterns in our experiments, and used the times of lights-on and lights-off as experimental analogs of dawn and dusk, respectively. We defined the day length (τ) as the total time the lights are on each day (Figure 1A). We found that the circadian rhythm in the cyanobacterium Synechococcus elongatus PCC 7942 (S. elongatus) follows a simple rule: the phase of clock-driven gene expression scales linearly with day length and remains fixed relative to the middle of the day over a wide range of day lengths. 10. 7554/eLife. 23539. 003Figure 1. Phase of the cyanobacterial circadian rhythm scales linearly with day length. (A) LED array device used to grow S. elongatus in programmable light-dark cycles. Cells grown in a 96-well plate on solid media (lower plate, green circles) are illuminated from above by LEDs (red circles). An Arduino microcontroller is used to dynamically change LED intensity in different columns of the plate (inset). Luminescence from the bottom plate is read out every 30 min on a plate reader. Drawing not to scale. (B) Drive-and-release strategy to measure phase of the circadian clock under light-dark (LD) cycling. Cells were exposed to five entraining LD cycles and then released into constant light. Bioluminescence signals (PkaiBC: : luxAB) from each well were separated into individual ‘day’ and ‘night’ windows. Data from night portions of the experiment were omitted from analysis (gray bars), and data from the day portions of the experiment were aligned to zero baseline and normalized to unit variance. Dashed lines indicate time of peak reporter signal calculated by parabolic fitting. See Computational methods for details. (C) Peak time of bioluminescence (PkaiBC: : luxAB) in light-dark cycles of different day length (red squares) was quantified by local parabolic fitting around the first maximum of the oscillation after release into constant light. Error bars represent standard deviations of peak time estimates from technical replicates (n = 4–8). Slope of the linear fit (red line, m = 0. 53 ± 0. 01) was determined by linear regression. Dashed and dotted black lines indicate scaling of phase with day length for dawn- and dusk-tracking oscillators; green line indicates midday-tracking behavior. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 00310. 7554/eLife. 23539. 004Figure 1—source data 1. Source data for Figure 1B. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 00410. 7554/eLife. 23539. 005Figure 1—source data 2. Source data for Figure 1C. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 00510. 7554/eLife. 23539. 006Figure 1—source data 3. Source data for bioluminescence trajectories in Figure 1—figure supplement 1. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 00610. 7554/eLife. 23539. 007Figure 1—source data 4. Source data for Kendall’s τ correlations in Figure 1—figure supplement 1. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 00710. 7554/eLife. 23539. 008Figure 1—source data 5. Source data for Figure 1—figure supplement 2A–E. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 00810. 7554/eLife. 23539. 009Figure 1—source data 6. Source data for Figure 1—figure supplement 3A, showing bioluminescence output from the purF repoter. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 00910. 7554/eLife. 23539. 010Figure 1—source data 7. Source data for Figure 1—figure supplement 3B. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 01010. 7554/eLife. 23539. 011Figure 1—figure supplement 1. Bioluminescence recordings from PkaiBC: : luxAB reporter in light-dark cycles. (left) Selected bioluminescence traces (PkaiBC: : luxAB, black) recorded from individual wells of the 96-well LED array device in conditions simulating day-night cycles of different day length (same dataset as in Figure 1B–C). Data recorded in the dark (gray bars) and in the first two hours after lights-on were omitted from analysis. Each trajectory was normalized to the mean and variance of the bioluminescence signal recorded after the release into constant light, and trajectories recorded after release into constant light were fit to sinusoids (green). For illustration purposes here, trajectories were aligned to zero baseline. See Computational methods for fitting details. (right) Similarity between the bioluminescence rhythms (PkaiBC: : luxAB) recorded during light-dark entrainment and after release into free running conditions. Nonparametric correlation coefficient (Kendall’s τ) between bioluminescence trajectories recorded in the light during entrainment (days 1–5) and the corresponding time period after release into constant light (e. g. a 16-hr window in LL for LD 16: 8). See Computational methods for details. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 01110. 7554/eLife. 23539. 012Figure 1—figure supplement 2. The circadian rhythm of S. elongatus rapidly entrains to 24 hr diurnal cycles with 8–16 hr of daylight. (A) Peak times of PkaiBC: : luxAB reporter after release into LL from 1 to 7 LD cycles of different day length (LD 8: 16, LD 12: 12 or LD 16: 8), as estimated by sinusoidal regression. Error bars represent the standard deviation (σ) of technical replicates (n = 6–8 replicates per condition). Lines mark linear fits to the data. (B) Slope m of clock phase scaling with day length stabilizes after three light-dark cycles. Slope of linear fits in (A) plotted against day length in diurnal cycles. Error bars mark uncertainty in fit slope (±σ). (C) Slope m of clock phase scaling with day length, as estimated by parabolic regression (see Computational methods). (D and E) Free-running period of PkaiBC: : luxAB reporter rhythms, as estimated from sinusoidal fits (D) or local parabolic fits (E), recorded in continuous light after entrainment to 24 hr cycles with 8,12, or 16 hr of daylight. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 01210. 7554/eLife. 23539. 013Figure 1—figure supplement 3. Bioluminescence recordings from PpurF: : luxAB reporter in light-dark cycles. (A) Rhythms in bioluminescence in continuous light recorded from a dawn gene reporter (PpurF: : luxAB) after entrainment to 24 hr light-dark cycles of different day length τ (LD 8: 16, LD 10: 14, LD 14: 10, LD 18: 6). Following the protocol in Figure 1, cells received five light-dark cycles and were then placed under continuous illumination. Bioluminescence trajectories from first 72 hr after release into constant conditions were aligned to zero baseline and normalized to unit variance. (B) Peak times of the PpurF: : luxAB reporter in constant light after entrainment with five LD cycles of different day length (LD 4: 20 to LD 20: 4). Peak times were determined by local parabolic regression (see Computational methods). Error bars represent standard errors of technical replicates (n = 4–8). Straight lines represent linear fits to data from wells entrained to day lengths at least 8 hr long. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 013 We biochemically simulated seasonal effects in the reconstituted KaiABC oscillator by driving the Kai proteins with pulses of nucleotides on a 24 hr cycle, simulating in vivo metabolic conditions during day and night. Remarkably, this in vitro system shows a similar linear scaling of clock phase with simulated day length, indicating that core mechanisms for seasonal adaptation are intrinsic to the Kai proteins. We developed a minimal mathematical model for oscillator entrainment based on rapid transitions between distinct day and night clock cycles. The model can account for the ability to track midday if the phase shifts caused by dawn and dusk depend linearly on clock time with appropriate slopes. By calibrating this model with independent measurements in vitro and in vivo, we were able to predict the results of phase response experiments, and the behavior of the clock in non-24 hr light-dark cycles. The essential feature of the model, the linear dependences of phase shifts on clock time, has a simple geometric interpretation in terms of the deformation of the clock orbit caused by a transition between light and dark. Thus, the entrained behavior of the circadian clock can be captured quantitatively by a mathematical framework with a small number of parameters. This framework in turn can be used to guide the design of experiments that probe the molecular origins of key mathematical features. The precisely defined nature of the cyanobacterial clock facilitates such experiments, but the model is general, so we expect it to be useful for other organisms as well.
