Non-sinusoidal Waveform in Temperature-Compensated Circadian Oscillations

Abstract

Time series of biological rhythms are of various shapes. Here, we investigated the waveforms of circadian rhythms in gene-protein dynamics using a newly developed, to our knowledge, index to quantify the degree of distortion from a sinusoidal waveform. In general, most biochemical reactions accelerate with increasing temperature, but the period of circadian rhythms remains relatively stable with temperature change, a phenomenon known as “temperature compensation.” Despite extensive research, the mechanism underlying this remains unclear. To understand the mechanism, we used transcriptional-translational oscillator models for circadian rhythms in the fruit fly Drosophila and mammals. Given the assumption that reaction rates increase with temperature, mathematical analyses revealed that temperature compensation required waveforms that are more nonsinusoidal at higher temperatures. We then analyzed a post-translational oscillator (PTO) model of cyanobacteria circadian rhythms. Because the structure of the PTO is different from that of the transcriptional-translational oscillator, the condition for temperature compensation would be expected to differ. Unexpectedly, the computational analysis again showed that temperature compensation in the PTO model required a more nonsinusoidal waveform at higher temperatures. This finding held for both models even with a milder assumption that some reaction rates do not change with temperature, which is consistent with experimental evidence. Together, our theoretical analyses predict that the waveform of circadian gene-activity and/or protein phosphorylation rhythms would be more nonsinusoidal at higher temperatures, even when there are differences in the network structures.

Introduction

In the constantly fluctuating environments found on Earth, organisms have evolved their own autonomous biological processes with a cycle that lasts ∼24 h, known as circadian rhythms. Many biological processes accelerate when the temperature increases, but an unusual property of circadian rhythms is that their period remains stable with temperature changes, a phenomenon known as “temperature compensation” (1, 2). In many literatures, temperature compensation is defined so that the period of biological rhythms is roughly constant at different but constant temperatures (0.85 < Q₁₀ < 1.15 (i.e., a change in temperature by 10°C to a new constant temperature results in less than 15% change in the period)) (3). Some experimental studies have shown that the circadian period is not perfectly compensated but rather that it increases slightly with temperature (1, 4, 5), referred to as “overcompensation.” What makes temperature compensation peculiar is that a system-level property (the period) is relatively stable to temperature even though individual components of the system are affected by temperature because of the nature of biochemical reactions. Temperature compensation is a well-conserved property of circadian rhythms across species. It has been studied extensively, both experimentally and theoretically, but its underlying mechanism remains unclear (4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19).

Circadian rhythms at the behavioral level are driven by cellular oscillations in gene-expression activity and protein phosphorylation levels (2). Transcriptional-translational oscillator (TTOs) have been shown to be essential for cellular circadian oscillations in Neurospora, Arabidopsis, Drosophila, and mammals (2, 20, 21, 22). In contrast, post-translational oscillators (PTOs) have been found to be sufficient for generating circadian rhythms (23, 24). In cyanobacteria, circadian oscillations can be reconstituted in vitro with three proteins (KaiA, KaiB, and KaiC) and ATP (23). Differences in network structures between TTOs and PTOs may suggest that the design principles on which temperature compensation is based may be species specific.

There have been proposed hypotheses to explain temperature compensation, including the balance hypothesis, the critical-reaction hypothesis, and the temperature-amplitude coupling hypothesis. The balance hypothesis proposes that stability of the circadian period with temperature arises from a balance between period-shortening and period-lengthening reactions (1, 7, 9). The critical-reaction hypothesis assumes that there are critical reactions that establish the circadian period (11, 12, 13); if those reaction rates are stable to temperature, the circadian period would be stable to temperature. The temperature-amplitude coupling hypothesis proposes that temperature-sensitive amplitudes in gene-activity rhythms can result in a stable period by generating larger amplitudes at higher temperatures (5, 6, 17). To verify these proposed hypotheses, quantification of cellular rhythms is useful. Indeed, the robustness of period length with respect to genetic mutations has been measured extensively for various species. Temperature compensation is sustained with many genetic mutations, suggesting that only a limited number of reactions underlie temperature compensation (12). Meanwhile, less attention has been paid to the shapes of the time series of gene-protein dynamics. These can be sinusoidal, nonsinusoidal, or rectangular depending on the molecular species, organ, and environmental conditions. The aim of this study is to explore how the shapes of the circadian time series are associated with temperature compensation. To do so, we introduce an index to describe the distortion from a sinusoidal time series of the waveforms in circadian gene-activity and protein phosphorylation rhythms. Using circadian TTO and PTO models, we demonstrate through mathematical and computational analyses that temperature sensitivity in the circadian waveform is essential for temperature compensation in the circadian rhythm.

