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del Olmo M, Schmal C, Mizaikoff C, Grabe S, Gabriel C, Kramer A, Herzel H. Are circadian amplitudes and periods correlated? A new twist in the story. F1000Res 2024; 12:1077. [PMID: 37771612 PMCID: PMC10526121 DOI: 10.12688/f1000research.135533.1] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Accepted: 02/14/2024] [Indexed: 09/30/2023] Open
Abstract
Three parameters are important to characterize a circadian and in general any biological clock: period, phase and amplitude. While circadian periods have been shown to correlate with entrainment phases, and clock amplitude influences the phase response of an oscillator to pulse-like zeitgeber signals, the co-modulations of amplitude and periods, which we term twist, have not been studied in detail. In this paper we define two concepts: parametric twist refers to amplitude-period correlations arising in ensembles of self-sustained, limit cycle clocks in the absence of external inputs, and phase space twist refers to the co-modulation of an individual clock's amplitude and period in response to external zeitgebers. Our findings show that twist influences the interaction of oscillators with the environment, facilitating entrainment, speeding upfastening recovery to pulse-like perturbations or modifying the response of an individual clock to coupling. This theoretical framework might be applied to understand the emerging properties of other oscillating systems.
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Affiliation(s)
- Marta del Olmo
- Institute for Theoretical Biology, Humboldt Universität zu Berlin and Charité Universitätsmedizin Berlin, Philippstr. 13, 10115 Berlin, Germany
| | - Christoph Schmal
- Institute for Theoretical Biology, Humboldt Universität zu Berlin and Charité Universitätsmedizin Berlin, Philippstr. 13, 10115 Berlin, Germany
| | - Camillo Mizaikoff
- Institute for Theoretical Biology, Humboldt Universität zu Berlin and Charité Universitätsmedizin Berlin, Philippstr. 13, 10115 Berlin, Germany
| | - Saskia Grabe
- Institute for Theoretical Biology, Humboldt Universität zu Berlin and Charité Universitätsmedizin Berlin, Philippstr. 13, 10115 Berlin, Germany
| | - Christian Gabriel
- Laboratory of Chronobiology, Institute for Medical Immunology, Charite Universitätsmedizin Berlin, Charitéplatz 1, 10117 Berlin, Germany
| | - Achim Kramer
- Laboratory of Chronobiology, Institute for Medical Immunology, Charite Universitätsmedizin Berlin, Charitéplatz 1, 10117 Berlin, Germany
| | - Hanspeter Herzel
- Institute for Theoretical Biology, Humboldt Universität zu Berlin and Charité Universitätsmedizin Berlin, Philippstr. 13, 10115 Berlin, Germany
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2
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Del Olmo M, Kalashnikov A, Schmal C, Kramer A, Herzel H. Coupling allows robust mammalian redox circadian rhythms despite heterogeneity and noise. Heliyon 2024; 10:e24773. [PMID: 38312577 PMCID: PMC10835301 DOI: 10.1016/j.heliyon.2024.e24773] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2023] [Revised: 12/06/2023] [Accepted: 01/14/2024] [Indexed: 02/06/2024] Open
Abstract
Circadian clocks are endogenous oscillators present in almost all cells that drive daily rhythms in physiology and behavior. There are two mechanisms that have been proposed to explain how circadian rhythms are generated in mammalian cells: through a transcription-translation feedback loop (TTFL) and based on oxidation/reduction reactions, both of which are intrinsically stochastic and heterogeneous at the single cell level. In order to explore the emerging properties of stochastic and heterogeneous redox oscillators, we simplify a recently developed kinetic model of redox oscillations to an amplitude-phase oscillator with 'twist' (period-amplitude correlation) and subject to Gaussian noise. We show that noise and heterogeneity alone lead to fast desynchronization, and that coupling between noisy oscillators can establish robust and synchronized rhythms with amplitude expansions and tuning of the period due to twist. Coupling a network of redox oscillators to a simple model of the TTFL also contributes to synchronization, large amplitudes and fine-tuning of the period for appropriate interaction strengths. These results provide insights into how the circadian clock compensates randomness from intracellular sources and highlight the importance of noise, heterogeneity and coupling in the context of circadian oscillators.
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Affiliation(s)
- Marta Del Olmo
- Institute for Theoretical Biology - Humboldt Universität zu Berlin and Charité Universitätsmedizin Berlin, Philippstraße 13, 10115 Berlin, Germany
| | - Anton Kalashnikov
- Institute for Theoretical Biology - Humboldt Universität zu Berlin and Charité Universitätsmedizin Berlin, Philippstraße 13, 10115 Berlin, Germany
| | - Christoph Schmal
- Institute for Theoretical Biology - Humboldt Universität zu Berlin and Charité Universitätsmedizin Berlin, Philippstraße 13, 10115 Berlin, Germany
| | - Achim Kramer
- Institute for Medical Immunology - Laboratory of Chronobiology, Charité Universitätsmedizin Berlin, Charitéplatz 1, 10117 Berlin, Germany
| | - Hanspeter Herzel
- Institute for Theoretical Biology - Humboldt Universität zu Berlin and Charité Universitätsmedizin Berlin, Philippstraße 13, 10115 Berlin, Germany
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3
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Motta FC, McGoff K, Moseley RC, Cho CY, Kelliher CM, Smith LM, Ortiz MS, Leman AR, Campione SA, Devos N, Chaorattanakawee S, Uthaimongkol N, Kuntawunginn W, Thongpiam C, Thamnurak C, Arsanok M, Wojnarski M, Vanchayangkul P, Boonyalai N, Smith PL, Spring MD, Jongsakul K, Chuang I, Harer J, Haase SB. The parasite intraerythrocytic cycle and human circadian cycle are coupled during malaria infection. Proc Natl Acad Sci U S A 2023; 120:e2216522120. [PMID: 37279274 PMCID: PMC10268210 DOI: 10.1073/pnas.2216522120] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2022] [Accepted: 05/09/2023] [Indexed: 06/08/2023] Open
Abstract
During infections with the malaria parasites Plasmodium vivax, patients exhibit rhythmic fevers every 48 h. These fever cycles correspond with the time the parasites take to traverse the intraerythrocytic cycle (IEC). In other Plasmodium species that infect either humans or mice, the IEC is likely guided by a parasite-intrinsic clock [Rijo-Ferreiraet al., Science 368, 746-753 (2020); Smith et al., Science 368, 754-759 (2020)], suggesting that intrinsic clock mechanisms may be a fundamental feature of malaria parasites. Moreover, because Plasmodium cycle times are multiples of 24 h, the IECs may be coordinated with the host circadian clock(s). Such coordination could explain the synchronization of the parasite population in the host and enable alignment of IEC and circadian cycle phases. We utilized an ex vivo culture of whole blood from patients infected with P. vivax to examine the dynamics of the host circadian transcriptome and the parasite IEC transcriptome. Transcriptome dynamics revealed that the phases of the host circadian cycle and the parasite IEC are correlated across multiple patients, showing that the cycles are phase coupled. In mouse model systems, host-parasite cycle coupling appears to provide a selective advantage for the parasite. Thus, understanding how host and parasite cycles are coupled in humans could enable antimalarial therapies that disrupt this coupling.
