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The circadian clock coordinates daily physiological, metabolic and behavioural rhythms. These endogenous oscillations are synchronized with external cues (‘zeitgebers’), such as daily light and temperature cycles. When the circadian clock is entrained by a zeitgeber, the phase difference ψ between the phase of a clock-controlled rhythm and the phase of the zeitgeber is of fundamental importance for the fitness of the organism. The phase of entrainment ψ depends on the mismatch between the intrinsic period τ and the zeitgeber period T and on the ratio of the zeitgeber strength to oscillator amplitude. Motivated by the intriguing complexity of empirical data and by our own experiments on temperature entrainment of mouse suprachiasmatic nucleus (SCN) slices, we present a theory on how clock and zeitgeber properties determine the phase of entrainment. The wide applicability of the theory is demonstrated using mathematical models of different complexity as well as by experimental data. Predictions of the theory are confirmed by published data on Neurospora crassa strains for different period mismatches τ − T and varying photoperiods. We apply a novel regression technique to analyse entrainment of SCN slices by temperature cycles. We find that mathematical models can explain not only the stable asymptotic phase of entrainment, but also transient phase dynamics. Our theory provides the potential to explore seasonal variations of circadian rhythms, jet lag and shift work in forthcoming studies.
The circadian clock has evolved to synchronize physiological and behavioural rhythms to periodically recurring environmental conditions termed zeitgebers. When entrained by a zeitgeber, the phase difference ψ between the zeitgeber phase and the phase of the circadian oscillator assumes a stable value. A ‘correct’, or well-adjusted, phase of entrainment ψ is of central importance for the fitness of organisms [1,2]. The so-called phase of entrainment ψ depends on several clock and zeitgeber properties.
In the laboratory, circadian rhythms can be entrained to variable zeitgeber periods T as reviewed in . A monotonic dependence of ψ on T has been found in 20 studied organisms and large variations of the entrainment phase were reported. Under natural conditions of T = 24 h, variations of intrinsic period τ lead to different entrained phases ψ, that are associated with ‘chronotypes’ . Short endogenous periods τ often lead to early phases (‘morning larks’) and long periods τ correspond to late phases (‘night owls’) . Mutations that affect the intrinsic period τ have been found to result in large shifts of the entrainment phase ψ. For example, a patient with familial advanced sleep phase syndrome has a short endogenous period of τ = 23.3 h leading to a phase advance of more than 3 h . Mutants with altered periods τ in hamsters , cyanobacteria  and fungi  obey pronounced shifts of the entrainment phase. All these data indicate that the phase of entrainment ψ is a monotonic function of the period mismatch τ − T.
There are other parameters that influence the entrainment phase ψ. For instance, zeitgeber strength has an effect on the phase of entrainment [10,11]. In particular, strong zeitgebers lead to narrower distributions of chronotypes [12,13]. Effective zeitgeber strength is intimately related to the amplitude of the intrinsic clock . The critical role of the oscillator amplitude for the entrainment has been demonstrated in chick pineal cells , in mouse mutants [16,17], and for human chronotypes . Those results suggest that in addition to the period mismatch τ − T, the ratio between the zeitgeber strength and the amplitude of circadian oscillations affects the entrainment phase. Furthermore, the phase of entrainment can depend on seasons and latitudes  and on additional clock parameters such as relaxation times and inter-cellular coupling .
Understanding the intriguing complexity of empirical data on the phase of entrainment requires quantitative theoretical concepts as pioneered by Wever , Winfree  and Kronauer et al. . Numerous more recent models of the circadian pacemaker were employed to study the phase of circadian entrainment [24–27].