Methods

Computation

Ordinary differential equations in this work were calculated by a fourth-order Runge-Kutta method with MATLAB (The MathWorks, Natick, MA). The time step size Δt for the modified Goodwin model ((2a), (2b), (2c), (2d)) and the van der Pol oscillator (Eq. 4) was 0.0001, and that for the mathematical model for realistic mammalian circadian rhythms (4) and cyanobacteria circadian rhythms (25) was 0.001.

Time-series analysis

We analyzed nonsinusoidal power (NS, see Eq. 1) by using the generalized harmonic analysis (GHA) method because it is difficult to accurately quantify NS by using Fourier analysis for some time series (26). The detailed procedure of GHA can be found in the Supporting Materials and Methods.

Results

Defining NS in circadian rhythms

To quantitatively discuss the role of the waveform in the robustness of circadian rhythms to temperature changes, we developed an index, the NS, defined as follows:

$$NS={\left[\frac{{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2{j}^m}{{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2{j}^q}\right]}^{\frac{1}{2}}\left(\mathrm{m>q}\ge \mathrm{0}\right)\text{,}$$ (1)

where a_j are the Fourier coefficients for the oscillatory time series $x\left(t\right)={\sum }_{j=-\infty }^{\infty }{a}_j\text{exp}\left(i\left(2\pi /\tau \right)jt\right)$ and m and q are integers. This index is to weight harmonics (m > q). If the time series of gene activity simply has a sinusoidal waveform, NS = 1 because only the coefficient for the fundamental component (|a₁|²) is nonzero, with the coefficients for the harmonics (|a_j|² (j ≥ 2)) all zero (Fig. 1 A, top). When a time series is nonsinusoidal in shape, the coefficients for the harmonics become nonzero, resulting in NS > 1. An example of this would be when the decay interval of a genetic product x(t) is longer (or shorter) than the synthesis interval (Fig. 1 A, middle and bottom). To our knowledge, the functional consequence of the waveform of circadian rhythms for temperature compensation has not been fully clarified.

Figure 1.

Examples of a nonsinusoidal waveform and transcriptional-translational oscillator (TTO) and post-translational oscillator (PTO) models for circadian rhythms. (A) Examples of waveform (left) and the corresponding power spectrum (right). From top to bottom, NS = 1.0, 2.0, 3.5 (m = 4, q = 2) (see text). (B) The TTO model. (C) The PTO model for cyanobacteria circadian rhythms. The functional consequence of the waveform of circadian rhythms for the temperature compensation was compared between the TTO and PTO models.

Role of the waveform in temperature compensation in a classical circadian oscillator

Seminal studies combining experimental and theoretical analyses showed that the waveform in the expression of the Drosophila clock component, timeless, was insensitive to temperature, whereas its average and amplitude changed with temperature; this suggested that the waveform may not be important for temperature compensation (14, 17). The relevance of the circadian waveform to temperature compensation can be discussed more clearly by using analytically tractable theoretical models. We therefore began this study by using a classical model for circadian rhythms, the Goodwin model with some modifications (27). The model includes negative feedback regulation of gene expression, which has been shown to be essential for the generation of TTO (2, 20, 21, 22). We modified the Goodwin model by incorporating the phosphorylation of proteins (Fig. 1 B), based on recent findings that showed the importance of protein phosphorylation for temperature compensation in mammals (4, 13). The dynamics of the model are as follows:

$$\frac{d{x}_1}{dt}=f\left(x_4\right)-\left(b+p+k_u\right)x_1,$$ (2a)

$$\frac{d{x}_2}{dt}=b{x}_1-k_o{x}_2,$$ (2b)

$$\frac{d{x}_3}{dt}=p{x}_1-\left(h+k_u\right)x_3,$$ (2c)

$$\frac{d{x}_4}{dt}=h{x}_3-k_t{x}_4.$$ (2d)

In the model, the variables are the concentration of nonphosphorylated (x₁), unstable type of monophosphorylated (x₂), stable type of monophosphorylated (x₃), and double-phosphorylated protein (x₄). Nonphosphorylated protein (x₁) is phosphorylated into unstable (x₂) or stable monophosphorylated protein (x₃). One of the monophosphorylated proteins (x₃) is further phosphorylated into the double-phosphorylated protein (x₄), which functions as a transcription factor; this corresponds to phosphorylation of the PER2 protein at familial advanced sleep phase sites (4, 28, 29). For simplicity, mRNA dynamics is ignored because the timescale is often faster than that of protein dynamics. The TTO is expressed by a single term f(x₄), which is a function of double-phosphorylated protein (x₄). The regulatory term of f(x₄) can take the form of any function provided that it results in oscillations that can be (but are not limited to) a decreasing function of transcription repressor (x₄), as in the Goodwin model (27). The model produces limit cycle oscillations for certain parameter values. To identify the condition for temperature compensation, we assume in the following that all biochemical reactions such as degradation and phosphorylation (represented by b, p, h, k_t, k_u, and k_o in the model) accelerate with temperature. When reaction rates are increased randomly (representing a higher temperature), the period can be unchanged or shortened depending on the choice of parameters (Fig. 2 A).

