Critical phase shifts slow down circadian clock recovery: Implications for jet lag

Journal of Theoretical Biology 333 (2013) 47–57

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Critical phase shifts slow down circadian clock recovery: Implications for jet lag Jean-Christophe Leloup n, Albert Goldbeter Faculté des Sciences, Université Libre de Bruxelles, Campus Plaine, C.P. 231, B-1050 Brussels, Belgium

H I G H L I G H T S








We study circadian clock recovery after a phase shift of the light–dark cycle. Re-entrainment of the clock is either orthodromic or antidromic. A computational model reveals a threshold between the two types of re-entrainment. Phase shifts near the threshold markedly slow down circadian clock recovery. The threshold could be responsible for severe disturbances associated with jet lag.

art ic l e i nf o

a b s t r a c t

Article history: Received 5 October 2012 Received in revised form 26 April 2013 Accepted 30 April 2013 Available online 10 May 2013

Advancing or delaying the light–dark (LD) cycle perturbs the circadian clock, which eventually recovers its original phase with respect to the new LD cycle. Readjustment of the clock occurs by shifting its phase in the same (orthodromic re-entrainment) or opposite direction (antidromic re-entrainment) as the shift in the LD cycle. To investigate circadian clock recovery after phase shifts of the LD cycle we use a detailed computational model previously proposed for the cellular regulatory network underlying the mammalian circadian clock. The model predicts the existence of a sharp threshold separating orthodromic from antidromic re-entrainment. In the vicinity of this threshold, resynchronization of the clock after a phase shift markedly slows down. The type of re-entrainment, the position of the threshold and the time required for resynchronization depend on multiple factors such as the autonomous period of the clock, the direction and magnitude of the phase shift, the clock biochemical kinetic parameters, and light intensity. Partitioning the phase shift into a series of smaller phases shifts decreases the impact on the recovery of the circadian clock. We use the phase response curve to predict the location of the threshold separating orthodromic and antidromic re-entrainment after advanced or delayed phase shifts of the LD cycle. The marked increase in recovery times predicted near the threshold could be responsible for the most severe disturbances of the human circadian clock associated with jet lag. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Circadian rhythms Computational model Phase shift Re-entrainment Jet lag

1. Introduction The circadian clock, characterized by an intrinsic period close to 24 h, is generally entrained by light–dark (LD) cycles, which correspond to the alternation of day and night. Circadian rhythms thus allow adaptation of most organisms to their periodically varying environment. In humans, flying across time zones disrupts circadian rhythms, which become re-entrained in a new LD system of reference after sufficient time. Advanced rhythms (corresponding to eastward flights) need generally more time to be re-entrained when compared with delayed rhythms (which correspond to westward flights). Re-entrainment rates are around 90 min per

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day for delayed phase shifts and 60 min per day for advanced phase shifts (Boulos et al., 1995; Aschoff et al., 1975). For small phase shifts of the LD cycle, the circadian clock resynchronizes in the same direction as the shift, while for larger shifts, the circadian clock sometimes resynchronizes in the opposite direction: the first type of resynchronization is called “orthodromic”, and the second “antidromic” (Aschoff et al., 1975; Klein and Wegmann, 1980). Although orthodromic readjustment remains the most frequent, opposite cases are not rare (Burgess et al., 2003). Antidromic readjustment occurs mostly for long advanced phase shifts; thus it was observed for 1 out of 7, 4 out of 8, and 7 out of 8 people after an eastward flight of 8 h (Arendt et al., 1987), 9 h (Klein et al., 1977) and 11 h (Takahashi et al., 2001), respectively. In view of the physiological disturbances, commonly referred to as “jet lag” (Waterhouse et al., 2007), that follow changes in the phase of the LD cycle, it is important to determine in a detailed

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manner the response of the circadian clock to such perturbations. This issue can be conveniently addressed by means of computational models for the mammalian circadian clock. Although Goodwin-type models (Leise and Siegelmann, 2006; Locke et al., 2008) and abstract models of the van der Pol type (Kostreva et al., 2002; Gundel and Spencer, 1999; Nakao et al., 2004; Dean et al., 2009) have been used to study the impact of jet lag on the circadian clock, the transition between orthodromic and antidromic re-entrainment after phase shifts of the LD cycle has not received much attention. Molecular models for the mammalian circadian clock (Leloup and Goldbeter, 2003, 2004; Forger and Peskin, 2003; Becker-Weimann et al., 2004; Mirsky et al., 2009) are available, but none of these has been used so far to address this issue. Here we resort to a detailed molecular model for the mammalian circadian clock (Leloup and Goldbeter, 2003) to investigate the conditions that result in either orthodromic or antidromic re-entrainment after a shift in the LD cycle. We examine the influence of several factors such as the autonomous period of the circadian clock in constant conditions, the magnitude of the phase shift achieved through changing the duration of a single light or dark phase of the LD cycle, and the effect of light intensity. Numerical simulations of the circadian clock model show that when it exists the switch between orthodromic and antidromic resynchronization takes place beyond a threshold near which the circadian clock takes extremely long times to synchronize with the new LD cycle. Thus, such critical phase shifts can markedly slow down circadian recovery. The prolongation of the resynchronization process near the threshold could well underlie some of the most pronounced physiological manifestations of jet lag. We show that the time needed for the system to resynchronize after a phase shift in the LD cycle can be minimized when the phase-shift in the LD cycle is partitioned in several, successive shorter phase shifts, a strategy that has been proposed to alleviate the consequences of jet lag (Eastman et al., 1995). Finally we use the phase response curve (PRC) of the circadian clock to explain and characterize the threshold separating the orthodromic and antidromic modes of resynchronization after a phase shift of the environmental cycle.

