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Waveform distortion for temperature compensation and synchronization in circadian rhythms: An approach based on the renormalization group method [1]

['Shingo Gibo', 'Riken Center For Interdisciplinary Theoretical', 'Mathematical Sciences', 'Ithems', 'Wako', 'Teiji Kunihiro', 'Yukawa Institute For Theoretical Physics', 'Yitp', 'Kyoto University', 'Kyoto']

Date: 2025-08 

Abstract Numerous biological processes accelerate as temperatures increase, but the period of circadian rhythms remains constant, known as temperature compensation, while synchronizing with the 24h light-dark cycle. We theoretically explore the possible relevance of waveform distortions in circadian gene-protein dynamics to the temperature compensation and synchronization. Our analysis of the Goodwin model provides a coherent explanation of most of temperature compensation hypotheses. Using the renormalization group method, we analytically demonstrate that the decreasing phase of circadian protein oscillations should lengthen with increasing temperature, leading to waveform distortions to maintain a stable period. This waveform-period correlation also occurs in other oscillators like Lotka-Volterra, van der Pol models, and a realistic model for mammalian circadian rhythms. A reanalysis of known data nicely confirms our findings on waveform distortion and its impact on synchronization range. Thus we conclude that circadian rhythm waveforms are fundamental to both temperature compensation and synchronization.

Author summary Our daily rhythms are underlain by gene regulatory and biochemical networks, called circadian clocks. Although most biochemical reactions accelerate as temperature increases, the period of circadian rhythms is almost constant even with increasing temperature. This phenomenon is called temperature compensation, and the mechanism is still unclear. By applying a method of theoretical physics, the renormalization group method to a biological problem, we revealed that the waveform of gene dynamics should be more distorted from sinusoidal wave at higher temperature when the circadian period is stable to changes in temperature. This prediction as for the importance of waveform in temperature compensation is verified by analyzing published experimental data of Drosophila and mice. Notably, the correlation between period and waveform distortion holds for other oscillator models, indicating the waveform distortion is important for determining the period in various types of oscillatory systems. Another important challenge in understanding circadian clocks is how they synchronize with environmental light-dark cycles. By theoretically analyzing a circadian clock model, we found that the frequency range for synchronization becomes narrower when the waveform is distorted.

Citation: Gibo S, Kunihiro T, Hatsuda T, Kurosawa G (2025) Waveform distortion for temperature compensation and synchronization in circadian rhythms: An approach based on the renormalization group method. PLoS Comput Biol 21(7): e1013246. https://doi.org/10.1371/journal.pcbi.1013246 Editor: Christian I. Hong, University of Cincinnati College of Medicine, UNITED STATES OF AMERICA Received: September 3, 2024; Accepted: June 17, 2025; Published: July 22, 2025 Copyright: © 2025 Gibo et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All code are available at https://github.com/gibo3/circadian_renormalization. Experimental data were previously published in Zhou et al. (2015) and Kidd et al. (2015). Funding: This work was supported by grants from the Japan Science and Technology Agency (JPMJCR1913 to G.K.), and from the Japanese Society for the Promotion of Science, and the Ministry of Education, Culture, Sports, Science, and Technology in Japan (21K06105 and 24H02025 to G.K., 19K03872 and 24K07049 to T.K.). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