Recently, we derived a generic model of entrainment using different approaches: (i) phase transition curves, (ii) Kuramoto phase equations and (iii) resonance theory . The core result of this theoretical framework is illustrated in figure 1: the phase of entrainment ψ is shown as a function of period mismatch τ − T and the ratio of zeitgeber strength to oscillator amplitude Z/A. According to this theory, the phase of entrainment can vary by about 12 h within the range of entrainment as predicted by Wever  and supported by empirical data . The range of entrainment increases with growing zeitgeber strength leading to a triangular region of entrainment termed ‘Arnold tongue’ . Figure 1 represents many of the empirical findings discussed above. It demonstrates that period mismatch τ − T and effective zeitgeber strength Z/A strongly affect the phase of entrainment ψ. Figure 1b shows that the phase of entrainment varies by 12 h, which leads to quite variable ψ for a narrow entrainment range (green line in figure 1b). Increasing zeitgeber strength can lead to both decrease or increase of ψ depending on the mismatch τ − T (figure 1c).
In this paper, we address the following questions:
The Kuramoto theory describes the dynamics of the phase difference between a self-sustained oscillator subjected to an external zeitgeber with arbitrary waveforms . This concept is related to the continuous approach of phase changes by Aschoff . The dynamics of the phase difference between the periodic zeitgeber and the oscillator is governed by the frequency mismatch Ω = ωz−ω between the zeitgeber frequency ωz and the oscillator frequency ω and the integral over the product of the zeitgeber Z(t) with the phase response curve (PRC):
The integral represents the cumulative action of the zeitgeber Z(t) over one period of the zeitgeber as ‘sensed’ by the PRC. The parameter K denotes the relative zeitgeber strength which is defined as a ratio of the zeitgeber strength to the oscillator amplitude. If PRC or zeitgeber are sinusoidal, the above equation simplifies to the so-called Adler  or Kuramoto equation
A stationary phase difference ψ can be obtained from the stationarity condition dψ/dt = 0, which leads to the following expression for the stable entrainment phase:
within the entrainment range Ω [−K,K]. Figure 1 shows the dependencies of the entrainment phase ψ on parameters. The frequency mismatch is related to the period mismatch by
Additionally to the simplest phase oscillator equation (2.1), we analysed two more detailed clock models: the Gonze model  and the Relogio model . Whereas the Gonze equations are a simplified model of a single self-inhibiting clock gene, the Relogio model consists of 19 differential equations that describe two interlocked translational feedback loops involving several genes.
The parameters in both models were chosen in such a way so as to produce stable oscillations with a period τ ≈ 24 h. Additionally, the ordinary differential equations were subjected to a periodic zeitgeber signal, characterized by an amplitude Z and a period T. With the help of AUTO continuation software , we computed the borders of the Arnold tongue in both models. Within the Arnold tongue, we calculated the isophases lines, i.e. the lines in the (Z,T) parameter planes along which the phase difference between the zeitgeber and the circadian oscillator was constant (cf. figure 2).
We analysed the data on the phase of circadian entrainment from the previously published paper with N. crassa experiments . There, the entrainment phases were measured in dependence on three experimental parameters: (i) the free running period for wild-type with τ = 22.5 h and two mutants with free running periods τ = 16 h and τ = 29 h, (ii) the period T of the zeitgeber in the range from 16 to 26 h in steps of 2 h, and (iii) the photoperiod, i.e. the relative duration of the light phase within a day, nine values in a range from 16 to 84%. The phase of circadian behaviour was determined as the time of conidiation onset. A total of 162 entrainment phases were thus available.
SCN slices of Per2::Luc mice were collected and cultured as described in . PER2-driven bioluminescence emitted from the SCN was recorded in temperature-adjustable light-tight boxes equipped with photomultiplier tubes . The following protocol for temperature entrainment experiments was performed: after several days of free run (i.e. at a constant temperature of 37°C), the slices were periodically subjected to cold phases in such a way that the total period of temperature cycles was either T = 22 h, T = 24 h or T = 26 h. After seven or 17 temperature cycles, the slices were again subjected to a constant temperature of 37°C for a few days in order to estimate the post-entrainment oscillation period of the slice. The bioluminescence data were detrended by dividing by a 24-h running average and after that fitted by series of trigonometric functions with slowly varying coefficients as described below. From those fits, the instantaneous phase and period of the oscillations were extracted.
The bioluminescence data from cultured PER2::LUC mouse SCN slices can be found as electronic supplementary material to this paper.