incorporating the regulation by REV-ERBα. The main conclusions obtained here hold for the two models both qualitatively and quantitatively; therefore, for the sake of clarity, we present only the results obtained for the 16-variable model. All figures are obtained by numerical integration of Eqs. (1)–(16) in Supporting Information in Leloup and Goldbeter (2003) for the parameter values listed there in Supporting Table 1 (see also Leloup and Goldbeter, 2004). To investigate the effect of phase shifts in the LD cycle on the re-entrainement of the circadian clock, we define the phase of a variable in the model by the time at which this variable reaches its maximum value. Similar results could be obtained by choosing its minimum or half-maximum value (onset and offset). For all our numerical simulations, we focus on the phase of a single variable of the model, i.e. the maximum in Per mRNA (MP). We consider that the system is fully resynchronized when it returns to 70.25 h of the final phase. Although this value was chosen arbitrarily, it falls within a reasonable range when compared to the precision of experimental data (Aschoff et al., 1975). Changing this cutoff value will also have an impact on the time for resynchronization: the system will need more time to be re-entrained for lower values and less time for higher values. In laboratory conditions or in silico, phase shifts of the LD cycle can be induced through changing the duration of either the dark or the light phase. Either one of these phases can be shortened or lengthened to achieve advances or delays, respectively. We applied phase shifts in the model by changing only the duration of the light phase. Numerical simulations generally yield similar patterns of resynchronization after changing the duration of either the dark or the light phase, even when the duration of the phase shift is important (around 8 or 10 h). Some differences can nevertheless be observed for very large phase shifts, of the order of 10 h or more.

3. Patterns of re-entrainment of the circadian clock after phase shift 3.1. Influence of the autonomous period of the circadian clock

2. Computational model for the mammalian circadian clock To study the dynamics of circadian clock re-entrainment after a phase shift in the LD cycle we use a computational model previously proposed for the mammalian circadian clock (see Leloup and Goldbeter, 2003, 2004 for a detailed description). This model is based on the main positive and negative feedback loops involved in the regulatory mechanism of circadian rhythms in mammals (see Fig. 1 in Leloup and Goldbeter, 2003). The product of the Clock and Bmal1 genes, the CLOCK and BMAL1 proteins, form a CLOCK–BMAL1 complex that activates the transcription of the Per and Cry genes and inhibits the transcription of the Bmal1 gene. The PER and CRY proteins, products of the Per and Cry genes, form a PER–CRY complex that inhibits the activity of the CLOCK– BMAL1 complex, thus creating an indirect, negative feedback loop on transcription. The different proteins are phosphorylated and then targeted for proteasomal degradation. Light induces the transcription of the Per genes. For simplicity a single copy of the Per and Cry genes is considered. To describe the effect of light, we consider that parameter vsP, which measures the rate of expression of the Per gene, varies as a square wave from a constant low value in the dark phase to a constant high value in the light phase. The latter value increases by the quantity δ with light intensity. For this study we consider the basal version of the model for the mammalian circadian clock, which contains 16 variables. We compared the results obtained with a 19-variable model

Circadian oscillations occur spontaneously in constant conditions, e.g., continuous darkness (DD), with an autonomous period τ. In human populations the value of τ ranges from 23.5 h to 24.6 h in men, and from 23.6 h to 24.5 h in women with a mean value close to 24.2 h and 24.1 h, respectively (Duffy et al., 2011). Such differences in τ could be responsible for the variability of 5–6 h observed in humans for the phase of the circadian clock with respect to the LD cycle (Kasukawa et al., 2012). To investigate the influence of τ on the resynchronization of the circadian system after a phase shift in the LD cycle, we first examine in Fig. 1 the effect of a delayed or advanced phase shift of 8 h in three cases that correspond either to an autonomous period of 24.2 h, close to the period observed in humans (Duffy et al., 2011), or to autonomous periods that are significantly shorter or longer than 24 h (τ¼ 22.7 or 25.8 h). These large differences in τ allow us to highlight the effect of the autonomous period on the dynamics of reentrainment of the clock. When τ is close to 24 h (middle panels), an advanced or delayed phase shift of 8 h leads to orthodromic re-entrainment (Fig. 1c and d). When the autonomous period is significantly shorter than 24 h (left panels), an advanced phase shift of 8 h leads to orthodromic re-entrainment (Fig. 1a), while a delayed phase shift results in antidromic synchronization (Fig. 1b). Finally, when τ is significantly larger than 24 h (right panels), an advanced phase shift of 8 h leads to antidromic re-entrainment (Fig. 1e), while a delayed phase shift results in orthodromic resynchronization (Fig. 1f). A

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comparison of the different situations in Fig. 1 indicates that the time needed for resynchronization is generally shorter in the case of orthodromic re-entrainment. Thus the system resynchronizes more easily upon delaying the clock when τ is greater than 24 h (Fig. 1f) while the opposite is true, i.e. re-entrainment upon advancing the clock is easier, when τ is shorter than 24 h (Fig. 1a). While Fig. 1 clearly shows the influence of the autonomous period on the appearance of ortho- or antidromic re-entrainment, it does not provide information about the transition between the two modes of resynchronization. To address this question we altered the autonomous period τ of the circadian clock by varying

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Fig. 1. Orthodromic versus antidromic re-entrainment for different autonomous periods of the circadian clock after an 8-h advanced or delayed phase shift. The phase of the daily peak of Per mRNA is represented by dots. White (gray) areas correspond to the 12 h light (dark) phases. White rectangles in the upper (lower) panels show the shortening (lengthening) of the light phase during an advanced (delayed) phase shift. In left, middle and right panels, the autonomous period τ of the circadian clock is smaller than 24 h (τ ¼22.7 h; a and b), close to 24 h (τ ¼24.2 h; c and d) or larger than 24 h (τ ¼ 25.8 h; e and f), respectively. The results are obtained by numerical integration of Eqs. (1)–(16) listed in Supporting Information in Leloup and Goldbeter (2003) for the parameter values listed there in Supporting Table 1 (see also Leloup and Goldbeter, 2004). The light-sensitive parameter vsP is increased from 1.5 nM h−1 in dark phase to 1.7 nM h−1 in light phase; moreover, KmB ¼ 0 nM h−1 (a and b), 0.55 nM h−1 (c and d) or 1.6 nM h−1 (e and f).