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Affiliation(s)
- Francis C. Motta
- Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL33431
| | - Kevin McGoff
- Department of Mathematics and Statistics, University of North Carolina, Charlotte, NC28223
| | | | - Chun-Yi Cho
- Department of Biochemistry and Biophysics, University of California, San Francisco, CA94143
| | - Christina M. Kelliher
- Department of Molecular & Systems Biology, Geisel School of Medicine at Dartmouth, Hanover, NH03755
| | | | | | | | | | | | - Suwanna Chaorattanakawee
- Department of Parasitology and Entomology, Faculty of Public Health, Mahidol University, Bangkok10400, Thailand
| | | | | | - Chadin Thongpiam
- US-Armed Forces Research Institute of Medical Sciences, Bangkok10400, Thailand
| | | | - Montri Arsanok
- US-Armed Forces Research Institute of Medical Sciences, Bangkok10400, Thailand
| | | | | | - Nonlawat Boonyalai
- US-Armed Forces Research Institute of Medical Sciences, Bangkok10400, Thailand
| | - Philip L. Smith
- U.S. Military HIV Research Program Walter Reed Army Institute of Research, Bethesda, MD20817
| | - Michele D. Spring
- US-Armed Forces Research Institute of Medical Sciences, Bangkok10400, Thailand
| | - Krisada Jongsakul
- US-Armed Forces Research Institute of Medical Sciences, Bangkok10400, Thailand
| | - Ilin Chuang
- US Naval Medical Research Center-Asia in Singapore, Assigned to Armed Forces Research Institute of Medical Sciences, Bangkok10400, Thailand
| | - John Harer
- Geometric Data Analytics, Durham, NC27701
| | - Steven B. Haase
- Department of Biology, Duke University, Durham, NC27708
- Department of Medicine Duke University, Durham, NC27710
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4
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Sanchez PGL, Mochulska V, Mauffette Denis C, Mönke G, Tomita T, Tsuchida-Straeten N, Petersen Y, Sonnen K, François P, Aulehla A. Arnold tongue entrainment reveals dynamical principles of the embryonic segmentation clock. eLife 2022; 11:79575. [PMID: 36223168 PMCID: PMC9560162 DOI: 10.7554/elife.79575] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2022] [Accepted: 08/23/2022] [Indexed: 11/13/2022] Open
Abstract
Living systems exhibit an unmatched complexity, due to countless, entangled interactions across scales. Here, we aim to understand a complex system, that is, segmentation timing in mouse embryos, without a reference to these detailed interactions. To this end, we develop a coarse-grained approach, in which theory guides the experimental identification of the segmentation clock entrainment responses. We demonstrate period- and phase-locking of the segmentation clock across a wide range of entrainment parameters, including higher-order coupling. These quantifications allow to derive the phase response curve (PRC) and Arnold tongues of the segmentation clock, revealing its essential dynamical properties. Our results indicate that the somite segmentation clock has characteristics reminiscent of a highly non-linear oscillator close to an infinite period bifurcation and suggests the presence of long-term feedbacks. Combined, this coarse-grained theoretical-experimental approach reveals how we can derive simple, essential features of a highly complex dynamical system, providing precise experimental control over the pace and rhythm of the somite segmentation clock.
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Affiliation(s)
| | | | | | - Gregor Mönke
- European Molecular Biology Laboratory (EMBL), Developmental Biology Unit
| | - Takehito Tomita
- European Molecular Biology Laboratory (EMBL), Developmental Biology Unit
| | | | - Yvonne Petersen
- European Molecular Biology Laboratory (EMBL), Transgenic Service
| | - Katharina Sonnen
- European Molecular Biology Laboratory (EMBL), Developmental Biology Unit
| | | | - Alexander Aulehla
- European Molecular Biology Laboratory (EMBL), Developmental Biology Unit
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5
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Liao G, Bose A. Entrainment within hierarchical circadian oscillator networks. Math Biosci 2022; 351:108883. [PMID: 35907509 DOI: 10.1016/j.mbs.2022.108883] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2021] [Revised: 07/15/2022] [Accepted: 07/18/2022] [Indexed: 11/24/2022]
Abstract
Circadian rhythms are endogenous oscillations, widely found across biological species, that have the capability of entraining to the 24-h light-dark cycle. Circadian systems often consist of both central oscillators that receive direct light-dark input and peripheral oscillators that receive input from the central oscillators. In this paper, we address questions related to what governs the time to and pattern of entrainment of these hierarchical circadian systems after an abrupt switch in the light-dark phasing. For a network consisting of a single central oscillator coupled to a chain of N feed-forward peripheral oscillators, we introduce a systematic way to derive an N-dimensional entrainment map whose fixed points correspond to entrained solutions. Using the map, we explain that the direction of reentrainment can involve fairly complicated phase advancing and delaying behavior as well as reentrainment times that depend sensitively on the nature of the perturbation. We also study the dynamics of a hierarchical system in which the peripheral oscillators are mutually coupled. We study how reentrainment times vary as a function of the degree to which the oscillators are desynchronized at the time of the change in light-dark phasing. We show that desynchronizing the peripheral oscillators can, in some circumstances, speed up their ultimate reentrainment following perturbations.
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Affiliation(s)
- Guangyuan Liao
- Key Laboratory of Intelligent Analysis and Decision on Complex Systems, School of Science, Chongqing University of Posts and Telecommunications, Chongwen Road, Nan'an, 400065, Chongqing, China
| | - Amitabha Bose
- Department of Mathematical Sciences, NJIT, Newark, NJ, 07102, USA.
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6
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Khan E, Saghafi S, Diekman CO, Rotstein HG. The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems. CHAOS (WOODBURY, N.Y.) 2022; 32:063137. [PMID: 35778129 DOI: 10.1063/5.0079198] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2021] [Accepted: 04/04/2022] [Indexed: 06/15/2023]
Abstract
Several distinct entrainment patterns can occur in the FitzHugh-Nagumo (FHN) model under external periodic forcing. Investigating the FHN model under different types of periodic forcing reveals the existence of multiple disconnected 1:1 entrainment segments for constant, low enough values of the input amplitude when the unforced system is in the vicinity of a Hopf bifurcation. This entrainment structure is termed polyglot to distinguish it from the single 1:1 entrainment region (monoglot) structure typically observed in Arnold tongue diagrams. The emergence of polyglot entrainment is then explained using phase-plane analysis and other dynamical system tools. Entrainment results are investigated for other slow-fast systems of neuronal, circadian, and glycolytic oscillations. Exploring these models, we found that polyglot entrainment structure (multiple 1:1 regions) is observed when the unforced system is in the vicinity of a Hopf bifurcation and the Hopf point is located near a knee of a cubic-like nullcline.
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Affiliation(s)
- Emel Khan
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
| | - Soheil Saghafi
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
| | - Casey O Diekman
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
| | - Horacio G Rotstein
- Federated Department of Biological Sciences, New Jersey Institute of Technology & Rutgers University, Newark, New Jersey 07102, USA
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7
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Jiménez A, Lu Y, Jambhekar A, Lahav G. Principles, mechanisms and functions of entrainment in biological oscillators. Interface Focus 2022; 12:20210088. [PMID: 35450280 PMCID: PMC9010850 DOI: 10.1098/rsfs.2021.0088] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2021] [Accepted: 03/07/2022] [Indexed: 12/12/2022] Open
Abstract
Entrainment is a phenomenon in which two oscillators interact with each other, typically through physical or chemical means, to synchronize their oscillations. This phenomenon occurs in biology to coordinate processes from the molecular to organismal scale. Biological oscillators can be entrained within a single cell, between cells or to an external input. Using six illustrative examples of entrainable biological oscillators, we discuss the distinctions between entrainment and synchrony and explore features that contribute to a system's propensity to entrain. Entrainment can either enhance or reduce the heterogeneity of oscillations within a cell population, and we provide examples and mechanisms of each case. Finally, we discuss the known functions of entrainment and discuss potential functions from an evolutionary perspective.
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Affiliation(s)
- Alba Jiménez
- Department of Systems Biology, Blavatnik Institute at Harvard Medical School, Boston, MA 02115, USA
| | - Ying Lu
- Department of Systems Biology, Blavatnik Institute at Harvard Medical School, Boston, MA 02115, USA
| | - Ashwini Jambhekar
- Department of Systems Biology, Blavatnik Institute at Harvard Medical School, Boston, MA 02115, USA
- Ludwig Center at Harvard, Boston, MA 02115, USA
| | - Galit Lahav
- Department of Systems Biology, Blavatnik Institute at Harvard Medical School, Boston, MA 02115, USA
- Ludwig Center at Harvard, Boston, MA 02115, USA
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8
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Systematic modeling-driven experiments identify distinct molecular clockworks underlying hierarchically organized pacemaker neurons. Proc Natl Acad Sci U S A 2022; 119:2113403119. [PMID: 35193959 PMCID: PMC8872709 DOI: 10.1073/pnas.2113403119] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 01/18/2022] [Indexed: 12/11/2022] Open
Abstract
In metazoan organisms, circadian (∼24 h) rhythms are regulated by pacemaker neurons organized in a master-slave hierarchy. Although it is widely accepted that master pacemakers and slave oscillators generate rhythms via an identical negative feedback loop of transcription factor CLOCK (CLK) and repressor PERIOD (PER), their different roles imply heterogeneity in their molecular clockworks. Indeed, in Drosophila, defective binding between CLK and PER disrupts molecular rhythms in the master pacemakers, small ventral lateral neurons (sLNvs), but not in the slave oscillator, posterior dorsal neuron 1s (DN1ps). Here, we develop a systematic and expandable approach that unbiasedly searches the source of the heterogeneity in molecular clockworks from time-series data. In combination with in vivo experiments, we find that sLNvs exhibit higher synthesis and turnover of PER and lower CLK levels than DN1ps. Importantly, light shift analysis reveals that due to such a distinct molecular clockwork, sLNvs can obtain paradoxical characteristics as the master pacemaker, generating strong rhythms that are also flexibly adjustable to environmental changes. Our results identify the different characteristics of molecular clockworks of pacemaker neurons that underlie hierarchical multi-oscillator structure to ensure the rhythmic fitness of the organism.