Figure 2.

A more nonsinusoidal waveform at higher temperature is sufficient for temperature compensation in a TTO model. (A) Numerical examples with temperature compensation (top, Q₁₀ = 0.9946) and without temperature compensation (bottom, Q₁₀ = 0.7336). The transcriptional-translational regulation function f(x₄) in the model was set to be the decreasing function v/(1 + (x₄/K)ⁿ) of transcription factor (x₄) with cooperativity n. The values of K and n were set as 0.0184 and 11, respectively. The reaction rates were changed according to the Arrhenius equation k_i = A_iexp(−E_i/RT), where k_i, E_i, A_i, R, and T are the rate constants, activation energies, frequency factors, gas constant (R = 8.314), and absolute temperature, respectively. The values of E_i and A_i are shown in Table S1 (top) and Table S2 (bottom). The initial values were (x₁, x₂, x₃, x₄) = (0.11, 0.0032, 0.025, 0.023) for all simulations. The blue and red lines represent the dynamics of x₄(t), which converged to a limit cycle, at 20 and 30°C, respectively. (B) Distribution of NS (m = 4 and q = 2) for x₄(t) as a function of period when reaction rates were increased. When the period was relatively maintained, the circadian waveform became always more nonsinusoidal. The parameter values of p, h, b, k_u, k_t, k_o, and v were increased by a random factor from the range 1.1 to 1.9. We plotted the two ratios (relative period and relative NS). The ratio is defined as the new period (or NS) with increased reaction speed divided by the basic period (or NS) with basic parameter sets. The gray lines are the upper (dashed) and lower (solid) limits of relative NS, predicted from Eq. 3. (C) Temperature dependency of the period and NS. The reaction rates were changed according to the Arrhenius equation k_i = A_iexp(−E_i/RT). The values of E_i and A_i are shown in Table S1. The values of K and n were the same as in (A). (D) Distribution of the synthesis interval of transcription factor (x₄) divided by the decay interval when the reaction rates were increased as in (B). The ratios (relative synthesis interval and relative decay interval) were calculated. When the period was relatively maintained (orange bars, relative period ≥ 0.8), the synthesis interval became shorter and/or the decay interval became longer as the reactions became faster. The black bars correspond to the sets for period-shortening cases (relative period < 0.8).

To understand the condition for temperature compensation, we derived the period from the model using signal-processing methods. By expressing the time series of the transcription factor (x₄) as a Fourier series $\left(x_4\left(t\right)={\sum }_{j=-\infty }^{\infty }{a}_j\text{exp}\left(i\left(2\pi /\tau \right)jt\right)\right)$, the period can be obtained as follows (30, 31) (see the Supporting Materials and Methods):

$$\tau ={\left[\frac{4{\pi }^2}{\left(b+p+k_u\right)\left(h+k_u\right)+\left(b+p+h+2k_u\right)k_t}\frac{{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2{j}^4}{{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2{j}^2}\right]}^{1/2}.$$ (3)

In this formula, the term ${\left[{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2{j}^4/{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2{j}^2\right]}^{1/2}$ corresponds to NS at m = 4 and q = 2 above. The rest of the formula $\left({\left[4{\pi }^2/\left\{\left(b+p+k_u\right)\left(h+k_u\right)+\left(b+p+h+2k_u\right)k_t\right\}\right]}^{1/2}\right)$ we refer to as the “direct effect” (D). The formula of Eq. 3 indicates that the period always becomes shorter at higher temperature as long as the circadian waveform remains the same (i.e., NS remains constant) because the direct effect always decreases with the reaction rates as the temperature increases. This result further indicates that the period remains constant with temperature in this model only when NS increases with temperature, which is therefore the necessary condition for temperature compensation.

Ruoff has shown theoretically that a balance between period-shortening and period-lengthening reactions is required for temperature compensation in any biochemical oscillator (1, 7, 9). The period formula of Eq. 3 above indicates that period-increasing reactions are always NS-increasing reactions. A mathematical analysis suggests that the balance between period-shortening and period-lengthening reactions can be achieved by modifying the circadian waveform (i.e., changing NS). In reality, some critical reactions for the circadian period (such as the ATPase activity of the KaiC protein in cyanobacteria and CK1ε/δ activity for PER2 phosphorylation in mammals) have been shown to be stable with changes in temperature (11, 13, 19). This is inconsistent with the assumption in our model that all the reactions (k_i, i = 1, 2,..., N) become faster as the temperature increases (i.e., dlnk_i/dT > 0), which raises the question of whether or not a nonsinusoidal waveform is important for circadian rhythms in actual biological systems. To be consistent with the experimental evidence (11, 13, 19), we considered the case in which some reactions, such as phosphorylation processes (e.g., b, p, or h), are stable to temperature (dlnk_i/dT = 0). As for the remaining reactions, the reaction rates are assumed to increase with temperature (dlnk_i/dT > 0, i = 1, 2,..., M (N > M)). Mathematical analysis shows that for temperature compensation, the waveform needs to be more nonsinusoidal (i.e., with a larger NS) at higher temperatures even in this setting because the “direct effect” term will always decrease with temperature because of the remaining reactions. If all the reactions were stable to temperature, modulation of the circadian waveform would no longer be required for temperature compensation; however, this is inconsistent with the notion that circadian rhythms can be entrained by temperature cycles (2). Furthermore, experimental studies have shown that the circadian period is not perfectly compensated but rather that it increases slightly as the temperature increases (overcompensation) in some species (1, 4, 5). To achieve temperature compensation or overcompensation, our mathematical analysis shows that a temperature-dependent circadian waveform would be necessary.