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Fig. 2. Influence of the autonomous period of the circadian clock on the pattern of re-entrainment. Lower and upper panels show, respectively, the effect of parameter KmB (Michaelis constant for degradation of Bmal1 mRNA) on the period of the oscillations and on the time for resynchronization after a phase shift. The curves for the time for resynchronization after delayed or advanced phase shifts show antidromic (dashed lines; “anti”) or orthodromic (continuous lines; “ortho”) re-entrainment. The duration of the phase shifts are 6 h (a), 8 h (b) or 10 h (c). The autonomous period at the threshold between anti- and orthodromic re-entrainment is indicated in the bottom part of the panels (arrows). Resynchronization times markedly increase in the vicinity of the thresholds. Parameter values are as in Fig. 1c and d.

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than 23.1 h while orthodromic re-entrainment is observed for larger autonomous periods (Fig. 2a). In the case of an 8 h phase shift (Fig. 2b), antidromic re-entrainment occurs following delayed phase shifts for autonomous periods shorter than 23.6 h and following advanced phase shifts for autonomous periods longer than 25.0 h, while orthodromic re-entrainment occurs otherwise. Finally, in the case of a 10 h change in the phase of the LD cycle (Fig. 2c), the threshold value of τ separating anti- and orthodromic re-entrainment is close to 24 h for delayed phase shifts and close to 24.6 h for advanced phase shifts. The results shown in Fig. 2 indicate that increasing the duration of the phase shift enlarges the domain of antidromic re-entrainment for both advanced and delayed phase shifts. The domain of antidromic re-entrainment increases from [22.7 h–23.1 h] to [22.7 h–23.6 h] and to [22.7 h– 24.0 h] for delayed phase shifts of 6 h (Fig. 2a), 8 h (Fig. 2b), and 10 h (Fig. 2c), respectively. The peak for delays moves to the right for larger phase shifts and is associated with increasing values of τ. Conversely, the peak for advances moves to the left for larger phase shifts and is associated with decreasing values of τ that also lead to larger domains of antidromic re-entrainment. The most striking observation pertains to the transition between the two types of re-entrainment. The results in Fig. 2 show that this transition occurs beyond a threshold through a peak corresponding to a large increase in the time taken for antidromic or orthodromic resynchronization. It is often observed that antidromic reentrainment takes more time than orthodromic re-entrainment (Arendt et al., 1987; Klein et al., 1977). Depending on the value of the autonomous period and the type and magnitude of the phase shift in the LD cycle, antidromic re-entrainment may in fact take less time than orthodromic re-entrainment. Our results predict that the “worst case scenario” is obtained near the threshold separating ortho- and antidromic re-entrainment: several tens of days are then needed for resynchronization of the circadian clock. For the sake of clarity we limit to 60 days the duration of resynchronization near the threshold in Fig. 2, but the time needed for re-entrainment may theoretically be much larger if the phase shift approaches the thresholds more closely. However, applying a critical phase shift with such precision might not be easily achieved, so that a quasipermanent block of the circadian clock after phase shift does not appear likely. To determine whether these results depend on the choice of the parameter used to modulate the autonomous period, we performed a similar numerical analysis for an 8 h phase shift by modulating τ via k1, the rate constant for entry of the PER–CRY complex into the nucleus (Fig. 3). Whereas τ increases with parameter KmB (Fig. 2), it decreases when k1 rises (Fig. 3). When comparing Fig. 3 with Fig. 2b established for the same phase shift magnitude, we observe the same qualitative behavior although the positions of the thresholds are slightly different (τ¼24.7 h instead of 25.0 h for advances, and τ¼23.7 h instead of 23.6 h for delays). The threshold for the transition between the two types of re-entrainment therefore depends not only on the autonomous period but also on the control parameter that modulates it. The actual time series showing the slowing down of resynchronization near the thresholds are shown in Fig. 4, which corresponds to the response to an 8 h advance for τ close to 24.7 h (in the vicinity of the left peak in Fig. 3). Just to the left of the peak for advances, i.e. for antidromic resynchronization (point 1 in Fig. 3), more than 60 days are needed to fully re-entrain (Fig. 4a and b). A similar time is needed for orthodromic re-entrainment (Fig. 4c and d) on the other side near the peak (point 2 in Fig. 3). In contrast, for this value of τ, which corresponds to the threshold for phase advances, the time needed to resynchronize after a phase delay of similar magnitude (point 3 in Fig. 3) is much shorter (Fig. 4e and f). A similar slowing down is also observed near the threshold separating orthodromic and antidromic resynchronization for delays (in the vicinity of

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Parameter k1 (h-1) Fig. 3. The position of the thresholds depends on the parameter used to modulate the autonomous period of the circadian clock. The influence of parameter k1 (rate constant for entry of the PER–CRY complex into the nucleus) on the period of the oscillations or on the time for resynchronization after a phase shift is shown in the lower and upper part of the panel, respectively. The curves for the time for resynchronization after delayed or advanced phase shifts show antidromic (dashed lines; “anti”) or orthodromic (continuous lines; “ortho”) re-entrainment. The duration of the phase shifts is 8 h. The autonomous period at the threshold between anti- and orthodromic re-entrainment is indicated in the bottom part of the panels (arrows). Points 1–3 in the upper panel refer to the three rows, from top to bottom in Fig. 4. Parameter values are as in Fig. 1c and d.

