Skip to main content
SpringerLink
Log in

Simple Kinetic Models in Molecular Chronobiology

  • Protocol
  • First Online:
Circadian Clocks

Part of the book series: Methods in Molecular Biology ((MIMB,volume 2130))

Abstract

Circadian rhythms are constituted by a complex dynamical system with intertwined feedback loops, molecular switches, and self-sustained oscillations. Mathematical modeling supports understanding available heterogeneous kinetic data, highlights basic mechanisms, and can guide experimental research. Here, we introduce the basic steps from a biological question to simple models providing insight into gene-regulatory mechanisms. We illustrate the general approach by three examples: modeling decay processes, clock-controlled genes, and self-sustained oscillations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Protocol
USD 49.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 149.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Wever R (1965) A mathematical model for circadian rhythms. Circadian Clocks 47: 47–63

    Google Scholar 

  2. Winfree AT (1970) Integrated view of resetting a circadian clock. J Theor Biol 28 (3): 327–374

    Article  CAS  PubMed  Google Scholar 

  3. 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

    Article  CAS  Google Scholar 

  4. Kaplan D, Glass L (2012) Understanding nonlinear dynamics. Springer Science & Business Media, New York

    Google Scholar 

  5. Segel LA (1984) Modeling dynamic phenomena in molecular and cellular biology. Cambridge University Press, Cambridge

    Google Scholar 

  6. Murray JD (2002) Mathematical biology I: an introduction. Interdisciplinary applied mathematics, vol 17. Springer, New York

    Google Scholar 

  7. Goldbeter A (1997) Biochemical oscillations and cellular rhythms. Cambridge University Press, Cambridge

    Google Scholar 

  8. Cornish-Bowden A, Cárdenas ML (2013) Control of metabolic processes, vol 190. Springer Science & Business Media, New York

    Google Scholar 

  9. Heinrich R, Schuster S (2012) The regulation of cellular systems. Springer Science & Business Media, New York

    Google Scholar 

  10. Ingalls BP (2013) Mathematical modeling in systems biology: an introduction. MIT Press, Cambridge

    Google Scholar 

  11. Bintu L, Buchler NE, Garcia HG, Gerland U, Hwa T, Kondev J, Phillips R (2005) Transcriptional regulation by the numbers: models. Curr Opin Genet Dev 15 (2): 116–124

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Chen WW, Niepel M, Sorger PK (2010) Classic and contemporary approaches to modeling biochemical reactions. Genes Dev 24 (17): 1861–1875

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  13. Westermark PO, Kotaleski JH, Björklund A, Grill V, Lansner A (2007) A mathematical model of the mitochondrial NADH shuttles and anaplerosis in the pancreatic β-cell. Am J Physiol Endocrinol Metab 292 (2): E373–E393

    Article  CAS  PubMed  Google Scholar 

  14. Westermark PO, Herzel H (2013) Mechanism for 12 hr rhythm generation by the circadian clock. Cell Rep 3 (4): 1228–1238

    Article  CAS  PubMed  Google Scholar 

  15. Ermentrout B (2002) Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students, vol 14. SIAM, Philadelphia

    Book  Google Scholar 

  16. Hucka M, Finney ABBJ, Bornstein BJ, Keating SM, Shapiro BE, Matthews J, Kovitz BL, Schilstra MJ, Funahashi A, Doyle JC, Kitano H (2004) Evolving a lingua franca and associated software infrastructure for computational systems biology: the systems biology markup language (SBML) project. Syst Biol 1 (1): 41–53

    Article  CAS  Google Scholar 

  17. Buckingham E (1914) On physically similar systems; illustrations of the use of dimensional equations. Phys Rev 4 (4): 345

    Article  Google Scholar 

  18. Vanselow K, Vanselow JT, Westermark PO, Reischl S, Maier B, Korte T, Herrmann A, Herzel H, Schlosser A, Kramer A (2006) Differential effects of PER2 phosphorylation: molecular basis for the human familial advanced sleep phase syndrome (FASPS). Genes Dev 20 (19): 2660–2672