Next, we used this TTO model to investigate how the circadian waveform affects the robustness of the circadian period to temperature. As an example, we set the transcriptional-translational regulation function f(x₄) to be Hill function of v/(1 + (x₄/K)ⁿ), in which v is the maximal transcription/translation rate in Hill function, K is the Michaelis constant, and n is the cooperativity. In the calculation, we measured the sensitivity of the period to reaction rates as follows. First, we prepared 30 basic parameter sets that yield oscillations in which b, p, h, k_u, k_o, k_t, v and K were assigned uniformly distributed random values from 0 to 10 and n was assigned a uniformly distributed random integer from 10 to 100, and we calculated the period (τ₁) for each parameter set. Second, we increased the values of b, p, h, k_u, k_o, k_t, and v by a random factor in the range 1.1–1.9, thereby obtaining 49 oscillatory parameter sets, and we calculated the new periods (τ₂). Finally, we used these results to calculate relative period (= τ₂/τ₁). In many cases, the period decreased with the faster reaction speeds. Out of a total of 1470 parameter sets, the period was relatively maintained (i.e., relative period ≧ 0.8) in 52 sets (Fig. 2 B). Notably, in these cases, NS tended to increase simultaneously with the faster reaction speeds (relative NS ≥ 1; relative period ≥ 0.8) (Fig. 2 B), consistent with the earlier discussion related to the formula for the period. The NS for the transcription factor (x₄) is quantified numerically by using GHA (26) (see Supporting Materials and Methods; Fig. S1). The gray lines in Fig. 2 B are the upper (dashed) and lower (solid) limits of relative NS, predicted from Eq. 3. When the parameter sets were increased by a random factor ranging from 1.1 to 1.9, the minimal and maximal change in direct effect should be 1/1.1 and 1/1.9, respectively. Then, the scatter plots’ relative NS should be above the slope of 1.1 and below that of 1.9. Again, these predicted lines show that the relative NS should be larger than one if the period is temperature compensated or overcompensated. By changing the reaction rates based on the Arrhenius equation, we can numerically show that the period can be stable to temperature change with larger values of NS at higher temperatures (Fig. 2 C).

If a more nonsinusoidal waveform at higher temperatures is important for temperature compensation, this raises the question of what the molecular basis is for such a waveform. By randomly increasing reaction rates from those in the basic parameter sets, we found that the ratio of synthesis interval/decay interval does not change on average. However, when the reaction rates increased but the period remained relatively stable, the synthesis interval became shorter and/or the decay interval became longer (Figs. 2 D and S2); this can also be seen in the example of temperature compensation in our calculation (Fig. 2 A, top). Indeed, a difference between the synthesis and decay intervals can result in distortion to the nonsinusoidal waveform, corresponding to large value for NS. Furthermore, a longer decay interval of a negative transcription factor should result in longer transcription inhibition and thus a longer period, which can cancel out the period-shortening effect of faster reaction rates at higher temperatures.

Next, we investigated which reactions were important for temperature compensation, possibly by modulating the circadian waveform. We computationally measured the sensitivity of period and NS to reaction rates when the reaction rates p, h, b, k_u, k_o, k_t, and v were each increased individually by 10% (see Supporting Materials and Methods). We found that an increase in the transcription/translation rate (v) lengthened the period, on average, with an increase in the value of NS, whereas increases in the other reactions shortened the period (see Supporting Materials and Methods; Fig. S3). Thus, our analysis of the modified classical circadian oscillator (TTO) predicted that higher temperature sensitivity of the transcription/translation rate would result in temperature compensation through a greater abundance of transcription factor and therefore a longer decay interval (Fig. 2 D) and larger NS (Fig. 2 B) at higher temperature.

Does the condition for temperature compensation differ between PTO and TTO?