the peak on the right, i.e. for τ¼23.7 h in Fig. 3), as well as near any of the thresholds shown in Fig. 2. 3.2. Influence of the magnitude of the phase shift As illustrated in Fig. 2, the magnitude of the phase shift greatly influences the range of ortho- and antidromic re-entrainment as well as the location of the threshold separating the two types of response. In Fig. 5 we examine more systematically the effect of the magnitude of the phase shift and vary it from 0 to 12 h, for three different autonomous periods of the circadian clock. The value of τ markedly influences the transition between orthodromic and antidromic re-entrainment upon varying the magnitude of delays or advances of the LD cycle. For phase advances and for autonomous periods smaller than 24 h (τ ¼22.7 h, Fig. 5a) only orthodromic re-entrainment is observed. For phase delays, however, a threshold appears around the duration of 4.5 h. Below this threshold magnitude of the phase shift, only orthodromic re-entrainment is observed while antidromic re-entrainment occurs above it. For an autonomous period close to 24 h (τ¼ 24.2 h, Fig. 5b), we observe the same situation but the threshold magnitude for delays is increased up to 11 h. Finally, when the autonomous period is greater than 24 h (τ¼ 25.8 h, Fig. 5c), antidromic re-entrainment appears for phase advances exceeding 7 h while phase delays now only give rise to orthodromic re-entrainment. For values of τ between 22.7 h and 24.2 h, we expect that the peak for delays in Fig. 5a will

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Fig. 4. Increase in resynchronization time around the threshold separating orthodromic from antidromic re-entrainment. Left panels show the phase of the daily peak of Per mRNA as in Fig. 1. Right panels show the corresponding time course of circadian oscillations in Per mRNA (MP) before and after phase shift applied on day 5. The situations correspond to points 1 (a and b), 2 (c and d) and 3 (e and f) in Fig. 3. The increase in resynchronization time for advances near the threshold in τ¼ 24.7 h (a–d) is not observed for delays (e and f) for which no threshold exists around this value of τ (see Fig. 3). Parameter values are as in Fig. 3 with k1 ¼ 0.369 nM h−1 (a and b) and k1 ¼ 0.370 nM h−1 (c–f).

move to the right, to a position intermediate with that in Fig. 5b. Likewise, for values of τ between 24.2 h and 25.8 h, the peak for advances in Fig. 5c should move to the right (it has disappeared in Fig. 5b). As observed in Fig. 1 for a single magnitude of the phase shift, the range of antidromic re-entrainment becomes larger at autonomous periods smaller than 24 h in the case of delays of the LD cycle (Fig. 5a) and in the case of phase advances at τ values larger than 24 h (Fig. 5c). For τ smaller than 24 h (Fig. 5a), regardless of the magnitude of the phase shift, the time needed for resynchronization is always shorter for advances compared to delays. Conversely, for τ larger than 24 h (Fig. 5c), the time needed for resynchronization is always shorter for delays compared to advances. For τ close to 24 h (Fig. 5b), the time needed for resynchronization is longer for advances compared to delays up to a magnitude of 10 h but for larger phase shifts the reverse becomes true. This inversion is due to the appearance of a threshold between orthodromic and antidromic re-entrainment, associated with a slowing down of resynchronization.

3.3. Influence of light intensity To examine the influence of light intensity on the pattern of reentrainment we varied the maximum value of the light-sensitive parameter vsP for advanced and delayed phase shifts, for an autonomous period of τ ¼24.2 h and for a magnitude of the phase shift of 2 h. Parameter vsP in the light phase is equal to its value in the dark phase, incremented by δ. In Fig. 5b, δ¼0.2 nM h−1 and the time for orthodromic resynchronization for 2 h phase shifts is close to 4 days (see dot in Fig. 6). We observe that a second domain of orthodromic re-entrainment in which the system needs larger times for resynchronization appears at smaller values of the light-sensitive parameter corresponding to reduced light intensity (Fig. 6). Such values of the light-sensitive parameter are apparently not sufficient for rapid reentrainment of the system, and the time needed for synchronization therefore increases. Upon further decreasing the light intensity, the latter becomes so low that entrainment by the LD cycle fails to occur.

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Magnitude of the phase shift (in h) Fig. 5. Influence of the magnitude of the phase shift on re-entrainment of the circadian clock for different autonomous periods. The time for resynchronization after delayed or advanced phase shifts corresponds to antidromic (dashed line; “anti”) or orthodromic (continuous line; “ortho”) re-entrainment. Parameter values are as in Fig. 1 with KmB ¼ 0 nM h−1 (a), 0.55 nM h−1 (b) or 1.6 nM h−1 (c); these values yield autonomous periods of 22.7 h, 24.2 h and 25.8 h, respectively.

3.4. Complex patterns of resynchronization In most of the cases that we investigated the circadian system displays a unidirectional response corresponding to either orthoor antidromic re-entrainment after delayed or advanced phase shifts. However, we also encountered during our study some cases in which the system displays a mix of both types of re-entrainment (Fig. 7). This situation is illustrated for a 10-h delayed phase shift in Fig. 7a, which shows a complex pattern of resynchronization where transient bouts of ortho- and antidromic re-entrainment alternate. Immediately after the phase shift the system shows

