Download PDF
Plant Biol (Stuttg) 2002; 4(3): 281-297
DOI: 10.1055/s-2002-32339
Review Article
© Georg Thieme Verlag Stuttgart · New York

Nonlinear Dynamics as a Tool for Modelling in Plant Physiology[1]

M.-Th. Hütt, U. Lüttge
  • Institute of Botany, Darmstadt University of Technology, Darmstadt, Germany
Further Information

Publication History

October 15, 2001

March 18, 2002

Publication Date:
20 June 2002 (online)

   Permissions and Reprints

Abstract

We show that recent methods from nonlinear dynamics and complexity theory developed in theoretical physics can and should be applied to the description and analysis of systems in plant physiology, in particular to interpret data series in time and space, i.e., rhythms and pattern formation processes. We start with a brief introduction into the key ideas of nonlinear dynamics, including a short outline of topics which are currently the focus of theoretical interest, in particular the concept, origin and possible function of noise in biological systems. The theoretical concepts are applied to a model system of circadian rhythmicity in plant physiology, crassulacean acid metabolism. With this application, we illustrate the main idea of our article: nonlinear dynamics serves as a remarkable tool for unveiling the internal mechanisms at work in a system's process of self-organization.

Key words

Bifurcation - cellular automata - chaos - crassulacean acid metabolism (CAM) - minimal model - noise - stochastic resonance - synchronization