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  19. Maiwald T, Timmer J (2008) Dynamical modeling and multi-experiment fitting with PottersWheel. Bioinformatics 24 (18): 2037–2043

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  20. Friedel CC, Dölken L, Ruzsics Z, Koszinowski UH, Zimmer R (2009) Conserved principles of mammalian transcriptional regulation revealed by RNA half-life. Nucleic Acids Res 37 (17): e115

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  21. Schwanhäusser B, Busse D, Li N, Dittmar G, Schuchhardt J, Wolf J, Chen W, Selbach M (2011) Global quantification of mammalian gene expression control. Nature 473 (7347): 337–342

    Article  PubMed  CAS  Google Scholar 

  22. Eden E, Geva-Zatorsky N, Issaeva I, Cohen A, Dekel E, Danon T, Cohen L, Mayo A, Alon U (2011) Proteome half-life dynamics in living human cells. Science 331 (6018): 764–768

    Article  CAS  PubMed  Google Scholar 

  23. Zhang R, Lahens NF, Ballance HI, Hughes ME, Hogenesch JB (2014) A circadian gene expression atlas in mammals: implications for biology and medicine. Proc Natl Acad Sci 111 (45): 16219–16224

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  24. Korenčič A, Bordyugov G, Rozman D, Goličnik M, Herzel H et al (2012) The interplay of cis-regulatory elements rules circadian rhythms in mouse liver. PLoS One 7 (11): e46835

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  25. Lück S, Thurley K, Thaben PF, Westermark PO (2014) Rhythmic degradation explains and unifies circadian transcriptome and proteome data. Cell Rep 9 (2): 741–751

    Article  PubMed  CAS  Google Scholar 

  26. Le Martelot G, Canella D, Symul L, Migliavacca E, Gilardi F, Liechti R, Martin O, Harshman K, Delorenzi M, Desvergne B, Herr W, Deplancke B, Schibler U, Rougemont J, Guex N, Hernandez N, Naef F, the CycliX consortium (2012) Genome-wide RNA polymerase ii profiles and RNA accumulation reveal kinetics of transcription and associated epigenetic changes during diurnal cycles. PLoS Biol 10 (11): e1001442

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  27. Ukai-Tadenuma M, Yamada RG, Xu H, Ripperger JA, Liu AC, Ueda HR (2011) Delay in feedback repression by Cryptochrome 1 is required for circadian clock function. Cell 144 (2): 268–281

    Google Scholar 

  28. Lamia KA, Storch K-F, Weitz CJ (2008) Physiological significance of a peripheral tissue circadian clock. Proc Natl Acad Sci 105 (39): 15172–15177

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  29. Eser P, Demel C, Maier KC, Schwalb B, Pirkl N, Martin DE, Cramer P, Tresch A (2014) Periodic mRNA synthesis and degradation co-operate during cell cycle gene expression. Mol Syst Biol 10 (1): 717

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  30. Thurley K, Herbst C, Wesener F, Koller B, Wallach T, Maier B, Kramer A, Westermark PO (2017) Principles for circadian orchestration of metabolic pathways. Proc Natl Acad Sci USA 114(7):1572–1577. https://doi.org/10.1073/pnas.1613103114

  31. Hughes ME, DiTacchio L, Hayes KR, Vollmers C, Pulivarthy S, Baggs JE, Panda S, Hogenesch JB (2009) Harmonics of circadian gene transcription in mammals. PLoS Genet 5 (4): e1000442

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  32. Locke JCW, Kozma-Bognár L, Gould PD, Fehér B, Kevei E, Nagy F, Turner MS, Hall A, Millar AJ (2006) Experimental validation of a predicted feedback loop in the multi-oscillator clock of Arabidopsis thaliana. Mol Syst Biol 2 (1): 59

    Article  PubMed  PubMed Central  Google Scholar 

  33. Blum ID, Zhu L, Moquin L, Kokoeva MV, Gratton A, Giros B, Storch K-F (2014) A highly tunable dopaminergic oscillator generates ultradian rhythms of behavioral arousal. eLife 3: e05105