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9
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Abstract
Parameter estimation from observable or experimental data is a crucial stage in any modeling study. Identifiability refers to one’s ability to uniquely estimate the model parameters from the available data. Structural unidentifiability in dynamic models, the opposite of identifiability, is associated with the notion of degeneracy where multiple parameter sets produce the same pattern. Therefore, the inverse function of determining the model parameters from the data is not well defined. Degeneracy is not only a mathematical property of models, but it has also been reported in biological experiments. Classical studies on structural unidentifiability focused on the notion that one can at most identify combinations of unidentifiable model parameters. We have identified a different type of structural degeneracy/unidentifiability present in a family of models, which we refer to as the Lambda-Omega (Λ-Ω) models. These are an extension of the classical lambda-omega (λ-ω) models that have been used to model biological systems, and display a richer dynamic behavior and waveforms that range from sinusoidal to square wave to spike like. We show that the Λ-Ω models feature infinitely many parameter sets that produce identical stable oscillations, except possible for a phase shift (reflecting the initial phase). These degenerate parameters are not identifiable combinations of unidentifiable parameters as is the case in structural degeneracy. In fact, reducing the number of model parameters in the Λ-Ω models is minimal in the sense that each one controls a different aspect of the model dynamics and the dynamic complexity of the system would be reduced by reducing the number of parameters. We argue that the family of Λ-Ω models serves as a framework for the systematic investigation of degeneracy and identifiability in dynamic models and for the investigation of the interplay between structural and other forms of unidentifiability resulting on the lack of information from the experimental/observational data.
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10
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Heltberg ML, Krishna S, Kadanoff LP, Jensen MH. A tale of two rhythms: Locked clocks and chaos in biology. Cell Syst 2021; 12:291-303. [PMID: 33887201 DOI: 10.1016/j.cels.2021.03.003] [Citation(s) in RCA: 20] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2020] [Revised: 01/19/2021] [Accepted: 03/17/2021] [Indexed: 12/16/2022]
Abstract
The fundamental mechanisms that control and regulate biological organisms exhibit a surprising level of complexity. Oscillators are perhaps the simplest motifs that produce time-varying dynamics and are ubiquitous in biological systems. It is also known that such biological oscillators interact with each other-for instance, circadian oscillators affect the cell cycle, and somitogenesis clock proteins in adjacent cells affect each other in developing embryos. Therefore, it is vital to understand the effects that can emerge from non-linear interaction between oscillations. Here, we show how oscillations typically arise in biology and take the reader on a tour through the great variety in dynamics that can emerge even from a single pair of coupled oscillators. We explain how chaotic dynamics can emerge and outline the methods of detecting this in experimental time traces. Finally, we discuss the potential role of such complex dynamical features in biological systems.
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Affiliation(s)
- Mathias L Heltberg
- Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark; Laboratoire de Physique Théorique, Ecole Normale Supérieure, 75 231 Paris Cedex 05, France
| | - Sandeep Krishna
- Simons Centre for the Study of Living Machines, National Centre for Biological Sciences TIFR, GKVK Campus, Bellary Road, Bangalore 560065, India
| | - Leo P Kadanoff
- The James Franck Institute, The University of Chicago, Chicago, IL 60637, USA
| | - Mogens H Jensen
- Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark; The James Franck Institute, The University of Chicago, Chicago, IL 60637, USA.
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11
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Androulakis IP. Circadian rhythms and the HPA axis: A systems view. WIREs Mech Dis 2021; 13:e1518. [PMID: 33438348 DOI: 10.1002/wsbm.1518] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/10/2020] [Revised: 11/20/2020] [Accepted: 11/30/2020] [Indexed: 12/26/2022]
Abstract
The circadian timing system comprises a network of time-keeping clocks distributed across a living host whose responsibility is to allocate resources and distribute functions temporally to optimize fitness. The molecular structures generating these rhythms have evolved to accommodate the rotation of the earth in an attempt to primarily match the light/dark periods during the 24-hr day. To maintain synchrony of timing across and within tissues, information from the central clock, located in the suprachiasmatic nucleus, is conveyed using systemic signals. Leading among those signals are endocrine hormones, and while the hypothalamic-pituitary-adrenal axis through the release of glucocorticoids is a major pacesetter. Interestingly, the fundamental units at the molecular and physiological scales that generate local and systemic signals share critical structural properties. These properties enable time-keeping systems to generate rhythmic signals and allow them to adopt specific properties as they interact with each other and the external environment. The purpose of this review is to provide a broad overview of these structures, discuss their functional characteristics, and describe some of their fundamental properties as these related to health and disease. This article is categorized under: Immune System Diseases > Computational Models Immune System Diseases > Biomedical Engineering.
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Affiliation(s)
- Ioannis P Androulakis
- Biomedical Engineering Department, Chemical & Biochemical Engineering Department, Rutgers University, New Brunswick, New Jersey.,Department of Surgery, Rutgers-Robert Wood Johnson Medical School, New Brunswick, New Jersey, USA
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12
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Ananthasubramaniam B, Schmal C, Herzel H. Amplitude Effects Allow Short Jet Lags and Large Seasonal Phase Shifts in Minimal Clock Models. J Mol Biol 2020; 432:3722-3737. [PMID: 31978397 DOI: 10.1016/j.jmb.2020.01.014] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2019] [Revised: 01/10/2020] [Accepted: 01/10/2020] [Indexed: 01/24/2023]
Abstract
Mathematical models of varying complexity have helped shed light on different aspects of circadian clock function. In this work, we question whether minimal clock models (Goodwin models) are sufficient to reproduce essential phenotypes of the clock: a small phase response curve (PRC), fast jet lag, and seasonal phase shifts. Instead of building a single best model, we take an approach where we study the properties of a set of models satisfying certain constraints; here, a 1h-pulse PRC with a range of 3h and clock periods between 22h and 26h is designed. Surprisingly, almost all these randomly parameterized models showed a 4h change in phase of entrainment between long and short days and jet lag durations of three to seven days in advance and delay. Moreover, intrinsic clock period influenced jet lag duration and entrainment amplitude and phase. Fast jet lag was realized in this model by means of an interesting amplitude effect: the association between clock amplitude and clock period termed "twist." This twist allows amplitude changes to speed up and slow down clocks enabling faster shifts. These findings were robust to the addition of positive feedback to the model. In summary, the known design principles of rhythm generation - negative feedback, long delay, and switch-like inhibition (we review these in detail) - are sufficient to reproduce the essential clock phenotypes. Furthermore, amplitudes play a role in determining clock properties and must be always considered, although they are difficult to measure.
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Affiliation(s)
| | - Christoph Schmal
- Institute for Theoretical Biology, Humboldt Universität zu Berlin, 10115 Berlin, Germany
| | - Hanspeter Herzel
- Institute for Theoretical Biology, Charité Universitätsmedizin Berlin, 10115 Berlin, Germany
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13
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Droin C, Paquet ER, Naef F. Low-dimensional Dynamics of Two Coupled Biological Oscillators. NATURE PHYSICS 2019; 15:1086-1094. [PMID: 32528550 PMCID: PMC7289635 DOI: 10.1038/s41567-019-0598-1] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/05/2019] [Accepted: 06/18/2019] [Indexed: 06/11/2023]
Abstract
The circadian clock and the cell cycle are two biological oscillatory processes that coexist within individual cells. These two oscillators were found to interact, which can lead to their synchronization. Here, we develop a method to identify a low-dimensional stochastic model of the coupled system directly from time-lapse imaging in single cells. In particular, we infer the coupling and non-linear dynamics of the two oscillators from thousands of mouse and human single-cell fluorescence microscopy traces. This coupling predicts multiple phase-locked states showing different degrees of robustness against molecular fluctuations inherent to cellular-scale biological oscillators. For the 1:1 state, the predicted phase-shifts upon period perturbations were validated experimentally. Moreover, the phase-locked states are temperature-independent and evolutionarily conserved from mouse to human, hinting at a common underlying dynamical mechanism. Finally, we detect a signature of the coupled dynamics in a physiological context, explaining why tissues with different proliferation states exhibited shifted circadian clock phases.