1. Introduction Humans exhibit sleep-wake cycles with an approximate 24h period, and these cycles persist under constant environmental conditions, a phenomenon termed the circadian rhythm. This temporal regulation exists in both humans and various organisms such as molds, plants, and insects [1–3]. Recent advances in genetic research through insects, molds, mammals, and plants have unveiled that genes and proteins are involved as integral components in the primary mechanism governing autonomous circadian rhythms [1,4–6]. Understanding circadian rhythms holds promise for deciphering a multitude of sleep patterns, including sleep disorders such as advanced sleep phase syndrome (characterized by early awakening around 4:00 am), delayed sleep phase syndrome (marked by late awakening), non-24h sleep-wake disorder, and narcolepsy [7]. Notably, advanced and delayed sleep phase syndromes are believed to be linked to the circadian rhythm period [8–13]. Ongoing studies explore possible correlations between genetic characteristics revealed by large-scale genetic analysis and various sleep patterns [14–16]. However, the nature of the system is so intricate that it remains a challenge to link sleep patterns to specific genes. In such a situation, it would be meaningful to have recourse to mathematical models and obtain possible hints for the linkage and hopefully suggestions for studies of genetic dynamics. One fundamental issue that remains to be understood in circadian rhythm research is temperature compensation [17–25], in which the period keeps constant despite temperature-induced changes in reaction rates. Temperature compensation of the period occurs not only in circadian rhythms but also in ultradian rhythms, such as yeast metabolic cycles [22,26]. Despite the extensive experimental and theoretical research on temperature compensation, the mechanism has remained elusive. Hypotheses have been proposed to explain temperature compensation, including the balance hypothesis, critical-reaction hypothesis, temperature-amplitude coupling hypothesis, and waveform hypothesis. The balance hypothesis proposes that the stability of the circadian period with temperature arises from a balance between period-lengthening and period-shortening reactions [17,18,27]. The critical-reaction hypothesis assumes that there should be critical reactions that determine the circadian period. If these reaction rates are stable against temperature variations, then the circadian period will similarly remain stable [28–30]. The temperature-amplitude coupling hypothesis suggests that temperature-sensitive amplitudes in gene activity rhythms should generate a stable period by generating larger amplitudes at higher temperatures [22,31]. Lastly, the waveform hypothesis proposes that temperature-sensitive waveforms in gene activity rhythms should be correlated with a stable period in a manner that their higher harmonic components become larger and the distortion of the waveform increases at higher temperatures [32]. These proposed mechanisms, along with other mechanisms discussed in [33], are not necessarily mutually exclusive and may work in combination to achieve temperature compensation. Another key issue in circadian rhythm research is synchronization with 24-hour environmental light-dark cycles. Previous theoretical and experimental studies on synchronization revealed that if the internal period of the oscillation closely matches the external period, then it is more likely to synchronize with the forced period [34–37]. Additionally, experimental studies on several species uncovered genes and proteins in circadian systems affected by a light pulse [38,39]. In reality, the circadian rhythm must adjust to the 24h light-dark cycle while maintaining a temperature-compensated circadian period. Therefore, multiple questions arise. (i) Given the significant temperature variations between seasons, how do organisms synchronize their circadian rhythms with the 24h light-dark cycle across various temperatures [20,36,40]? (ii) if the gene activity rhythm of the circadian rhythms becomes more distorted as temperatures increase to achieve temperature compensation, how does the ease of synchronization change with