1 Dedicated to Professor Dr. Freder Beck on the occasion of his 75th birthday.

References

  • 1 Albino,  A. M.,, Rapp,  P. E.,, Abraham,  N. B.,, and Passamante,  A.. (1996) Measures of spatiotemporal dynamics. Physica D. 96
    Search in Google Scholar
  • 2 Anishchenko,  V.,, Moss,  F.,, Neiman,  A.,, and Schimansky-Geier,  L.. (1999);  Stochastic resonance: noise induced order.  Sov. Phys. Usp.. 42 (1) 7-36
    PubMedSearch in Google Scholar
  • 3 Bak,  P.,, Tang,  C.,, and Wiesenfeld,  K.. (1988);  Self-organized criticality.  Phys. Rev. . A 38 364-371
    PubMedSearch in Google Scholar
  • 4 Bar-Yam, Y.. (1997) Dynamics of complex systems. Reading, Mass.; Addison-Wesley
    Search in Google Scholar
  • 5 Beck,  F.,, Blasius,  B.,, Lüttge,  U.,, Neff,  R.,, and Rascher,  U.. (2001);  Stochastic noise interferes coherently with a model biological clock and produces specific dynamic behaviour.  Proc. Roy. Soc. Lond. B.. 268 1307-1313
    PubMedSearch in Google Scholar
  • 6 Bezrukov,  S. M., and Vodyanoy,  I.. (1995);  Noise-induced enhancement of signal transduction across voltage-dependent ion channels.  Nature. 378 362-364
    CrossrefPubMedSearch in Google Scholar
  • 7 Blasius,  B.,, Beck,  F.,, and Lüttge,  U.. (1997);  A model for photosynthetic oscillations in crassulacean acid metabolism (CAM).  J. Theor. Biol.. 184 345-351
    CrossrefPubMedSearch in Google Scholar
  • 8 Blasius,  B.,, Beck,  F.,, and Lüttge  U.. (1998);  Oscillatory model of crassulacean acid metabolism: structural analysis and stability boundaries with a discrete hysteresis switch.  Plant, Cell and Environment. 21 775-784
    CrossrefPubMedSearch in Google Scholar
  • 9 Blasius,  B.,, Huppert,  A.,, and Stone,  L.. (1999 a);  Complex dynamics and phase synchronization in spatially extended ecological systems.  Nature. 399 354-359
    CrossrefPubMedSearch in Google Scholar
  • 10 Blasius,  B.,, Neff,  R.,, Beck,  F.,, and Lüttge,  U.. (1999 b);  Oscillatory model of crassulacean acid metabolism with a dynamic hysteresis switch.  Proc. Roy. Soc. Lond. B.. 266 93-101
    PubMedSearch in Google Scholar
  • 11 Borland,  A. M.,, Hartwell,  J.,, Jenkins,  G. I.,, Wilkins,  M. B.,, and Nimmo,  H. G.. (1999);  Metabolic control overrides circadian regulation of phosphoenolpyruvate carboxylase kinase and CO2 fixation in crassulacean acid metabolism.  Plant Physiology. 121 889-896
    CrossrefPubMedSearch in Google Scholar
  • 12 Busch,  H.,, Hütt,  M. Th.,, and Kaiser,  F.. (2001);  The effect of colored noise on networks of nonlinear oscillators.  Phys. Rev. E.. 64 021105
    CrossrefPubMedSearch in Google Scholar
  • 13 Cermakian,  N., and Sassone-Corsi,  P.. (2001);  Rhythmes biologiques: les secrets d'une horloge.  La Recherche. 338 38-42
    PubMedSearch in Google Scholar
  • 14 Cladis,  P. E.,, Palffy-Muhoray,  P., eds.. (1995) Spatio-Temporal Patterns in Nonequilibrium Complex Systems. Reading, Mass.; Addison-Wesley
    Search in Google Scholar
  • 15 De Pury,  D. G. G., and Farquhar,  G. D.. (1997);  Simple scaling of photosynthesis from leaves to canopies without errors of big-leaf models.  Plant, Cell and Environment. 20 537-557
    CrossrefPubMedSearch in Google Scholar
  • 16 Douglass,  J. K.,, Wilkins,  L.,, Pantazelou,  E.,, and Moss  F.. (1993);  Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance.  Nature. 365 337-340
    CrossrefPubMedSearch in Google Scholar
  • 17 Ermentrout,  G. B., and Edelstein-Keshet,  L.. (1993);  Cellular automata approaches to biological modelling.  J. Theor. Biol.. 160 97-133
    CrossrefPubMedSearch in Google Scholar
  • 18 Gammaitoni,  L.,, Hänggi,  P.,, Jung,  P.,, and Marchesoni,  F.. (1998);  Stochastic resonance.  Rev. Mod. Physics. 70 223-287
    PubMedSearch in Google Scholar
  • 19 Garcia-Ojalvo,  J., and Sancho,  J. M.. (1999) Noise in spatially extended systems. Berlin; Springer
    Search in Google Scholar
  • 20 Gering,  W. J.. (1992) Entwicklung der Gene. Weinheim; Spektrum der Wissenschaft Verlagsgesellschaft
    Search in Google Scholar
  • 21 Gierer,  A.. (1998) Im Spiegel der Natur erkennen wir uns selbst. Wissenschaft und Menschenbild. Reinbek bei Hamburg; Rowohlt
    Search in Google Scholar
  • 22 Glass,  L.. (1997) Dynamical disease - the impact of nonlinear dynamics and chaos on cardiology and medicine. The impact of chaos on science and society. Grebogi, C. and Yorke, J. A., eds. Tokyo; United Nations University Press
    Search in Google Scholar
  • 23 Guevara,  M. R.,, Glass,  L.,, and Shrier,  A.. (1981);  Phase locking, period-doubling bifurcations and irregular dynamics in periodically stimulated cardiac cells.  Science. 214 1350-1352
    PubMedSearch in Google Scholar