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14
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Kim JK. Protein sequestration versus Hill-type repression in circadian clock models. IET Syst Biol 2018; 10:125-35. [PMID: 27444022 DOI: 10.1049/iet-syb.2015.0090] [Citation(s) in RCA: 59] [Impact Index Per Article: 9.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
Abstract
Circadian (∼24 h) clocks are self-sustained endogenous oscillators with which organisms keep track of daily and seasonal time. Circadian clocks frequently rely on interlocked transcriptional-translational feedback loops to generate rhythms that are robust against intrinsic and extrinsic perturbations. To investigate the dynamics and mechanisms of the intracellular feedback loops in circadian clocks, a number of mathematical models have been developed. The majority of the models use Hill functions to describe transcriptional repression in a way that is similar to the Goodwin model. Recently, a new class of models with protein sequestration-based repression has been introduced. Here, the author discusses how this new class of models differs dramatically from those based on Hill-type repression in several fundamental aspects: conditions for rhythm generation, robust network designs and the periods of coupled oscillators. Consistently, these fundamental properties of circadian clocks also differ among Neurospora, Drosophila, and mammals depending on their key transcriptional repression mechanisms (Hill-type repression or protein sequestration). Based on both theoretical and experimental studies, this review highlights the importance of careful modelling of transcriptional repression mechanisms in molecular circadian clocks.
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Affiliation(s)
- Jae Kyoung Kim
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro Yuseong-gu, Daejeon, 34141, Korea.
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15
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Tiana-Alsina J, Quintero-Quiroz C, Panozzo M, Torrent MC, Masoller C. Experimental study of modulation waveforms for entraining the spikes emitted by a semiconductor laser with optical feedback. OPTICS EXPRESS 2018; 26:9298-9309. [PMID: 29715883 DOI: 10.1364/oe.26.009298] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2018] [Accepted: 03/26/2018] [Indexed: 06/08/2023]
Abstract
The entrainment phenomenon, by which an oscillator adjusts its natural rhythm to an external periodic signal, has been observed in many natural systems. Recently, attention has focused on which are the optimal conditions for achieving entrainment. Here we use a semiconductor laser with optical feedback, operating in the low-frequency fluctuations (LFFs) regime, as a testbed for a controlled entrainment experiment. In the LFF regime the laser intensity displays abrupt spikes, which can be entrained to a weak periodic signal that directly modulates the laser pump current. We compare the performance of three modulation waveforms for producing 1:1 locking (one spike is emitted in each modulation cycle), as well as higher order locking regimes. We characterize the parameter regions where high-quality locking occurs, and those where the laser emits spikes which are not entrained to the external signal. The role of the modulation amplitude and frequency, and the role of the dc value of the laser pump current (that controls the natural spike frequency) in the entrainment quality are discussed.
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16
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Abstract
Circadian oscillators found across a variety of species are subject to periodic external light-dark forcing. Entrainment to light-dark cycles enables the circadian system to align biological functions with appropriate times of day or night. Phase response curves (PRCs) have been used for decades to gain valuable insights into entrainment; however, PRCs may not accurately describe entrainment to photoperiods with substantial amounts of both light and dark due to their reliance on a single limit cycle attractor. We have developed a new tool, called an entrainment map, that overcomes this limitation of PRCs and can assess whether, and at what phase, a circadian oscillator entrains to external forcing with any photoperiod. This is a 1-dimensional map that we construct for 3 different mathematical models of circadian clocks. Using the map, we are able to determine conditions for existence and stability of phase-locked solutions. In addition, we consider the dependence on various parameters such as the photoperiod and intensity of the external light as well as the mismatch in intrinsic oscillator frequency with the light-dark cycle. We show that the entrainment map yields more accurate predictions for phase locking than methods based on the PRC. The map is also ideally suited to calculate the amount of time required to achieve entrainment as a function of initial conditions and the bifurcations of stable and unstable periodic solutions that lead to loss of entrainment.
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Affiliation(s)
- Casey O. Diekman
- Department of Mathematical Sciences, New Jersey Institute of
Technology, Newark, New Jersey
| | - Amitabha Bose
- Department of Mathematical Sciences, New Jersey Institute of
Technology, Newark, New Jersey
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17
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Schmal C, Myung J, Herzel H, Bordyugov G. A theoretical study on seasonality. Front Neurol 2015; 6:94. [PMID: 25999912 PMCID: PMC4423511 DOI: 10.3389/fneur.2015.00094] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2015] [Accepted: 04/16/2015] [Indexed: 11/20/2022] Open
Abstract
In addition to being endogenous, a circadian system must be able to communicate with the outside world and align its rhythmicity to the environment. As a result of such alignment, external Zeitgebers can entrain the circadian system. Entrainment expresses itself in coinciding periods of the circadian oscillator and the Zeitgeber and a stationary phase difference between them. The range of period mismatches between the circadian system and the Zeitgeber that Zeitgeber can overcome to entrain the oscillator is called an entrainment range. The width of the entrainment range usually increases with increasing Zeitgeber strength, resulting in a wedge-like Arnold tongue. This classical view of entrainment does not account for the effects of photoperiod on entrainment. Zeitgebers with extremely small or large photoperiods are intuitively closer to constant environments than equinoctial Zeitgebers and hence are expected to produce a narrower entrainment range. In this paper, we present theoretical results on entrainment under different photoperiods. We find that in the photoperiod-detuning parameter plane, the entrainment zone is shaped in the form of a skewed onion. The bottom and upper points of the onion are given by the free-running periods in DD and LL, respectively. The widest entrainment range is found near photoperiods of 50%. Within the onion, we calculated the entrainment phase that varies over a range of 12 h. The results of our theoretical study explain the experimentally observed behavior of the entrainment phase in dependence on the photoperiod.
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Affiliation(s)
- Christoph Schmal
- Institute for Theoretical Biology, Charité Universitätsmedizin, Berlin, Germany
| | | | - Hanspeter Herzel
- Institute for Theoretical Biology, Humboldt Universität zu Berlin, Berlin, Germany
| | - Grigory Bordyugov
- Institute for Theoretical Biology, Charité Universitätsmedizin, Berlin, Germany
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18
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Velocity response curves demonstrate the complexity of modeling entrainable clocks. J Theor Biol 2014; 363:307-17. [PMID: 25193284 DOI: 10.1016/j.jtbi.2014.08.044] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/09/2014] [Revised: 08/20/2014] [Accepted: 08/26/2014] [Indexed: 12/22/2022]
Abstract
Circadian clocks are biological oscillators that regulate daily behaviors in organisms across the kingdoms of life. Their rhythms are generated by complex systems, generally involving interlocked regulatory feedback loops. These rhythms are entrained by the daily light/dark cycle, ensuring that the internal clock time is coordinated with the environment. Mathematical models play an important role in understanding how the components work together to function as a clock which can be entrained by light. For a clock to entrain, it must be possible for it to be sped up or slowed down at appropriate times. To understand how biophysical processes affect the speed of the clock, one can compute velocity response curves (VRCs). Here, in a case study involving the fruit fly clock, we demonstrate that VRC analysis provides insight into a clock׳s response to light. We also show that biochemical mechanisms and parameters together determine a model׳s ability to respond realistically to light. The implication is that, if one is developing a model and its current form has an unrealistic response to light, then one must reexamine one׳s model structure, because searching for better parameter values is unlikely to lead to a realistic response to light.
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19
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Tiedemann HB, Schneltzer E, Zeiser S, Wurst W, Beckers J, Przemeck GKH, Hrabě de Angelis M. Fast synchronization of ultradian oscillators controlled by delta-notch signaling with cis-inhibition. PLoS Comput Biol 2014; 10:e1003843. [PMID: 25275459 PMCID: PMC4196275 DOI: 10.1371/journal.pcbi.1003843] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2014] [Accepted: 08/03/2014] [Indexed: 01/09/2023] Open
Abstract
While it is known that a large fraction of vertebrate genes are under the control of a gene regulatory network (GRN) forming a clock with circadian periodicity, shorter period oscillatory genes like the Hairy-enhancer-of split (Hes) genes are discussed mostly in connection with the embryonic process of somitogenesis. They form the core of the somitogenesis-clock, which orchestrates the periodic separation of somites from the presomitic mesoderm (PSM). The formation of sharp boundaries between the blocks of many cells works only when the oscillators in the cells forming the boundary are synchronized. It has been shown experimentally that Delta-Notch (D/N) signaling is responsible for this synchronization. This process has to happen rather fast as a cell experiences at most five oscillations from its 'birth' to its incorporation into a somite. Computer simulations describing synchronized oscillators with classical modes of D/N-interaction have difficulties to achieve synchronization in an appropriate time. One approach to solving this problem of modeling fast synchronization in the PSM was the consideration of cell movements. Here we show that fast synchronization of Hes-type oscillators can be achieved without cell movements by including D/N cis-inhibition, wherein the mutual interaction of DELTA and NOTCH in the same cell leads to a titration of ligand against receptor so that only one sort of molecule prevails. Consequently, the symmetry between sender and receiver is partially broken and one cell becomes preferentially sender or receiver at a given moment, which leads to faster entrainment of oscillators. Although not yet confirmed by experiment, the proposed mechanism of enhanced synchronization of mesenchymal cells in the PSM would be a new distinct developmental mechanism employing D/N cis-inhibition. Consequently, the way in which Delta-Notch signaling was modeled so far should be carefully reconsidered.
