2.1. Index for waveform distortion

Let the time dependence of a certain variable in the circadian rhythm system be expressed in a Fourier series as , with a_j being the Fourier coefficients of the oscillatory time series. Then, we introduce an index for describing the distortion of x(t) from a sinusoidal shape as

(1)

where m and q are integers. Termed the “non-sinusoidal power (NS)”, this index is designed to emphasize higher harmonics (m > q) as discussed in a previous paper [32]. For instance, we have NS = 1 when the time series has a sinusoidal waveform in which only the coefficients for the fundamental component are non-zero (). Conversely, for non-sinusoidal time series, the coefficients for higher harmonics are non-vanishing, resulting in NS > 1. The previous theoretical work demonstrated that a more distorted waveform (larger NS) at higher temperature is necessary for temperature compensation in the four-variable negative-feedback model [32]. However, it is unclear whether the relevance of waveforms to temperature compensation found in previous research has general validity not restricted to some specific model.

To explore the possible relevance of the waveform characteristics to temperature compensation and synchronization, we consider the simplest model for circadian rhythms, known as the Goodwin model (Fig 1A) [62]. This model incorporates negative-feedback regulation of gene expression, a mechanism established as essential for transcriptional-translational oscillations. The three-component Goodwin model reads:

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Fig 1. (A) The circadian clock model and (B) the relevance of waveform characteristics to temperature compensation (magenta) and synchronization (cyan).

The transcriptional-translational function is defined as , with the values of K and n set to be 0.0184 and 20, respectively. Assuming that the parameters k₁, k₂, k₃, p₁, p₂, and r increase with increasing temperature, we model the reactions rates using the Arrhenius law , where b_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 detailed in S1 Table. The synchronization region depends on the period of light-dark cycles and light intensity, known as the Arnold tongue [37]. The size the Arnold tongue is defined by the difference between the lower limit of synchronization, , and the upper limit of synchronization, . When the period is constant in a different temperature, the waveform distortion (NS) becomes larger, and the synchronization range (size of the Arnold tongue) becomes narrower with increasing temperature as shown in (B) in the case of the light intensity, I = 0.001.

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(2)

(3)

(4)

where x₁(t) represents mRNA abundance, and x₂(t) and x₃(t) denote protein abundance. The function f(x₃) in the model signifies transcriptional regulation, and the parameters p₁ and p₂ denote protein synthesis and phosphorylation rates, respectively, and k_i () represent degradation rates (Fig 1A). In fact, there are two versions of the Goodwin model: (i) one in which the degradation rate is expressed as a linear function of the substrate, as in Eqs (2)–(4), and which requires high cooperativity (n) for oscillations; and (ii) another formulation in which the degradation rate follows a Michaelis-Menten equation, where a cooperativity of one is sufficient to generate oscillations [63–65]. In the present paper, we assume that the degradation rate follows a linear function of the substrate, for the sake of simplicity and analytical tractability. Forger derived the period of Goodwin model with linear degradation (2)–(4) by applying signal processing methods [66], while a similar analysis has also been conducted for other negative feedback oscillators [67]. First, Eqs (2)–(4) were transformed into a single equation with higher-order derivatives (see Eq (47) in S2 Text). Next, the nonlinear term f(x₃) was eliminated by multiplying the transformed equation by dx₃/dt and integrating for one period. Then, by expressing the periodic solution in Fourier series as , the period was derived as follows:

(5)

where a_j is the Fourier coefficient of x₃(t). If x₃(t) is distorted from a sinusoidal wave, the higher-order components a_j () become large, causing NS to be large. Subsequently, two of the present authors showed that this formula implies that temperature compensation of the period in this model occurs only when the waveform (NS) is distorted as temperature increases [32]. Suppose that all reactions become faster as temperature increases in the model. Then, one can numerically demonstrate that the waveform tends to be more distorted (larger NS) at higher temperatures for temperature compensation (Fig 1B, magenta line). We call this mechanism the waveform hypothesis for temperature compensation. This result, based on the Goodwin model [32], which demonstrated that a more distorted waveform at higher temperatures is both necessary and sufficient for temperature compensation, aligns with a previous study [67] showing that an alternative oscillator model with sinusoidal oscillations alone cannot achieve temperature compensation.