temperature variations? Theoretical analyses incorporating the findings of light pulse experiments might provide further insights into these questions. In the present paper, we investigate possible roles of the waveform distortion in temperature compensation based on analytical and numerical analyses of the Goodwin model for circadian rhythms and clarify how the waveform in gene activity rhythms tends to be more distorted at higher temperatures (e.g., steeper rise, longer tail) for temperature compensation. To this end, we employ the renormalization group (RG) method [41–46] adapted for global and asymptotic analysis of differential equations on the basis of the perturbation theory [47–53]. As elucidated in a historical review by Shirkov [54], there are several distinct approaches that have been referred to collectively as the renormalization group (RG) method. Among these approaches, concepts such as asymptotic functional self-similarity, reminiscent of RG techniques used in extracting critical exponents in statistical physics [44–46], were notably applied by Feigenbaum [55] to derive universal constants characterizing bifurcation points in certain iterative maps encountered in population biology [56,57]. In contrast, the RG method utilized in this current work [47,48,51] is an adaptation of the approach originally developed in quantum field theory [42,43]. This approach provides a straightforward resummation technique for perturbative solutions, thus offering practical approximate solutions valid over global time domain. The method also serves as a powerful reduction theory of the dynamics in the asymptotic region. The method has been applied to various models, including ordinary and partial differential equations, discrete maps, and stochastic equations [48–50,52,53,58]. The origin of the powerfulness of the RG method as a tool of the global analysis can be intuitively understood by reformulating the method in terms of the classical theory of envelopes [48,53]: The envelope of a set of perturbative solutions which are only valid only locally around the arbitrary initial time t 0 can be set up to constitute a valid solution to the equation in a global domain. Combining an index for waveform distortion, namely non-sinusoidal power (NS) introduced by two of the present authors (KG) [32], with the result of the RG method, we can obtain both a unified picture of the above mentioned theoretical hypotheses (balance hypothesis, critical-reaction hypothesis, temperature-amplitude coupling hypothesis, and waveform hypothesis) and quantify previous experimental data on Drosophila [59]. Our analyses demonstrate that the fundamental role of the waveform distortions in temperature compensation from both theoretical and experimental perspectives in accordance with the previous findings [32]. Moreover, we reveal for the first time the mechanism by which the synchronization of circadian rhythms changes with temperature if the waveform in gene activity rhythms is more distorted at higher temperatures. We theoretically prove that the frequency range of the external force that synchronizes circadian rhythms becomes narrower if the waveform of gene activity rhythms is more distorted. This indicates that it is more difficult to synchronize with light-dark cycles at higher temperatures. The present result of synchronization is consistent with the previous experimental and numerical studies demonstrating that the magnitude of the phase shift caused by light pulses was smaller at higher temperature [32,60,61]. This paper is organized as follows. In Sect 2, we introduce the index for waveform distortion and the Goodwin model, and summarize the main results regarding temperature compensation and synchronization of circadian rhythms. In Sect 3, we investigate the conditions for temperature compensation. We present our results based on the renormalization group (RG) method in Sects 3.1, 3.2 and 3.3, followed by experimental verification of the model’s predictions in Sect 3.4. In Sect 4, we examine the conditions for synchronization. Sect 5, the final section, is devoted to discussions. Some detailed technical account of our method of analysis and numerical analyses are given in S1 Text–S5 Text.

