Skip to main content

Oscillations, Synchrony and Deterministic Chaos

  • Chapter
Progress in Botany

Part of the book series: Progress in Botany ((BOTANY,volume 70))

The coherence and robustness of biological systems is an astonishing phenomenon that depends on oscillations, synchronous behaviour and, in some instances, deterministic chaos. Understanding of dynamic interactions on an extended range of timescales involves homeodynamic rather than homeostatic concepts. Thereby, oscillations produce highly complex processes of intracellular as well as intercellular synchrony and have led to the evolutionary emergence of responsiveness, motility, and developmental change. “Creative destruction” is as important as biosynthesis in the elaboration and maintenance of cellular structure. Synchronisation of oscillators even of the simplest physical kind is still incompletely understood. In a huge population of oscillators it involves the idea of coupling strength and sudden cohesion of a small cluster of oscillators as the spread of natural frequencies is decreased below a threshold. This represents an example of Prigogine’s theory of time order in which spontaneous transitions give spatio-temporal patterns in non-equilibrium open systems. Clearly, the ordered complexity of biological systems out-performs the simplicity of physical or chemical systems and its basic understanding remains a challenge despite recent successes in imaging and the increasing power of analytical chemistry. Network dynamics provides a promising tool for handling large data sets in such a way as to provide interpretation of behaviour of the whole system.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aon MA, Cortassa S, Lloyd (2000). Chaotic dynamics and fractal space in biochemistry: simplicity underlies complexity? Cell Biol Int 24: 581–587

    Google Scholar 

  • Aon MA, Cortassa S, Marban E, O’Rourke (2003). Synchronized whole-cell oscillations in mitochondrial metabolism triggered by a local release of reactive oxygen species in cardiac myocytes. J Biol Chem 278: 4475–44744

    Google Scholar 

  • Aon MA, Cortassa S, O’Rourke B (2006a). The fundamental organization of cardiac mitochondria as a network of coupled oscillators. Biophys J 91: 4317–4327

    CAS  Google Scholar 

  • Aon MA, Cortassa S, O’Rourke B (2006b). Mitochondrial oscillations in physiology and pathophysiology. Landes Bioscience, Austin.

    Google Scholar 

  • Aon MA, Cortassa S, Lemar KM, Hayes AJ, Lloyd D (2007a). Single and cell population respiratory oscillations in yeast: a 2-photon scanning laser microscopy study. FEBS Lett 581: 8–14

    CAS  Google Scholar 

  • Aon MA, Roussel MR, Cortassa S, O’Rourke B, Murray DB, Beckmann M and Lloyd D (2007b). The scale-free network organization of yeast and heart.

    Google Scholar 

  • Aserinsky E, Kleitman N (1955). Regularly occurring periods of eye motility and concomitant phenomena during sleep. Science 118: 273–274

    Google Scholar 

  • Baum P, Zewail AH (2006). Breaking resolution limits in ultrafast electron diffraction and microscopy. Proc Nat Acad Sci USA 103: 16105–16110

    PubMed  CAS  Google Scholar 

  • Beilby MJ (2007). Modeling oscillations of membrane potential difference. In Mancuso S and Shebala S (eds.) Rhythms in Plants. Springer, Berlin.

    Google Scholar 

  • Bernard C (1927). Introduction à l’Etude de la Medicine Experimentale. Green HC (trans.) McMillan, New York.

    Google Scholar 

  • Beynon JR (2005). The proteome as a dynamic entity – is yesterday’s proteome the same as today’s. Biochem Soc Trans, abstract SA038

    Google Scholar 

  • Brodsky VY (1975). Protein synthesis rhythm. J Theor Biol 55: 167–200

    Google Scholar 

  • Brodsky VY (2005). Direct cell–cell communication: a new approach derived from recent data on the nature and self-organization of ultradian (circahoralian) intracellular rhythms. Biol Rev Cambs Philos Soc 81: 143–162

    Google Scholar 

  • Brodsky VY, Lloyd D (2008). Self-organised intracellular ultradian rhythms provide direct cell-cell communication. In Lloyd D, Rossi ER (eds.) Ultradian Rhythms from Molecules to Mind: A New Vision of Life. Springer, Berlin.