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Affiliation(s)
- Hendrik B. Tiedemann
- Institute of Experimental Genetics, Helmholtz Zentrum München - German Research Center for Environmental Health, Neuherberg, Germany
| | - Elida Schneltzer
- Institute of Experimental Genetics, Helmholtz Zentrum München - German Research Center for Environmental Health, Neuherberg, Germany
| | | | - Wolfgang Wurst
- Institute of Developmental Genetics, Helmholtz Zentrum München - German Research Center for Environmental Health, Neuherberg, Germany
- Technische Universität München, Center of Life and Food Sciences Weihenstephan, Chair of Developmental Genetics, Freising, Germany
| | - Johannes Beckers
- Institute of Experimental Genetics, Helmholtz Zentrum München - German Research Center for Environmental Health, Neuherberg, Germany
- Technische Universität München, Center of Life and Food Sciences Weihenstephan, Chair of Experimental Genetics, Freising, Germany
| | - Gerhard K. H. Przemeck
- Institute of Experimental Genetics, Helmholtz Zentrum München - German Research Center for Environmental Health, Neuherberg, Germany
| | - Martin Hrabě de Angelis
- Institute of Experimental Genetics, Helmholtz Zentrum München - German Research Center for Environmental Health, Neuherberg, Germany
- Technische Universität München, Center of Life and Food Sciences Weihenstephan, Chair of Experimental Genetics, Freising, Germany
- * E-mail:
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20
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Abstract
The circadian timekeeping system appears more complex in birds than in mammals. In mammals, the main pacemaker is centralized in the suprachiasmatic nuclei, whereas in birds, the pacemaker involves the interplay between the pineal and hypothalamic oscillators. In order to investigate the consequence of this complex mechanism, we propose here a mathematical model for the bird circadian clock. The model is based on the internal resonance between the pineal and hypothalamic oscillators, each described by Goodwin-like equations. We show that, consistently with experimental observations, self-sustained oscillations can be generated by mutual inhibitory coupling of the 2 clocks, even if individual oscillators present damped oscillations. We study the effect of constant and periodic administrations of melatonin, which, in intact birds, acts as the coupling variable between the pineal and the hypothalamus, and compare the prediction of the model with the experiments performed in pinealectomized birds. We also assess the entrainment properties when the system is subject to light-dark cycles. Analyses of the entrainment range, resynchronization time after jet lag, and entrainment phase with respect to the photoperiod lead us to formulate hypotheses about the physiological advantage of the particular architecture of the avian circadian clock. Although minimal, our model opens promising perspectives in modeling and understanding the bird circadian clock.
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Affiliation(s)
- Aurore Woller
- Unité de Chronobiologie Théorique, CP 231, Faculté des Sciences, Université Libre de Bruxelles, Bruxelles, Belgium
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21
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Woller A, Gonze D, Erneux T. The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment. Phys Biol 2014; 11:045002. [PMID: 25075916 DOI: 10.1088/1478-3975/11/4/045002] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Abstract
The three-variable Goodwin oscillator is a minimal model demonstrating the emergence of oscillations in simple biochemical feedback systems. As a prototypical oscillator, this model was extensively studied from a theoretical point of view and applied to various biological systems, including circadian clocks. Here, we reexamine this model, derive analytically the amplitude equation near the Hopf bifurcation and investigate the effect of a periodic modulation of the oscillator. In particular, we compare the entrainment performance when the free oscillator displays either self-sustained or damped oscillations. We discuss the results in the context of circadian oscillators.
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Affiliation(s)
- Aurore Woller
- Unité Mixte de Recherche 1011, Université Lille 2, INSERM, Institut Pasteur de Lille, Lille, France. Laboratoire de Physique des Lasers, Atomes, Molécules, Unité Mixte de Recherche 8523, Université Lille 1, CNRS, Villeneuve d'Ascq, France
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22
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Novel recurrent neural network for modelling biological networks: Oscillatory p53 interaction dynamics. Biosystems 2013; 114:191-205. [DOI: 10.1016/j.biosystems.2013.08.004] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2012] [Revised: 08/07/2013] [Accepted: 08/28/2013] [Indexed: 12/12/2022]
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23
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Zlotnik A, Chen Y, Kiss IZ, Tanaka HA, Li JS. Optimal waveform for fast entrainment of weakly forced nonlinear oscillators. PHYSICAL REVIEW LETTERS 2013; 111:024102. [PMID: 23889405 DOI: 10.1103/physrevlett.111.024102] [Citation(s) in RCA: 42] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2013] [Revised: 04/02/2013] [Indexed: 05/18/2023]
Abstract
For many biological and engineered systems, a central function or design goal is to abbreviate the time required to synchronize a rhythmic process to an external forcing signal. We present a theory for deriving the input that effectively minimizes the average transient time required to entrain a phase model, which enables a practical technique for constructing fast entrainment waveforms for general nonlinear oscillators. This result is verified in numerical simulations using the Hodgkin-Huxley neuron model, and in experiments on an oscillatory electrochemical system.
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Affiliation(s)
- Anatoly Zlotnik
- Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, Missouri 63130, USA.
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24
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Schmal C, Reimann P, Staiger D. A circadian clock-regulated toggle switch explains AtGRP7 and AtGRP8 oscillations in Arabidopsis thaliana. PLoS Comput Biol 2013; 9:e1002986. [PMID: 23555221 PMCID: PMC3610657 DOI: 10.1371/journal.pcbi.1002986] [Citation(s) in RCA: 51] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2012] [Accepted: 01/29/2013] [Indexed: 12/25/2022] Open
Abstract
The circadian clock controls many physiological processes in higher plants and causes a large fraction of the genome to be expressed with a 24h rhythm. The transcripts encoding the RNA-binding proteins AtGRP7 (Arabidopsis thaliana Glycine Rich Protein 7) and AtGRP8 oscillate with evening peaks. The circadian clock components CCA1 and LHY negatively affect AtGRP7 expression at the level of transcription. AtGRP7 and AtGRP8, in turn, negatively auto-regulate and reciprocally cross-regulate post-transcriptionally: high protein levels promote the generation of an alternative splice form that is rapidly degraded. This clock-regulated feedback loop has been proposed to act as a molecular slave oscillator in clock output. While mathematical models describing the circadian core oscillator in Arabidopsis thaliana were introduced recently, we propose here the first model of a circadian slave oscillator. We define the slave oscillator in terms of ordinary differential equations and identify the model's parameters by an optimization procedure based on experimental results. The model successfully reproduces the pertinent experimental findings such as waveforms, phases, and half-lives of the time-dependent concentrations. Furthermore, we obtain insights into possible mechanisms underlying the observed experimental dynamics: the negative auto-regulation and reciprocal cross-regulation via alternative splicing could be responsible for the sharply peaking waveforms of the AtGRP7 and AtGRP8 mRNA. Moreover, our results suggest that the AtGRP8 transcript oscillations are subordinated to those of AtGRP7 due to a higher impact of AtGRP7 protein on alternative splicing of its own and of the AtGRP8 pre-mRNA compared to the impact of AtGRP8 protein. Importantly, a bifurcation analysis provides theoretical evidence that the slave oscillator could be a toggle switch, arising from the reciprocal cross-regulation at the post-transcriptional level. In view of this, transcriptional repression of AtGRP7 and AtGRP8 by LHY and CCA1 induces oscillations of the toggle switch, leading to the observed high-amplitude oscillations of AtGRP7 mRNA. The circadian clock organizes the day in the life of a plant by causing 24h rhythms in gene expression. For example, the core clockwork of the model plant Arabidopsis thaliana causes the transcripts encoding the RNA-binding proteins AtGRP7 and AtGRP8 to undergo high amplitude oscillations with a peak at the end of the day. AtGRP7 and AtGRP8, in turn, negatively auto-regulate and reciprocally cross-regulate their own expression by causing alternative splicing of their pre-mRNAs, followed by rapid degradation of the alternatively spliced transcripts. This has led to the suggestion that they represent molecular slave oscillators downstream of the core clock. Using a mathematical model we obtain insights into possible mechanisms underlying the experimentally observed dynamics, e.g. a higher impact of AtGRP7 protein compared to the impact of AtGRP8 protein on the alternative splicing explains the experimentally observed phases of their transcript. Previously, components that reciprocally repress their own transcription (double negative loops) have been shown to potentially act as a toggle switch between two states. We provide theoretical evidence that the slave oscillator could be a bistable toggle switch as well, operating at the post-transcriptional level.
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Affiliation(s)
- Christoph Schmal
- Condensed Matter Theory, Faculty of Physics, Bielefeld University, Bielefeld, Germany.