2.2. Theories of temperature compensation

The waveform hypothesis and the other three hypotheses for temperature compensation (balance hypothesis I, critical-reaction hypothesis II, and amplitude hypothesis III) are not mutually exclusive, which can be understood in a unified way through Eq (5):

1. I. The balance hypothesis, previously explored theoretically by Ruoff [18], suggests that the temperature compensation of the period is caused by a balance between the effects of reactions that shorten the period and those that lengthen the period. Eq (5) illustrates that the balance between the effect of shortening the period and that of lengthening the period can be caused by changing the distortion of the waveform.
2. II. Eq (5) demonstrates that even if some of the governing reactions in circadian rhythms are temperature-insensitive [28,30] as discussed in the critical-reaction hypothesis. Still, the temperature compensation requires waveform distortion at high temperatures if reactions other than some of the governing reactions accelerate at higher temperatures.
3. III. Our numerical analysis of the circadian model (Fig 1B) show that when temperature compensation occurs, the waveform is more distorted at higher temperatures, and the amplitude of the oscillation is larger (S1 Fig). This tendency for the amplitude to increase at high temperatures is consistent with the amplitude hypothesis. According to Eq (5), a greater distortion of the waveform at higher temperatures is necessary, but not sufficient, for temperature compensation.

2.3. Synchronization in circadian rhythms

If the waveform is more distorted at higher temperatures for temperature compensation, then it would be intriguing to explore whether the temperature- dependent waveform also affects the synchronization of circadian rhythms with environmental light-dark cycles at various temperatures. Theoretical and experimental studies of synchronization and circadian rhythms illustrated that the oscillation is more likely to synchronize with forcing cycles if the internal period is sufficiently close to the period of the forcing cycles [34–37]. In mammals and Neurospora, a light pulse is known to increase Per1 and frq mRNA expression [38,39]. To incorporate gene activation during the light phase, we employ a model in which Eq (2) for the change in mRNA expression is modified to

(6)

where I represents the light intensity and is the angular frequency of the light-dark cycles.

2.4. Main results

In Sects 3 and 4, we provide detailed discussions on how the waveform in gene activity rhythms should be distorted at higher temperatures using the RG method, as well as the synchronization in circadian rhythms. The main results are pictorially summarized in Fig 1B. The magenta line indicates that the waveform of the gene activity rhythms should be more distorted at higher temperatures for temperature compensation, whereas the cyan line indicates that synchronization with the light-dark cycles should become more difficult at higher temperatures because of the larger waveform distortion.

3.1. Numerical simulation of the waveform-period correlation

Eq (5) and numerical simulations indicate that NS tends to be larger when the period is relatively stable even with increased parameter values (see Fig 1B). To quantitatively reveal the correlation between waveform and period, we conduct numerical simulations using a circadian clock model. In the analysis of the circadian clock model, the transcription function was considered for simplicity. We first search for parameter sets in which oscillations occur. We define those parameter sets as the reference parameter sets. Because many biochemical parameters have not yet been measured, we prepared 100 random reference parameter sets for the oscillations. k₁, k₂, k₃, p₁, p₂, and r were assigned uniformly distributed random values ranging from 0 to 10, and n was assigned a uniformly distributed random integer ranging from 9 to 15. The period obtained with each reference parameter set was denoted as . Next, reaction rates often follow the Arrhenius equation, which states that a 10°C rise in temperature increases the reaction rate by a factor of 2-3. To incorporate the effect of high temperature, instead of using the Arrhenius equation, each parameter in the model’s reference parameter set was randomly multiplied by a factor of 1.1-1.9, and the period and waveform were examined when the oscillation behavior persists. The period obtained by increasing the parameters from each reference parameter set was denoted as , and the ratio, , was called the relative period.