  S. elongatus is a photosynthetic bacterium whose physiology is closely tied to light and dark. In constant light, the circadian clock exerts pervasive control over gene expression: most transcripts cycle in abundance with ≈24 hr period, with the majority of transcripts peaking either near subjective morning (dawn genes), or nearly 12 hr later, at subjective nightfall (dusk genes) (Liu et al. , 1995; Ito et al. , 2009). In the dark, however, growth stops, and most gene expression is highly repressed (Hosokawa et al. , 2011; Tomita et al. , 2005; Pattanayak et al. , 2015). Thus, portions of the circadian gene expression program that fall into nighttime hours are strongly attenuated. Viewed in this way, the hours between dawn and dusk provide a limited window for gene expression, so that winter months at high latitude provide fewer daylight hours for this cyanobacterium to accomplish biosynthetic tasks relative to summer months. We surmised that an important function for the circadian clock in this organism might be to schedule gene expression appropriately during daylight hours, when biosynthetic resources are available. We thus expected that asymmetric light-dark cycles mimicking days in different seasons would realign the clock cycle. To systematically study how circadian gene expression in S. elongatus adjusts to asymmetric light-dark schedules, we built a microcontroller-driven LED array device and used it to grow cells in 24 hr day-night cycles with photoperiod (day length) varying from 4 to 20 hr (Figure 1A). This LED array was coupled to a plate reader that allowed us to monitor clock output—gene expression from clock-driven promoters—using strains engineered with luminescent reporters of representative dusk (kaiBC) and dawn (purF) genes. In these experiments, the circadian clock stably entrained to light-dark cycles with a wide range of day lengths (τ = 8–16 hr) within roughly 3 days (Figure 1—figure supplements 1 and 2). We found that the phase of the oscillation, that is, when the peak clock output signal occurred relative to dawn, varied systematically as a function of day length. In general, longer days resulted in the peak reporter signal occurring later in the day (Figure 1B). By plotting the time of peak reporter signal versus day length, we found that the driven behavior of the clock follows a simple scaling law: the clock phase is proportional to day length (Figure 1C). The proportionality constant, m, falls in an intermediate range (Table 1) between the limits corresponding to either dawn (m = 0) or dusk (m = 1) fully resetting the clock. The linear relationship between clock phase and day length with slope m ≈ 0. 5 implies that cyanobacteria set their clocks to reach the same internal time at the middle of the daylight hours, independent of day length. This entrainment behavior is not unique to the kaiBC promoter; a reporter for purF, a representative of the dawn class of genes, shows similar scaling behavior (Figure 1—figure supplement 3). 10. 7554/eLife. 23539. 014Table 1. Summary of biologically independent in vivo experiments measuring entrainment to 24 hr light-dark cycles of varying day length and corresponding estimates of m, the proportionality coefficient between the peak time of PkaiBC: : luxAB reporter and day length during light-dark entrainment. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 014FigureDriving period T (hr) Day length τ (hr) Slope m ± SD of estimateFigure 1C244,6, 8,9, 10,11,12,13,14,16,18,200. 55 ± 0. 02 (sinusoidal fitting) 0. 53 ± 0. 01 (parabolic fitting) Figure 1—figure supplement 2248,12,160. 47 ± 0. 03 (sinusoidal fitting) 0. 57 ± 0. 02 (parabolic fitting) Figure 5C22,23,24,25,268,10,12,140. 51 ± 0. 11 (sinusoidal fitting) The fact that the core oscillator of the cyanobacterial circadian clock can be reconstituted in vitro from purified Kai proteins (Nakajima et al. , 2005; Kageyama et al. , 2006) naturally led to the question of whether the ability to track midday in vivo is intrinsic to the core oscillator, or if it requires additional factors present in the cell. We thus sought to extend the reconstituted system such that we could drive the biochemical oscillator with rhythmic input signals. A purified mixture of the KaiA, KaiB, and KaiC proteins spontaneously generates a stable circadian rhythm in KaiC phosphorylation and in the formation of KaiB-KaiC complexes (Nakajima et al. , 2005; Kageyama et al. , 2006). Although the Kai proteins are not light-sensitive, recent work has shown that they are sensitive to metabolite pools that shift in response to changes in photosynthetic activity in the cell: KaiA is sensitive to the redox state of quinones, and KaiC phosphorylation is sensitive to the ATP/ADP ratio (Rust et al. , 2011; Kim et al. , 2012; Phong et al. , 2013). We thus developed a protocol to mimic repeated light-dark cycles in vitro by cycling the ATP/ADP ratio between physiologically relevant levels experienced during the day and night (Pattanayak et al. , 2014), using ADP addition to simulate nightfall ([ATP]/ ([ATP]+[ADP]) ≈ 25%) followed by buffer exchange into an ATP-only buffer to simulate dawn ([ATP]/ ([ATP]+[ADP]) ≈ 100%) (Figure 2A). 10. 7554/eLife. 23539. 015Figure 2. Reconstitution of the seasonal clock response in vitro. (A) Buffer exchange protocol to simulate metabolic driving of the clock. To mimic daytime in vitro, purified Kai proteins (green, blue and red symbols) were incubated in ‘day’ reaction buffer containing ATP. ADP was added to mimic nightfall ([ATP]/ ([ATP]+[ADP]) ≈ 25%). At simulated dawn, reactions were returned to ‘day’ buffer via buffer exchange. (B) Example traces of KaiC phosphorylation rhythm from in vitro reactions mimicking LD 18: 6 (top), LD 12: 12 (middle), and LD 6: 18 (bottom). (C) Phase of KaiABC oscillation scales linearly with simulated day length (time spent in ‘day’ buffer), as assessed by the peak time of KaiC phosphorylation (black squares) or peak time of fluorescence polarization (cyan squares and circles, for two replicates) of fluorescently labeled KaiB. Peak times of fluorescence polarization were estimated from sinusoidal fits to oscillations recorded in free-running conditions after entraining the oscillator with three metabolic cycles. Peak time of %P-KaiC was estimated by fitting sinusoids to KaiC phosphorylation time series from the third day of reactions. Error bars represent uncertainty of fit phase from sinusoidal regression. Lines of best fit were determined by linear regression (cyan squares: m = 0. 39 ± 0. 06, cyan circles: m = 0. 36 ± 0. 04, black: m = 0. 51 ± 0. 04). In vivo data (from Figure 1C) is plotted in red. Scaling of entrained phase was measured once via KaiC phosphorylation analysis (black) and twice using the fluorescence polarization probe (cyan squares and circles) with an independent preparation of proteins. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 01510. 7554/eLife. 23539. 016Figure 2—source data 1. Source data for Figure 2B. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 01610. 7554/eLife. 23539. 017Figure 2—source data 2. Source data for Figure 2C. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 01710. 7554/eLife. 23539. 018Figure 2—source data 3. Source data for Figure 2—figure supplement 1. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 01810. 7554/eLife. 23539. 019Figure 2—figure supplement 1. Validation of KaiB fluorescence polarization reporter against KaiC phosphorylation rhythm. (top) Fluorescence polarization of KaiABC mixture probed with fluorescently labeled KaiB exhibits ≈24 hr rhythms (black and blue squares). (bottom) KaiC phosphorylation rhythm of the same reaction mixtures as in the top panel (black and blue squares). Black vs blue markers show data from two identical reactions started at different times. Curves show best-fit sinusoids, where all fits share a common period; the phase offset between out-of-phase reactions was also a shared parameter across fits to polarization and phosphorylation datasets. This experiment was performed once. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 019 We found that the KaiC phosphorylation rhythm readily responded to this metabolite pulsing protocol (Figure 2B). Stepping down to lower ATP/ADP conditions promoted dephosphorylation, in accord with previously published observations (Pattanayak et al. , 2014). Conversely, stepping up to higher ATP/ADP conditions favored phosphorylation (Figure 2B). Pulsing the ATP/ADP ratio in this way caused the KaiC phosphorylation rhythm to synchronize to the external cycle. By the third cycle, the oscillator appeared to have stably entrained, and, similar to the clock behavior that we observed in live cells, longer times in daytime buffer led to later peak phosphorylation times. We estimated peak times for these oscillator reactions during the third entraining cycle and found that the phase of the in vitro rhythm scales linearly in proportion with simulated day length (Figure 2C), with KaiC phosphorylation rhythm peaking roughly 3 hr after the midpoint of simulated daytime, suggesting that the reconstituted Kai oscillator is capable of tracking the approximate midday point of an externally imposed metabolic rhythm. To more accurately measure the scaling of entrained phase with day length in KaiABC oscillator reactions, we turned to a fluorescence polarization probe that enables automated measurement of oscillator state with high temporal resolution (Figure 2—figure supplement 1) (Chang et al. , 2012; Heisler et al. , 2017). We used this assay to determine clock phases in free-running conditions after entraining the Kai proteins with three metabolic cycles, analogous to the design of the in vivo experiments. We again measured linear scaling of the entrained phase, albeit with a lower slope than the estimate from sparser gel-based measurements of KaiC phosphorylation (m = 0. 38 ± 0. 07 for polarization probe vs. m = 0. 51 ± 0. 04 for phosphorylation data, Figure 2C). Despite the variability in our estimates of m, these measurements suggest that the in vitro oscillator successfully captures the essential feature of seasonal entrainment we observed in vivo: linear scaling of entrained phase with an intermediate slope. In the cell, the Kai proteins interact with histidine kinases and a network of other factors absent from the reconstituted system, ultimately leading to rhythms in transcription across the genome, including the kai genes themselves (Takai et al. , 2006; Gutu et al. , 2013; Markson et al. , 2013). These additional factors may account for the differences we observe between proportionality constants in vitro and in vivo, and the time delay between KaiC phosphorylation and luminescent output of the kaiBC expression reporter is likely responsible for the offset in Figure 2C. While these considerations suggest that care must be taken in connecting the in vitro and in vivo results, they also underscore the significance of our observation that the Kai oscillator proteins by themselves are sufficient to yield the linear response of the system to altered day length. We therefore sought to use this in vitro model of seasonality to uncover the biochemical basis of the linear seasonal clock response. In the purified system, the entraining cues are steps between high and low ATP/ADP that simulate dawn and dusk. We therefore sought to decompose the seasonal adaptation of the clock into step responses following each metabolic transition. This approach is related to the limit cycle theory of oscillators: if we assume that a unique stable orbit exists for both the day and night conditions, then a metabolic transition forces the system to adjust from its current cycle to one associated with the new condition. If relaxation to the new orbit occurs rapidly relative to the length of the day—that is, if the light and dark orbits are strongly attracting—then the state of the oscillator at each transition can be specified simply by a phase angle on that orbit. In this limit, the response of the system is fully determined by instantaneous phase shifts caused by the transitions. Mathematically, we describe this limit as an oscillator comprised of a single phase variable, θ, that runs along a fixed limit cycle trajectory at constant speed (see Appendix 1). The responses to dark-to-light or light-to-dark transitions are then represented by phase shifts that are specified by the functions L (θ) and D (θ), respectively. To determine whether this fast-relaxation approximation can describe phase shifts in the core Kai oscillator, and to study their biochemical basis, we measured the L (θ) and D (θ) shift functions in our reconstituted system. To measure the D (θ) function, we incubated the Kai proteins in daytime buffer, then transferred them into nighttime buffer at various times throughout the circadian cycle and studied how the oscillation was affected. As above, we assayed oscillator response in separate experiments using either SDS-PAGE to measure KaiC phosphorylation or the fluorescence polarization probe to achieve high temporal resolution. Example time series from the fluorescence polarization measurements are shown in Figure 3A–B. 10. 7554/eLife. 23539. 020Figure 3. Clock responses to metabolic steps mimicking dawn (step up) and dusk (step down). (A) Phase shift in fluorescence polarization (red curve) caused by a shift to a buffer that mimics the nucleotide pool at night ([ATP]/ ([ATP]+[ADP]) ≈ 25%, gray bar). The control reaction remained in the original buffer (black curve). (B) Phase shift in fluorescence polarization (blue curve) caused by a shift from night buffer back to day buffer ([ATP]/ ([ATP]+[ADP]) ≈ 100%, beige bar). (C, D) Summary of phase shifts caused by metabolic step-down (C) or step-up (D) perturbations throughout the clock cycle. Simulated day-night or night-day steps were administered as in (A) and (B). Different colors represent independent measurements. To estimate the phase of each reaction, trajectories were fit to sinusoids. Phase shifts were determined relative to the respective control reactions. The times at which buffer steps were administered were converted to circadian time (CT 0 corresponds to the estimated trough of KaiC phosphorylation based on Figure 2—figure supplement 1). Colored arrows indicate clock phases when metabolic shifts occur in entrained conditions. (E, F) Analogs of (C, D) for the gel-based phosphorylation measurements on an independent preparation of Kai proteins. Error bars represent standard deviations calculated by bootstrapping (see Computational methods). Horizontal error bars are smaller than marker widths. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 02010. 7554/eLife. 23539. 021Figure 3—source data 1. Source data for Figure 3A–B. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 02110. 7554/eLife. 23539. 022Figure 3—source data 2. Source data for Figure 3C–D. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 02210. 7554/eLife. 23539. 023Figure 3—source data 3. Source data for Figure 3E–F. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 02310. 7554/eLife. 23539. 024Figure 3—figure supplement 1. Example calculation of phase shifts in response to metabolic step transitions. Phase shifts are computed from the difference in phase of the control reaction and the perturbed reaction evaluated at the time of the step. Phase of each reaction at the time of the step (green dashed line) is calculated based on the sinusoidal fit to the normalized fluorescence polarization trajectory for that reaction. Periods of the oscillator in day and night conditions were fit globally to all reactions measured in one experiment. See Computational methods for fitting details. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 02410. 7554/eLife. 23539. 025Figure 3—figure supplement 2. Experimentally measured step-response functions predict entrainment to driving periods near 24 hr. (A) Entrainment simulations with driving periods 4–48 hr were performed for 1000 cycles, and phases at the end of nighttime (immediately before the action of L) of the last 920 cycles were plotted. Simulation results are double-plotted along the vertical axis for clarity. In each simulation, the oscillator runs at constant angular speed in the light and dark, and experiences instantaneous phase shifts at dawn and dusk, according to the values of L and D. Refer to Figure 4A for an illustration of a single entrainment simulation. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 025 We proceeded to calculate phase shifts from these measurements, taking into consideration that a switch to nighttime buffer conditions could affect the oscillator in two ways: the phase of the oscillation can shift, and the oscillator period may change, both of which can change the relative timing of peaks. To separate these effects, it is necessary to collect enough data to accurately determine the period, and then extrapolate backwards to infer the instantaneous phase shift at the time of buffer exchange. We detail our procedure for analyzing these data to extract phase shifts in Figure 3—figure supplement 1. We used an analogous design to measure the reverse steps from nighttime to daytime buffer (Figure 3B). Plotted together in Figure 3C–D, the extrapolated phase shifts measure the sensitivity of the clock to individual light and dark steps throughout the 24 hr day. The measured L (θ) and D (θ) step-response functions show that the phase of the purified KaiABC oscillator is responsive to both step-up and step-down metabolic changes. For example, a day-to-night shift lengthens the period of the clock by about 2 hr, and, when it is applied at subjective morning, leads to an instantaneous phase delay of about 3 hr (Figure 3—figure supplement 1). On the other hand, a night-to-day transition in the middle of subjective night shortens the period and causes a phase advance of about 1. 