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25
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Granada AE, Bordyugov G, Kramer A, Herzel H. Human chronotypes from a theoretical perspective. PLoS One 2013; 8:e59464. [PMID: 23544070 PMCID: PMC3609763 DOI: 10.1371/journal.pone.0059464] [Citation(s) in RCA: 76] [Impact Index Per Article: 6.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2012] [Accepted: 02/14/2013] [Indexed: 11/18/2022] Open
Abstract
The endogenous circadian timing system has evolved to synchronize an organism to periodically recurring environmental conditions. Those external time cues are called Zeitgebers. When entrained by a Zeitgeber, the intrinsic oscillator adopts a fixed phase relation to the Zeitgeber. Here, we systematically study how the phase of entrainment depends on clock and Zeitgeber properties. We combine numerical simulations of amplitude-phase models with predictions from analytically tractable models. In this way we derive relations between the phase of entrainment to the mismatch between the endogenous and Zeitgeber period, the Zeitgeber strength, and the range of entrainment. A core result is the “180° rule” asserting that the phase varies over a range of about 180° within the entrainment range. The 180° rule implies that clocks with a narrow entrainment range (“strong oscillators”) exhibit quite flexible entrainment phases. We argue that this high sensitivity of the entrainment phase contributes to the wide range of human chronotypes.
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Affiliation(s)
- Adrián E. Granada
- Institute for Theoretical Biology, Humboldt University, Berlin, Germany
| | - Grigory Bordyugov
- Institute for Theoretical Biology, Humboldt University, Berlin, Germany
| | - Achim Kramer
- Laboratory of Chronobiology, Charité Universitätsmedizin, Berlin, Germany
| | - Hanspeter Herzel
- Institute for Theoretical Biology, Humboldt University, Berlin, Germany
- * E-mail:
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26
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Bordyugov G, Westermark PO, Korenčič A, Bernard S, Herzel H. Mathematical modeling in chronobiology. Handb Exp Pharmacol 2013:335-57. [PMID: 23604486 DOI: 10.1007/978-3-642-25950-0_14] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
Abstract
Circadian clocks are autonomous oscillators entrained by external Zeitgebers such as light-dark and temperature cycles. On the cellular level, rhythms are generated by negative transcriptional feedback loops. In mammals, the suprachiasmatic nucleus (SCN) in the anterior part of the hypothalamus plays the role of the central circadian pacemaker. Coupling between individual neurons in the SCN leads to precise self-sustained oscillations even in the absence of external signals. These neuronal rhythms orchestrate the phasing of circadian oscillations in peripheral organs. Altogether, the mammalian circadian system can be regarded as a network of coupled oscillators. In order to understand the dynamic complexity of these rhythms, mathematical models successfully complement experimental investigations. Here we discuss basic ideas of modeling on three different levels (1) rhythm generation in single cells by delayed negative feedbacks, (2) synchronization of cells via external stimuli or cell-cell coupling, and (3) optimization of chronotherapy.
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Affiliation(s)
- G Bordyugov
- Institute for Theoretical Biology, Humboldt University, Invalidenstr. 43, 10115, Berlin, Germany.
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27
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Webb AB, Taylor SR, Thoroughman KA, Doyle FJ, Herzog ED. Weakly circadian cells improve resynchrony. PLoS Comput Biol 2012; 8:e1002787. [PMID: 23209395 PMCID: PMC3510091 DOI: 10.1371/journal.pcbi.1002787] [Citation(s) in RCA: 40] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2012] [Accepted: 10/05/2012] [Indexed: 12/05/2022] Open
Abstract
The mammalian suprachiasmatic nuclei (SCN) contain thousands of neurons capable of generating near 24-h rhythms. When isolated from their network, SCN neurons exhibit a range of oscillatory phenotypes: sustained or damping oscillations, or arrhythmic patterns. The implications of this variability are unknown. Experimentally, we found that cells within SCN explants recover from pharmacologically-induced desynchrony by re-establishing rhythmicity and synchrony in waves, independent of their intrinsic circadian period We therefore hypothesized that a cell's location within the network may also critically determine its resynchronization. To test this, we employed a deterministic, mechanistic model of circadian oscillators where we could independently control cell-intrinsic and network-connectivity parameters. We found that small changes in key parameters produced the full range of oscillatory phenotypes seen in biological cells, including similar distributions of period, amplitude and ability to cycle. The model also predicted that weaker oscillators could adjust their phase more readily than stronger oscillators. Using these model cells we explored potential biological consequences of their number and placement within the network. We found that the population synchronized to a higher degree when weak oscillators were at highly connected nodes within the network. A mathematically independent phase-amplitude model reproduced these findings. Thus, small differences in cell-intrinsic parameters contribute to large changes in the oscillatory ability of a cell, but the location of weak oscillators within the network also critically shapes the degree of synchronization for the population. Circadian rhythms are daily, near 24-h oscillations in biological processes that nearly all organisms on Earth experience. Single cells contain a molecular clock that drives circadian rhythms in physiology and, when many cells synchronize in a population, daily behaviors. We hypothesized that small differences in intrinsic cellular properties allow for a diversity of circadian periods and amplitudes across cells. We observed circadian cells and their synchrony before, during, and after limiting communication between cells and then compared their intrinsic properties to their resynchronization behavior. We found that arrhythmic, weakly oscillating, and self-sustained circadian cells rejoined the rhythmic population independent of their cell-intrinsic oscillations. Using a mechanistic computational model of circadian cells, we found that resynchronization could be enhanced by including more weak oscillators or by placing weak oscillators at more connected nodes in the network. We conclude that intrinsic properties (e.g. oscillator weakness and responsiveness) and network structure (e.g. positions of weak oscillators) can independently buffer tissue rhythms from perturbations. This reveals how cellular and network properties impose rules on systems of circadian cells that must achieve synchrony from a desynchronized state, for example during perinatal development or when forced to overcome societal constraints on sleep-wake behavior, such as working early or late shifts.
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Affiliation(s)
- Alexis B. Webb
- Department of Biology, Washington University, St. Louis, Missouri, United States of America
| | - Stephanie R. Taylor
- Department of Computer Science, Colby College, Waterville, Maine, United States of America
- * E-mail:
| | - Kurt A. Thoroughman
- Department of Biomedical Engineering, Washington University, St. Louis, Missouri, United States of America
| | - Francis J. Doyle
- Department of Chemical Engineering, University of California Santa Barbara, Santa Barbara, California, United States of America
| | - Erik D. Herzog
- Department of Biology, Washington University, St. Louis, Missouri, United States of America
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28
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Bodenstein C, Gosak M, Schuster S, Marhl M, Perc M. Modeling the seasonal adaptation of circadian clocks by changes in the network structure of the suprachiasmatic nucleus. PLoS Comput Biol 2012; 8:e1002697. [PMID: 23028293 PMCID: PMC3447953 DOI: 10.1371/journal.pcbi.1002697] [Citation(s) in RCA: 38] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/21/2012] [Accepted: 08/08/2012] [Indexed: 11/18/2022] Open
Abstract
The dynamics of circadian rhythms needs to be adapted to day length changes between summer and winter. It has been observed experimentally, however, that the dynamics of individual neurons of the suprachiasmatic nucleus (SCN) does not change as the seasons change. Rather, the seasonal adaptation of the circadian clock is hypothesized to be a consequence of changes in the intercellular dynamics, which leads to a phase distribution of electrical activity of SCN neurons that is narrower in winter and broader during summer. Yet to understand this complex intercellular dynamics, a more thorough understanding of the impact of the network structure formed by the SCN neurons is needed. To that effect, we propose a mathematical model for the dynamics of the SCN neuronal architecture in which the structure of the network plays a pivotal role. Using our model we show that the fraction of long-range cell-to-cell connections and the seasonal changes in the daily rhythms may be tightly related. In particular, simulations of the proposed mathematical model indicate that the fraction of long-range connections between the cells adjusts the phase distribution and consequently the length of the behavioral activity as follows: dense long-range connections during winter lead to a narrow activity phase, while rare long-range connections during summer lead to a broad activity phase. Our model is also able to account for the experimental observations indicating a larger light-induced phase-shift of the circadian clock during winter, which we show to be a consequence of higher synchronization between neurons. Our model thus provides evidence that the variations in the seasonal dynamics of circadian clocks can in part also be understood and regulated by the plasticity of the SCN network structure. Circadian clocks drive the temporal coordination of internal biological processes, which in turn determine daily rhythms in physiology and behavior in the most diverse organisms. In mammals, the 24-hour timing clock resides in the suprachiasmatic nucleus (SCN) of the hypothalamus. The SCN is a network of interconnected neurons that serves as a robust self-sustained circadian pacemaker. The electrical activity of these neurons and their synchronization with the 24-hour cycle is established via the environmental day and night cycles. Apart from daily luminance changes, mammals are exposed to seasonal day length changes as well. Remarkably, it has been shown experimentally that the seasonal adaptations to different photoperiods are related to the modifications of the neuronal activity of the SCN due to the plasticity of the network. In our paper, by developing a mathematical model of the SCN architecture, we explore in depth the role of the structure of this important neuronal network. We show that the redistribution of the neuronal activity during winter and summer can in part be explained by structural changes of the network. Interestingly, the alterations of the electrical activity patterns can be related with small-world properties of our proposed SCN network.