To quantitatively analyze the correlation between period and waveform when temperature compensation occurs, we consider the case of the relative period because it has been experimentally confirmed that the circadian rhythm frequency at high temperature divided by that at low temperature of the wild-type ranges between 0.85 and 1.15 when the temperature is increased by 10°C and temperature compensation occurs [19]. In the present numerical analysis, the period is relatively stable (relative period ) in 34 of the 4900 parameter sets, in qualitative agreement with previous theoretical analyses that the period often shortens with increasing reaction rates [18,27]. Because the range of parameter variation is 1.1-1.9 and the average value is 1.5, the reaction rate is accelerated by a factor of 1.5 on average, and the average relative period is approximately . Fig 2 indicates that when temperature compensation occurs, there is a clear correlation between the period and waveform.

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Fig 2. (A) Distribution of relative NS for of the circadian clock model as a function of the relative period when reaction rates are increased.

We first generated reference parameter sets. To incorporate the effect of the increase of temperature, instead of using the Arrhenius equation, each parameter k₁, k₂, k₃, p₁, p₂, and r in the model’s reference parameter set was randomly multiplied by a factor of 1.1-1.9. The insertion indicates the region of temperature compensation (). (B) Examples of the waveform in Goodwin model when the period was relatively unchanged. The blue and red lines represent slow and fast reactions, respectively.

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3.2. Brief introduction of the renormalization group method

In this subsection, we give a brief introduction to the renormalization group method for obtaining a useful approximate solution that is valid in a global domain from the perturbative solution in a way presented in [48,49,52].

Given a differential equation with t being the independent variable, naive perturbative solutions to it generically contain secular terms proportional to polynomials of t, such as or with n being an integer when the unperturbed solution is given by a harmonic oscillation with an angular velocity . Thus the naive perturbative solution would show a divergent behavior and goes far away from the exact solution for large t because of the secular terms. The RG method gives a simple resummation method of the perturbation series and hence a globally valid solution utilizing the initial condition with the integral constants at arbitrary time t₀.

To show this, let us take the following linear damped oscillator as a simplest example [48]:

(7)

with . The exact solution is given by

(8)

where and with : Owing to the velocity-dependent friction , the amplitude has a time-dependence and the angular velocity is shifted from the unperturbed value 1; note that the solution (Eq (8)) can be expressed as an infinite power series of although the enters the Eq (7) linearly.

Now we dare to apply the perturbation theory to get a solution x(t;t₀) of the linear equation (Eq (7)) with the initial value W(t₀) at an arbitrary time t = t₀,

which is unknown but yet to be determined by the following procedure. The perturbation solution x(t;t₀) is expressed as a power series of as . In the RG method, we assume that the initial value W(t₀) is on an exact solution, and is also expressed as a power series of as with (.

The zeroth-order equation reads

the solution to which may be expressed as with . Here we have made explicit the fact that the amplitude A(t₀) and phase depend on the initial time t = t₀. Then, the first-order equation is given by . Since the inhomogeneous term has the same angular velocity as the unperturbed one, the first-order solution necessarily contains a resonance term, and can be given in the following form

which is made vanish at t = t₀ by adding an unperturbed solution, and hence . This setting proposed in [48] is found to make the procedure of the RG method simple and transparent. In the present linear case, the first-order correction can be made vanish at the initial time, but when the equation has nonlinear terms, some terms would remain there. Similarly, the second-order solution can be given by , which vanishes at t = t₀, and hence . Up to the third order of , the perturbative solution is given by with the initial value It is apparent that our approximate solution is diverging as due to the secular terms, although it gives a sensible and hence valid solution locally around t = t₀. To obtain a valid solution in the global domain, we utilize the fact that the approximate solution should be valid around an arbitrary time t = t₀, and try to construct the envelope curve of the perturbative solutions, each contact point of which with the envelope should be chosen at t = t₀, such that the envelope function is given by the initial value . This can be accomplished through the following envelope equation, which is also identified with the RG equation:

(9)

This equation turns out to give the dynamical equations of the amplitude A(t) and the initial phase . In fact, substituting the perturbation solution up to second-order to the RG Eq (9), we have

(10)

To satisfy this condition for any time t, the amplitude A(t) and the initial phase should satisfy the following dynamical equations

(11)

respectively. These equations are readily solved to yield and , respectively. Inserting these expressions into the envelope function , we arrive at

(12)

which is an approximate but globally valid solution to Eq (7): Indeed it has an infinite power series of . It is also noteworthy that the angular velocity of Eq (12) gives a good approximation of the exact one up to the third order, as . For more details with a nonlinear equations, see S1 Text.