5 hr. These measurements characterize how the oscillator responds to a metabolic step-change throughout the entire clock cycle. However, when the clock is stably synchronized to a cycling environment, dawn and dusk fall only in a limited window of clock phases. To determine this window, we returned to our measurements in Figure 2C. We identified phases that correspond to dawn or dusk in entrained conditions mimicking LD 6: 18 to LD 18: 6 and marked these phases on the L (θ) and D (θ) functions (Figure 3C–D, colored arrows). These points fell on gradually changing, approximately linear, regions of both curves, spanning roughly 6 hr windows near subjective dawn (for L (θ) ) and subjective dusk (for D (θ) ). Because these regions of L (θ) and D (θ) represent phase shifts comparable in magnitude but opposite in sign, we hypothesized that their opposing forces could enable oscillator entrainment to light-dark cycles. To check this hypothesis, we simulated a simple (phase-only) oscillator that runs at constant frequencies in the light and dark and adjusts its phase in response to dawn and dusk according to the phase shift functions that we measured in Figure 3C–D (see Computational methods). This model maps the one-dimensional phase variable from one cycle to the next according to L (θ) and D (θ). We analyzed the stability properties of this map and found that the oscillator could entrain with a unique phase to driving periods of within 4–5 hr of its natural circadian period. Outside this range, the oscillator failed to entrain or exhibited more complex dynamics (Figure 3—figure supplement 2). We proceeded to test whether the measured L (θ) and D (θ) step-response functions could successfully reproduce the entrainment of the oscillator to repeated 24 hr metabolic cycles of varying day length, shown in Figure 2. When subjected to light-dark cycles in simulations, the oscillator model stably entrains to the diurnal schedule within two-to-five cycles (Figure 4A–B, Figure 4—figure supplement 1A). Importantly, the simulated entrained phase scales linearly with day length with a slope similar to the experimental data, indicating that the driven clock response can indeed be decomposed into a series of step responses to environmental transitions (Figure 4C). 10. 7554/eLife. 23539. 026Figure 4. Entrainment of the phase oscillator model to a driving cycle. (A) Schematic of a phase-only oscillator that responds to dawn and dusk with instantaneous phase shifts. The oscillator runs at constant velocities ωL during the day and ωD at night (green lines), except for dawn and dusk where sudden shifts occur (red and blue arrows). This simulation illustrates entrainment to a LD 12: 12 cycle for an oscillator with ωL (1/23. 7 hr), ωD (1/25. 7 hr), and L (θ) and D (θ) as experimentally measured for the KaiABC oscillator (see main text, Computational Methods). Refer to Figure 4—figure supplement 1B for illustrations of L (θ) and D (θ) used in this simulation. (B) Simulated approach to stable entrainment in the model (as in Figure 4A) for light-dark cycles of different day length (τ = 8–14 hr). (C) Simulated seasonal response of an oscillator that responds rapidly to light-dark cues according to the phase shift functions in Figure 3C (see text). In simulations, tpk is defined as the time when oscillator phase θ equals π rad (0. 5 cycles), corresponding to the peak of KaiC phosphorylation. Shaded areas correspond to standard deviations of entrainment simulations using the four possible combinations of L and D functions shown in Figure 3 (C–D). Blue squares and circles indicate experimentally determined entrained phases measured using the fluorescence polarization reporter in Figure 2C. Peak times in polarization data were converted to equivalent peak KaiC phosphorylation times using the measured phase offset for the polarization reporter (Figure 2—figure supplement 1). Error bars on in vitro data show uncertainty of fit phase. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 02610. 7554/eLife. 23539. 027Figure 4—source data 1. Source data for Figure 4C. This file contains data from in vitro entrainment measurements shown as blue circles and squares. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 02710. 7554/eLife. 23539. 028Figure 4—source data 2. Source data for Figure 4—figure supplement 2. This file contains in vitro entrainment measurements shown as blue squares in Figure 4—figure supplement 2B. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 02810. 7554/eLife. 23539. 029Figure 4—figure supplement 1. Example simulation of a phase oscillator governed by one set of experimentally determined L (θ) and D (θ) functions and subjected to a driving cycle. (A) The phase oscillator reaches stable entrainment within 3–5 light-dark cycles (τ = 10–14 hr) for a wide range of starting phases. Simulation parameters same as in Figure 4B; L (θ) and D (θ) as shown in (B) below. (B) Example pair of step response functions, denoted L^ (θ^) and D^ (θ^), measured using the fluorescence-polarization reporter of KaiB-KaiC binding (magenta and blue curves in Figure 3C–D). Colored circles mark phases corresponding to dawn (on L^ (θ^) ) and dusk (on D^ (θ^) ) in seasonal entrainment simulations of different day length τ (see color bar). θ = 0 refers to the phase when the minimum of the KaiC phosphorylation rhythm occurs. See Computational methods for details. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 02910. 7554/eLife. 23539. 030Figure 4—figure supplement 2. Simulations of seasonal entrainment for a phase oscillator driven by linearized step-response functions (Llin and Dlin). (A) Simulated seasonal response of phase oscillators governed by the four possible combinations of nonlinear L^ (θ^) and D^ (θ^) step-response functions in Figure 3 (C–D) (blue shaded areas) and their linearized versions L^lin (θ^) and D^lin (θ^) (shaded green areas). In simulations, tpk is defined as the time when oscillator phase θ equals π rad (0. 5 cycles), corresponding to the peak of KaiC phosphorylation. In vitro measurements overlaid in blue as in Figure 4C. (B) Simulated seasonal response of phase oscillators governed by step-response functions measured for an independent preparation of clock proteins using SDS-PAGE analysis of KaiC phosphorylation Figure 3 (E–F). Shaded areas correspond to standard deviations determined by bootstrapping (see Computational methods). Blue squares show seasonal response measured in the purified KaiABC oscillator using the same preparation of proteins and measured via SDS-PAGE (Figure 2C). tpk is defined as in (B) above. Error bars represent uncertainty of fit phase from sinusoidal regression. (C) Example pair of step-response functions L^ (θ^) and D^ (θ^) and their linearized versions, L^lin (θ^) and D^lin (θ^), used in simulations in (A) above and measured using the fluorescence-polarization reporter of KaiB-KaiC binding (magenta and blue curves in Figure 3C–D). Colored circles mark stably entrained phases corresponding to dawn (for L^ (θ^) ) and dusk (for D^ (θ^) ) in simulations of different day length τ (see color bar). θ = 0 refers to the minimum of the KaiC phosphorylation rhythm. See Computational methods for details. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 03010. 7554/eLife. 23539. 031Figure 4—figure supplement 3. Dependence of the slope of entrained phase on the slopes of step response functions. Heat map of slope m, describing the scaling of oscillator peak time with day length, as a function of the slopes l and d of linear L (θ) and D (θ) step response functions and oscillator frequencies in light and dark (ωD and ωL) (see main text and Appendix 1). This relationship is described by m (l, d) =1−− (1−l) × (ωD/ωL) d+l−ld−1d+l−ld. The ratio of oscillator frequencies in light and dark was estimated to be ωD/ωL=0. 93 based on the analysis of datasets in Figure 3 (C–D). Dashed isolines connect all points with m = 0. 53 (left) and m = 0. 