Coupling governs entrainment range of circadian clocks

Abstract

Circadian clocks are endogenous oscillators driving daily rhythms in physiology and behavior. Synchronization of these timers to environmental light-dark cycles ('entrainment') is crucial for an organism's fitness. Little is known about which oscillator qualities determine entrainment, i.e., entrainment range, phase and amplitude. In a systematic theoretical and experimental study, we uncovered these qualities for circadian oscillators in the suprachiasmatic nucleus (SCN-the master clock in mammals) and the lung (a peripheral clock): (i) the ratio between stimulus (zeitgeber) strength and oscillator amplitude and (ii) the rigidity of the oscillatory system (relaxation rate upon perturbation) determine entrainment properties. Coupling among oscillators affects both qualities resulting in increased amplitude and rigidity. These principles explain our experimental findings that lung clocks entrain to extreme zeitgeber cycles, whereas SCN clocks do not. We confirmed our theoretical predictions by showing that pharmacological inhibition of coupling in the SCN leads to larger ranges of entrainment. These differences between master and the peripheral clocks suggest that coupling-induced rigidity in the SCN filters environmental noise to create a robust circadian system.

Figure 1

Basic concepts of entrainment. (A) Schematic representation of a circadian rhythm entrained by 24-h temperature cycles. The quasi-square-wave temperature cycle represents the zeitgeber cycle used in the experiments of this study (see Figures 4 and 6). Note that upon complete entrainment the phase angle between rhythmic variable and zeitgeber cycle is constant. (B) Schematic representation of the entrainment region. The entrainment region is dependent on zeitgeber period (T) and zeitgeber strength and is also known as 1:1 Arnold tongue. Small-amplitude oscillators exhibit a broader range of entrainment than large-amplitude oscillators. For a constant zeitgeber strength, the entrainment region is confined between its lower (T_Low) and upper (T_High) limit.

Figure 2

Entrainment range and amplitude depend on the oscillator relaxation rate. (A) Numerically calculated entrainment region for a Poincaré oscillator with radius 1 plotted as a function of zeitgeber period and zeitgeber strength. The entrainment range is broader for weak oscillators with low relaxation rates λ. (B) Entrained amplitude of weak and rigid oscillators within the entrainment range. Weak oscillators exhibit a strong amplitude expansion for zeitgeber periods close to the oscillator's intrinsic period (24 h), and an amplitude reduction for zeitgeber periods close to both upper and lower limit of entrainment. Rigid oscillators remain almost unperturbed along the entrainment range. Numerical simulations results (dots) and analytically derived curves (lines) are in good agreement.

Figure 3

The ratio of zeitgeber strength to oscillator amplitude determines entrainment range. The lower limit of entrainment is plotted as a function of the ratio of zeitgeber strength to oscillator amplitude. The area under the analytically derived curve represents the entrainment region. The dots are numerically simulated lower limits of entrainment for a Poincaré oscillator with λ=1 h⁻¹. Computational details are given in Material and methods, and theoretical details are provided in Supplementary Information.

Figure 4

Lung tissue behaves like a weak and the SCN like a rigid circadian oscillator. (A, C) Temperature entrainment experiment with SCN and lung slices from PER2::LUC knockin mice. Tissues were cultured in luciferin-containing medium at 37°C, and bioluminescence derived from rhythmic PER2-LUC abundance was continuously monitored. Between days 4 and 8 or 9, tissues were subjected to 20 or 28-h temperature cycles with 10 or 14 h 35.5°C (blue boxes) and 10 h or 14 h 37°C before releasing them in constant 37°C conditions. Periods before, during and after entrainment are given in Supplementary Table S1. (B, D) Double plots of the peak times (±s.d.; n=4) of PER2-LUC bioluminescence. The circadian clock of lung tissue entrains to the ΔT=1.5°C temperature cycles (blue: low temperature), while the SCN clock does not: The bioluminescence peaks of the lung slices adopt a stable phase relation to the temperature cycle, while the SCN-derived peaks ‘run through’. After release in constant 37°C (from day 9 on), lung-derived bioluminescence rhythms continue with a phase that is predicted from the previous temperature cycles, thus excluding temperature-driven, so-called, masking effects. Source data is available for this figure at www.nature.com/msb.