    Google Scholar 

  • Brunori M, Cutruzzola F, Savino C, Travaglini-Allocatelli C, Voellone B, Gibson QH (1999). Does picosecond protein dynamics have survival value? Trends Biochem Sci 24: 253–255

    PubMed  CAS  Google Scholar 

  • Buck J (1988). Synchronous rhythmic flashing of fireflies II. Q Rev Biol 63: 265–289

    PubMed  CAS  Google Scholar 

  • Buck J, Buck E (1976). Synchronous fireflies. Sci Am 234 (May): 74–85

    PubMed  CAS  Google Scholar 

  • Cannon WB (1932). The Wisdom of the Body. Knopf, Norton, New York

    Google Scholar 

  • Chance B, Pye K, Higgins J (1967). Waveform generation by enzymatic oscillators. IEEE Spectrum 4: 79–86

    CAS  Google Scholar 

  • Chance B, Williamson G, Lee IY, Mela L, DeVault D, Ghosh A, Pye EK (1973). Synchronization Phenomena in Oscillations of Yeast Cells and Isolated Mitochondria. Academic Press, New York

    Google Scholar 

  • Cortassa S, Aon MA, Winslow RL, O’Rourke B (2004). A mitochondrial oscillator depends on reactive oxygen species. Biophys J 87: 2060–2073

    PubMed  CAS  Google Scholar 

  • Crawford JD, Davies KTR (1999). Synchronisation of globally coupled phase oscillations: singularities and scaling for general couplings. Physica D 125: 1–46

    Google Scholar 

  • Davey HM, Davey CL, Woodward AM, Edmonds AN, Lee AW, Kell DB (1996). Oscillatory, stochastic and chaotic growth rate fluctuations in permittistatically controlled yeast cultures. BioSystems 39: 43–61

    PubMed  CAS  Google Scholar 

  • Edwards SE, Lloyd D (1980). Oscillations in protein and RNA during synchronous growth of Acanthamoeba castellanii: evidence for periodic turnover of macromolecules during the cell cycle. FEBS Lett 109: 21–26

    PubMed  CAS  Google Scholar 

  • Eigen M, Biebricher CK, Grebinoga M, Gardiner WC (1991). The hypercycle. Coupling of RNA and protein biosynthesis in the infection cycle of an RNA bacteriophage. Biochem 30: 11005–11018

    CAS  Google Scholar 

  • Fenchel T (2002). The Origin and Early Evolution of Life. Oxford University Press, Oxford.

    Google Scholar 

  • Feigenbaum MJ (1983). Universal behaviour in non-linear systems. Physica D 7: 16–39

    Google Scholar 

  • Gilbert DA and Lloyd D (2000). The living cell: a complex autodynamic multi-oscillator system? Cell Biol Int 24: 569–580

    PubMed  CAS  Google Scholar 

  • Gilbert DA, Visser G, Ferreira GMN, Hammond KD (2000). Transient chaos in intracellular dynamics? Cell Biol Int 24: 589–591

    PubMed  CAS  Google Scholar 

  • Glandsdorff P, Prigogine I (1971). Thermodynamic Theory of Structure, Stability and Fluctuations. Wiley, New York

    Google Scholar 

  • Hastings JW, Sweeney B (1957). On the mechanism of temperature compensation in a biological clock. Proc Natl Acad Sci USA 43: 804–811

    PubMed  CAS  Google Scholar 

  • Hess B, Boiteux A (1971). Oscillatory phenomena in biochemistry. Ann Rev Biochem 40: 237–258

    PubMed  CAS  Google Scholar 

  • Hopkins FG (1913). The dynamic side of biochemistry. Brit Med J 2: 13–24

    Google Scholar 

  • Hütt M-T, Lüttge U (2002). Non linear dynamics as a tool for modelling in plant physiology. Plant Biol 4: 281–297

    Google Scholar 

  • Hütt M-T, Lüttge U (2005). Network dynamics in plant biology: current progress and perspectives. Prog Bot 66: 277–309

    Google Scholar 

  • Hütt M-T, Lüttge U (2007). Noise-induced phenomena and complex rhythms: theoretical considerations, modelling and experimental evidence. In Mancuso S and Shabala S (eds.) Rhythms in Plants. Springer-Verlag, Berlin, Heidelberg, New York

    Google Scholar 

  • Iglesia H, Meyer J, Carpino A, Schwartz W (2000). Antiphase oscillation of the left and right suprachiasmatic nucleus. Science 290: 799–801

    PubMed  Google Scholar 

  • Ishimatsu K, Horikawa K, Takeda H (2007). Coupling cellular oscillations: a mechanism that maintains synchrony against developmental noise in the segmentation clock. Dev Dyn 236: 1416–1421