In order to extract the period and phase information from the bioluminescence data, we used the following procedure. First, the time series were fitted by the Fourier-like series:
where N was the number of Fourier modes, 24 h is the base period of the oscillations and x0 is the offset. The Fourier coefficients aj(t) and bj(t) were chosen as polynomials in time of order P − 1:
with parameters to be found by the fitting procedure. The total number of fitting parameters was hence P × N + 1, which included and the offset x0. The fitting procedure was realized as a minimization problem for the squared difference between the experimental data and the model described above. Numerically, the Levenberg–Marquardt algorithm was used as implemented in MINPACK library accessed through the Python package scipy.optimize.leastsq.
The instantaneous phase of the oscillations ϕ(t) was extracted from the fitting parameters a1(t) and b1(t) as
where the standard arctan2 function computes the arctangent in the proper quadrant depending on the signs of both arguments a1(t) and b1(t). Having calculated the phase, we calculated the instantaneous period τ(t) as
with the time derivative of the phase calculated through finite differences. As a consistency check for our calculations, we used the same fitting procedure to otherwise analytically known zeitgeber signal to extract its phase and period.
Figure 1 shows the zone of entrainment (termed ‘Arnold tongue’) together with the phase of entrainment ψ derived from the Kuramoto equation (2.3). This model is based on the assumption that the integral over the product of the zeitgeber Z(t) with the PRC is a sine function. Note that the integration generates sine-like waveforms even for PRCs or zeitgeber profiles Z(t) deviating strongly from a sine-curve .
In the following, we illustrate that the behaviour shown in figure 1 is indeed generic in more realistic clock models such as a modified Goodwin model  and a two-loop model . Figure 2 shows the entrainment phases derived from these models within the Arnold tongue. It is evident that the basic features of the entrainment of the Kuramoto equation displayed in figure 1 also hold for more complex models: there is a triangular zone of entrainment in all cases and the phases of entrainment depend on zeitgeber period, zeitgeber strength, and period mismatch between the zeitgeber and the circadian oscillator. Quite similar results have been obtained for non-sinusoidal zeitgeber profiles Z(t) .
Quantitative representations of the entrainment phase as a function of period mismatch and relative zeitgeber strength as shown in figures figures11 and and22 allow generic predictions that can be compared with empirical data:
The theoretical predictions from figures figures11 and and22 state the following: (i) the distribution of the entrainment phase ψ spreads over the range of 180° across the width of the Arnold tongue, (ii) larger sensitivity of ψ to the detuning τ − T is to be expected for weak zeitgebers and, accordingly, smaller sensitivity is to be expected for strong zeitgebers, and (iii) entrainment phase ψ decreases for increasing zeitgeber strength for τ > T and entrainment phase ψ increases for increasing zeitgeber strength for τ < T.
We verified those predictions using the published data on entrainment of N. crassa from . Neurospora crassa is a filamentous fungus serving as a model organism in chronobiology. The production of asexual spores (conidia) is regulated by the circadian clock. Conidiation patterns in the so-called ‘race tubes’ can be analysed to extract period and phases of their circadian oscillations. The wild-type strain of N. crassa has a period of τ = 22.5 h and there are well-studied short- and long-period mutants. Here, we assessed the claim that the period detuning τ − T determines the phase of entrainment. From our theory, we expect that larger (smaller) free-running periods τ result in later (earlier) entrainment phases, respectively. Indeed, in figure 3a, we see this pattern: the distribution of entrainment phase shifts further to the right for larger values of τ. We found the differences between the short-period mutants and the wild-type to be small. This observation indicates that other factors such as light-sensitivity or oscillator strength affect the entrainment phase as well.
We find the following ranges of entrainment phase: 165° for the short-period strain, 185° for the wild-type and 135° for the long-period strain (here, 360° correspond to 24 h). For the short-period mutants and the wild-type, those ranges of entrainment phases are close to 180°, which has been predicted to be the maximal distribution width of the entrainment phase . From this, we conclude that for the strains with τ = 16 and τ = 22.5 h, zeitgeber periods cover most of the width of the Arnold tongue.