38 (right), corresponding to our estimates of scaling coefficients in vivo and in vitro in Figure 2 (C). The approximate range of values of l and d estimated from experimental measurements in Figure 3 (C–D) is indicated by the white crosshair. Crosshair lengths represent mean±standard deviation of estimates of l and d determined by bootstrapping, as described in Computational methods. Negative values of m are colored in grey. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 031 As described above, we also measured L (θ) and D (θ) step-response functions with a separate preparation of Kai proteins using a gel-based assay to read out the phase of the KaiC phosphorylation rhythm (Figure 3E–F). Although the absolute magnitude of the step responses in these measurements was larger, they still predict linear scaling of entrained phase in simulations for day lengths between 6 and 14 hr (Figure 4—figure supplement 2B). The discrepancy in magnitude between these measurements may point to differences in sensitivity to input cues between different preparations of Kai proteins, or to a slight perturbative effect of fluorescently labeled KaiB. We found that simulated entrainment of the phase oscillator model was particularly sensitive to the regions surrounding subjective dusk (for D (θ) ) and subjective dawn (for L (θ) ) (Figure 3E–F, colored arrows). Because the fluorescence polarization approach allows us to measure many conditions in an automated way over many days, and thus to disentangle phase shifts from period differences (Figure 3C–D, Figure 3—figure supplement 1), these higher time resolution measurements better constrain the portions of the response functions critical for entrainment. Given the observation that oscillator entrainment was sensitive to local shapes of L (θ) and D (θ), we wondered if midday tracking requires a specific form of these response functions or if certain generic features are sufficient. Based on our observation that L (θ) and D (θ) in the KaiABC system are approximately linear in the regions used during metabolic entrainment, we asked whether linear L (θ) and D (θ) would result in a linear dependence of clock phase on day length. Consider a phase oscillator driven by light-dark cycles of period T and day length τ (Figure 7). When the oscillator is entrained (phase-locked) to the light-dark cycle, the oscillator returns to the same state by the end of a full cycle. Starting from an initial phase θ0, the clock accumulates phase ωLτ during the day, then experiences a phase shift of magnitude D (θτ) at dusk, accumulates phase ωD (T−τ) at night, and finally responds to dawn with a phase shift L (θT): 10. 7554/eLife. 23539. 032Figure 7. Phase oscillator entrainment. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 032 Here all angles are measured in units of cycles (1 cyc=2π), and ωL and ωD are the oscillator frequencies in the light and dark, respectively. For stable entrainment, the effects of L (θ) at dawn and D (θ) at dusk must balance the phase accumulated by the oscillator, so that the phase returns to the same point at the end of each cycle. This expression for the entrained clock phase scales linearly with day length if its derivative with respect to τ is constant. The simplest way to achieve this condition is if L (θ) and D (θ) are themselves linear functions of θ, such that D (θ) ≈−d (θ−θD) and L (θ) ≈−l (θ−θL) over the relevant range of clock times. If oscillator frequency is the same in light and dark, as is approximately true for the Kai oscillator (ωD/ωL= 0. 93 ± 0. 01, see Computational methods), the peak time of the oscillation (measured in hours after dawn) can be expressed as tpk= mτ+C, with the slope m (l, d) =d (1−l) / (d+l−ld) determined by the slopes of the linear L (θ) and D (θ) functions. If the day and night oscillator frequencies differ, we still obtain a linear dependence on day length, but with an altered expression for the proportionality constant m (see Appendix 1). This expression for m imposes constraints on values of l and d required for an oscillator to track different portions of the day-night cycle. Figure 4—figure supplement 3 highlights the requirements on l and d for a midday-tracking clock. To determine whether L (θ) and D (θ) for the Kai oscillator are in line with these mathematical requirements, we examined the linear portions of the step-response functions in Figure 3. Indeed, the slopes of L (θ) and D (θ) from the fluorescence polarization assay (l = 0. 34 ± 0. 03, d = 0. 38 ± 0. 05) predict an m value consistent with our measurements of the entrained in vitro oscillator using the same method (m (l, d) = 0. 34 ± 0. 04, calculated in the linear model vs. m = 0. 38 ± 0. 07, measured) (Figure 4—figure supplement 3). Because L (θ) and D (θ) are periodic functions on a circle, they cannot be linear everywhere with non-integer slope. However, maintaining the linear scaling of phase with day length only requires that both L (θ) and D (θ) be linear over the range of clock times when dawn and dusk occur, respectively, with the nonlinearities required to satisfy periodicity appearing at other times. The exact width of the linear region of L (θ) and D (θ) depends on m and the range of day lengths that the oscillator is required to track. For example, an oscillator capable of tracking midday (m = 0. 5) over a 12 hr range of day lengths requires that L (θ) and D (θ) be linear over at least one quarter of the cycle. This criterion is met by the measured L (θ) and D (θ) (Figures 3C-D). To further test whether entrainment of the Kai system can be described by this framework, we returned to our phase oscillator simulations (Figure 4). When we replaced our experimentally measured L (θ) and D (θ) functions with linear approximations, the simulated oscillator exhibited a similar scaling of entrained phase over a wide range of day lengths (Figure 4—figure supplement 2A–B). The linear approximations work because the regions where the step response functions deviate strongly from linearity are avoided in our entrainment simulations (Figure 4—figure supplement 2C). Together, these results indicate that the driven behavior of the cyanobacterial circadian clock in 24 hr cycles can be approximated as a simple oscillator that shifts in response to light-dark transitions with a sensitivity that varies linearly with phase. To test whether this linear mathematical framework holds more generally and to understand how the cyanobacterial clock responds to a broader range of environments, we sought to measure clock response to a variety of external conditions with both single and repeated light-dark transitions. The phase oscillator model described above was inspired by the observation of linear scaling of clock phase in 24 hr days with various day lengths. The key model assumptions are (i) a unique clock cycle in the light and in the dark, (ii) rapid relaxation from one cycle to another when conditions change, and (iii) sensitivity to environmental changes that varies linearly with clock time. While these assumptions hold, the model should be capable of describing the behavior of a biological clock in arbitrary fluctuating environments. For example, the response to a single dark pulse can be decomposed into sequential step-down and step-up responses. To test the range of validity of this mathematical model, we used our LED array system (Figure 1A) to collect data on S. elongatus clock function in response to dark pulses administered at different clock times. This corresponds to a classical phase response curve (PRC) analysis, a commonly used tool in circadian biology for characterizing the response of a biological clock to perturbations. We also probed clock responses to dark pulses of varying lengths (a so-called ‘wedge’ analysis), and to repeated light-dark cycles with periods different from 24 hr. Our phase oscillator model predicts that the quantitative response to these various perturbations should all be related through the step-up and step-down functions (see Appendix 1). One specific prediction of the model is that changing the time at which a dark pulse begins and the duration of a dark pulse should have separable linear effects on the oscillator, which would manifest as linear curves in phase response and wedge analyses. These experimental data and a fit of our linear model to the results are presented in Figure 5. The lines of best fit in Figure 5A–C were obtained from a global fit to all of the datasets, with two fitting parameters that fix the slopes of the regression lines across all conditions (Appendix 1). The overall agreement between the model and data suggest that limited data on clock response can be successfully extrapolated to other conditions using this approach. 10. 7554/eLife. 23539. 