  • 24 Haefner,  J. W.,, Buckley,  T. N.,, and Mott , K. A.. (1997);  A spatially explicit model of patchy stomatal responses to humidity.  Plant, Cell and Environment. 20 1087-1097
    CrossrefPubMedSearch in Google Scholar
  • 25 Hütt,  M. Th.. (2001) Datenanalyse in der Biologie. Heidelberg; Springer
    Search in Google Scholar
  • 26 Hütt,  M. Th., and Neff,  R.. (2001);  Quantification of spatio-temporal phenomena by means of cellular automata techniques.  Physica A.. 289 498-516
    PubMedSearch in Google Scholar
  • 27 Hütt,  M. Th.,, Rascher,  U.,, Beck,  F.,, and Lüttge,  U.. (2002);  Period-2 cycles and 2 : 1 phase locking in a biological clock driven by temperature pulses.  J. Theor. Biol.. in press
    PubMedSearch in Google Scholar
  • 28 Kadar,  S.,, Wang,  J.,, and Showalter,  K.. (1998);  Noise-supported travelling waves in subexcitable media.  Nature. 39 770-772
    PubMedSearch in Google Scholar
  • 29 Kapitaniak,  T., and Bishop,  S. R.. (1999) The illustrated dictionary of nonlinear dynamics and chaos. Chichester; Wiley
    Search in Google Scholar
  • 30 Kaplan,  D., and Glass,  L.. (1995) Nonlinear Dynamics and Chaos. New York; Springer
    Search in Google Scholar
  • 31 Kendall,  B. E.,, Schaffer,  W. M.,, Tidd,  C. W.,, and Olsen,  L. F.. (1997) The impact of chaos on biology: promising directions for research. The impact of chaos on science and society. Grebogi, C. and Yorke, J. A., eds. Tokyo; United Nations Universtiy Press
    Search in Google Scholar
  • 32 Kohl,  P.,, Noble,  D.,, Winslow,  R. L.,, and Hunter,  P. J.. (2000);  Computational modelling of biological systems: tools and visions.  Philosophical Transactions of the Royal Society A. 358 579-610
    PubMedSearch in Google Scholar
  • 33 Lee,  S.-G.,, Neiman,  A.,, and Kim,  S.. (1998);  Coherence resonance in a Hodgkin-Huxley neuron.  Phys. Rev. E. 57 3292-3297
    CrossrefPubMedSearch in Google Scholar
  • 34 Longtin,  A.,, Bulsara,  A.,, and Moss,  F.. (1991);  Time interval sequences in bistable systems and the noise induced transmission of information by sensory neurons.  Phys. Rev. Lett.. 67 656-660
    CrossrefPubMedSearch in Google Scholar
  • 35 Lorenz,  E.. (1963);  Deterministic nonperiodic flow.  J. Atmos. Sci.. 2 130-141
    PubMedSearch in Google Scholar
  • 36 Ludwig,  D.,, Jones,  D. D.,, and Holling,  C. S.. (1978);  Qualitative analysis of insect outbreak systems: the spruce budworm and forest.  J. Anim. Ecol.. 47 315-323
    PubMedSearch in Google Scholar
  • 37 Lüttge,  U., and Beck,  F.. (1992);  Endogenous rhythms and chaos in crassulacean acid metabolism.  Planta. 188 28-38
    CrossrefPubMedSearch in Google Scholar
  • 38 Lüttge,  U.. (2000);  The tonoplast functioning as a master switch for circadian regulation of crassulacean acid metabolism.  Planta. 211 761-769
    CrossrefPubMedSearch in Google Scholar
  • 39 Mackey,  M. C.. (1997) Mathematical models of hematopoietic cell replication and control. The Art of Mathematical Modelling: Case Studies in Ecology, Physiology and Cell Biology. Othmer, H. G., Adler, F. R., Lewis, M. A., and Dallon, J. C., eds. Eaglewood Cliffs, NJ; Prenctice Hall
    Search in Google Scholar
  • 40 Mackey,  M. C., and Glass,  L.. (1977);  Oscillations and chaos in physiological control systems.  Science. 197 287-289
    CrossrefPubMedSearch in Google Scholar
  • 41 Magnani,  F.,, Leonardi,  S.,, Tognetti,  R.,, Grace,  J.,, and Borghetti,  M.. (1998);  Modelling the surface conductance of a broad-leaf canopy: effects of partial decoupling from the atmosphere.  Plant, Cell and Environment. 21 867-879
    CrossrefPubMedSearch in Google Scholar
  • 42 Martiel,  J. L., and Goldbeter,  A.. (1987);  A model based on receptor desensitization for cyclic AMP signaling in Dictyostelium cells.  Biophys. J.. 52 807-828
    PubMedSearch in Google Scholar
  • 43 May,  R. M.. (1976);  Simple mathematical models with very complicated dynamics.  Nature. 26 459-467
    PubMedSearch in Google Scholar
  • 44 Mayer,  U.,, Torres Ruiz,  R. A.,, Berleth,  T.,, Miséra,  S., and Jürgens,  G.. (1991);  Mutations affecting body organization in the Arabidopsis embryo.  Nature. 353 402-407
    CrossrefPubMedSearch in Google Scholar
  • 45 Medlyn,  B. E.,, Badeck,  F.-W.,, De Pury,  D. G. G.,, Barton,  C. V. M.,, Broadmeadow,  M.,, Ceulemans,  R.,, De Angelis,  P.,, Forstreuter,  M.,, Jach,  M. E.,, Kellomäki,  S.,, Laitat,  E.,, Marek,  M.,, Philippot,  S.,, Rey,  A.,, Strassemeyer,  J.,, Laitinen,  K.,, Liozon,  R.,, Portier,  B.,, Roberntz,  P.,, Wang,  K.,, and Jstbid,  P. G.. (1999);  Effects of elevated [CO2] on photosynthesis in European forest species: a meta-analysis of model parameters.  Plant, Cell and Environment. 22 1475-1495
    CrossrefPubMedSearch in Google Scholar