In cyanobacteria, three proteins (KaiA, KaiB, and KaiC) with ATP are sufficient for diurnal KaiC phosphorylation rhythms as well as accounting for temperature compensation (23); this suggests that the post-translational network (the PTO) underlies temperature compensation. Because of the differences in the network structures of TTOs and PTOs, the condition for temperature compensation can differ between them, and this can be explored computationally. We, therefore, analyzed one of the published mathematical models for cyanobacteria circadian rhythms (16, 18, 25, 32, 33, 34). In this model (25), the variables are the protein concentration of nonphosphorylated KaiC (V₁), monophosphorylated KaiC (V₂), double-phosphorylated KaiC (V_new), double-phosphorylated KaiC bound to KaiA (V₃), and double-phosphorylated KaiC bound to both KaiA and KaiB (V₄) (Fig. 1 C). An increase in the abundance of phosphorylated KaiC results in an increase in a complex of KaiC and KaiA, causing the sequestration of free KaiA because the total amount of KaiA is conserved. This sequestration of free KaiA functions as negative feedback for KaiC phosphorylation, potentially resulting in limit cycles (Fig. 1 C; see also Supporting Materials and Methods). We used this PTO model to investigate the condition for temperature compensation, assuming as before that all the reactions become faster with increased temperature. When the reaction rates for KaiC phosphorylation rhythms are increased randomly (representing a higher temperature), the period can either remain unchanged or become shorter (Fig. 3 A).

Figure 3.

A more nonsinusoidal waveform at higher temperature is sufficient for temperature compensation in a cyanobacteria PTO model. (A) Numerical examples of the model with temperature compensation (top, Q₁₀ = 0.9894) and without temperature compensation (bottom, Q₁₀ = 0.6912). The reaction rates k_phos, k_new, k_CpA, k_CpAB, and k_dephos and the Michaelis constant k_m were changed according to the Arrhenius equation as in Fig. 2A. The values for k_i, E_i, A_i, R, and T are shown in Table S3 (top) and Table S4 (bottom). The values of the following parameters were fixed: a = 253, a₁ = 123, a₂ = 0, b = 131, s = 0.391, and ∑V_i = 233. The initial values were (V₁, V₂, V_new, V₃, V₄) = (87.53, 6.78, 1.70, 123.81, 12.95) for all simulations. The blue and red lines represent the dynamics of phosphorylated KaiC, which converged to a limit cycle, at 20 and 30°C, respectively. (B) Distribution of relative NS (m = 4 and q = 2) for phosphorylated KaiC as a function of relative period when reaction rates were increased. When the period was relatively maintained, the waveform of KaiC phosphorylation rhythms always became more nonsinusoidal. In the calculation, we increased the reaction rates k_phos, k_new, k_CpA, k_CpAB, and k_dephos by a random factor in the range 1.1–1.9 with Michaelis constant (k_m) fixed (green). For the case of stable phosphorylation speed to temperature, we increased the reaction rates as well as the Michaelis constant k_m by a random factor in the range 1.1–1.9 (purple). (C) Temperature dependency of the period and NS. Reaction rates k_phos, k_new, k_CpA, k_CpAB, and k_dephos and the Michaelis constant k_m were changed according to the Arrhenius equation k_i = A_iexp(−E_i/RT) as in Fig. 2C. The values for k_i, E_i, A_i, R, and T are shown in Table S5. The values of the following parameters were fixed: a = 374, a₁ = 103, a₂ = 0, b = 194, s = 0.166, and ∑V_i = 573. (D) Distribution of the relative synthesis interval of phosphorylated KaiC divided by the relative decay interval when reaction rates were increased as in (B). When the period was relatively maintained (orange bars, relative period ≥ 0.8), the synthesis interval became shorter and/or the decay interval of phosphorylated KaiC became longer. The black bars correspond to the parameter sets for period-shortening cases (relative period < 0.8).

To understand the condition for temperature compensation, we first computationally analyzed the sensitivity of period to temperature change in the PTO model in the same way as for the earlier analysis using the TTO models (see Supporting Materials and Methods). As all the reactions became faster with increased temperature, the calculated period often shortened, as was the case with the TTO models. The computational analysis showed that when the reaction rates increased but the period remained relatively unchanged (relative period ≥ 0.8), the waveform of the KaiC phosphorylation rhythms became more nonsinusoidal (with a larger NS), which was consistent with the TTO models (Fig. 3 B, green).

As mentioned earlier, the critical reaction of cyanobacterial circadian rhythms—the ATPase activity of KaiC—has been shown to be stable to temperature change (11, 16). To be consistent with this experimental evidence, we next increased not only the phosphorylation rate (k_phos) with temperature but also the Michaelis constant (k_m) with temperature for which effective phosphorylation speed (k_phosg₁(V₃,V₄)V₁/(k_m + V₁)) can be stable to temperature. As with the earlier analysis, we randomly increased the reaction rates and also the Michaelis constant k_m by 1.1–1.9-fold (see Supporting Materials and Methods). Again, the computational analysis showed that the NS for the KaiC phosphorylation rhythms always increased when the period remained relatively unchanged (Fig. 3 B, purple). When the reaction rates for KaiC phosphorylation were changed based on the Arrhenius equation, the period could be stable to temperature change when NS was larger at higher temperature (Fig. 3 C).

