Temperature Scaling in Some Simple Examples.

We begin by developing a simple mathematical example of the pathway model. The model features three main components: an mRNA X that is translated into a protein Y, which has an alternate modification state Z. Each main component actually consists of a pair of reversibly interconverted isoforms, X/X*, Y/Y*, and Z/Z* (diagrammed in Fig. 1A). All of the kinetics are linear except the repression of X/X* by Z/Z*. In the simple limit where each isoform is replaced by a single species, the model is equivalent to the Goodwin oscillator model (47) and is described by the following equations:X˙=kXZn−dXXY˙=kYX−dYYZ˙=kZY−dZZ,[1]for some Hill coefficient n, production rates ki, and degradation rates di. Note we have assumed that the Michaelis–Menten kinetics in the first term in X˙ are totally unsaturated, a slight modification from ref. 47 that does not affect the behavior because Zn is large. François et al. (45) showed that by a simple rescaling of the units for X, Y, and Z, one can obtain the modified systemx˙=1zn−dXxy˙=x−dYyz˙=y−dZz,[2]in which the production coefficients ki have vanished without changing the di and without rescaling time. This result implies that a change in any of the ki can be reversed by a rescaling of concentration units, leaving the shape of the orbit and its period unchanged. The Goodwin oscillator can thus be made into a temperature-compensated/temperature-entrainable oscillator by making the di independent of temperature and at least one of the ki temperature-sensitive. The effect of changes in temperature will then be described by the simple rescaling given above, just as predicted by the evolved models in ref. 45. To see how the degradation rates di can be rendered temperature-independent in a natural way, we restore the isoform pairs shown in Fig. 1A. This adjustment leads to a pair of equations for X/X*:X˙=kX(Z+Z*)n−dX−fXX+rXX*X˙*=kX(Z+Z*)n−d*X*−rXX*+fXX,[3]where fX and rX represent the forward and reverse rates of conversion between X and X*, respectively (assumed much faster than the period-determining circadian rates). Similar equations can be written for Y/Y* and Z/Z*. Adding the above equations together gives back the simplified form in 1, except with X and Z replaced by sums (X+X*) and (Z+Z*), and an effective degradation rate given bydX=dK1+K+d*11+K,[4]where K=rX/fX is the equilibrium constant describing the balance between X and X*. The effective rate dX can be rendered temperature-independent by letting d*/K≫d and having K increase with temperature, so that increases in d and d* are cancelled by a shift in the equilibrium toward the more slowly degraded isoform. In particular, we can set d=0 always and describe the temperature dependence of rates d* and K=rX/fX by an Arrhenius form exp(−E/kT) with energies Ed* and EK=Er−Ef, respectively. The effective rate dX will be temperature-independent as long as K≫1 at room temperature and EK=Ed*.

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Fig. 1.

(A) Diagram of a simple model for oscillator temperature compensation. mRNA isoforms X/X* are translated into proteins Y/Y*, which are converted into a second form Z/Z*, which represses transcription of X/X*. Each half of an isoform pair can be reversibly converted into and plays the same role as the other but has a different degradation rate. (B) Circadian oscillation of TIM protein, with various phases of the daily circadian cycle marked on corresponding locations on the oscillation. In red, a hypothetical scenario is shown in which a change in temperature causes a shortening of nuclear translocation time and a compensating increase in transcription time, leading to a change in shape of the oscillation.

In this case, the model will behave exactly as the temperature-compensated Goodwin oscillator discussed above and will show a simple scaling of oscillation shape with temperature. Additionally, although the model is abstract, it shares a number of features with the real circadian clock. In particular, it is known that in Drosophila, per and tim mRNA are present in multiple isoforms (48) and that PER and TIM proteins are subject to reversible biochemical modifications that alter the proteins’ degradation rates (49, 50). The model serves as a useful paradigmatic example of the pathway model and additionally shows that independent temperature compensation of all reactions in a biological oscillator can be accomplished without an excessively complicated network. Of course, temperature compensation in the Goodwin model can be achieved by a constraint imposed on the di, but in this case, the shape of the orbits will change with temperature, as is easily seen by simulation.