    PubMed  Google Scholar 

  • Jiang Z, Morré DM, Morré DJ (2006). A rate for copper in biological timekeeping. J Inorg Biochem 100: 2140–2149

    PubMed  CAS  Google Scholar 

  • Kaiser F (2000). External signals and internal oscillation dynamics: principal aspects and responses stimulated rhythmic processes. In Walleczek J (ed.) Cambridge University Press, Cambridge UK, pp. 15–43

    Google Scholar 

  • Kim D (2004). A spiking neuron model for synchronous flashing of fireflies. Biosystems 76: 7–20

    PubMed  Google Scholar 

  • Kiss IZ, Zhai Y, Hudson JL (2002). Emerging coherence in a population of chemical oscillators. Science 296: 1676–1678

    PubMed  CAS  Google Scholar 

  • Klevecz RR (1992). A precise circadian clock from chaotic cell cycle oscillations. In Lloyd D, Rossi EL (eds.) Ultradian Rhythms in Life Processes. Springer, London, pp. 41–70

    Google Scholar 

  • Klevecz RR, Bolen J, Forrest G, Murray DB (2004). A genome wide oscillation in transcription gates the cell cycle. Proc Natl Acad Sci USA 101: 1200–1205

    PubMed  CAS  Google Scholar 

  • Kori H, Kuramoto Y (2001). Slow switching in globally coupled oscillators? Rubustness and occurrence through delayed coupling. Phys Rev E 63: 046–214

    Google Scholar 

  • Krebs HA (1981). Reminiscences and Reflections. Clarendon Press, Oxford

    Google Scholar 

  • Kuramoto Y (1975). Self-entrainment of a population of coupled non-linear oscillators, Araki H (ed). Lect Notes Theoret Phys 39: 420–422

    Google Scholar 

  • Kuramoto Y (1984). Chemical Oscillations, Waves and Turbulence. Springer, Berlin

    Google Scholar 

  • Kuramoto Y, Nishikawa I (1987). Dynamics of order parameters for globally coupled oscillators. J Stats Phys 49: 569–572

    Google Scholar 

  • Landau L (1946). On the vibrations of electronic plasmas. J Phys USSR 10: 25–34

    Google Scholar 

  • Lara-Aparicio M, Barriga-Montoya C, Fuentes-Pardo B (2006). A brief history of circadian rhythms: from Wigglesworth to Winfree. Sci Math Japn 2: 357–370

    Google Scholar 

  • Lloyd D (1997). Circadian and ultradian clock-controlled rhythms in unicellular microorganisms. Adv Microbial Physiol Biochem 39: 291–338

    Google Scholar 

  • Lloyd D (2007a) Rhythms, clocks and deterministic chaos in unicellular organisms. In Mancuso S and Shabala S (eds.) Rhythms in Plants. Springer, Berlin, pp. 267–294

    Google Scholar 

  • Lloyd D (2007b) Oscillations in Yeasts. Landes Books, Austin.

    Google Scholar 

  • Lloyd D, Gilbert DA (1997). Temporal organization of the cell division cycle in unicellular microorganisms. Soc Gen Microbiol Symp 56: 271–278

    Google Scholar 

  • Lloyd AL, Lloyd D (1993). Hypothesis: the central oscillator of the circadian clock is a controlled chaotic attractor. BioSystems 29: 77–85

    PubMed  CAS  Google Scholar 

  • Lloyd AL, Lloyd D (1995). Chaos: its significance and detection in biology. Biol Rhythm Res 26: 233–252

    Google Scholar 

  • Lloyd D, Murray DB (2005). Ultradian metronome: timekeeper for the orchestration of cellular coherence. Trends Biochem Sci 30: 373–377

    PubMed  CAS  Google Scholar 

  • Lloyd D, Murray DB (2006). The temporal architecture of eukaryotic growth. FEBS Lett 580: 2830–2835

    PubMed  CAS  Google Scholar 

  • Lloyd D, Murray DB (2007). Redox rhythmicity: clocks at the core of temporal coherence. Bioessays 29: 465–473

    PubMed  CAS  Google Scholar 

  • Lloyd D, Volkov EI (1990). Quantized cell cycle times: interaction between a relaxation oscillator and ultradian clock pulses. Bio Systems 23: 305–310

    PubMed  CAS  Google Scholar 

  • Lloyd D, Volkov EI (1991). The ultradian clock: timekeeping for intracellular dynamics. In E Mosekilde (ed.) Complex Dynamics and Biological Evolution. Plenum, New York.