The next prediction by figure 1 is that the sensitivity of the entrainment phase ψ to the detuning τ − T depends itself on the zeitgeber strength. For a stronger zeitgeber, the average slope of the sensitivity ψ(τ − T) is smaller than for a weak one (figure 1b). In figure 3b, we show the slopes of the linear regression of ψ(τ − T) calculated for the N. crassa data. To distinguish between strong and weak zeitgebers, we used photoperiod as a proxy for the zeitgeber strength. Short days can be interpreted as light pulses, whereas short nights represent dark pulses. The zeitgeber strength perceived by N. crassa could be a complicated function of the day length (photoperiod). We make here a simplifying assumption that the overall standard deviation of the zeitgeber signal is proportional to zeitgeber strength. This implies that short days are considered as a weak zeitgeber, whereas 12 L : 12 D cycles are a strong zeitgeber. This assumption has been proposed earlier  and was recently explored theoretically in a study on circadian seasonality . It seems reasonable to assume that longer photoperiods represent stronger zeitgebers, as N. crassa is sensitive to the total amount of the received light . Under this assumption, we indeed observe that weak zeitgebers result in larger slopes of ψ(τ − T) regression (figure 3b).
We finally analyse the dependence of the entrainment phase ψ on the zeitgeber strength. The theoretically calculated figure 1c predicts that extremely early and extremely late entrainment phases move closer to each other with increasing zeitgeber strength. Figure 3c mirrors this prediction, supporting the above suggestion that the photoperiod can indeed be thought of as a surrogate of zeitgeber strength in N. crassa. From figure 3c, we see that early entrainment phases become later for entrainment with a slower zeitgeber (i.e. τ − T < 0) and late entrainment phases become earlier for entrainment with a faster zeitgeber (i.e. τ − T > 0). This behaviour of the entrainment phase coincides with the intuition that stronger zeitgebers diminish the difference between ‘morning larks’ and ‘night owls’.
Summarizing our analysis of Rémi et al.  data, we found that the predicted behaviour of the entrainment phase within the Arnold tongue can explain the measured entrainment phases in N. crassa. In particular, the dependence of the entrainment phase on the detuning τ − T and on the zeitgeber strength (as approximated by photoperiod) agrees with our theory. The detailed statistical analysis of the above results can be further found in .
In mammals, the SCN has been identified as the master circadian clock . This paired nucleus receives photic input via the retino-hypothalamic tract, which allows for synchronization with the environmental light–dark cycle . The SCN orchestrates circadian rhythms in peripheral tissues and, thereby, generate a synchronized circadian output in physiology and behaviour. When explanted, the SCN is no longer sensitive to light, but can respond to temperature cycles [20,46]. In contrast to the ‘weak oscillators’ of peripheral tissues, explanted SCN has been distinguished as a ‘strong oscillator’ due to its relative robustness towards entrainment by temperature cycles [20,46]. As a result, the entrainment range of the SCN has been found to be narrower than that of peripheral tissues: temperature cycles with the period of T = 20 h and T = 28 h could not entrain the SCN (in contrast to peripheral tissues), whereas temperature cycles with the period of T = 22 h lead to entrainment, given the zeitgeber was strong enough (6° and 8°C temperature variation).
Here we investigate what entrainment and transient phase dynamics result from the strong oscillator properties of the SCN. Figure 1 predicts that for strong oscillators (or, equivalently, small relative zeitgeber strength Z/A), the phase of entrainment should vary a lot under variations of detuning τ − T and relative zeitgeber strength. To explore the phase dynamics of the SCN oscillations, we performed experiments on explanted SCN that carry a bioluminescence reporter under control of the clock gene PERIOD2 (PER2::LUC), see , and hence allow us to monitor in real time the response of the molecular clock of the SCN to environmental stimuli. As explanted SCN are not light-sensitive, we used temperature as a means to non-invasively simulate zeitgeber cycles. We recorded clock gene-driven bioluminescence rhythms in real time during constant temperature for 5 days, followed by seven or 17 temperature cycles (alterations of low and high temperatures), and another 5 days of constant temperature.