Results

5. Discussion We theoretically explored the conditions for clarifying the temperature compensation of the biological clock and its synchronization to light-dark cycles with a particular focus on waveform distortion. To investigate these conditions, we focus on the Goodwin model, which is widely used as a mathematical model that simulates various properties of biological clocks, including temperature compensation, synchronization to temperature cycles, and phase resetting by temperature steps [72]. The theoretical analysis of the Goodwin model revealed that waveform distortion of gene activity rhythms with increasing temperature is necessary for temperature compensation. Furthermore, we derived an approximate but globally valid solution to the waveform of the time profiles using the RG method as a powerful tool for global analysis. This allowed us to investigate, based on the analytical solution, whether there is a universal law for the mechanism by which the waveform changes with temperature variation. Notably, the relation between the period and waveform distortion holds not only for the Goodwin model but also for more realistic models. We have numerically showed the NS is larger when the period is stable to changes in temperature by using Zhou model, which includes 190 variables [21,32]. Additionally, we numerically analyzed the period and NS of Relógio model, which includes 19 variables [73], when all reaction rates increased by a factor of 1.5–2.5 (see S3 Text for details). When the period is stable or increases with increasing reaction rates, geometric mean of NS becomes larger (S5 Fig). This suggests that the waveform distortion is important for temperature compensation in realistic circadian clock models. The results indicated that temperature compensation is more likely to occur when the waveform is distorted, especially if the decreasing duration of circadian protein oscillation elongates as temperature increases. Although theoretical predictions based on a model might not always be realized in real organisms, we quantified the gene activity rhythms of published experimental data using Drosophila and mice. This quantification confirmed that the waveform is distorted at high temperatures, in accordance with our theoretical predictions. It is notable that the systematic wave distortion governed by Eq (5), which we have found to hold in the Goodwin model, also applies to a wide class of non-linear oscillators used for biological phenomena different from biological rhythms, including the Lotka-Volterra model [74], which is commonly used in ecology, and the van der Pol model, as presented in the S4 Text and S5 Text, S6 Fig and S7 Fig. (see also [32]). This suggests that exploring the possible significance of waveform distortion in other mathematical models, such as the Fitz-Hugh-Nagumo model in neuroscience, would be intriguing [74–76]. To the best of our knowledge, this is the first study to apply the RG method, a powerful resummation method of the perturbation series first developed in physics, to circadian rhythm problems. In the RG method, secular terms appearing in the naïve perturbation series are renormalized into the ’integral constants’ , which thus acquire the nature of the slow modes, making it a powerful tool for global and asymptotic analysis. Unlike the naive perturbation theory, the solutions given by the RG method provide a time evolution close to numerical simulations in the relevant global domain of time, offering an approximate solution for the period and waveform. The analytical results predict that longer tails of gene activity rhythms at higher temperatures occur for temperature compensation. This study also investigated the synchronization with environmental light-dark cycles at various temperatures [35,36]. The numerical simulations and theoretical analysis predict that as the distortion of the gene activity rhythms for achieving a temperature-compensated period increases, it becomes more difficult to synchronize with the light-dark cycle. This prediction aligns with the reported temperature-dependent variation in response to light pulses in Drosophila and Neurospora, displaying smaller phase shifts at higher temperatures [60,61]. These results suggest that the waveform of gene and/or protein activity may vary seasonally between hot summers and cold winters, potentially affecting the entrainment of circadian clocks. Further studies are needed to investigate this possibility, particularly in light of seasonal variations in both temperature and light intensity. To compare the role of waveform with that of amplitude in temperature compensation and synchronization, we also analyzed amplitude. Specifically, we numerically showed that the amplitude also becomes larger at higher temperatures when the period is stable to changes in temperatures (S1 Fig), which is consistent with previous study [22,31]. The correlation between period and amplitude is called “twist" [77]. It has been known that large amplitudes lead to narrow synchronization region [36,78]. These might indicate that both the large amplitudes and distorted waveform lead to similarly narrow synchronization region. As mentioned in the Introduction, previous theoretical and experimental studies, such as those using Drosophila [59], suggested that waveforms in gene activity rhythms do not change under temperature variations, although they are blurred by error bands. These findings are apparently inconsistent with our current conclusion. We believe that this discrepancy stems from two main factors: differing assumptions about the temperature sensitivity of degradation rates and differing interpretations of experimental results regarding gene activity rhythms at distinct temperatures. First, the previous study assumed that all degradation rates are temperature-insensitive. Thus, the waveform of gene activity rhythms does not need to change with temperature. By contrast, we assume that some degradation rates at least should accelerate with temperature, leading to a more distorted waveform of gene activity rhythms at higher temperatures. Second, the previous study [59] interpreted their experimental results as indicating that gene activity rhythms at different temperatures can be collapsed onto each other by rescaling, supporting their prediction that temperature compensation occurs because of rescaling and the temperature insensitivity of degradation rates. Conversely, our quantification of their experimental data indicated that the waveform tends to be more distorted at higher temperatures, whereas variation in NS values was noted. Thus, the present result is consistent with our theoretical prediction. Our theoretical prediction can be tested in circadian organisms such as mice and Neurospora [79]. We believe that further systematic quantification of the waveforms of gene activity and/or protein activity rhythms in various circadian organisms will be essential for clarifying the importance of the waveform in circadian rhythms in the future.

Acknowledgments We thank H. Nakao, H. Chiba, Y. Kawahara, A. Mochizuki for useful comments on this study.

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