3.3. RG analysis of waveform-period correlations

In Sects 2.1 and 3.1, we demonstrated that the temperature compensation of the period in the Goodwin model is accompanied by an increase of the index NS and waveform distortion correspondingly occurs as temperatures increase. This raises the following question: Is there a universal law governing the waveform distortion occurring when temperature increases? To answer this question, we derive an approximate solution for the time evolution of the Goodwin model for the circadian rhythm using the renormalization group (RG) method [47]. The solution obtained using the RG method can be interpreted as the envelope of the set of solutions given in the perturbation theory, which has been applied to various models, including (but not limited to) ODE, PDE, discrete systems, and stochastic equations [48,49,52,53,58]. To apply the RG method, we again set the transcriptional regulation function f(x₃) to be . In this function, n is the cooperativity of the transcriptional regulation, which is a Hopf bifurcation parameter. The approximate solution of the phosphorylated protein of the circadian clock reads (see S2 Text)

(13)

where we have

(14)

(15)

(16)

with the angular velocity and the phase parameter of the second-order term

(17)

(18)

as well as the amplitudes

(19)

(20)

The RG method provides an approximate but globally valid solution, and thus enable us to make a detailed investigation of the waveform distortion when temperature compensation occurs in the Goodwin model. For example, we analyzed how the value of A₁ in Eq (20) depends on the reaction rates. This parameter is associated with the amplitude of the second harmonic component. To quantify its sensitivity to changes in the reaction rates, we used elasticity, defined as , where q_i represents the parameters k₁, k₂, k₃, p₁, p₂, and r. We found that A₁ increased or decreased with increasing degradation rates (k₁, k₂, and k₃), and consistently decreased with increasing phosphorylation and transcription rates (p₁, p₂, and r). The sum of the elasticities, , was always zero, which reflects the fact that A₁ remains unchanged when all reaction rates are scaled by the same factor (S2 Fig).

The numerical analysis using the same parameter sets as used in Fig 2A, in which the relative period in the model remains stable and within the interval against the temperature variations, shows that the phase parameter in the 2nd-order frequency tends to decrease with increasing reaction rates (see Fig 3A). When the increase in reaction rates is small, the change in the phase of the second-order frequency scatters around zero and is negligible. However, with a significant increase in the reaction rates, the phase of the second order always tends to decrease as the reaction rates increase.

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Fig 3. Change in the phase of the second-order frequency in phosphorylated protein oscillation under varying circadian clock model parameters.

(A) The parameters k₁, k₂, k₃, p₁, p₂, and r were randomly increased from 1.1 to 1.9. The horizontal axis represents the arithmetic mean of the fast-case values of k₁, k₂, and k₃ divided by the slow-case values. The parameters p₁, p₂, and r are omitted because they do not affect the period in Eq (17). The vertical axis presents changes in the phase of the second-order frequency obtained via generalized harmonic analysis of numerically calculated periodic time series. When the period length is maintained (relative period within 0.85-1.0), the phase of of x₃(t) tends to decrease as the reaction rate increases. (B, C) Time series for the increased reaction rates (1.1-1.9) while maintaining the period. The red and blue lines represent the time series for increased reaction rates and those for the reference parameter set, respectively. The parameter values are provided in S2 Table. For comparison, the period length is standardized to 1. The time of the minimum value is set to 0, and the maximum phosphorylated protein oscillation is indicated by the dotted line. Increased reaction rates tend to advance the timing of the maximum oscillation.