We expected two qualitatively different outcomes of the entrainment experiments: if an SCN slice gets entrained by the zeitgeber, the phase difference between the SCN slice bioluminescence signal and the zeitgeber will approach a stable entrainment phase. If otherwise no entrainment occurs, the phase difference between the SCN slice bioluminescence signal and the zeitgeber will show no stationary, but rather transient dynamics, drifting to larger or smaller values depending on the period difference τ − T.
Figure 4 shows representative time courses of these experiments. Amplitude expansion (or ‘resonance effects’, see also ) and phase changes indicate the influence of temperature cycles on the cultured SCN slices. For T = 22 h, we expect late entrainment phases as the SCN slice has a longer period compared with the period of the temperature cycle, thus mimicking a ‘night owl’ chronotype. Contrarily, for the temperature cycle period T = 26 h, the phase should move towards early values. We indeed find that for T = 22 h, the oscillations peak later during the warm phase and for T = 26 h, the oscillations peak earlier near the end of the cold phase.
A more detailed discussion of SCN phase dynamics under temperature cycles requires quantification of the periods, amplitudes and phases from the time series of the SCN slice bioluminescence, which we present below. In figure 5, we illustrate the regression of the time series of the SCN slice bioluminescence (see Material and methods for the details) in entrainment experiments with 24 h temperature cycles. The onset of temperature cycles leads to an amplitude expansion of the signal, a stabilization of the period close to 24 h and a stable phase difference between the signal and zeitgeber phases within a range of ±1 h. Next, we apply the regression technique to temperature cycles with periods T = 22 h and T = 26 h. The free-running period of most slices has been found to be above 24 h, and thus entrainment to T = 26 h should be more easy than entrainment to T = 22 h. Our theory additionally predicts that rhythms with small amplitudes, for which a given zeitgeber hence appears stronger, should exhibit shorter transients and, consequently, fast entrainment. Two examples shown in figure 6a,b confirm these predictions. Entrainment to T = 26 h is achieved almost immediately with a stable entrainment phase (figure 6b,e).
The other six experiments (the dynamics of the phase difference is displayed in figure 6c,f) are characterized by a drifting phase over the seven temperature cycles. An explanation for the lack of immediate entrainment is a narrow entrainment range, which implies that temperature cycles can shift phases by only up to 2 h per cycle . According to our theory, phase adjustment can be as large as 12 h and thus within seven cycles, we expect primarily transient phase changes. Still theoretical predictions can be tested. For τ > T, the phase of entrainment should increase towards late phases, whereas for τ < T we predict the opposite trend in the phase dynamics. Figure 6c,f confirm these predictions: for T = 22 h, phase differences increase monotonously (figure 6c) and for T = 26 h, we found decreasing phase differences (figure 6f). These gradual phase drifts are also visible in the time series presented in figure 4.
In circadian biology, the phase of entrainment is of central importance for the adaptation of organisms to environmental cycles. Numerous empirical data reveal that the entrainment phase depends on oscillator characteristics such as period and amplitude as well as on zeitgeber properties. The Arnold tongue diagram in figure 1 predicts that the period mismatch τ − T and the effective zeitgeber strength Z/A control the phase of entrainment.
The critical role of the detuning τ − T has been studied intensively [3,47]. Figures 1c and and33c show that the ratio of zeitgeber strength to oscillator amplitude also affects entrainment phase shifts. The relevant amplitude of the autonomous oscillator is often difficult to quantify. Reporter signals or actograms are only indirect measurements of core clock oscillation amplitudes. Nevertheless, the effect of amplitudes on entrainment properties have been demonstrated in several systems [14–16,18,19]. In our SCN slice experiments, rhythms with small amplitudes are entrained almost instantaneously (e.g. figure 6).
The Kuramoto theory of entrainment illustrated in figure 1 assumes that the integral of the product of the zeitgeber Z(t) with the PRC can be approximated by a sine-like function (cf. Material and methods). In order to test the relevance of this assumption in mechanistic models, we analysed entrainment in a modified Goodwin model  and in a sophisticated core clock model . We found that the dependencies of the entrainment phase ψ on the detuning τ − T and the relative zeitgeber strength Z/A are quite similar to those predicted by the Kuramoto model. A re-analysis of the data on the entrainment phase of N. crassa published in  shows that the entrainment phase dependencies are consistent with the theoretical predictions.