    Google Scholar 

  • Lloyd D, Rossi EL (1992). Ultradian Rhythms in Life Processes: an Inquiry into Fundamental Principles of Chronobiology and Psychobiology. Springer-Verlag, London.

    Google Scholar 

  • Lloyd D, Edwards SW, Fry JC (1982). Temperature-compensated oscillations in respiration and cellular protein content in synchronous cultures of Accanthamoeba castellanii. Proc Natl Acad Sci USA 79: 3785–3788

    PubMed  CAS  Google Scholar 

  • Lloyd D, Lloyd AL, Olsen LF (1992). The cell division cycle: a physiologically plausible dynamic model can exhibit chaotic solutions. BioSystems 27: 17–24

    PubMed  CAS  Google Scholar 

  • Lloyd D, Aon MA, Cortassa S (2001). Why homeodynamics not homeostasis? Sci World 1: 133–145

    CAS  Google Scholar 

  • Lorenz EN (1963). Deterministic nonperiodic flow. J Atmos Sci 20: 130–141

    Google Scholar 

  • Lorenz EN (1993). The Essence of Chaos. University College London Press, London.

    Google Scholar 

  • Lüttge U (2003). Circadian rhythmicity: is the “biological clock” hardware or software? Prog Bot 64: 277–319

    Google Scholar 

  • Lüttge U, Hütt M-T (2004). High frequency or ultradian rhythms in plants. Prog Bot 65: 235–263

    Google Scholar 

  • Lüttge U, Hütt M-T (2007). Spatio-temporal patterns and distributed computation – a formal link between CO2 signalling, diffusion and stomatal regulation. Prog Bot 68: 242–260

    Google Scholar 

  • Luzikov VN (1981). Control over assembly of the mitochondrial inner membrane: selection by a performance criterion. FEBS Lett 125: 131–133

    PubMed  CAS  Google Scholar 

  • Luzikov VN (2004). Quality control of proteins and organelles. Biochemistry (Moscow) 67: 171–183

    Google Scholar 

  • Mancuso S, Shabala S (2007). Rhythms in Plants. Springer, Berlin.

    Google Scholar 

  • Masamizu Y, Ohtsuka T, Takashima Y, Nagahara H, Takenaka Y, Yoshikawa K, Okamura H, Kageyama R (2006). Real-time imaging of the somite segmentation clock: revelation of unstable oscillators in the individual presonitic mesoderm cells. Proc Natl Acad Sci USA 103: 1313–1318

    PubMed  CAS  Google Scholar 

  • Maroto M, Pourquié O (2001). A molecular clock involved in somite segmentation. Furr Top Dev Biol 51: 221–248

    CAS  Google Scholar 

  • Maroto M, Dale JK, Dequéant ML, Petit AC, Pourquié O (2005). Synchronised cycling gene oscillations in presomitic mesoderm cells require cell-cell contact. Int J Dev Biol 49: 309–315

    PubMed  CAS  Google Scholar 

  • May RM (1976). Simple mathematical models with very complicated dynamics. Nature 261: 459–467

    PubMed  CAS  Google Scholar 

  • Mitsui A, Kumazawa S, Takahashi A, Ikemoto H, Cao S, Arai T (1986). Strategy by which nitrogen-fixing unicellular cyanobacteria grow photoantotrophically. Nature 323: 720–722

    CAS  Google Scholar 

  • Moreno N, Colaço, Feijo JA (2007). The pollen tube oscillator: integrating biophysics and biochemistry. In Mancuso S and Shabala S (eds.) Rhythms in Plants. Springer, Berlin.

    Google Scholar 

  • Morré DJ, Church PJ, Pletcher T, Tang X, Wu LY, Morré DM (2002). Biochemical basis for the biological clock. Biochemistry 41: 11941–11945

    PubMed  Google Scholar 

  • Morré DJ, Healds M, Coleman J, Orezyk J, Jiang Z, Morré DM (2007a). Structural observations of time dependent oscillatory behaviour of CuIICl2 solutions measured via extended x-ray absorption fine structure. J Inorg Biochem 101: 715–726

    Google Scholar 

  • Morré DJ, Jiang Z, Marjonovic M, Orezyk J, Morré DM (2007b). Response to electromagnetic fields of copper II containing ECTO-NOX proteins and CuII Cl2 in solution related to equilibrium dynamics of ortholpara hydrogen spins isomers of water. J Inorg Biochem

    Google Scholar 

  • Murray DB, Lloyd D (2007). A tuneable attractor underlies yeast respiratory dynamics. BioSystems 90: 287–294