033Figure 5. Phase oscillator model with linear phase shift functions predicts entrainment of the cyanobacterial clock to different light-dark (LD) patterns. In all panels, error bars represent standard deviations (n = 4–8 technical replicates per point). Lines are fit globally to all three datasets in (A) - (C). See Computational methods for details. (A) Phase resetting analysis. Phase shifts of bioluminescence rhythm (PkaiBC: : luxAB) due to 12 hr dark pulses (δ = 12 hr) administered throughout the circadian cycle. The experimental protocol is represented schematically above the graph. Cells were exposed to one 12 hr dark pulse and released into constant light; 12 hr dark pulses were administered at the indicated times. θ (tDP) is the clock phase at beginning of the dark pulse, with θ = 0 defined as clock phase at the trough of the bioluminescence rhythm. (B) Wedge analysis. Phase shifts of bioluminescence rhythm (PkaiBC: : luxAB) due to dark pulses of varied length (δ = 8–16 hr) administered near subjective dusk (36 or 39 hr after an initial 12 hr dark pulse). Clock phases at the beginning of the dark pulse are listed in parentheses; θ = 0 is defined as clock phase at the trough of the bioluminescence rhythm. The experimental protocol is represented schematically above the graph. (C) Seasonal response in non-24 hour environmental cycles. Cells were grown in LD cycles with period T = 22–26 hr and day length τ = 8–14 hr (see schematic on left). After five entraining cycles, cells were released into LL and the phase of the circadian rhythm was estimated by sinusoidal regression. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 03310. 7554/eLife. 23539. 034Figure 5—source data 1. Source data for Figure 5A–C. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 03410. 7554/eLife. 23539. 035Figure 5—figure supplement 1. Simulation of a phase-resetting curve. (left) Simulated phase-resetting curve due to a 12 hr dark pulse for a phase oscillator governed by linear step-response functions L^lin (θ^) and D^lin (θ^) (right), as in in Figure 4—figure supplement 2. Colored circles indicate phases at the beginning of the dark pulse (see color bar). θ = 0 refers to the trough of the KaiC phosphorylation rhythm. DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 035 To understand the implications of these results, consider the phase response curve (PRC) for a 12 hr dark pulse (Figure 5A). Although phase response analyses frequently separate PRCs into regions of low sensitivity to perturbation (‘dead zones’) followed by highly nonlinear regions with large phase shifts, our model suggests a different interpretation (Pattanayak et al. , 2014; Johnson, 1999; Schmitz et al. , 2000; Pfeuty et al. , 2011). Clock response is weakest at times when nightfall is expected (t = 36 and 60 hr) and increases gradually with clock time. The slowly changing regions of the PRC are well described by line segments with the same slope, in agreement with our model based on linear step-response functions L (θ) and D (θ). The ‘breakpoint’ in the PRC near 50 hr is a consequence of the fact that these periodic step-response functions cannot be linear everywhere with non-integer slope, as discussed earlier. In other words, a circadian clock capable of tracking midday over a wide range of day lengths is expected to have a PRC that is approximately linear over many hours except for a narrow region of rapid change that is required to satisfy periodicity. Indeed, in simulations where L (θ) and D (θ) are linear everywhere except for a discontinuous jump, the PRC for a long dark pulse appears as a straight line except for a single breakpoint (Figure 5—figure supplement 1). Near the breakpoint, small changes in the timing of darkness result in large differences in phase shifts, and we would expect to find this region of the PRC at a clock time when such a perturbation is least likely to occur naturally. In the dark-pulse PRC for S. elongatus, the breakpoint near 50 hr occurs when prolonged darkness is improbable (Figure 5A). The maximal phase shift in a PRC can be a useful tool to characterize clock mutants, but our analysis highlights that these regions are unlikely to be experienced in natural conditions and may exist only as consequences of periodicity constraints. Finally, we asked how step response functions with linearly increasing sensitivity are related to the mathematical structure of the oscillator and ultimately to the underlying molecular mechanism. Do step response functions with these properties arise generically, or do they require fine-tuned choices of parameters? To address these issues, we considered the simplest possible dynamical representation that allows for distinct day and night clock cycles. In this model, the clock cycle during the day is represented by a circular limit cycle with unit radius, and the clock time is defined by the angular coordinate of the oscillator on this limit cycle (Figure 6A). The effect of darkness is to deform the limit cycle, transforming the daytime orbit to a nighttime orbit. For simplicity, we assume this night cycle lies in the same plane as the day cycle and is also circular, but may be offset relative to the day cycle and have a distinct radius (Figure 6A). 10. 7554/eLife. 23539. 036Figure 6. Nearly linear step response functions can arise from the relative geometry of day and night limit cycles. (A) Geometric model of oscillator phase resetting. During the day, the oscillator runs with constant angular velocity along the daytime orbit (yellow), which has unit radius and is centered at the origin. At dusk, the oscillator transits to the nighttime orbit (black), which has radius R and is displaced from the daytime orbit by X units. In the limit where the nighttime orbit is strongly radially attracting, we can approximate oscillator response to the light-dark transition (D (θτ), red arrow) as an instantaneous jump from phase θτ on the daytime orbit toward the center of the nighttime cycle, resulting in phase θτ+D (θτ) on the night orbit. (B) Simulation of oscillator phase shifts due to light-dark transitions at different phases on the day orbit (red arrows) for R = 2, X = 2. For geometries with X ≈ R, phase angles on the day orbit are compressed to an arc on the night limit cycle that subtends a smaller angle. See Computational methods for calculation details. (C and D) Simulations of L (θ) and D (θ) step response functions arising from the geometric arrangement of day and night cycles in (B). Linear regions of L (θ) and D (θ) are marked with black dashes. See Computational methods and Appendix 1 for calculation details. (E) Heat map of the slope m of the approximately linear relationship between entrained phase and day length, plotted as a function of X and R. In white regions, the oscillator does not entrain stably or the oscillator does not show linear scaling of phase with day length. Slope determined from simulations of oscillator entrainment to 24 hr driving cycles of day length τ = 6–18 hr. See Computational methods for details. (F) Limit cycles traversed by the KaiABC oscillator in vitro in metabolic conditions mimicking day (yellow, [ATP]/ ([ATP]+[ADP]) ≈ 100%) and night (black, [ATP]/ ([ATP]+[ADP]) ≈ 25%). Oscillations in KaiC phosphorylation on Ser431 and Thr432 are replotted from data in Phong et al. (2013). DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 03610. 7554/eLife. 23539. 037Figure 6—figure supplement 1. Illustrations of limit cycle geometries that give rise to step-response functions L (θ) and D (θ) with different slopes, resulting in dusk-, dawn-, or midday-tracking entrainment. In all schematics, the day orbit (yellow) is centered at the origin and has radius 1. The night orbit (black) has radius R and is displaced from the day orbit (logX=0. 5) units). Light-dark (D (θ) ) and dark-light (L (θ) ) transitions are indicated by red and blue arrows, respectively. (A) When the night cycle is much smaller than the day cycle (R << 1), both dawn and dusk are strongly resetting (l≈1, d≈1). Because all phases on the night orbit are mapped to a small range of phases on the day orbit, oscillator phase at dawn is independent of day length during entrainment (m≈0, dawn-tracking). (B) When the sizes of day and night orbits are comparable (R ≈ 1), the slopes of L (θ) and D (θ) depend on the fine-tuned arrangement of the orbits. Generally, both L (θ) and D (θ) exert entraining effects in this regime (l > 0, d > 0). The slope of entrained phase with day length depends on both l and d, according to m (l, d) =d (1−l) / (d+l−ld) (see Figure 4—figure supplement 3 and Appendix 1). (C) When the night cycle is much bigger than the day cycle (R >> 1), dusk is strongly resetting, but dawn transitions have little effect on clock phase (l≈0, d≈1). Entrainment to repeated light-dark cycles maps clock phases on the night orbit to a shrinking range of angles, leading to dusk-tracking entrainment (m≈1). DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 03710. 