rapid antidromic re-entrainment (arrow 1; the resynchronization rate is close to 4 h per day). This antidromic movement is followed by an orthodromic re-entrainment for several days (arrow 2), which becomes antidromic again (arrow 3). At first the latter reentrainment is very slow (of the order of only several minutes per day) but it becomes more and more rapid (of the order of several hours per day; arrow 1′). Again this antidromic re-entrainment ends (arrow 2′) and switches to an orthodromic one (arrow 4). Finally, around the asymptotic state of the system, the phase displays damped oscillations (arrow 5) until it settles to its final value. The effect of the phase shift can also be seen on the amplitude of the oscillations (Fig. 7b) where the different types of re-entrainment are associated with successive changes in the amplitude. The changes of direction in the re-entrainment pattern in Fig. 7a occur around the dark-to-light transition in the new system of reference. This holds with the fact that the light-sensitive parameter is increased at this moment, which can lead to steep changes in the concentrations of the variables of the system. What is more surprising is that the changes of direction occur just before the transition and thus precede the actual increase in the lightsensitive parameter. 3.5. Partitioning the phase shift It is useful to compare the effect of a single phase shift of given magnitude with the situation where it is divided into a series of successive, daily partial phase shifts of the same global magnitude (Fig. 8). Such a strategy has been proposed to minimize jet lag (Eastman et al., 1995). For illustrative purpose let us consider the case of an advanced phase shift of 8 h in the conditions of Fig. 2b for KmB ¼1.01 nM (in the vicinity of the threshold); the time evolution of the phase and the corresponding time series for Per mRNA during resynchronization are shown in Fig. 8a and b, respectively. We then divide the 8 h phase shift into either two phase shifts of 4 h on two consecutive days (Fig. 8c and d) or four phase shifts of 2 h on four consecutive days (Fig. 8e and f). First we note a switch from antidromic (Fig. 8a) to orthodromic (Fig. 8c and e) re-entrainment when partial phase shifts are given on several consecutive days instead of a single large phase shift. We also observe an important reduction in the time needed to resynchronize the circadian system: from 55 days (Fig. 8a) to 23 days (Fig. 8c) and finally 16 days (Fig. 8e). The effect of partitioning the phase shift can

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Fig. 7. Complex patterns of re-entrainment: (a) after a 10-h delayed phase shift, the system during resynchronization shows complex evolution of its phase before reaching its final, stable phase: it displays successively episodes of rapid antidromic re-entrainment (marked 1, 1′), sudden switch from anti- to orthodromic re-entrainment (2, 2′), slow switch from ortho- to antidromic re-entrainment (3), slow orthodromic re-entrainment (4) and low-amplitude oscillations around the final phase (5). White (gray) areas correspond to the 12 h light (dark) phases. The white rectangle shows the lengthening of the light phase during the delayed phase shift. (b) The effect of the phase shift can also be seen on the amplitude of the oscillations. Numbers correspond to those in panel (a). Parameter values are as in Fig. 1 with vsP ¼ 2.0 nM h−1 in light phase and KmB ¼0.4 nM h−1.

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Fig. 8. Partitioning the phase shift: comparing resynchronization after an advanced phase shift given as an 8 h shortening of the light phase in a single day (a and b), two 4 h shortenings on two consecutive days (c and d), or four 2 h shortenings on 4 consecutive days (e and f). Upper panels show the phase of the daily peak of Per mRNA represented by dots. Lower panels show the oscillations of Per mRNA (MP) prior and during resynchronization. Re-entrainment to the correct phase is obtained after 55 days in the case of a single 8-h delayed phase shift, after 23 days for two consecutive 4-h delayed phase shifts, and after 16 days for 4 consecutive 2-h delayed phase shifts. The arrows above the lower panels indicate the single, double or quadruple phase shifts. Parameter values are as in Fig. 2b with KmB ¼1.01 nM h−1.

also be seen on the amplitude of the oscillations (Fig. 8, lower panels): the change in the amplitude of circadian oscillations is significantly reduced upon partitioning the phase shift of the LD cycle into a series of smaller phase shifts (compare Fig. 8b with Fig. 8d and f).

4. Using the phase response curve to predict patterns of resynchronization To further understand the different patterns of resynchronization after a phase shift, the question arises as to whether we can resort to the phase response curve (PRC) that shows the effect of a light pulse on delaying or advancing the oscillations as a function of the phase at which the perturbation is applied. A direct comparison is, however, difficult since the PRC is obtained in

constant (dark) conditions whereas a system subjected to a phase shift remains driven by LD cycles in the course of re-entrainment. The process of resynchronization after the phase shift can be seen as a repetitive, daily perturbation by a 12 h light pulse which corresponds to the light phase of the LD cycle. It could be represented by a PRC that would slightly change every day because the amplitude and phase of the oscillations are progressively changing until full resynchronization is reached (see Fig. 9c and d). In the following discussion we will use the PRC established for a 12 h light pulse (upper panels in Fig. 9a and b) applied at different phases of the oscillations (bottom panels in Fig. 9a and b) in the absence of permanent phase shift. Depending on when the light phase/pulse begins during a 24 h oscillation, either advances (negative values of the phase shift Δϕ) or delays (positive values of Δϕ) are obtained (see upper panels in Fig. 9a and b). Even though the use of the PRC established for unperturbed oscillations represents

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Fig. 9. Using the phase response curve to predict patterns of resynchronization: (a and b) top panels show the phase response curve (PRC) obtained for a 12 h pulse of light. Bottom panels show the oscillation of Per mRNA in unperturbed LD conditions (i.e., before phase shift of the LD cycle). The PRC presents regions of advances (negative values of the phase change, Δϕ) and delays (positive values of Δϕ). To obtain the PRC, we first entrain the system by LD cycles and then put it in constant darkness conditions (DD). We apply a pulse of 12 h of light at times varying from 0 to 24 h after the last L phase before DD. Time 0 in Fig. 9a and b corresponds to the beginning of DD. The fourth peak of the oscillation is then compared to the corresponding peak of the non-perturbed oscillation. The difference between the two represents the phase shift, Δϕ, plotted in the PRC. Each point on the oscillation curve can be associated with a phase shift on the PRC. In (c) and (d) are shown the oscillations of Per mRNA before and after a phase shift of the LD cycle. The dots in (c) and (d) show the level of Per mRNA when the light phase/pulse begins. These dots are redrawn at nearly the same relative location with respect to maximum and minimum on the curve of the unperturbed oscillations [bottom panels of (a) and (b)] and the corresponding phase shifts are indicated on the PRC by the dots in the top panels of (a) and (b). Such a procedure represents an approximation because the waveform of the oscillations changes slightly during resynchronization. The left column shows the effect of advanced phase shifts of 6 h, 10 h and 12 h while the right column shows the effect of delayed phase shifts of the same durations. Black squares on the PRC mark the phase of Per mRNA giving rise to a null phase shift corresponding to a stable entrainment, and on the oscillation curve they show the level of Per mRNA at the beginning of the L phase when the clock is stably entrained by the LD cycle. Black triangles on the PRC mark the phase of Per mRNA giving rise to a null phase shift corresponding to the threshold between ortho- and antidromic re-entrainment; on the oscillation they show the level of Per mRNA at the beginning of the L phase that follows a phase shift of threshold magnitude. Parameter values are as in Fig. 5b.