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The significance of the phase parameter given in the second-order term on the waveform of the time series can be understood intuitively as follows: when the phase is large, the increasing duration tends to become longer because of the less overlap of the time profiles given by and (Fig 3B and 3C, blue line). Conversely, a smaller tends to result in a shorter increasing duration because of an additive effect of the two terms, which leads to a steeper slope on total, as presented in Fig 3B and 3C, red line. Therefore, the numerical results in Fig 3A suggest that the decreasing duration of the time series elongates with as the reaction rate increases when the period is relatively stable despite the increasing reaction rate.

For a theoretical confirmation of the numerical result that the phase parameter given in the second-order term tends to become smaller as the temperature increases when the period is temperature-compensated, we analyze the sensitivity of the angular frequency and the phase to the reaction rates by utilizing the results of the RG method. With use of Eqs (17) and (18), we calculate and . In Fig 4A, we present the parameter regions given by the constraints (red surface) and (yellow surface) for the cooperativity n = 12 of the transcription regulation. We can see that the region for (outside yellow surface) includes that given by the constraint for all k_i (). This implies that if the period is robust against a change in the parameters k_i (), then the phase in the second-order term always becomes smaller with increasing parameters.

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Fig 4. The parameter space for the constant period (, red surface) with increasing degradation rates k₁, k₂, and k₃, and that for the constant phase of second-order frequency (, yellow surface).

The phase of second-order frequency should decrease with increasing k₁, k₂, and k₃ outside the yellow surface (). The cooperativity n is n = 12 (A), and n = 20 (B), and the variations of the parameters are set to dk₁ = 0.3, dk₂ = 0.2, and dk₃ = 0.1. (A) When n is small (n = 12), surface is included in space , which means that should become smaller if does not change with increasing k₁, k₂, and k₃. (B) When n is large (n = 20), surface can intersect although high cooperativity is unrealistic.

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Next, let us examine how the parameter regions given by the constraints and change with variations of the cooperativity n. The numerical calculation shows that the region corresponding to includes that given by for n = 13 and 14. However, in the case of an exceedingly high cooperativity of transcription regulation n, which is not biologically realistic, can be positive when . For instance, for n = 20, although is negative for most of the parameter space, there is a region in which when (see Fig 4B). These results indicate that if circadian rhythms are stable under temperature variations, the slope in the increasing phase of phosphoproteins should become sharper as temperature increases. Thus, we conclude that (i) the waveform of the gene activity rhythm should be more distorted at higher temperatures, and (ii) the rate of the increase in phosphoprotein levels should be greater at higher temperatures if temperature compensation is achieved. In principle, these features can be tested experimentally.

3.4. Verification of the theoretical analysis of temperature compensation using published experimental data

The period formula Eq (5) of the Goodwin model indicates that the non-sinusoidal index NS of the waveform of the circadian rhythms becomes larger, implying greater distortion of the waveform when all reactions are faster at higher temperatures during temperature compensation. To test this theoretical prediction of circadian gene activity in actual organisms, we analyze the waveform of the activity rhythms of the timeless gene in Drosophila at 18 and 29°C using published experimental data [59]. First, we extract the time series of the average curve from Fig 3C in a prior study [59]. Then, we add uniformly distributed noise between -0.4 and 0.4 to the extracted data to consider data errors (Fig 5A). The Fourier coefficients are estimated using generalized harmonic analysis (GHA) [32,68–70] (see Methods for details). The resultant NS of the activity rhythms of the timeless gene, as defined by Eq (1), at a higher temperature (29 °C) tend to be larger than that at a lower temperature (18 °C), whereas the NS values are somewhat varied (Fig 5B), which is consistent with the prediction.

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Fig 5. Analysis of the circadian waveform of wild-type Drosophila at different temperatures using previously reported experimental data [59].

(A) Re-plot of Fig 3C in [59]. The average curves of tim-luc at 18 and 29 °C are extracted, and uniformly distributed noise between –0.4 and 0.4 is added to the extracted data, generating 100 time series datasets. Spline interpolation is applied to set the sampling interval to 0.1 h. The interpolated time series data at 18 (cyan) and 29 °C (magenta) are plotted. (B) Distribution of the quantified waveform distortion (NS) from the data at 18 (cyan) and 29 °C (magenta). The time series data with noise are detrended by multiplying an exponential function so that the positions of local minima of the oscillations are approximately equal. The Fourier coefficients of the detrended time series are quantified using GHA. The NS values are estimated from the coefficients up to the third harmonics. Dots represent the NS values for each dataset, and the horizontal lines represent their average values. The estimated NS values increased significantly at higher temperatures (t = 9.5755, df = 193, p < 0.001).