    PubMed  CAS  Google Scholar 

  • Murray DB, Klevecz RR, Lloyd D (2003). Generation and maintenance of synchrony in saccharomyces cerevisiae in continuous culture. Exp Cell Res 287: 10–15

    PubMed  CAS  Google Scholar 

  • Murray DB, Beckmann M, Kitano H (2007). The regulation of yeast oscillatory dynamics. Proc Natl Acad Sci USA 104: 2241–2246

    PubMed  CAS  Google Scholar 

  • Nityananda V, Balakrishnan R (2007). Synchrony during acoustic interactions in the bush cricket Mecopoda ‘Chirpen’ (Tettigoniidae Orthoptera) is generated by a combination of chirp by chirp resetting and change in intrinsic chirp rate. J Comp Physiol A Neuroethol Sens Neunal Behav Physiol 193: 51–65

    Google Scholar 

  • O’Rourke B (2000). Pathophysiological and protective roles of mitochondrial ion channels. J Physiol 529: 23–36

    PubMed  CAS  Google Scholar 

  • O’Rourke B, Ramza BM, Marban E (1994). Oscillations of membrane current and excitability driven by metabolic oscillations in heart cells. Science 265: 962–966

    PubMed  CAS  Google Scholar 

  • Ott E, Grebogic C, Yorke JA (1990). Controlling chaos. Phys Rev Lett 64: 1196–1199

    PubMed  Google Scholar 

  • Peel A, Akam M (2003). Evolution of segmentation: rolling back the clock. Curr Biol 13: R708–R710

    PubMed  CAS  Google Scholar 

  • Pittendrigh CS, Pikovsky A (1993). Temporal organization: reflections of a Darwinian clock-watcher. Annu Rev Physiol 55: 16–54

    PubMed  CAS  Google Scholar 

  • Pourquié O (2003). The segmentation clock: converting embryonic time into spatial pattern. Science 301: 328–330

    PubMed  Google Scholar 

  • Prigogine I, Nicolis G (1971). Biological order, structure and instabilities. Quart Rev Biophys 4: 107–148

    CAS  Google Scholar 

  • Prytz G (2001). A biophysical study of oscillatory water regulation in plants measurements and Models. DSc Thesis, NTNU: Trondheim.

    Google Scholar 

  • Pyragas K (1992). Continuous control of chaos by self-controlling feed-back. Phys Lett 170: 421–428

    Google Scholar 

  • Queiroz-Claret C, Valon C, Queiroz O (1988). Are spontaneous conformational interconversions a molecular basis for long-period oscillations in enzyme activity? Chronobiol Int 5: 301–309

    PubMed  CAS  Google Scholar 

  • Rascher V, Hütt M-T, Siebka K, Osmond B, Beck F, Lüttge U (2001). Spatiotemporal variation of metabolism in a plant circadian rhythm: the biological clock as an assembly of coupled individual oscillators. Proc Natl Acad Sci USA 98: 11801–11805

    PubMed  CAS  Google Scholar 

  • Reich JG, Sel’kov EE (1981). Energy Metabolism of the Cell. Academic Press, London.

    Google Scholar 

  • Rittenberg D (1948). Turnover of proteins. J Mount Sinai Hosp 14: 891–901

    CAS  Google Scholar 

  • Romashko DN, Marban E, O’Rourke B (1998). Sub-cellular metabolic transients and mitochondrial redox waves in heart cells. Proc Natl Acad Sci USA 95: 1618–1623

    PubMed  CAS  Google Scholar 

  • Roussel MR, Lloyd D (2007). Observation of a chaotic multioscillatory metabolic attractor by real-time monitoring of a yeast continuous culture. FEBS 274: 1011–1018

    CAS  Google Scholar 

  • Sagan D (1994). On the physics of Landau damping. Am J Phys 62: 450–462

    Google Scholar 

  • Satroutdinov AD, Kuriyama H, Kobayashi H (1992). Oscillatory metabolism of Saccharomyces cerevisiae in continuous culture. FEMS Microbiol Lett 98: 261–268

    CAS  Google Scholar 

  • Schneider ED, Sagan D (2005). Into the Cool: Energy Flow, Thermodynamics and Life. University of Chicago Press, Chicago.