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Affiliation(s)
| | - Marko Gosak
- Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia
| | - Stefan Schuster
- Department of Bioinformatics, Friedrich Schiller University Jena, Jena, Germany
| | - Marko Marhl
- Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia
- * E-mail:
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29
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Méndez JM, Mindlin GB, Goller F. Interaction between telencephalic signals and respiratory dynamics in songbirds. J Neurophysiol 2012; 107:2971-83. [PMID: 22402649 DOI: 10.1152/jn.00646.2011] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022] Open
Abstract
The mechanisms by which telencephalic areas affect motor activities are largely unknown. They could either take over motor control from downstream motor circuits or interact with the intrinsic dynamics of these circuits. Both models have been proposed for telencephalic control of respiration during learned vocal behavior in birds. The interactive model postulates that simple signals from the telencephalic song control areas are sufficient to drive the nonlinear respiratory network into producing complex temporal sequences. We tested this basic assumption by electrically stimulating telencephalic song control areas and analyzing the resulting respiratory patterns in zebra finches and in canaries. We found strong evidence for interaction between the rhythm of stimulation and the intrinsic respiratory rhythm, including naturally emerging subharmonic behavior and integration of lateralized telencephalic input. The evidence for clear interaction in our experimental paradigm suggests that telencephalic vocal control also uses a similar mechanism. Furthermore, species differences in the response of the respiratory system to stimulation show parallels to differences in the respiratory patterns of song, suggesting that the interactive production of respiratory rhythms is manifested in species-specific specialization of the involved circuitry.
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Affiliation(s)
- Jorge M Méndez
- Dept. of Biology, Univ. of Utah, Salt Lake City, UT 84112, USA
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30
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Heiland I, Bodenstein C, Hinze T, Weisheit O, Ebenhoeh O, Mittag M, Schuster S. Modeling temperature entrainment of circadian clocks using the Arrhenius equation and a reconstructed model from Chlamydomonas reinhardtii. J Biol Phys 2012; 38:449-64. [PMID: 23729908 DOI: 10.1007/s10867-012-9264-x] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2011] [Accepted: 01/17/2012] [Indexed: 10/28/2022] Open
Abstract
Endogenous circadian rhythms allow living organisms to anticipate daily variations in their natural environment. Temperature regulation and entrainment mechanisms of circadian clocks are still poorly understood. To better understand the molecular basis of these processes, we built a mathematical model based on experimental data examining temperature regulation of the circadian RNA-binding protein CHLAMY1 from the unicellular green alga Chlamydomonas reinhardtii, simulating the effect of temperature on the rates by applying the Arrhenius equation. Using numerical simulations, we demonstrate that our model is temperature-compensated and can be entrained to temperature cycles of various length and amplitude. The range of periods that allow entrainment of the model depends on the shape of the temperature cycles and is larger for sinusoidal compared to rectangular temperature curves. We show that the response to temperature of protein (de)phosphorylation rates play a key role in facilitating temperature entrainment of the oscillator in Chlamydomonas reinhardtii. We systematically investigated the response of our model to single temperature pulses to explain experimentally observed phase response curves.
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Affiliation(s)
- Ines Heiland
- Department of Bioinformatics, School of Biology and Pharmacy, Friedrich-Schiller University Jena, E.-Abbe-Platz 2, 07743 Jena, Germany
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Biochemical frequency control by synchronisation of coupled repressilators: an in silico study of modules for circadian clock systems. COMPUTATIONAL INTELLIGENCE AND NEUROSCIENCE 2011; 2011:262189. [PMID: 22046179 PMCID: PMC3199195 DOI: 10.1155/2011/262189] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/18/2011] [Revised: 07/06/2011] [Accepted: 07/06/2011] [Indexed: 11/17/2022]
Abstract
Exploration of chronobiological systems emerges as a growing research field within bioinformatics focusing on various applications in medicine, agriculture, and material sciences. From a systems biological perspective, the question arises whether biological control systems for regulation of oscillatory signals and their technical counterparts utilise similar mechanisms. If so, modelling approaches and parameterisation adopted from building blocks can help to identify general components for frequency control in circadian clocks along with gaining insight into mechanisms of clock synchronisation to external stimuli like the daily rhythm of sunlight and darkness. Phase-locked loops could be an interesting candidate in this context. Both, biology and engineering, can benefit from a unified view resulting from systems modularisation. In a first experimental study, we analyse a model of coupled repressilators. We demonstrate its ability to synchronise clock signals in a monofrequential manner. Several oscillators initially deviate in phase difference and frequency with respect to explicit reaction and diffusion rates. Accordingly, the duration of the synchronisation process depends on dedicated reaction and diffusion parameters whose settings still lack to be sufficiently captured analytically.
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Tsumoto K, Kurosawa G, Yoshinaga T, Aihara K. Modeling light adaptation in circadian clock: prediction of the response that stabilizes entrainment. PLoS One 2011; 6:e20880. [PMID: 21698191 PMCID: PMC3116846 DOI: 10.1371/journal.pone.0020880] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2011] [Accepted: 05/11/2011] [Indexed: 11/18/2022] Open
Abstract
Periods of biological clocks are close to but often different from the rotation period of the earth. Thus, the clocks of organisms must be adjusted to synchronize with day-night cycles. The primary signal that adjusts the clocks is light. In Neurospora, light transiently up-regulates the expression of specific clock genes. This molecular response to light is called light adaptation. Does light adaptation occur in other organisms? Using published experimental data, we first estimated the time course of the up-regulation rate of gene expression by light. Intriguingly, the estimated up-regulation rate was transient during light period in mice as well as Neurospora. Next, we constructed a computational model to consider how light adaptation had an effect on the entrainment of circadian oscillation to 24-h light-dark cycles. We found that cellular oscillations are more likely to be destabilized without light adaption especially when light intensity is very high. From the present results, we predict that the instability of circadian oscillations under 24-h light-dark cycles can be experimentally observed if light adaptation is altered. We conclude that the functional consequence of light adaptation is to increase the adjustability to 24-h light-dark cycles and then adapt to fluctuating environments in nature.
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Affiliation(s)
- Kunichika Tsumoto
- Aihara Complexity Modelling Project, ERATO, Japan Science and Technology Agency, Tokyo, Japan
| | - Gen Kurosawa
- Aihara Complexity Modelling Project, ERATO, Japan Science and Technology Agency, Tokyo, Japan
- * E-mail:
| | - Tetsuya Yoshinaga
- Institute of Health Biosciences, University of Tokushima, Tokushima, Japan
| | - Kazuyuki Aihara
- Aihara Complexity Modelling Project, ERATO, Japan Science and Technology Agency, Tokyo, Japan
- Institute of Industrial Science, University of Tokyo, Tokyo, Japan
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Pfeuty B, Thommen Q, Lefranc M. Robust entrainment of circadian oscillators requires specific phase response curves. Biophys J 2011; 100:2557-65. [PMID: 21641300 PMCID: PMC3117189 DOI: 10.1016/j.bpj.2011.04.043] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2010] [Revised: 04/14/2011] [Accepted: 04/18/2011] [Indexed: 01/29/2023] Open
Abstract
The circadian clocks keeping time in many living organisms rely on self-sustained biochemical oscillations entrained by external cues, such as light, to the 24-h cycle induced by Earth's rotation. However, environmental cues are unreliable due to the variability of habitats, weather conditions, or cue-sensing mechanisms among individuals. A tempting hypothesis is that circadian clocks have evolved so as to be robust to fluctuations in the signal that entrains them. To support this hypothesis, we analyze the synchronization behavior of weakly and periodically forced oscillators in terms of their phase response curve (PRC), which measures phase changes induced by a perturbation applied at different times of the cycle. We establish a general relationship between the robustness of key entrainment properties, such as stability and oscillator phase, on the one hand, and the shape of the PRC as characterized by a specific curvature or the existence of a dead zone, on the other hand. The criteria obtained are applied to computational models of circadian clocks and account for the disparate robustness properties of various forcing schemes. Finally, the analysis of PRCs measured experimentally in several organisms strongly suggests a case of convergent evolution toward an optimal strategy for maintaining a clock that is accurate and robust to environmental fluctuations.