only an approximation, we will see that it nevertheless provides useful insights into the different patterns of resynchronization after a phase shift of the LD cycle. To understand the origin of the threshold between ortho- and antidromic re-entrainment, let us consider, for example, the situation illustrated in Fig. 5b where advanced phase shifts only lead to orthodromic re-entrainment while delayed phase shifts give rise to ortho- or antidromic patterns of resynchronization, separated by a threshold phase shift around 11 h. We tested the effect of advanced (Fig. 9a and c) and delayed (Fig. 9b and d) phase shifts of 6 h, 10 h and 12 h. The phase shift in the LD cycle is indicated in Fig. 9c and d as a shortening (Fig. 9c) or lengthening

(Fig. 9d) of the L phase on the third day of entrainment by a LD cycle (vertical arrow). The dots in Fig. 9c and d show the level of Per mRNA when the 12 h light pulse/phase begins on the next 3 days. These dots are then redrawn on the unperturbed oscillation curve at nearly the same relative location with respect to maximum and minimum (bottom panels in Fig. 9a and b) and the corresponding phase shifts of the circadian clock are marked on the PRC (top panels in Fig. 9a and b). Let us stress that this procedure is not very accurate from a quantitative point of view because the waveform of the oscillation slightly changes after the initial phase shift of the LD cycle in the course of resynchronization (Fig. 9c and d). All the advanced phase shifts fall in the region

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of advances on the PRC and thus lead only to orthodromic reentrainment (Fig. 9a). In contrast, while some delayed phase shifts (6 h and 10 h) fall in the region of delays on the PRC and thus give rise to orthodromic re-entrainment, larger delayed phase shifts (12 h) fall in the region of advances on the PRC and lead to antidromic re-entrainment. In spite of the fact that the use of the PRC for unperturbed oscillations represents a rough approximation, these results indicate that the threshold between ortho- and antidromic re-entrainment corresponds to delayed phase shifts of a duration between 10 h and 12 h, in agreement with the results presented in Fig. 5b. While Fig. 9c and d only shows the first three days of resynchronization after the phase shift, we investigate in Fig. 10 the full resynchronization patterns for delayed phase shifts of 6 h, 10 h, 12 h and 16 h to further characterize the dynamics of orthoand antidromic re-entrainment. We follow the evolution of a single 24 h-oscillation and its phase every four days for almost one month. Longer delays lead to a longer time for resynchronization if re-entrainement is orthodromic (Fig. 10a and b) but to a shorter time if re-entrainment is antidromic (Fig. 10c and d). Surprisingly, antidromic re-entrainment does not necessarily lead to a longer time of resynchronization, as it is often observed (Arendt et al., 1987; Klein et al., 1977). It is near the threshold between ortho- and antidromic re-entrainment that the system takes the longest time to resynchronize. By determining the phase of the oscillations at which the 12 h light phase/pulse is given after an advanced or delayed phase shift, we can thus predict the type of resynchronization (i.e., ortho- or antidromic) and the rate of re-entrainment. If this phase of the oscillations corresponds to the region of delays (advances) of the PRC after a delayed (advanced) phase shift, then re-entrainment should occur in the same direction and give rise to orthodromic re-entrainement. On the other hand, if it corresponds on the PRC to the region of advances (delays) after a delayed (advanced) phase shift, then re-entrainment is antidromic. Two particular cases are encountered when the PRC crosses the zero on the Y-axis (corresponding to a null phase shift). In one case (around 12.5 h in Fig. 9a and b) the system reaches a stable state because all the shifts around this point converge towards it (e.g. on the left of this point, phase shifts give rise to advances, thus moving the system to the right on both curves in Fig. 9a and b,

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while on the right of this point, phase shifts give rise to delays, thus moving the system to the left). This point corresponds to stable entrainment of the circadian clock by the LD cycle: starting the L phase in this point (black squares in Fig. 9a–d) does not induce any phase shift of the oscillations. In the other case (around 21.5 h in Fig. 9a and b), the null phase shift point (black triangles) represents an unstable state because all the shifts around this point move the system away from it. This phase corresponds to the threshold observed between ortho- and antidromic reentrainment (see Fig. 9b; the threshold can also be reached in Fig. 9a for advanced phase shifts larger than 12 h). When the system is very close to this point, phase shifts induced by 12 h light pulses tend to zero and the time for re-entrainment increases considerably. The system will not seem to resynchronize at first and will be “stuck” for a very long time with nearly the same phase, as observed in Fig. 4a–d. The PRC can also yield useful information in regard to the rate of resynchronization. The larger the phase shifts predicted by the PRC, the faster the resynchronization. As an example, a 10-h delayed phase shift (Fig. 10b) initially gives rise to a slow resynchronization (from day 1 to day 9), which then accelerates (from day 9 to 13), and finally slows down again (from day 13 to 17) before reaching a stable pattern of entrainment (from day 17). Such variation in the rates of resynchronization can also be explained by examining the PRC in Fig. 9b. A 10-h delayed phase shift initially gives rise to small phase shifts of several minutes (dots on the far right) which subsequently increase up to 1.15 h (maximum delay predicted by the PRC), to finally decrease again (moving further to the left of the maximum delay on the PRC) before reaching stable entrainment corresponding to the point (black square) on the PRC where the phase shift vanishes. The shape of a PRC is highly sensitive not only to the duration or intensity of the light pulse but also to the amplitude, phase and period of the oscillations. Changing one of these will modify the shape of the PRC and will affect the range and the magnitude of delays or advances, as well as the time for resynchronization. Such modification of the PRC could therefore affect both the direction (orthodromic or antidromic) and the rate of resynchronization (slower or faster). The complex pattern of re-entrainment observed in Fig. 7 likely provides an example where the shape of the PRC is changing from day to day, giving rise to alternating phases of ortho-