    Google Scholar 

  • Schoenheimer R (1942). The Dynamic State of Body Constituents. Harvard University Press, Cambridge, MA

    Google Scholar 

  • Schuster HG (1988). Deterministic Chaos, An Introduction, 2nd edn. Physik-Verlag, Weinheim

    Google Scholar 

  • Smith HM (1935). Synchronous flashing of fireflies. Science 82: 151–152

    PubMed  Google Scholar 

  • Strogatz SH (2000). From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D 143: 1–20

    Google Scholar 

  • Strogatz SH (2003). Synch. The Emerging Science of Spontaneous Order. Hyperion, New York.

    Google Scholar 

  • Strogatz SH, Mirello RE (1991). Stability of incoherence in a population of coupled oscillators. J Stat Phys 63: 613–635

    Google Scholar 

  • Strogatz SH, Mirello RE, Matthews PC (1992). Coupled non-linear oscillators below the synchronization threshold: relaxation by generalized Landau damping. Phys Rev Lett 68: 2730–2733

    PubMed  Google Scholar 

  • Sweeney BM, Tuffli CF , Rubin RH (1967). The circadian rhythm in photosynthesis in Acetabularia in the presence of actinomycin. D. J Gen Physiol 50: 647–659

    CAS  Google Scholar 

  • Toth R, Taylor RE (2006). Loss of coherence in a population of diffusively coupled oscillators. J Chem Phys 125: 224708

    PubMed  Google Scholar 

  • Trimmer BA, Apritle JR, Dudzinski DM, Lagace CJ, Lewis SM, Michel T, Oazis S, Zagas RM (2001). Nitric oxide and the control of firefly flashing. Science 292: 2486–2488

    PubMed  CAS  Google Scholar 

  • Van de Pol B, Van der Mark (1928). The heartbeat considered as a relaxation oscillation and an electrical model of the heart. Phil Mag 6: 763–775

    Google Scholar 

  • Visser GR, Reinten C, Coplam P, Gilbert DA, Hammond KD (1990). Oscillations in cell morphology and redox state. Biophys Chem 37: 383–394

    PubMed  CAS  Google Scholar 

  • Von Bertalanffy L (1952). Problems of Life. Harper, New York.

    Google Scholar 

  • Weber G (1976). Practical application and philosophy of optical spectroscopic probes. In Kasha M and Pullman (eds.) Horizons in Biochemistry and Biophysics, vol. 2. Addison-Wesley, Reading, MA, pp. 163–198

    Google Scholar 

  • Wheatley DN (1989). Protein turnover in relation to growth and cell cycle stage of cultured mammalian cells. In Grisda S, Knecht E (eds.) Current Trends in Intracellular Protein Degradation Bilbao. Springer, Berlin, pp. 377–400

    Google Scholar 

  • Wiener N (1961). Cybernetics, 2nd edn. MIT Press, Cambridge, MA

    Google Scholar 

  • Wiley DA, Strogatz SH, Girvan M (2006). The size of the synch basin. Chaos. 16(1): 015103

    PubMed  Google Scholar 

  • Winfree AT (1967). Biological rhythms and the behaviour of populations of coupled oscillators. J. Theoret Biol 16: 15–42

    CAS  Google Scholar 

  • Winfree AT (2001). The Geometry of Biological Time, 2nd ed. Springer, Berlin

    Google Scholar 

  • Yates FE (1982). Outline of a physical theory of physiological systems. J Physiol Pharmacol 60: 217–248

    CAS  Google Scholar 

  • Yates FE (ed.) (1987). Self-Organizing Systems: The Emergence of Order. Plenum, New York.

    Google Scholar 

  • Yates FE (1992). Fractal applications in biology:scaling time in biochemical networks. Methods Enzymol 210: 636–675

    PubMed  CAS  Google Scholar 

  • Yates FE, Yates LB (2008). Ultradian rhythms as the dynamic signature of life. In Lloyd D, Rossi ER (eds.) Ultradian Rhythms from Molecules to Mind: A New Vision of Life. Springer, Berlin.

    Google Scholar 

  • Yuan Z, Medina MA, Boiteux A, Müller SC, Hess B (1990). The role of fructose 2, 6-bisphosphate in glycollytic oscillations in extracts and cells of Saccharomyces cerevisae. Eur J Biochem 192: 791–795

    PubMed  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to D. Lloyd .

Editor information

Ulrich Lüttge Wolfram Beyschlag Burkhard Büdel Dennis Francis

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Lloyd, D. (2009). Oscillations, Synchrony and Deterministic Chaos. In: Lüttge, U., Beyschlag, W., Büdel, B., Francis, D. (eds) Progress in Botany. Progress in Botany, vol 70. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68421-3_4

Download citation

Publish with us

Policies and ethics