For the simpler TTO model, we computationally demonstrated that a longer decay interval for gene product at a higher temperature would result in more nonsinusoidal waveform (larger NS), potentially providing a mechanism for temperature compensation. We investigated whether there was a similar potential molecular basis for waveform modulation for temperature compensation in the PTO model, in which the protein abundances are conserved. The computational analysis showed that when the period remained relatively unchanged (relative period ≥ 0.8), the synthesis interval of phosphorylated KaiC became shorter and/or the decay interval became longer (Fig. 3 D). This result may indicate that asymmetry in the circadian waveform resulting in larger NS at higher temperatures should result in a stronger negative feedback through a greater abundance of phosphorylated KaiC via the sequestration of free KaiA, canceling out the period-shortening effect at higher temperature.

Together, our analysis suggested that a more nonsinusoidal waveform (larger NS) of phosphorylation rhythms at higher temperatures can be the mechanism for temperature compensation in cyanobacteria and that the temperature compensation mechanism may be the same for TTO and PTO.

Role of the waveform in a classical nonlinear oscillator

The mathematical and numerical analyses in this work have shown that the modulation of the circadian waveform to be nonsinusoidal is necessary for temperature compensation in the TTO and PTO models. To assess whether the shape of the waveform was relevant for the robustness of the other oscillators, we considered one of the classical nonlinear oscillators:

$$\frac{d^{2}x}{d{t}^2}+f\left(x\right)\frac{dx}{dt}+{\omega }_0^{2}x=0,$$ (4)

where ω₀ is the natural angular velocity and f(x) can take the form of any function that results in oscillations. When f(x) = 0, the model becomes a simple harmonic oscillator. When f(x) = μ(x² − 1), the model is known as a van der Pol oscillator. By expressing the periodic orbit of x(t) as the Fourier series $x\left(t\right)={\sum }_{j=-\infty }^{\infty }{a}_j\text{exp}\left(i\left(2\pi /\tau \right)jt\right)$, as we used in the TTO model, the following formula for period can be obtained:

$$\tau ={\left[\frac{{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2{j}^2}{{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2}\right]}^{\frac{1}{2}}{\tau }_0,$$ (5)

where τ₀ = 2π/ω₀ is the period of the linear harmonic oscillator (i.e., with f(x) = 0), and a_j are the Fourier coefficients of x(t). The term ${\left[{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2{j}^2/{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2\right]}^{1/2}$ is NS at m = 2 and q = 0 in Eq. 1 and also indicates a nonsinusoidal waveform. In the case of the linear harmonic oscillator, the time series given by Eq. 4 as a sinusoidal waveform and the term above is unity and minimal, resulting in τ = τ₀. Similar to the TTO and PTO models, the period of the nonlinear oscillator increases as the waveform becomes nonsinusoidal. For example, in the case of the van der Pol oscillator, the period increases as the parameter μ increases, and this is always accompanied by an increase in ${\left[{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2{j}^2/{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2\right]}^{1/2}$ and NS (Fig. 4 A). In Fig. 4 A, the period was 6.28, 7.63, and 10.20, and ${\left[{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2{j}^2/{\sum }_{j=1}^{\infty }{\left|a_j\right|}^2\right]}^{1/2}$ was 1.00, 1.21, and 1.62 when μ was set to be 0, 2, and 4. An analysis of the classical nonlinear oscillator suggested that the waveform of the time series is critical for determining the period not only in circadian oscillators but also in many other types of oscillator.

Figure 4.

Time series for a van der Pol oscillator and a TTO. (A) Numerical examples of the van der Pol oscillator, $d^{2}x/d{t}^2+\mu \left(x^2-1\right)dx/dt+{\omega }_0^{2}x=0$. In the van der Pol oscillator, the period also increases with NS. For the time series of x(t) and the corresponding NS (m = 4, q = 2), the values of μ were 0, 2, and 4 (top to bottom). The value of ω₀ was set to be 1. (B) Time series of the TTO oscillator. Mathematical results suggest that the period-lengthening effect due to a more nonsinusoidal waveform at higher temperature should cancel out the period-shortening effect and underlie temperature compensation. To see this figure in color, go online.

Discussion

In this study, we considered theoretically how the temperature compensation of circadian rhythms occurs. Based on mathematical and computational approaches using TTO and PTO models with different network structures, we found that larger distortions of the circadian waveform from being sinusoidal, indicated by higher values of the NS at higher temperatures, are necessary for temperature compensation (Fig. 4 B). Also, we confirmed that a larger value of NS at higher temperature was necessary for temperature compensation for various cases of the TTO model such as a TTO model with mRNA dynamics, Michaelis-Menten degradation, and a published realistic mammalian model (4); these analyses are described in the Supporting Materials and Methods. We also showed that the period of classical nonlinear oscillators, including the van der Pol oscillator, could be elongated through increasing NS without a change in natural angular velocity ω₀ (Fig. 4 A). Thus, NS may be an index for determining the period not only of circadian oscillators but also of other oscillators.

