Abstract
One of the essential characteristics of an authentic circadian clock is that the free-running period sustains an approximately 24-hour cycle. When organisms are exposed to an external stimulus, the endogenous oscillators synchronize to the cycling environment signal in a process known as entrainment. These environmental cues perform an important role in resetting the phase and period of the circadian clock. A “generalized assumption” states that when an organism has a short period, it will experience a phase advance, while an organism with a long period experiences a phase delay. Despite widespread use, this positive relationship relating period to the phase of entrainment does not describe all known experimental data. We developed a two-step entrainment model to explain a broader range of results as well as provide more quantitative analysis. We prove existence and stability of periodic orbits and given analytical solutions of the range of entrainment, fit the phase trajectory over the entire entrainment process to data from a published study for 12 subjects in extended day cycles, i.e., longer than 24 h. Our simulations closely replicated the phase data and predicted correctly the phase of entrainment. We investigate the factors related to the rate of entrainment (ROE) and present the three-dimensional parameter spaces, illustrating the various behaviors of the phase of entrainment and ROE. Our findings can be applied to diagnostics and treatments for patients with sleep disorders caused by shift work or jet lag.
Similar content being viewed by others
References
Abraham U, Granada AE, Westermark PO, Heine M, Kramer A, Herzel H (2010) Coupling governs entrainment range of circadian clocks. Mol Syst Biol 6(1):438
Aschoff J (1981) Freerunning and entrained circadian rhythms. In: Aschoff J (ed) Biological rhythms, Springer, Berlin, pp 81–93
Aschoff J, Pohl H (1978) Phase relations between a circadian rhythm and its zeitgeber within the range of entrainment. Naturwissenschaften 65(2):80–84
Bordyugov G, Abraham U, Granada A, Rose P, Imkeller K, Kramer A, Herzel H (2015) Tuning the phase of circadian entrainment. J Royal Soc Interface 12(108):20150282
Darrah C, Taylor BL, Edwards KD, Brown PE, Hall A, McWatters HG (2006) Analysis of phase of luciferase expression reveals novel circadian quantitative trait loci in arabidopsis. Plant Physiol 140(4):1464–1474
Duffy JF, Czeisler CA (2002) Age-related change in the relationship between circadian period, circadian phase, and diurnal preference in humans. Neurosci Lett 318(3):117–120
Duffy JF, Dijk DJ, Klerman EB, Czeisler CA (1998) Later endogenous circadian temperature nadir relative to an earlier wake time in older people. Am J Physiol-Regul Integr Comp Physiol 275(5):R1478–R1487
Duffy JF, Cain SW, Chang AM, Phillips AJ, Münch MY, Gronfier C, Wyatt JK, Dijk DJ, Wright KP, Czeisler CA (2011) Sex difference in the near-24-hour intrinsic period of the human circadian timing system. Proc Natl Acad Sci 108(Supplement 3):15602–15608
Dunlap JC, Loros JJ (2017) Making time: conservation of biological clocks from fungi to animals. Microbiol Spect 5(3):515–534
Granada AE, Bordyugov G, Kramer A, Herzel H (2013) Human chronotypes from a theoretical perspective. PLoS One 8(3):e59464
Gronfier C, Wright KP, Kronauer RE, Czeisler CA (2007) Entrainment of the human circadian pacemaker to longer-than-24-h days. Proc Natl Acad Sci 104(21):9081–9086
Heintzen C, Liu Y (2007) The neurospora crassa circadian clock. Adv Genet 58:25–66
Hida A, Ohsawa Y, Kitamura S, Nakazaki K, Ayabe N, Motomura Y, Matsui K, Kobayashi M, Usui A, Inoue Y et al (2017) Evaluation of circadian phenotypes utilizing fibroblasts from patients with circadian rhythm sleep disorders. Trans Psychiatr 7(4):e1106
Hoffmann K (1963) Zur beziehung zwischen phasenlage und spontanfrequenz bei der endogenen tagesperiodik. Z für Naturforschung B 18(2):154–157
Jones CR, Campbell SS, Zone SE, Cooper F, DeSano A, Murphy PJ, Jones B, Czajkowski L, Ptček LJ (1999) Familial advanced sleep-phase syndrome: a short-period circadian rhythm variant in humans. Nat Med 5(9):1062–1065
Kandalepas PC, Mitchell JW, Gillette MU (2016) Melatonin signal transduction pathways require e-box-mediated transcription of per1 and per2 to reset the scn clock at dusk. PloS one 11(6):e0157824