We can extend this example using a simple qualitative argument, which is depicted in Fig. 1B. Consider a time course of TIM protein levels over the course of 24 h in constant darkness. TIM protein will be at a minimum at subjective midday and a maximum at subjective midnight. The rising phase of TIM concentration corresponds to transcription and translation of tim mRNA during the late day, the peak phase corresponds to nuclear translocation of the PER-TIM dimer, and the falling phase corresponds to repression of TIM production and degradation of the dimer. If all of these subprocesses of the clock have temperature-dependent rates, such that a shortening of one process is compensated by a lengthening in another, then we should expect the shape of the curve of TIM protein oscillation to change with temperature. For example, if at higher temperatures, a shortening in nuclear translocation time is canceled by a lengthening in transcription/translation times, then we should expect a narrowing of the portion of the curve where TIM levels are high (because of more rapid repression of tim transcription) and a corresponding widening of the part of the curve where TIM levels are low (because of slowing in TIM production). This possibility is shown in Fig. 1B.

Western Blot and Luciferase Measurements.

To test for the shape invariance of the circadian oscillations, we measured the oscillations of different components of the circadian clock in D. melanogaster at three different temperatures: 18°C, 25°C, and 29° C. We chose to measure the oscillation in TIM protein levels (by Western blot) and transcription rate (by observing appearance of a luciferase reporter protein) to get a picture of both the positive (transcriptional) and negative (posttranslational) sides of the clock feedback loop (Fig. 1B). In both cases, WT flies were prepared by synchronization in at least 3 d of a light-dark (LD) cycle at the requisite temperature, followed by transfer into constant darkness (DD) at the same temperature. Measurements were made on the second day in constant darkness to minimize the residual effects of light while not allowing too much desynchronization of the population. For Western blots, yw flies were sampled by flash freezing every hour, and TIM concentrations were measured relative to cadherin (CDH). Results are shown in Fig. 2. For luciferase measurements, groups of 30–50 flies carrying a timeless luciferase (tim-luc) reporter (shown to recapitulate the dynamics of tim expression reported in ref. 51) were put into a 35-mm dish filled with luciferin-containing cornmeal food and observed in a Hamamatsu top-counting luminescence detector. Results are shown in Fig. 3.

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Fig. 2.

TIM protein Western blots at three different temperatures. Representative blot images are shown on the left. In A, bands correspond to time points 2 h apart as shown, covering a full day at the indicated temperature. B shows bands from the first 6 h of the day at both 18° C and 29° C (as labeled) run side-by-side. C shows quantification of TIM protein concentrations relative to CDH, normalized so that the mean of the 18° C time series is 1. D shows that the same curves effectively coincide after rescaling to a mean of 0 and an rms deviation of 1. All error bars are SEM for three biological replicates.

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Fig. 3.

Luminescence from tim-luc flies at 18°C, 25°C, and 29° C. (A) Sample of raw luminescence data plotted in millions of photon cpm. (B) The same data after detrending and smoothing. (C) Rescaled luminescence curves from tim-luc flies at different temperatures, taken from the second day in constant darkness. Shaded areas indicate SEM across three repetitions of the experiment.

Representative blot images are shown in Fig. 2A, from which it is clear that TIM levels increase at higher temperatures. Fig. 2B shows samples from two different temperatures (18° C and 29° C) run side-by-side on the same gel. A quantification combining the data from each of these gels (with three biological replicates) is shown in Fig. 2C. This quantification shows that both the amplitude and average of TIM oscillations increase with temperature. In Fig. 2D, the curves are shown after a rescaling by subtracting a linear trend (so that the mean is 0 throughout) and normalizing the root-mean-square (rms) deviation from the mean to 1. This process leads to a precise collapse of the curves onto each other, confirming the simple scaling prediction of the model discussed above.

In Fig. 3A, representative raw luciferase data at 25° C are shown. The decaying trend is attributable to depletion of the luciferin substrate (52) but can be easily subtracted. Fig. 3B shows the same data after detrending by subtracting a 30-h moving average and smoothing in a 1-h window (time points are taken every 15 min). Fig. 3C shows the oscillations at the three temperatures, on the second day in constant darkness, after rescaling by mean subtraction and normalization of the rms deviation. Again the curves collapse onto each other, confirming the simple scaling prediction. One might object that luciferase activity and stability are temperature dependent. We performed quantitative PCR (qPCR) measurements of tim mRNA levels at two different temperatures to show that mRNA oscillations scale similarly to luminescence oscillations and that the delay between peak mRNA concentration and luminescence levels is independent of temperature. Data are shown in Fig. S1, confirming that luminescence and qPCR data are consistent, with a delay of about 4–5 h at both 18° C and 25° C.