7554/eLife. 23539. 038Figure 6—figure supplement 2. The relative size (R) and center-to-center distance (X) of day and night limit cycles are major determinants of entrained behavior. Heat maps of m, the slope of the approximately linear relationship between entrained phase and day length, are plotted as a function of X and R on the same color scale as in Figure 6E. See Computational methods for simulation details. (A) Entrainment simulations with non-instantaneous jumps between day and night limit cycles. The half-times t1/2 for transition between the circular orbits are indicated above the heat maps. Schematic on the left illustrates the evolution of the oscillator from a point on the night limit cycle to the day limit cycle in two scenarios with different relaxation times. Each arrow represents the displacement of the oscillator in 1 hr. (B) Entrainment simulations for day and night limit cycles of varying ellipticity. The ratios of the major axis length to the minor axis length of the day and night orbits (ρD and ρN) are indicated above the heatmaps. In these simulations, we considered strongly attracting orbits (t1/2=0. 069 hr) oriented with their major axes perpendicular to the separation between their centers. (C) Entrainment simulations for day and night limit cycles with non-constant angular velocities. In these simulations, we considered circular orbits for both day and night limit cycles. Orbit attraction timescale was set to t1/2=0. 69 hours. The variability in angular speed θ˙ throughout the cycle is given by θ. =ω (1+εωsin⁡ωt). Schematic on the left illustrates the evolution of the oscillator along day limit cycle in two scenarios with different values of εω. Each arrow represents the displacement of the oscillator in 1 hr; size of the arrowheads illustrates changes in angular velocity throughout the cycle (not to scale). DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 03810. 7554/eLife. 23539. 039Figure 6—figure supplement 3. Interpretation of m, the slope of the approximately linear relationship between entrained phase and day length. (A) The value of m dictates whether the circadian rhythm aligns to dawn (m = 0), dusk (m = 1), or an intermediate point of the day-night cycle (e. g. midday for m = 0. 5). Orange and gray curves show sinusoidal fits to average transcriptional profiles of dawn and dusk genes in S. elongatus based on data from Vijayan et al. (2009). Gray bars indicate night in light-dark cycles for m = 0,0. 5, and 1. (B) Tracking midday as the day length varies may be a strategy to balance biosynthetic resources between dawn and dusk transcriptional programs. For each value of m, the corresponding curve shows a numerical estimate of the relative fraction of dusk and dawn gene expression that occurs during the day in S. elongatus (arbitrary units, see Computational methods). DOI: http: //dx. doi. org/10. 7554/eLife. 23539. 039 After a transition from one condition to the other, the system state must evolve to the new limit cycle. We suppose that each cycle is very strongly radially attracting. That is, when the system is in a state off the limit cycle, it is rapidly pulled to the closest point on the cycle. Under these conditions, the step transitions from one cycle to another are determined purely by geometry (Figure 6A), and we can connect this picture with the phase-only description we used to analyze the experimental data. Here, linearly increasing sensitivity of a step response has a simple geometric interpretation: when the two cycles are displaced from each other, a step transition maps an arc of one cycle onto an arc of the other cycle that subtends a smaller angle than the original (Figure 6B). Thus, step transitions compress or expand angular distance when mapping one circle onto another. The slopes of the step response functions are given by the compression factor in this mapping (Figure 6C–D, Figure 6—figure supplement 1; see Appendix 1). To determine how the entrained clock phase depends on geometry in this model, we simulated step transitions and then calculated the slope m of clock phase versus day length (Figure 6E). This calculation indicates that midday tracking (m ≈ 0. 5) requires that the separation between the light and dark cycles is comparable to the radius of the cycle (R ≈ X). This requirement can be understood intuitively by considering three cases where orbits have the same center-to-center separation (log X = 0. 5) but different sizes (Figure 6—figure supplement 1). Dusk transitions are strongly resetting for this choice of X, but the strength of dawn resetting varies with the relative size of the orbits. Dawn-tracking entrainment results when the night orbit is much smaller than the day orbit (l ≈ 1) and dusk-tracking entrainment results when the night orbit is much larger (l ≈ 0, d ≈ 1), in accord with how m varies as a function of l and d (Figure 4—figure supplement 3). When the size of both orbits is similar to their center separation, the oscillator can track intermediate phases, such as midday. Although this model makes a simple connection between attractor geometry and entrainment, it also makes a number of simplifying assumptions about oscillator dynamics that may not hold true for real biological clocks, such as instant transitions between cycles, perfectly circular orbits and constant angular frequencies along each cycle. To test the consequences of relaxing these assumptions, we used a dynamical model where the evolution of the system is described explicitly (see Computational methods). Our simulations in Figure 6—figure supplement 2 show that in these more complicated scenarios the geometric arrangement of day and night cycles remains a key determinant of the slope of entrained clock phase as a function of day length. Indeed, in all cases we studied, midday tracking was only possible for geometries where the center-to-center distance between the day and night cycles was comparable to the radius of the larger orbit (R ≈ X > 1). This dynamical systems perspective allows us to reframe conditions on the underlying biochemical mechanisms that can produce the observed midday tracking behavior. Changes in the external environment caused by transitions between night and day should affect the oscillator in such a way that the period remains close to 24 hr, but that the limit cycle is shifted by an amount comparable to its radius. The relative geometry of the two limit cycles, which is determined by the mechanisms that couple the environment to the clock, must be fine-tuned to give a specific slope for the entrained phase. Indeed, when we plotted experimentally determined orbits of the purified KaiABC oscillator on axes showing the extent of phosphorylation of two key sites, we observed an arrangement similar to the expected geometry (R ≈ X) using nucleotide conditions that simulate either day or night (Figure 6F) (Rust et al. , 2007; Phong et al. , 2013).
  Although circadian clocks are defined in part by their ability to continue to cycle in constant environments, the defects associated with clock mutants are often most apparent when organisms are faced with fluctuating environments (Ouyang, 1998; Woelfle et al. , 2004; Pittendrigh, 1972; Spoelstra et al. , 2016; Pittendrigh and Minis, 1972). Thus, an important challenge is to understand how biological clocks respond to the cycling environments found in nature, and how they function to appropriately schedule gene expression and behavior. For most organisms, there is an asymmetry between day and night, in terms of food availability, predation risk, etc. , so that the need to carry out certain activities diurnally or nocturnally presents changing demands as the length of the day varies throughout the year. The situation is especially dramatic for cyanobacteria because there is an extreme metabolic contrast between day and night. We found that S. elongatus contends with these challenges using a clock that tracks the middle of the day. The ability of circadian clocks to keep track of the phase of the light-dark cycle has been long recognized in plants, insects, rodents and higher mammals (de Montaigu et al. , 2015; Daan et al. , 2001; Edwards et al. , 2010; Hut et al. , 1999, Hut et al. , 2013; Wehr, 2001). Although molecular mechanisms that give rise to these entrainment behaviors are still being uncovered, analysis of circadian clock models has found that the presence of multiple feedback loops in complex clocks determines the number of points in the driving cycle that the oscillator can track simultaneously, by allowing different internal phase relationships between the clock components (Rand et al. , 2004). For example, the multi-feedback loop clocks in plants are able to track phases of both dawn and dusk (Edwards et al. , 2010). Consistent with this picture, we find that the core circadian oscillator in cyanobacteria, which relies on a single posttranslational feedback loop, keeps track of a single phase—the midpoint of the day portion of the cycle, a property described in our mathematical framework as a linear scaling of entrained phase with day length with slope m ≈ 0. 