Ruoff has proposed that temperature compensation for any biochemical oscillators requires a balance between period-increasing and period-decreasing reactions (the balance hypothesis) (7). It has been also suggested that the robustness to temperature change of specific critical reaction(s) should underlie temperature compensation (the critical-reaction hypothesis) (11, 12, 13). Our findings for the waveform are consistent with both these hypotheses. First, using the simpler TTO model, we showed mathematically that the elongation of the period with increases in the reaction rate was always accompanied by the waveform becoming more nonsinusoidal (larger NS), indicating that a reaction associated with a longer period is also associated with increased NS. The results suggested that a balance between period-increasing and period-decreasing reactions should be achieved through modulating the waveform (i.e., changing NS). Second, using the same model, we discussed the condition of temperature compensation for parameter settings in which some reactions are stable to temperature and showed temperature compensation again required a more nonsinusoidal waveform (larger NS) at higher temperatures. Using the PTO model, we numerically realized temperature compensation for a circadian period in which NS increased with temperature but in which the effective phosphorylation speed was roughly stable to temperature change, qualitatively consistent with experimental evidence on the stability of KaiC ATPase activity (11). The results suggest that the circadian waveform should be more nonsinusoidal (larger NS) at higher temperatures even when the critical reactions for the period are stable to temperature change. We therefore consider our hypothesis of waveform for temperature compensation to be complementary to the critical-reaction hypothesis.

To quantitatively discuss the role of waveforms, we introduced the index, named NS. In fact, the form of NS for the Goodwin and that for the van der Pol model are not the same. We numerically tested the consistency between 1) NS for the Goodwin model 2), NS for the van der Pol model, and 3) total power divided by power of the fundamental frequency using the Goodwin model. Then, we confirmed that these were consistent (see Supporting Materials and Methods; Fig. S9). NS for the Goodwin and the van der Pol should be appropriate for experimentally quantifying waveform because NS for the Goodwin and the van der Pol tended to be more sensitive to change in waveform than total power divided by power of the fundamental frequency. We think that NS for the van der Pol model is the best for experimental measurements because it is one of the simplest forms.

To our knowledge, this is the first article to discuss the functional consequence of detailed waveform for temperature compensation. Numerical analysis using the simple TTO model showed that with larger NS at higher temperatures, temperature compensation could be achieved through the decay interval increasing at higher temperature. How is the longer decay interval at higher temperature generated? Notably, a numerical analysis of the simpler and realistic TTO models showed that when the period is relatively stable to temperature, NS and the amplitude of variables increased simultaneously (5) (Figs. S7 F and S10, relative period ≥ 0.9). This result suggests that greater amplitude of oscillations at higher temperature are likely to result in a longer decay interval as well as a more nonsinusoidal circadian waveform, resulting in temperature compensation.

In this work, we predict that nonsinusoidal waveform at higher temperature should be observed in temperature-compensated oscillators. Simultaneously, our period formula (Eq. 3) tells us when temperature compensation can occur without changing the waveform. First, if all the reactions are insensitive to temperature, a temperature-dependent waveform is not required for temperature compensation. Second, if all the degradation and the phosphorylation rates are insensitive to temperature but the remaining other reactions are sensitive to temperature, a temperature-dependent waveform is not required for temperature compensation. Indeed, Kidd and his colleagues showed experimentally that the waveform of the TIMELESS protein in Drosophila did not change with temperature. Also, they theoretically realized this phenomenon by using the Goodwin model, suggesting that circadian waveform is not important for temperature compensation (17). Although we cannot exclude the possibility that our hypothesis does not hold for the circadian rhythm of Drosophila, we think that the discrepancy may be due to differences in model assumptions. Their model assumed that degradation rates were stable to temperature (17), whereas these accelerated with temperature in our TTO models. A temperature-dependent waveform is not required for temperature compensation in the Goodwin model if degradation rates are stable to temperature. However, our formula for the period (Eq. 3) showed that without modulating the waveform, the Goodwin model with the temperature-insensitive degradation rates could not realize temperature overcompensation, which is often observed experimentally (5), although it can realize temperature compensation. Furthermore, if phosphorylation processes are incorporated into the Goodwin model as in our simpler model ((2a), (2b), (2c), (2d)) and at least some of the phosphorylation rates increase with temperature, then temperature compensation (and overcompensation) require a more nonsinusoidal waveform at higher temperatures.