Kim DW, Chang C, Chen X, Doran AC, Gaudreault F, Wager T, DeMarco GJ, Kim JK (2019) Systems approach reveals photosensitivity and per 2 level as determinants of clock-modulator efficacy. Mol Syst Biol 15(7):e8838
Kronauer RE, Czeisler CA, Pilato SF, Moore-Ede MC, Weitzman ED (1982) Mathematical model of the human circadian system with two interacting oscillators. Am J Physiol-Regul, Integr Comp Physiol 242(1):R3–R17
Kuramoto Y (1984) Cooperative dynamics of oscillator communitya study based on lattice of rings. Prog Theor Phys Suppl 79:223–240
Kurien P, Hsu PK, Leon J, Wu D, McMahon T, Shi G, Xu Y, Lipzen A, Pennacchio LA, Jones CR et al (2019) Timeless mutation alters phase responsiveness and causes advanced sleep phase. Proc Natl Acad Sci 116(24):12045–12053
Lee K, Shiva Kumar P, McQuade S, Lee JY, Park S, An Z, Piccoli B (2017) Experimental and mathematical analyses relating circadian period and phase of entrainment in neurospora crassa. J Biol Rhythm 32(6):550–559
Lewy AJ, Hasler BP, Emens JS, Sack RL (2001) Pretreatment circadian period in free-running blind people may predict the phase angle of entrainment to melatonin. Neurosci Lett 313(3):158–160
Loros JJ, Dunlap JC (2001) Genetic and molecular analysis of circadian rhythms in n eurospora. Ann Rev Physiol 63(1):757–794
Lowrey PL, Shimomura K, Antoch MP, Yamazaki S, Zemenides PD, Ralph MR, Menaker M, Takahashi JS (2000) Positional syntenic cloning and functional characterization of the mammalian circadian mutation tau. Science 288(5465):483–491
Michael TP, Salome PA, Hannah JY, Spencer TR, Sharp EL, McPeek MA, Alonso JM, Ecker JR, McClung CR (2003) Enhanced fitness conferred by naturally occurring variation in the circadian clock. Science 302(5647):1049–1053
Phillips AJ, Vidafar P, Burns AC, McGlashan EM, Anderson C, Rajaratnam SM, Lockley SW, Cain SW (2019) High sensitivity and interindividual variability in the response of the human circadian system to evening light. Proc Natl Acad Sci 116(24):12019–12024
Pittendrigh CS, Kyner WT, Takamura T (1991) The amplitude of circadian oscillations: temperature dependence, latitudinal clines, and the photoperiodic time measurement. J Biol Rhythm 6(4):299–313
Roenneberg T, Merrow M (2016) The circadian clock and human health. Current Biol 26(10):R432–R443
Roenneberg T, Daan S, Merrow M (2003) The art of entrainment. J Biol Rhythm 18(3):183–194
Roenneberg T, Hut R, Daan S, Merrow M (2010) Entrainment concepts revisited. J Biol Rhythm 25(5):329–339
Saunders D, Gillanders S, Lewis R (1994) Light-pulse phase response curves for the locomotor activity rhythm in period mutants of drosophila melanogaster. J Insect Physiol 40(11):957–968
Schmal C, Myung J, Herzel H, Bordyugov G (2015) A theoretical study on seasonality. Front Neurol 6:94
Serkh K, Forger DB (2014) Optimal schedules of light exposure for rapidly correcting circadian misalignment. PLoS Comput Biol 10(4):e1003523
Stelling J, Gilles ED, Doyle FJ (2004) Robustness properties of circadian clock architectures. Proc Natl Acad Sci 101(36):13210–13215
To TL, Henson MA, Herzog ED, Doyle FJ III (2007) A molecular model for intercellular synchronization in the mammalian circadian clock. Biophys J 92(11):3792–3803
Tokuda IT, Schmal C, Ananthasubramaniam B, Herzel H (2020) Conceptual models of entrainment, jet lag, and seasonality. Front Physiol 11:334
Wright KP, Hughes RJ, Kronauer RE, Dijk DJ, Czeisler CA (2001) Intrinsic near-24-h pacemaker period determines limits of circadian entrainment to a weak synchronizer in humans. Proc Natl Acad Sci 98(24):14027–14032
Xu Y, Padiath QS, Shapiro RE, Jones CR, Wu SC, Saigoh N, Saigoh K, Ptáček LJ, Fu YH (2005) Functional consequences of a cki\(\delta \) mutation causing familial advanced sleep phase syndrome. Nature 434(7033):640–644
Zehring WA, Wheeler DA, Reddy P, Konopka RJ, Kyriacou CP, Rosbash M, Hall JC (1984) P-element transformation with period locus dna restores rhythmicity to mutant, arrhythmic drosophila melanogaster. Cell 39(2):369–376
Acknowledgements
We gratefully acknowledge Dr. Hanspeter Herzel for his constructive advice and suggestions. We also thank Dr. Sean T. McQuade and Caleb Robelle for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
An, Z., Merrill, N.J., Lee, K. et al. A Two-Step Model of Human Entrainment: A Quantitative Study of Circadian Period and Phase of Entrainment. Bull Math Biol 83, 12 (2021). https://doi.org/10.1007/s11538-020-00829-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11538-020-00829-5