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Experimental studies have demonstrated that temperature compensation can be impaired by genetic mutations. In the Drosophila mutant perL, the period increases with increasing temperature [59,71]. Eq (5) implies that if the period increases with temperature, then the waveform of the circadian rhythm should be non-sinusoidal and more distorted at higher temperatures. Thus, it is predicted that the waveform of the circadian gene activity in perL should become more non-sinusoidal with higher temperatures. Again, we can quantify the waveform of perL using experimental data [59] (S3 Fig). The waveform of circadian gene activity tends to be more non-sinusoidal at higher temperatures in perL, as observed in the wild-type, whereas the NS values varied, in line with the prediction. To be precise, we performed a t-test to compare the two groups at higher and lower temperatures. The difference in estimated NS values between the two temperature conditions was statistically significant in the wild-type (t = 9.5755, df = 193, p < 0.001) (Fig 5). In the perL mutant, the difference was also statistically significant (t = 14.7971, df = 191, p < 0.001) (S3 Fig), supporting our theoretical predictions.

To examine the generality of the results, we also tested our conclusion using mouse embryonic fibroblasts (MEFs) data [21]. According to our prediction, the waveform of the circadian gene activity (Per2-luc) in wildtype mice should become more non-sinusoidal at higher temperatures. Indeed, the NS value was significantly higher at elevated temperatures (t = 2.9793, df = 189, p = 0.0016), supporting the generality of our findings (S4 Fig).

Computation of ODEs

The 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 for the Goodwin model (Eqs (2)–(4)) was 0.001, that for the Lotka-Volterra model (Eqs (104)–(105)) and the van der Pol oscillator (Eq (114)) was 0.0001, and that for Relógio model [73] was 0.01.

Numerical analysis of synchronization with light-dark cycles

The angular frequency of the light-dark cycle in Eq (6) was changed by 0.0001. We considered that the model was synchronized with the light-dark cycle when the change in the amplitude was smaller than the rounding error as in Fig 1B.

Estimation of waveform distortion from simulated time-series

We estimated waveform distortion, i.e., non-sinusoidal power (NS, see Eq (1)) from the simulated time-series by using the generalized harmonic analysis (GHA) method as in Figs 1B and 2A. The GHA method estimates the amplitudes A_j and B_j, and the frequencies f_j, which minimize the squared residual . The detailed procedure of GHA can be found in [32,69].

Estimation of phases of Fourier components from simulated time-series

By using GHA method, we computed the Fourier series . This form was converted into , where and .

Computation of parameter space for temperature compensation using the RG method

The frequency change and the phase change were calculated from Eqs (17) and (18) by using Maple. The parameter space for and in Fig 4A and 4B were numerically generated by Maple.

Estimation of waveform distortion from published experimental data

The average curves of experimental time-series of timeless luciferase (tim-luc) reporter (shown to recapitulate the dynamics of timeless gene expression) at 18 and 29 °C of Figs 3C and figure54B in the published literature [59] were extracted using WebPlotDigitizer at 1 h intervals. Uniformly distributed noise between –0.4 and 0.4 was added to the extracted data, generating 100 time series datasets so that the experimental noise is roughly reproduced as in Figs 5A and S3 FigA. Spline interpolation was applied to set the sampling interval to 0.1 h to smooth time series data. The time series data with noise were detrended by multiplying an exponential function so that the positions of local minima of the oscillations are approximately reproduced. The width of the window for the analysis is set to one period. The Fourier coefficients of the detrended time series were quantified using GHA. The NS values were estimated from the coefficients up to the third harmonics using Eq (1) with m = 4 and q = 2, as shown in Fig 5B and S3 FigB. To determine the significance of the differences, Welch’s t-test was used.