    Article  PubMed Central  Google Scholar 

  34. Hardin PE, Hall JC, Rosbash M (1990) Feedback of the Drosophila Period gene product on circadian cycling of its messenger RNA levels. Nature 343 (6258): 536–540

    Article  CAS  PubMed  Google Scholar 

  35. Goodwin BC (1965) Oscillatory behavior in enzymatic control processes. Adv Enzyme Regul 3: 425–437

    Article  CAS  PubMed  Google Scholar 

  36. Pett JP, Korenčič A, Wesener F, Kramer A, Herzel H (2016) Feedback loops of the mammalian circadian clock constitute repressilator. PLoS Comput Biol 12 (12): 1–15

    Article  CAS  Google Scholar 

  37. Zeng H, Qian Z, Myers MP, Rosbash M (1996) A light-entrainment mechanism for the Drosophila circadian clock. Nature 380 (6570): 129–135

    Article  CAS  PubMed  Google Scholar 

  38. Myers MP, Wager-Smith K, Rothenfluh-Hilfiker A, Young MW (1996) Light-induced degradation of TIMELESS and entrainment of the Drosophila circadian clock. Science 271 (5256): 1736

    Article  CAS  PubMed  Google Scholar 

  39. Gallego M, Virshup DM (2007) Post-translational modifications regulate the ticking of the circadian clock. Nat Rev Mol Cell Biol 8 (2): 139–148

    Article  CAS  PubMed  Google Scholar 

  40. Meinhardt H (1982) Models of biological pattern formation. Academic, London

    Google Scholar 

  41. Tsai TY-C, Choi YS, Ma W, Pomerening JR, Tang C, Ferrell JE Jr (2008) Robust, tunable biological oscillations from interlinked positive and negative feedback loops. Science 321: 126–129

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  42. Stricker J, Cookson S, Bennett MR, Mather WH, Tsimring LS, Hasty J (2008) A fast, robust and tunable synthetic gene oscillator. Nature 456: 516–519

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  43. Ananthasubramaniam B, Herzel H (2014) Positive feedback promotes oscillations in negative feedback loops. PLoS One 9: e104761

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  44. Clodong S, Dühring U, Kronk L, Wilde A, Axmann I, Herzel H, Kollmann M (2007) Functioning and robustness of a bacterial circadian clock. Mol Syst Biol 3 (1): 90

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  45. Zimmerman WF, Pittendrigh CS, Pavlidis T (1968) Temperature compensation of the circadian oscillation in Drosophila pseudoobscura and its entrainment by temperature cycles. J Insect Physiol 14 (5): 669–684

    Article  CAS  PubMed  Google Scholar 

  46. Rensing L, Ruoff P (2002) Temperature effect on entrainment, phase shifting, and amplitude of circadian clocks and its molecular bases. Chronobiol Int 19 (5): 807–864

    Article  CAS  PubMed  Google Scholar 

  47. 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

    Article  PubMed  PubMed Central  Google Scholar 

  48. Buhr ED, Yoo S-H, Takahashi JS (2010) Temperature as a universal resetting cue for mammalian circadian oscillators. Science 330 (6002): 379–385

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  49. Ruoff P, Rensing L (1996) The temperature-compensated Goodwin model simulates many circadian clock properties. J Theor Biol 179 (4): 275–285

    Article  Google Scholar 

  50. Hatakeyama TS, Kaneko K (2012) Generic temperature compensation of biological clocks by autonomous regulation of catalyst concentration. Proc Natl Acad Sci 109 (21): 8109–8114

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  51. Goldbeter A (1995) A model for circadian oscillations in the Drosophila PERIOD protein (PER). Proc R Soc Lond B Biol Sci 261 (1362): 319–324

    Article  CAS  Google Scholar 

  52. Tyson JJ, Hong CI, Thron CD , Novak B (1999) A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM. Biophys J 77 (5): 2411–2417