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Day 1 Day 5 Day 9 Day 13 Day 17 Day 21 Day 25 Day 29 Fig. 10. Dynamics of resynchronization below and above the threshold separating orthodromic from antidromic re-entrainment. All panels display the time evolution of a single oscillation of Per mRNA from day 1 to day 29, every four days, after a delayed phase shift of 6 h (a), 10 h (b), 12 h (c) or 16 h (d). Bold lines track the evolution of the peak of an oscillation. Re-entrainment is either orthodromic (a and b) or antidromic (c and d). The time for resynchronization markedly increases near the threshold separating the two modes of re-entrainment, i.e. for a delayed phase shift between 11 h and 12 h. Parameter values are as in Fig. 5b.

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and antidromic re-entrainment, accompanied by changes in the rate of resynchronization (from several minutes to several hours per day) until the stable pattern of re-entrainment is reached.

5. Discussion In this work we examined the patterns of resynchronization of the mammalian circadian clock after a phase shift of the LD cycle. Using a detailed model for the transcription–translation feedback loops controlling the circadian clock in mammalian cells (Leloup and Goldbeter, 2003) we focused on the conditions leading to orthodromic (when the clock goes in the direction of the shift) or antidromic (when it moves in the opposite direction) resynchronization. Values of the autonomous period τ of the circadian clock larger than 24 h favor the appearance of antidromic re-entrainment after advanced phase shifts, while shorter autonomous periods favor antidromic synchronization after delayed phase shifts (Figs. 1 and 5). For both delayed and advanced phase shifts, our results predict the existence of a sharp threshold between orthoand antidromic re-entrainment (Figs. 2, 3 and 5). The threshold corresponds to a critical phase shift of the LD cycle for which the time needed for resynchronization increases exponentially (Figs. 2–5). The slowing down of resynchronization near the “switch point” between ortho- and antidromic re-entrainment was noted in the study of a model for circadian rhythms which incorporated the coupling of six oscillators identical in structure but possessing different parameter values (Leise and Siegelmann, 2006); each circadian oscillator was described by a two-variable model involving time delays (Scheper et al., 1999). While strong coupling between these oscillators leads to rapid resynchronization, the authors observed that weak coupling could induce a very slow resynchronization process near the “switch point” (Leise and Siegelmann, 2006). The threshold in this study largely originates from the coupling between weak oscillators. In contrast our results obtained in a detailed molecular model for the circadian clock characterize the factors that impinge on the threshold and show that it might already occur at the level of a single oscillating cell. When varying the magnitude of the phase shift we observed that larger phase shifts more readily produce antidromic reentrainment after delays when the intrinsic period τ of the circadian clock is smaller than 24 h (Fig. 5a) and after advances when τ is greater than 24 h (Fig. 5c). Upon decreasing light intensity by diminishing the rate of Per expression in the light phase we observed the appearance of a second domain of orthodromic re-entrainment associated with the slowing down of resynchronization (Fig. 6). The prediction of such a phenomenon could be tested in animal experiments and raises the possibility that exposure to a light phase of insufficient strength may have perturbing effects on the resynchronization of the human circadian system after phase shifts of the LD cycle. This theoretical study further revealed the possibility of highly complex patterns of re-entrainment after a phase shift of the LD cycle (Fig. 7). Several changes of directions can alternate during the resynchronization process, leading to multiple, successive switches between ortho- and antidromic re-entrainment. This phenomenon is reminiscent of resynchronization by partition, in which different organs synchronize in opposite directions (Aschoff, 1978), although here the same cell or group of cells exhibits an alternation between the two modes of resynchronization of the circadian clock. We investigated the effect of partitioning the phase shift into a series of smaller phases shifts given on consecutive days (Fig. 8). Such a protocol was used to progressively shift the phase of the circadian clock before transmeridian flights to avoid antidromic re-entrainment and thus reduce the effect of jet lag (Eastman