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Affiliation(s)
- Benjamin Pfeuty
- Laboratoire de Physique des Lasers, Atomes, Molécules, and Institut de Recherche Interdisciplinaire, Université Lille 1 Sciences et Technologies, CNRS, F-59655 Villeneuve d'Ascq, France.
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Coupling governs entrainment range of circadian clocks. Mol Syst Biol 2011; 6:438. [PMID: 21119632 PMCID: PMC3010105 DOI: 10.1038/msb.2010.92] [Citation(s) in RCA: 237] [Impact Index Per Article: 18.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2010] [Accepted: 10/07/2010] [Indexed: 11/08/2022] Open
Abstract
Circadian clock oscillator properties that are crucial for synchronization with the environment (entrainment) are studied in experiment and theory. The ratio between stimulus (zeitgeber) strength and oscillator amplitude, and the rigidity of the oscillatory system (relaxation rate upon perturbation) determine entrainment properties. Coupling among oscillators affects both qualities resulting in increased amplitude and rigidity. Uncoupled lung clocks entrain to extreme zeitgeber cycles, whereas the coupled oscillator system in the suprachiasmatic nucleus (SCN) does not; however, when coupling in the SCN is inhibited, larger ranges of entrainment can be achieved.
Daily rhythms in physiology, metabolism and behavior are controlled by an endogenous circadian timing system, which has evolved to synchronize an organism to periodically recurring environmental conditions, such as light–dark or temperature cycles. In mammals, the circadian system relies on cell-autonomous oscillators residing in almost every cell of the body. Cells of the SCN in the anterior hypothalamus are able to generate precise, long-lasting self-sustained circadian oscillations, which drive most rhythmic behavioral and physiological outputs, and which are believed to originate from the fact that the SCN tissue consists of tightly coupled cells (Aton and Herzog, 2005). In contrast, peripheral oscillators, such as lung tissue, exhibit seemingly damped and usually less precise oscillations, which are thought to be brought about by the lack of intercellular coupling. Precise synchronization of these rhythms within the organism, but also with the environment (so-called entrainment), is an essential part of circadian organization. Entrainment is one of the cornerstones of circadian biology (Roenneberg et al, 2003). In evolution, the phase of a rhythmic variable is selective rather than its endogenous period. Thus, the synchronization of endogenous rhythms to zeitgeber cycles of the environment (resulting in a specific phase of entrainment) is fundamental for the adaptive value of circadian clocks. In this study, we systematically investigated the properties of circadian oscillators that are essential for entrainment behavior and describe coupling as a primary determinant. As an experimental starting point of this study, we found that the circadian oscillators of lung tissue have a larger range of entrainment than SCN tissue—they readily entrained to extreme experimental temperature cycle of 20 or 28 h, whereas SCN tissue did not (Figure 4). For this purpose, we cultured SCN and lung slices derived from mice that express luciferase as fusion protein together with the clock protein PERIOD2 (Yoo et al, 2004). The detection of luciferase-driven bioluminescence allowed us to follow molecular clock gene activity in real-time over the course of several days. In theoretical analyses, we show that both the ratio of amplitude and zeitgeber strength and, importantly, inter-oscillator coupling are major determinants for entrainment. The reason for coupling being critical is twofold: (i) Coupling makes an oscillatory system more rigid, i.e., it relaxes faster in response to a perturbation, and (ii) coupling increases the amplitude of the oscillatory system. Both of these consequences of coupling lead to a smaller entrainment range, because zeitgeber stimuli affect the oscillatory system less if the relaxation is fast and the amplitude is high (Figure 1). From these theoretical considerations, we conclude that the lung clock probably constitutes a weak oscillatory system, likely because a lack in coupling leads to a slow amplitude relaxation. (Circadian amplitude is not particularly low in lung (Figure 4).) In contrast, the SCN constitutes a rigid oscillator, whereby coupling and its described consequences probably are the primary causes for this rigidity. We then tested these theoretical predictions by experimentally perturbing coupling in the SCN (with MDL and TTX; O'Neill et al, 2008; Yamaguchi et al, 2003) and find that, indeed, reducing the coupling weakens the circadian oscillatory system in the SCN, which results in an enlargement of the entrainment range (Figure 6). Why is the SCN designed to be a stronger circadian oscillator than peripheral organs? We speculate that the position of the SCN—as the tissue that conveys environmental timing information (i.e., light) to the rest of the body—makes it necessary to create a circadian clock that is robust against noisy environmental stimuli. The SCN oscillator needs to be robust enough to be protected from environmental noise, but flexible enough to fulfill its function as an entrainable clock even in extreme photoperiods (i.e., seasons). By the same token, peripheral clocks are more protected from the environmental zeitgebers due to intrinsic homeostatic mechanisms. Thus, they do not necessarily need to develop a strong oscillatory system (e.g., by strengthening the coupling), rather they need to stay flexible enough to respond to direct or indirect signals from the SCN, such as hormonal, neural, temperature or metabolic signals. Such a design ensures that only robust and persistent environmental signals trigger an SCN resetting response, while SCN signals can relatively easily be conveyed to the rest of the body. Thus, the robustness in the SCN clock likely serves as a filter for environmental noise. In summary, using a combination of simulation studies, analytical calculations and experiments, we uncovered critical features for entrainment, such as zeitgeber-to-amplitude ratio and amplitude relaxation rate. Coupling is a primary factor that governs these features explaining important differences in the design of SCN and peripheral oscillators that ensure a robust, but also flexible circadian system. Circadian clocks are endogenous oscillators driving daily rhythms in physiology and behavior. Synchronization of these timers to environmental light–dark cycles (‘entrainment') is crucial for an organism's fitness. Little is known about which oscillator qualities determine entrainment, i.e., entrainment range, phase and amplitude. In a systematic theoretical and experimental study, we uncovered these qualities for circadian oscillators in the suprachiasmatic nucleus (SCN—the master clock in mammals) and the lung (a peripheral clock): (i) the ratio between stimulus (zeitgeber) strength and oscillator amplitude and (ii) the rigidity of the oscillatory system (relaxation rate upon perturbation) determine entrainment properties. Coupling among oscillators affects both qualities resulting in increased amplitude and rigidity. These principles explain our experimental findings that lung clocks entrain to extreme zeitgeber cycles, whereas SCN clocks do not. We confirmed our theoretical predictions by showing that pharmacological inhibition of coupling in the SCN leads to larger ranges of entrainment. These differences between master and the peripheral clocks suggest that coupling-induced rigidity in the SCN filters environmental noise to create a robust circadian system.
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Granada AE, Cambras T, Díez-Noguera A, Herzel H. Circadian desynchronization. Interface Focus 2010; 1:153-66. [PMID: 22419981 DOI: 10.1098/rsfs.2010.0002] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2010] [Accepted: 10/21/2010] [Indexed: 01/18/2023] Open
Abstract
The suprachiasmatic nucleus (SCN) coordinates via multiple outputs physiological and behavioural circadian rhythms. The SCN is composed of a heterogeneous network of coupled oscillators that entrain to the daily light-dark cycles. Outside the physiological entrainment range, rich locomotor patterns of desynchronized rhythms are observed. Previous studies interpreted these results as the output of different SCN neural subpopulations. We find, however, that even a single periodically driven oscillator can induce such complex desynchronized locomotor patterns. Using signal analysis, we show how the observed patterns can be consistently clustered into two generic oscillatory interaction groups: modulation and superposition. In seven of 17 rats undergoing forced desynchronization, we find a theoretically predicted third spectral component. Combining signal analysis with the theory of coupled oscillators, we provide a framework for the study of circadian desynchronization.
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Affiliation(s)
- Adrián E Granada
- Institute for Theoretical Biology, Humboldt University Berlin, Invalidenstraße 43, 10115 Berlin, Germany
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Harada T, Tanaka HA, Hankins MJ, Kiss IZ. Optimal waveform for the entrainment of a weakly forced oscillator. PHYSICAL REVIEW LETTERS 2010; 105:088301. [PMID: 20868133 DOI: 10.1103/physrevlett.105.088301] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2010] [Indexed: 05/18/2023]
Abstract
A theory for obtaining a waveform for the effective entrainment of a weakly forced oscillator is presented. Phase model analysis is combined with calculus of variation to derive a waveform with which entrainment of an oscillator is achieved with a minimum power forcing signal. Optimal waveforms are calculated from the phase response curve and a solution to a balancing condition. The theory is tested in chemical entrainment experiments in which oscillations close to and farther away from a Hopf bifurcation exhibited sinusoidal and higher harmonic nontrivial optimal waveforms, respectively.
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Affiliation(s)
- Takahiro Harada
- Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan
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