SCN slices exhibit a relatively narrow entrainment range . According to our theory, we expect large phase changes close to the border of the entrainment range using temperature cycles of 22 and 26 h. Within 7 days of temperature cycles, we observed primarily transient phase dynamics. In order to quantify instantaneous phases, we employed a novel regression method using trigonometric functions with weights being polynomial functions of time. We found that temperature cycles with 22 h lead to increasing phases, whereas long temperature cycles with T = 26 h induce decreasing phases consistent with oscillator theory.
Most of the theoretical results in this manuscript refer to the long-time behaviour of the phase difference between zeitgeber and circadian oscillator. Our SCN experiments illustrate that the dynamics of transients can last many days. Consequently, it is not easy to assess the true asymptotic phase of entrainment from short time series. A quantification of instantaneous periods and phases together with oscillator modelling can help to distinguish transients, such as jet lag and after effects, beating (also termed ‘relative coordination’) and more complex temporal patterns such as m/n frequency locking (aka frequency de-multiplication ) and deterministic chaos .
The Kuramoto theory discussed in this paper is valid in a strict sense only for relatively weak interactions of zeitgeber and the circadian oscillator, i.e. near the tip of the Arnold tongue. Here, the boundaries of the Arnold tongue are given by saddle-node bifurcations of the limit cycles. As mammals have typically PRCs with only a few hours of phase shifts  and narrow entrainment ranges , the theory shows promise of being applicable. Neurospora crassa, however, is quite sensitive to light  and thus the good agreement of the circadian surface with our theoretical predictions was not a priori expected.
A key result of this paper is the high variability of entrainment phases near the tip of the Arnold tongue, i.e. for narrow entrainment ranges. This situation is relevant for vertebrates exhibiting ‘strong oscillators’ typically with type 1 PRC [3,49]. Within an entrainment range of 4 h, we predict entrainment phase variations of 12 h. This implies that small differences in period mismatch τ − T lead to large differences in entrainment phase. Indeed in , slopes of the ψ(τ) dependencies were found to range from 2.64 to 10.43. Also in studies with humans, quite steep dependencies ψ(τ) have been reported [5,50]. Furthermore, it has been shown for animals  and humans  that lowering zeitgeber strength widens the distribution of ψ . These observations are consistent with the predictions by the Kuramoto theory (cf. figure 1).
In summary, robustness with respect to external inputs (small PRCs) implies quite variable entrainment phases. Consequently, in strong circadian oscillators such as vertebrates or SCN slices, minor changes in period mismatch τ − T and in effective zeitgeber strength induce large phase changes. This contributes to a broad chronotype distribution in human populations with a standard deviation of midsleep-time of about 1.5 h  despite a small variation in τ with a standard deviation of only 0.2 h .
Another implication of variable entrainment phases is the flexibility of seasonal adaptation . The effects of varying photoperiod have been studied in several models [26,51–54]. Seasonal variations induce major changes of effective zeitgeber strength and thus the entrainment phases can vary by several hours. In particular, phases can parallel dusk or dawn as observed in the N. crassa data , which has been re-analysed in this study. For vertical variations within the Arnold tongue, our model also predicts that changes in zeitgeber strength keep the entrainment phase almost constant. This could explain observations that entrainment in N. crassa follows midnight. The extension of the presented theoretical framework including seasonal variations is discussed in a parallel theoretical study . The quantification of entrainment using our novel regression technique potentially allows the simulation of jet lag and shift-work schedules in forthcoming studies.
We acknowledge fruitful discussions with Christoph Schmal, Bharath Ananthasubramaniam and Julia Katharina Schlichting.
A.K. and H.H. designed the study; U.A. performed experiments with SCN explants; P.R. analysed the N. crassa data; A.G., K.I. and G.B. performed numerical studies of the models; H.H. and G.B. wrote the manuscript.
We declare we have no competing interests.
We acknowledge the financial support from the DFG through the grants SPP InKomBio, GRK 1772 and BO 3612/2-1, and from the BMBF through the grant no. 01GQ1001C.