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  53. Toh KL, Jones CR, He Y, Eide EJ, Hinz WA, Virshup DM, Ptáček LJ, Fu Y-H (2001) An hPer2 phosphorylation site mutation in familial advanced sleep phase syndrome. Science 291 (5506): 1040–1043

    Article  CAS  PubMed  Google Scholar 

  54. 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

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  55. Gallego M, Eide EJ, Woolf MF, Virshup DM, Forger DB (2006) An opposite role for tau in circadian rhythms revealed by mathematical modeling. Proc Natl Acad Sci 103 (28): 10618–10623

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  56. Zhou M, Kim JK, Eng GWL, Forger DB, Virshup DM (2015) A Period2 phosphoswitch regulates and temperature compensates circadian period. Mol Cell 60 (1): 77–88

    Article  PubMed  CAS  Google Scholar 

  57. Ferrell JE (1996) Tripping the switch fantastic: how a protein kinase cascade can convert graded inputs into switch-like outputs. Trends Biochem Sci 21 (12): 460–466

    Article  CAS  PubMed  Google Scholar 

  58. Legewie S, Blüthgen N, Herzel H (2006) Mathematical modeling identifies inhibitors of apoptosis as mediators of positive feedback and bistability. PLoS Comput Biol 2 (9): e120

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  59. Legewie S, Schoeberl B, Blüthgen N, Herzel H (2007) Competing docking interactions can bring about bistability in the MAPK cascade. Biophys J 93 (7): 2279–2288

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  60. Jolley CC, Ode KL, Ueda HR (2012) A design principle for a posttranslational biochemical oscillator. Cell Rep 2 (4): 938–950

    Article  CAS  PubMed  Google Scholar 

  61. Nakajima M, Imai K, Ito H, Nishiwaki T, Murayama Y, Iwasaki H, Oyama T, Kondo T (2005) Reconstitution of circadian oscillation of cyanobacterial KaiC phosphorylation in vitro. Science 308 (5720): 414–415

    Article  CAS  PubMed  Google Scholar 

  62. Qin X, Byrne M, Xu Y, Mori T, Johnson CH (2010) Coupling of a core post-translational pacemaker to a slave transcription/translation feedback loop in a circadian system. PLoS Biol 8 (6): e1000394

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  63. Brettschneider C, Rose RJ, Hertel S, Axmann IM, Heck AJR, Kollmann M (2010) A sequestration feedback determines dynamics and temperature entrainment of the KaiABC circadian clock. Mol Syst Biol 6 (1): 389

    Article  PubMed  PubMed Central  Google Scholar 

  64. Axmann IM, Legewie S, Herzel H (2007) A minimal circadian clock model. Genome Inform 18: 54–64

    PubMed  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Berlin, Germany

    J. Patrick Pett

  2. Leibniz Institute for Farm Animal Biology, Institute of Genetics and Biometry, Dummerstorf, Germany

    Pål O. Westermark

  3. Institute for Theoretical Biology, Charité-Universitätsmedizin Berlin, Berlin, Germany

    Hanspeter Herzel

Authors
  1. J. Patrick Pett
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pål O. Westermark
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hanspeter Herzel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hanspeter Herzel .

Editor information

Editors and Affiliations

  1. Chronobiology and Sleep Research Group, Institute of Pharmacology and Toxicology, University of Zurich, Zürich, Switzerland

    Steven A. Brown

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Science+Business Media, LLC, part of Springer Nature

About this protocol

Check for updates. Verify currency and authenticity via CrossMark

Cite this protocol

Pett, J.P., Westermark, P.O., Herzel, H. (2021). Simple Kinetic Models in Molecular Chronobiology. In: Brown, S.A. (eds) Circadian Clocks. Methods in Molecular Biology, vol 2130. Humana, New York, NY. https://doi.org/10.1007/978-1-0716-0381-9_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-0716-0381-9_7

  • Published:

  • Publisher Name: Humana, New York, NY

  • Print ISBN: 978-1-0716-0380-2

  • Online ISBN: 978-1-0716-0381-9

  • eBook Packages: Springer Protocols

Publish with us

Policies and ethics

Search

Navigation