5. Why might keeping track of midday be useful for a photosynthetic organism with a simple clock? Clock-controlled gene expression in S. elongatus tends to be bimodal, with most genes falling into subjective dawn or dusk classes. Because the biosynthetic capacity of S. elongatus is severely limited in darkness, the midday tracking effect we describe here could be a mechanism to ensure that biosynthetic resources are partitioned in a balanced way between the dawn and dusk genes, even as the day length changes with the seasons (Figure 6—figure supplement 3). In particular, clock-driven transcription in this organism has been shown to implement a switch between anabolic and catabolic carbon metabolism, suggesting that a role for the midday tracking we observe here is to ensure balanced growth by timing this switch appropriately in days of different length (Diamond et al. , 2015). The ability to reconstitute this effect in vitro by delivering metabolic pulses to the purified Kai proteins indicates that midday tracking is not necessarily achieved through additional feedback mechanisms in the cell, but appears to be a property of the clock proteins themselves. The purified clock responds to metabolic steps with phase shifts that are linear functions of the previous phase. The slopes of these response functions are presumably tuned to give an appropriately entrained clock phase. Notably, linear responses have also been observed for the Kai oscillator following temperature steps, suggesting that this is a general reaction of the system to inputs (Yoshida et al. , 2009). The mathematical framework that we describe here has deep similarities to the theory of nonparametric entrainment developed by Colin Pittendrigh (Pittendrigh and Daan, 1976; Daan, 2000; Pittendrigh and Minis, 1964). His work motivated a theory of entrainment to diurnal cycles mediated by instantaneous phase shifts at dawn and dusk, which can be summarized by a phase response curve. Daan and Pittendrigh suggested that the ability of the clock to track specific phases of the day-night cycle in different seasons depends on the shape of the phase response curve as well as the difference between the free-running period of the clock and the period of the day-night cycle (Pittendrigh and Daan, 1976; Johnson et al. , 2003). Our decomposition of driven behavior of the KaiABC oscillator into individual step responses is in the spirit of this classic paradigm. We note that a mismatch between the free-running clock period and the external cycle is not required for stable entrainment in our phase oscillator model (see Appendix 1). Instead, entrainment can be achieved from the opposing effects of the step-up and step-down phase shifts, which occur at different clock times in days of different length. The step-response curves underlying entrainment in our model are nonlinear functions of clock phase, but they can be successfully approximated by linear functions over the interval of clock phases used during entrainment. Our simulations suggest that the slopes of these locally linear functions are key determinants of entrained phase, along with changes in clock period in daytime and nighttime conditions. Successful prediction of how the in vitro oscillator entrains to rhythmic environments is due in large part to the ability to map out the step-response curves with high temporal resolution. In this study, we achieve this by measuring free-running oscillations in an automated way using a fluorescence polarization probe. Even though there are likely other effects at play in natural environments, as long as the system can be described by fast relaxation back to distinct limit cycles in day and night, instantaneous step-response descriptions of the kind used here should be applicable. The data reduction achieved in our linear model holds the promise of predicting the behavior of the circadian rhythms in many time-varying environments from a minimal data set that characterizes oscillator response, and may be applicable to clocks in many organisms. Although the biochemistry of even the simplest circadian clock is complex, our analysis suggests that a key determinant of entrained behavior is how the clock limit cycle is deformed by coupling to the environment. Viewed in this way, the entrainment properties of circadian oscillators arise from simple features of the geometry of the limit cycle attractor that could be measured in any organism. The concept that the phase-shifting of oscillators can be studied in terms of their geometric properties was initially developed by Winfree (Winfree, 1973). In this context, it would be informative to analyze the geometric arrangement of day and night limit cycles of clock gene transcripts in other species. An important goal for the future is to understand how the biochemical properties of the clock components in cyanobacteria and other organisms achieve the effect of shifting the limit cycle without changing the period, allowing us to use dynamical systems theory to bridge the gap between molecular detail and systems-level clock phenotypes.
summary: All life forms on the surface of our planet face an environment that cycles between day and night as the Earth rotates around its axis. To deal with these regular changes, organisms from bacteria to humans have internal rhythms, called circadian clocks, that coordinate different aspects of these organisms’ lives, from growth to the urge to sleep, with the day-night cycle. The simplest of all known circadian clocks is found in bacteria called cyanobacteria, which live in bodies of water all around the world. Like plants, these microorganisms can harness the energy in sunlight to fuel their growth. The internal clock of cyanobacteria is remarkable because it can be rebuilt in the laboratory using just three components, proteins called KaiA, KaiB and KaiC. When mixed together in a test tube, these proteins spontaneously generate a 24-hour cycle in which KaiC gets chemically modified by the addition and removal of a phosphate group. These chemical modifications on KaiC are like the hands of the clock that tell the time of day. Much work on the internal clock of cyanobacteria has focused on how the clock proteins generate rhythms. What has been less clear is how this rhythm works in an environment that changes between day and night and from season to season. For example, away from the equator, the length of the day changes throughout the year. Changes in day length are important for cyanobacteria because they can grow only during the day and must rest at night. Yet, it was not understood how this microorganism' s clock adjusts to days of different length. To address this question, Leypunskiy et al. made a device to grow cyanobacteria in many different light-dark cycles while monitoring their circadian rhythm. The experiments revealed a fairly simple result: the internal clock time is always the same at the middle of the day, regardless of how long the day is. Next, Leypunskiy et al. developed a way to study how the clock proteins in the test tube respond to chemical signals mimicking day and night. Unexpectedly, this showed that the Kai proteins themselves have the ability to track the middle of the day. It turns out that these proteins are not only able to generate a 24-hour rhythm, but also to correctly align it to the day-night cycle. Together, these findings will guide other researchers to better understand the molecular origins of how circadian clocks align to the day-night cycle. Moreover, by showing that the effects of a light-dark cycle on a circadian clock can be mimicked in a test tube, these results open the doors to using the tools of biochemistry to understand how circadian clocks work in natural environments.
keywords: computational and systems biology
  microbiology and infectious disease
title: The cyanobacterial circadian clock follows midday in vivo and in vitro