Figure 5

Coupling constrains the entrainment range of oscillators. (A) Coupling increases the amplitude of the synchronized, coupled system. Given are numerically calculated results of two coupled Poincaré oscillators (see Equation 3). Coupling is quantified by the coupling strength parameter K. The amplitude increase upon coupling is more pronounced, if the individual oscillators are weak, i.e., for small relaxation rates γ. (B) Coupling makes oscillators more rigid by increasing the relaxation rate λ of the coupled system. Shown are numerically calculated results of two coupled Poincaré oscillators. Note that the effect of the coupling strength on the rigidity of the coupled system is not dependent on the amplitude relaxation rates γ of the individual oscillators. As coupling slightly affects the period of the coupled system, the relaxation rate λ was normalized by multiplying with the coupling-dependent period. (C) Coupling makes an oscillatory system harder to entrain. Two coupled Poincaré oscillators were entrained by adding a periodic square-wave forcing of period T, alternating with equal duration and a forcing between 0 and 0.1 (term F in Equation 3). The lower limit of entrainment T_Low of the coupled system was calculated for different coupling strengths K and different amplitude relaxation rates γ of individual oscillators. As the coupling affects the intrinsic (non-forced) period itself, the lower limit of entrainment was normalized with respect to a constant intrinsic period of 24 h to visualize the effects on the entrainment range that are entirely due to the effects on the rigidity and amplitude of the coupled system.

Figure 6

Decoupling of SCN cells broadens the entrainment range of SCN tissue. (A) Circadian rhythms of individual SCN cells can be monitored. The digital image shows a 10-min exposure of bioluminescence emitted from a medial section through the SCN of a PER2::LUC mouse. Bioluminescence reporting abundance of the clock protein PER2 was quantified from 130 single cells (marked by white circles) over the course of several days. Scale bar, 500 μm. (B) Representative bioluminescence traces from 10 different single cells. Typical for SCN slices, individual cells oscillate with a circadian period, and the majority of cells have their bioluminescence peaks at similar times. After addition of the adenylyl cyclase inhibitor MDL on day 6, bioluminescence recording was resumed as shown for 10 representative cells (note that representative cells before and after MDL treatment were not identical). While MDL addition reduced the magnitude of PER2 abundance and lead to random phasing of peak bioluminescence, relative amplitudes of untreated and MDL-treated cells were similar. (C) Polar plots showing the peak times of PER2 expression of individual SCN cells from the experiment shown in (B) during the third day of slice recording. Each dot represents the phase of a single cell. Single-cell phases in an untreated SCN slice (left) clustered significantly at 12 h (Rayleigh test, r=0.8, P<0.0001, n=49), but were randomly distributed upon MDL treatment (right, Rayleigh test, r=0.26, P=0.07, n=40). Arrows in polar plots represent the mean vectors, the direction denotes the mean phase, and length measures the tendency of the data to cluster based upon a Rayleigh test, in which r-values range from 0 (randomly phased) to 1 (all cells peak at the same time). (D) The relative amplitudes of single-cell oscillations do not vary with treatment (Mann–Whitney test, one-tailed, P=0.47, n=169 for untreated cells; n=94 for MDL-treated cells). Displayed are normalized median amplitudes±quartiles±minima/maxima from three independent imaging experiments. For each experiment, individual amplitude values were normalized to the mean amplitudes of untreated cells. (E, F) SCN cultured in the presence of MDL (10 μM final concentration) or TTX (2-4 μM final concentration), respectively, entrain to a 20-h temperature cycle consisting of 10 h cold (35.5°C, blue boxes) and 10 h warm (37°C). Given are double plots of the peak times (±s.d.; n=8 for MDL, and n=6 for TTX) of PER2-LUC bioluminescence. Periods before, during and after entrainment are given in Supplementary Table S1. Thus, the range of entrainment for SCN slices is increased (compare with Figure 4) when SCN oscillators are decoupled (see B and C). Source data is available for this figure at www.nature.com/msb.