Previous theoretical studies showed that several harmonic oscillators cannot be temperature compensated by using the model for the p53/Mdm2 system and the Goodwin model with a zero-inhibition constant when reactions become faster with temperature (35, 36). These results possibly suggest that temperature compensation can occur only in nonlinear (nonsinusoidal) oscillators. What we showed theoretically in this article is that waveform of the variables in nonlinear oscillators should change with temperature, and the sinusoidal or nonsinusoidal waveform should become more nonsinusoidal along with the temperature increase for temperature compensation. The condition for temperature compensation that we derived is a necessary condition. Indeed, a larger nonsinusoidal waveform at higher temperature is not sufficient for temperature compensation in the TTO and PTO models as depicted in Figs. 2 B and 3 B. By using the Goodwin model with a zero-inhibition constant (36), we also numerically confirmed that a more nonsinusoidal waveform at higher temperature is not sufficient but necessary for temperature compensation (see Supporting Materials and Methods; Fig. S11).

From mathematical and computational approaches, we predicted that a more nonsinusoidal waveform (larger NS) should be observed at higher temperatures. How can we test this prediction experimentally? In our analyses, the waveforms (NS) of specific variable(s) (i.e., transcription factor and protein phosphorylation levels) were focused. In reality, it would be difficult to specify which variable of the waveform (NS) is essential for period and temperature compensation. Previously, Pett and his colleagues numerically showed that CRY1, REV-ERBα, and PER2 should be essential for generating oscillations in their circadian model (37, 38). It raises the possibility that the choice of output variables for measuring NS might be important. By comparing NSs of Per2, Cry1, and Rev-erb mRNAs and proteins, and also other variables in the mammalian realistic model (Figs. S7, A–E and S8), we found that faster reactions increased the calculated NS of all these variables when the period was robust (relative period ≥ 0.8), whereas the sensitivity of NS to reactions was not perfectly the same. We think that this result suggests that a more nonsinusoidal waveform at higher temperature could be observed in many output variables and the choice of the output variables might not be essential. In cyanobacteria, KaiC phosphorylation and ATPase activity are known to be essential for the circadian period (11, 23, 39). Our prediction can be tested directly if the temperature sensitivity of NS for KaiC phosphorylation and/or ATPase activity rhythms can be quantified. In experimental study, the time series of circadian gene expression is often measured by bioluminescence. If the synthesis and/or degradation rates of the reporter change with temperature, NS of the reporter time series might be different from that of the original time series. In fact, we numerically confirmed that the waveform of the reporter became more nonsinusoidal when the reporter was set to be more unstable at higher temperature (Fig. S12). Also, if the reporter half-life was 30 min, for example, the NS of the reporter was similar to that of gene expression time series in our calculation (NS = 1.105 and 1.119 for reporter and original time series). Moreover, we mathematically showed that if the temperature sensitivity of the reporter half-life is experimentally measured, NS of gene expression time series can be explicitly estimated from the reporter time series (see Supporting Materials and Methods). Our prediction could also be tested by phase response curve experiments. A numerical analysis showed that a more nonsinusoidal waveform at higher temperatures is likely to result in a phase response curve of smaller magnitude in response to a pulse of light (Fig. S13). Studies have shown that the magnitude of phase shifts by light pulses was smaller at higher temperatures in both Neurospora and Drosophila (40, 41).

The type of bifurcations often helps us to understand the dynamics, especially near the bifurcation point. In several theoretical models, circadian oscillations occur via Hopf bifurcation (27, 42). Recently, one experimental study reported that low temperature nullifies the cyanobacterial circadian oscillations in vitro through Hopf bifurcation (43). It may raise the question of whether low temperature nullifies the cyanobacterial oscillations also by decreasing NS of KaiC phosphorylation time series or not. In this article, we introduced the modified Goodwin model for which bifurcation type is not necessarily a Hopf bifurcation, and it can be a Hopf or saddle-node bifurcation, for example, depending on the functional form (i.e., f(x₄)). Althoug our period formula states that the waveform of circadian gene expression should be more nonsinusoidal at higher temperature regardless of the type of bifurcations, it would be interesting to compare theoretically the effect of NS on temperature compensation in the vicinity of the bifurcation points between various bifurcation types.

In this article, we focused on the functional consequences of the waveform for the robustness of circadian periods to changes in temperature. If natural selection favors temperature-compensated oscillators, the waveform of circadian gene expression should evolve to have become more nonsinusoidal at higher temperature. However, the waveform of the time series may have other functional consequences for organisms. For example, the waveforms (NS) of circadian time series may also be important for the entrainment of circadian rhythms (44). Indeed, previous theoretical studies have shown that when an autonomous time series includes harmonics as well as sinusoidal waveform, the autonomous oscillation can be entrained by periodic forcing with the period of its harmonics; this was realized recently in acoustic engineering using an organ pipe (45, 46, 47). A circadian time series with a nonsinusoidal waveform might also enhance the entrainment into various forms of environmental cycles that undergo considerable change according to seasons, latitudes, and climates. Future theoretical and experimental studies may reveal a variety of rich functional consequences of waveforms for circadian rhythms.

Author Contributions

G.K. designed the research. S.G. performed the numerical simulations. S.G. and G.K. analyzed the data and wrote the article.

Editor: James Sneyd.

Supporting Citations

References (48, 49) appear in the Supporting Material.