It is useful in assessing these data to have a control experiment in which one would expect to see changes in the shape of the curve of TIM oscillation at different temperatures. Our arguments in Temperature Scaling in Some Simple Examples imply that any mutant with a temperature-dependent change in rate for some circadian process should show a corresponding change in the shape of its oscillations. The perL mutant has a substantial defect in temperature compensation [the period increases from 27 h at 18°C to 31 h at 29° C (53)]. Fig. 4A shows the results of Western blots from perL flies, and Fig. 4B shows luciferase measurements from perL; tim-luc flies, after rescaling in both cases. A shift in the peak of the curves with temperature is evident and has a magnitude significantly larger than what would be expected from the difference in periods alone. It is worth pointing out that molecular circadian oscillations in perL are noisier and have lower relative amplitude than in WT (Fig. S2 and data in refs. 52 and 54). However, the shift in peaks is statistically significant.

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Fig. 4.

(A) TIM protein oscillations in perL flies, rescaled to have mean 0 and rms deviation 1. CT corresponds to a 30-h period with hour 0 being subjective morning on the second day in DD. Error bars are SEM for three biological replicates. (B) Rescaled luminescence curves from perL; tim-luc flies at different temperatures, taken from the second day in constant darkness, showing a change in shape. Shaded areas indicate SEM across three different experiments. (C) Peaks of protein (solid lines) and luminescence (dotted lines) oscillations show a change in relative phase in perL. Each wedge is centered on the peak phase, with the width of the wedges giving the error in the location the peak (determined by a sinusoidal fit). Wedges on the inner circle come from perL flies, and wedges on the outer circle come from WT flies. Phase labels around the two concentric circles correspond to fractions of the period for each strain. The wedges corresponding to WT are offset slightly to aid visibility.

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Fig. S2.

Raw data from circadian oscillation measurements in perL flies. (A) Representative Western blot images at three different temperatures. Time points span a 30-h period starting at CT1 on the second day in DD, at 3-h intervals as indicated. (B) Raw luminescence from perL; tim-luc flies, taken at 25° C. Note that the oscillations are relatively noisy and low in amplitude compared with WT (Fig. 3A), as noted in Western Blot and Luciferase Measurements.

Fig. 4C shows the position of the peaks in protein and luminescence oscillations for both WT and perL at all three temperatures. Peak times are measured relative to the natural period for each strain, so for WT flies, 0.5 corresponds to 12 h. Wedges are centered at the position of the peaks, with widths corresponding to the uncertainty in position of the peaks found from fitting a sine function. From these data, it is clear that in the perL background, there is not only a shift in the peaks at different temperatures, but the phase difference between luminescence and protein peaks is also temperature-dependent. This finding confirms that a change in shape is present, as predicted by our model, and that our method is able to detect such changes reliably. It is important to keep in mind that (in WT flies) the peak in luminescence lags peak mRNA concentration by about 5 h (Fig. S1), so that the true tim mRNA peak precedes the TIM protein peak at all temperatures.

It is interesting that a yeast two-hybrid experiment (55), as well as stains of circadian neurons (32), suggests that the defect in perL temperature compensation may be attributable to a temperature-dependent effect on PER-TIM binding, which in turn leads to a delay in nuclear translocation. The increasing delay in nuclear translocation likely explains the progressive delay in the peak of protein and mRNA expression and explains the narrowing of the gap between protein concentration and transcription levels, because a defect in repression of tim transcription by PER/TIM protein would allow relatively high levels of protein and mRNA to be present simultaneously. The effect of PER/TIM binding on TIM stability (56) may also play a role in determining how temperature influences the shape of the curves in Fig. 4, but in any case it is clear that strong temperature effects are present in the perL background that are absent in WT.

The other prediction of our model is the presence of a specific signaling pathway for coupling the circadian oscillation to temperature. As briefly mentioned earlier, a significant amount of evidence for this proposition has already been found. Glaser et al. (41) and Sehadova et al. (43) have found that mutations associated with the function of mechanosensitive chordotonal organs affect circadian temperature entrainment. In particular, morning anticipation in hot–cold cycles is eliminated and luciferase reporter oscillations damp out rapidly under temperature entrainment in mutants of the genes nocte and norpA. Animals lacking functional temperature-sensitive TrpA channels have been shown to entrain more slowly to temperature cycles and to have noisier molecular rhythms under temperature entrainment (40). One experiment has shown that flies carrying a null mutation in the photoreceptor cryptochrome (cryb) show reduced phase shifts in response to a 30-min 37° C heat shock (42). However, experiments performed in our own laboratory (Fig. S3) have failed to find this effect, and subsequent work by other groups has cast doubt on the role of cryptochrome in circadian temperature entrainment (57, 58). New data from the laboratory of Patrick Emery (39) suggest that calcium signaling is essential for circadian temperature responses and that the above-mentioned neural pathways converge by calcium-dependent degradation of TIM.