et al., 1995). In agreement with these observations, the predictions of the model and those of a previous theoretical study (Leise and Siegelmann, 2006) show that this protocol not only decreases the occurrence of antidromic re-entrainment and shortens the time needed for resynchronization, but also reduces the impact of the phase shift on the amplitude of circadian oscillations. Although partition always gave rise to a shorter time of resynchronization in our numerical simulations, it might, however, be possible, theoretically, to obtain a lengthening of this time if the last (partitioned) phase shift is given near the threshold, at the critical point of zero phase shift in the PRC (black triangle in Fig. 9a and b). This situation should nevertheless be very rare and would only be encountered when several particular conditions are fulfilled. For example, in the case of Fig. 5b with an autonomous period of 24.2 h close to the mean human circadian clock period, lenghthening of the recovery time upon partitioning in two might be expected when the initial phase shift is around 22 h, since the threshold between ortho- and antidromic re-entrainment is around 11 h. The phase response curve (PRC) provides a major tool for the study of biological oscillators, and has been used for long in the field of circadian rhythms. Although the PRC is generally used to determine the phase shift of the circadian clock elicited by a single pulse of light of given duration and magnitude, we showed how it can predict the pattern of resynchronization of the circadian clock after a permanent phase shift in the LD cycle. The use of the PRC for determining the effect of a phase shift of the LD cycle is by no means straightforward. The reason is that the waveform of oscillations after such a phase shift progressively changes in the course of resynchronization. As shown in Fig. 9, using the PRC established for non-perturbed oscillations nevertheless represents a useful approximation as it permits to foresee whether reentrainment will be orthodromic or antidromic, and allows us to locate roughly the critical phase shift that corresponds to the threshold separating the two modes of resynchronization. If switching from anti- to orthodromic resynchronization generally helps to decrease the time necessary for the system to reentrain (Boulos et al., 1995; Klein et al., 1977; Winfree, 1980), the present results indicate that this is not always the case, if only because of the existence of a threshold. For example, the results in Fig. 2b predict for advances of the LD cycle that a phase shift immediately to the left of the threshold, causing orthodromic reentrainment, may lead to slower resynchronization than a phase shift of the same magnitude to the right of the threshold, but further away from it, that leads to antidromic re-entrainment. Another example is shown in Fig. 10. Orthodromic re-entrainment after a 10 h delay (Fig. 10b) is faster than antidromic re-entrainment after a 12 h delay (Fig. 10c), but slower than antidromic resynchronization after a 16 h delay (Fig. 10d). In the model, the transition from anti- to orthodromic reentrainment depends on multiple factors such as the autonomous period of the clock, the biochemical kinetic parameters that control it, the direction or the magnitude of the phase shift, the light intensity, or even the partition of a single phase shift of the LD cycle into successive partial shifts. Numerical simulations of a realistic molecular model allow a detailed, systematic study of the role played by these various factors in a complex physiological response. The results on the dynamics of resynchronization were obtained in a model developed for the mammalian circadian clock at the level of single cells. The question arises as to whether the results on the dynamical behavior of single cells can be extrapolated to the dynamics of a tissue or even an organism. Given that circadian cellular oscillations underlie the collective circadian behavior of cells at the physiological level (Liu et al., 1997; Pagani et al., 2010), can we use the present results on cellular re-

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entrainment of the circadian clock after a phase shift in the LD cycle to investigate the dynamical bases of jet lag? In regard to the link between cellular clock properties and circadian behavior in humans, it is noteworthy to recall the example of the familial advanced sleep-phase syndrome (FASPS). A mutation in the hPER2 gene is accompanied by a decrease in the autonomous period of the circadian clock that is associated with a phase advance of several hours in LD conditions, resulting in premature onset/ offset of the sleep phase (Jones et al., 1999; Toh et al., 2001). This example, which was also studied by means of computational models (Leloup and Goldbeter, 2003, 2011; Vanselow et al., 2006), shows how a change in the oscillatory dynamics at the cellular level can affect the dynamics of the sleep–wake cycle at the physiological level in humans. The finding in a cellular model for the mammalian circadian clock that certain conditions greatly expand the time needed to resynchronize after a phase shift of the LD cycle could therefore bear on the dynamical bases of longlasting physiological disturbances associated with jet lag in humans. Acknowledgments This work was supported by Grant no. 3.4607.99 from the Fonds de la Recherche Scientifique Médicale (F.R.S.M., Belgium). J.-C.L. is Chercheur qualifié du Fonds National de la Recherche Scientifique (FRS-F.N.R.S., Belgium). References Arendt, J., Aldous, M., English, J., Marks, V., Arendt, J.H., 1987. Some effects of jet lag and their alleviation by melatonin. Ergonomics 30, 1379–1393. Aschoff, J., 1978. Problems of re-entrainment of circadian rhythms: asymmetry effect, dissociation and partition. In: Assenmacher, I., Farner, DS (Eds.), Environmental Endocrinology. Springer, Berlin, Heidelberg, New York, pp. 185–195. Aschoff, J., Hoffman, K., Pohl, H., Wever, R., 1975. Re-entrainment of circadian rhythms after phase-shifts of the zeitgeber. Chronobiologia 2, 23–78. Becker-Weimann, S., Wolf, J., Herzel, H., Kramer, A., 2004. Modeling feedback loops of the mammalian circadian oscillator. Biophys. J. 87, 3023–3034. Boulos, Z., Campbell, S.S., Lewy, A.J., Terman, M., Dijk, D.J., Eastman, C.I., 1995. Light treatment for sleep disorders: consensus report. VII. Jet lag. J. Biol. Rhythms 10, 167–176. Burgess, H.J., Crowley, S.J., Gazda, C.J., Fogg, L.F., Eastman, C.I., 2003. Preflight adjustment to eastward travel: 3 days of advancing sleep with and without morning bright light. J. Biol. Rhythms 18, 318–328. Dean II, D.A., Forger, D.B., Klerman, E.B., 2009. Taking the lag out of jet lag through model-based schedule design. PLoS Comput. Biol. 5 (6), e1000418. Duffy, J.F., Cain, S.W., Chang, A.M., Phillips, A.J., Münch, M.Y., Gronfier, C., Wyatt, J.K., Dijk, D.J., Wright Jr, K.P., Czeisler, C.A., 2011. Sex difference in the near-24-hour intrinsic period of the human circadian timing system. Proc. Natl. Acad. Sci. USA 108, 15602–15608. Eastman, C.I., Boulos, Z., Terman, M., Campbell, S.S., Dijk, D.J., Lewy, A.J., 1995. Light treatment for sleep disorders: consensus report. VI. Shift work. J. Biol. Rhythms 10, 157–164. Forger, D.B., Peskin, C.S., 2003. A detailed predictive model of the mammalian circadian clock. Proc. Natl. Acad. Sci. USA 100, 14806–14811.

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