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Fig. S3.

Phase shift of circadian activity resulting from a 30-min heat pulse at 37° C, applied at CT 15. (Upper) Average activity of WT flies for a heat-shocked population (green) and a control population held at 25° C (blue). (Lower) The same data for cryb flies. The heat shock was applied at hour 0 on the plots, and in all cases, the results are an average of at least 19 flies. The phase shifts, computed by cross-correlation, are 3.1±0.24 for WT and 2.9±0.23 for cryb (SEM) and are thus statistically indistinguishable.

Given that there are clearly multiple pathways present for communicating temperature to the clock, it may be difficult to eliminate circadian temperature entrainment in an intact mutant animal. However, virtual elimination of temperature phase-shifting has been achieved in mouse fibroblasts subjected to pharmacological inhibition of the heat-shock pathway (38). It has also been shown that a knockout of the HSF1 transcription factor affects circadian expression in fibroblasts under temperature cycles and that there is a physical association between HSF1 and the mammalian circadian transcription factor BMAL1 (44). We were therefore interested in looking for an effect of the heat-shock pathway on circadian temperature sensation in Drosophila.

The Drosophila heat-shock response is mediated by the single transcription factor HSF. HSF is required for embryonic development, but viable heterozygous knockouts and a temperature-sensitive mutant (inactive above 30° C) are available (59). We performed phase-resetting curve (PRC) experiments on these mutants, using a temperature step from 18°C to 29° C as the stimulus. The results are shown in Fig. 5A. The phase shifts are computed relative to the circadian time (CT) 13 sample, rather than a control population. This is because the shape of activity oscillations in WT flies is significantly different at 18°C and 29° C, making a principled phase comparison between temperatures impossible. The CT 13 time point was chosen as a reference because at that time point the phase shift relative to a control appeared small by eye, but the choice is essentially arbitrary. Both heterozygous and temperature-sensitive heat-shock mutants show increased phase-shifting in response to temperature steps. This is different from what is seen in mammalian fibroblasts, and the difference is likely explained by the presence of neural temperature signaling pathways in Drosophila, which presumably interact with the cell autonomous heat-shock pathway in complex ways.

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Fig. 5.

(A) Results of a PRC experiment applying a temperature step from 18° C to 29° C to three strains of flies. HSF/- indicates a heterozygous knockout of the heat-shock transcription factor, and HSFTS indicates a temperature-sensitive HSF mutant. Phase shifts are plotted against the hour of the subjective day at which the step occurred and are calculated relative to the phase of the group at CT 13. Error bars are SEM. (B and C) Effect of temperature on sleep profile in WT and heat-shock mutant flies. Sleep per 30 min is shown over the course of the circadian day at 18° C and 29° C in WT flies (B) and heterozygous heat-shock transcription factor knockouts (C). Shaded regions indicate SEM across at least 15 individuals.

We performed a control experiment by repeating the above procedure but replacing the stimulus with a temperature step down, from 25°C to 18° C. In this case (Fig. S4), the phase shifts are comparable in WT and heterozygous HSF knockouts, as one might expect, because the heat-shock pathway should not be activated at these temperatures. It is interesting to note that the shape of the curves are different in the two experiments, suggesting that the mechanisms of circadian temperature sensation are different for warm and cold temperatures. This was also suggested by experiments in ref. 60, which found that a specifically cold-sensitive RNA-binding protein influences circadian transcription in mice, as well as prior work examining circadian responses to cold temperatures (61, 62).

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Fig. S4.

Control PRC experiment (Fig. 5A) applying a temperature step from 25° C to 18° C to two strains: WT and a heterozygous knockout of the heat-shock transcription factor (HSF/-). The HSFTS strain is not tested because the HSFTS molecule is expected to have normal activity at both 18° C and 25° C. Phase shifts are plotted against the hour of the subjective day at which the step occurred and are calculated relative to the phase of the group at CT 13. Error bars are SEM.

Intriguingly, the heat-shock pathway also appears to have an effect on the daily distribution of sleep at different temperatures. WT Drosophila tend to sleep more during the day and less at night at high temperatures (63), as shown in Fig. 5B. However, heterozygous HSF knockouts show a similar distribution of sleep at both 18°C and 29° C (Fig. 5C). This finding suggests that, in Drosophila at least, the heat-shock pathway may also have an unexpected role in regulation of sleep by temperature.