  • 46 Meinhardt,  H.. (1995) The alogrithmic beauty of sea shells. Berlin; Springer
    Search in Google Scholar
  • 47 Meinhardt,  H.. (1997);  Biological pattern formation as a complex dynamic phenomenon.  Int. J. Bifurc. Chaos. 7 1-26
    CrossrefPubMedSearch in Google Scholar
  • 48 Mosekilde,  E., and Mouristen,  O. G., eds.. (1995) Modelling the dynamics of living systems: nonlinear phenomena and pattern formation. New York; Springer
    Search in Google Scholar
  • 49 Moss,  F.,, Pierson,  D.,, and O'Gorman,  D.. (1994);  Stochastic resonance: tutorial and update.  Int. J. Bifurc. Chaos. 4 1383-1397
    CrossrefPubMedSearch in Google Scholar
  • 50 Moss,  F.. (2000) Stochastic resonance: looking forward:. Self-organized biological dynamics and nonlinear control. Wallaczek, J., ed. Cambridge; Cambridge University Press
    Search in Google Scholar
  • 51 Mulquiney,  P. J., and Kuchel,  P. W.. (1999);  Model of 2,3-bisphosphoglycerate metabolism in the human erythrocyte based on detailed enzyme kinetic equations: equations and parameter refinement.  Biochem. J.. 342 581-596
    CrossrefPubMedSearch in Google Scholar
  • 52 Murray,  J.. (1989) Mathematical Biology. New York; Springer [Chapter 6]: 140-178
    Search in Google Scholar
  • 53 Nature Medicine Editorial. (2000);  The changing face of biomedical research.  Nature Medicine. 6 113
    CrossrefPubMedSearch in Google Scholar
  • 54 Neff,  R.,, Blasius,  B.,, Beck,  F.,, and Lüttge,  U.. (1998);  Thermodynamics and energetics of the tonoplast membrane operating as a hysteresis switch in an oscillatory model of crassulacean acid metabolism.  J. Memb. Biol.. 165 37-43
    PubMedSearch in Google Scholar
  • 55 Niinemets,  Ü., and Tenhunen,  J. D.. (1997);  A model separating leaf structural and physiological effects on carbon gain along light gradients for the shade-tolerant species Acer saccharum.  Plant, Cell and Enviroment. 20 845-866
    CrossrefPubMedSearch in Google Scholar
  • 56 Niinemets,  Ü.,, Tenhunen,  J. D.,, and Steinbrecher,  R.. (1999);  A model of isoprene emission based on energetic requirements for isoprene synthesis and leaf photosynthetic properties for Liquidambar and Quercus.  Plant, Cell and Environment. 22 1319-1335
    CrossrefPubMedSearch in Google Scholar
  • 57 Nimmo,  H. G.. (2000);  The regulation of phosphoenolpyruvate carboxylase in CAM plants.  Trends in Plant Science. 5 75-80
    CrossrefPubMedSearch in Google Scholar
  • 58 Nijhout,  H. F.,, Nadel,  L.,, and Stein,  D. L., eds.. (1997) Pattern Formation in the Physical and Biological Sciences. Reading, Mass.; Addison-Wesley
    Search in Google Scholar
  • 59 Pearcy,  R. W.,, Gross,  L. J.,, and He,  D.. (1997);  An improved dynamic model of photosynthesis for estimation of carbon gain in sunfleck light regimes.  Plant, Cell and Enviroment. 20 411-424
    PubMedSearch in Google Scholar
  • 60 Pikovsky,  A. S., and Kurths,  J.. (1997);  Coherence resonance in a noise driven excitable system.  Phys. Rev. Lett.. 78 775-778
    CrossrefPubMedSearch in Google Scholar
  • 61 Poolman,  M. G.,, Fell,  D. A.,, and Thomas,  S.. (2000);  Modelling photosynthesis and its control.  J. Exp. Botany. 51 319-328
    PubMedSearch in Google Scholar
  • 62 Pyke,  K.. (1994);  Arabidopsis - its use in the genetic and molecular analysis of plant morphogenesis.  New Phytol.. 128 19-37
    PubMedSearch in Google Scholar
  • 63 Rascher,  U.. (2001) Der endogene CAM-Rhythmus von Kalanchoë daigremontiana als nicht-lineares Modellsystem zum Verständnis der raum-zeitlichen Dynamik einer biologischen Uhr. Darmstadt; PhD thesis
    Search in Google Scholar
  • 64 Rascher,  U.,, Hütt,  M.-T.,, Siebke,  K.,, Osmond,  B.,, Beck,  F.,, and 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
    PubMedSearch in Google Scholar
  • 65 Running,  S. W.. (1990) Estimating terrestrial primary productivity by combining remote sensing and ecosystem simulation. Remote sensing of biopshere functioning. Ecological Studies, Vol. 79. Hobbs, R. J. and Mooney, H. A., eds. Berlin, Heidelberg, New York; Springer
    Search in Google Scholar
  • 66 Schreiber,  T.. (1999);  Nonlinear time series analysis.  Physics Reports. 308 1-64
    CrossrefPubMedSearch in Google Scholar
  • 67 Solé,  R. V.,, Manrubia,  S. C.,, Luque,  B.,, Delgado,  J.,, and Bascompte,  J.. (1996);  Phase transitions and complex systems.  Complexity. 2 13-29
    CrossrefPubMedSearch in Google Scholar
  • 68 Strogatz,  S.. (1994) Nonlinear dynamics and chaos with applications to physics, biology, chemistry, and engineering. Reading, Mass.; Addison-Wesley
    Search in Google Scholar
  • 69 Taylor,  D. K.,, Rank,  D. R.,, Keiser,  D. R.,, Smith,  B. N.,, Criddle,  R. S.,, and Hansen,  L. D.. (1998);  Modelling temperature effects on growth-respiration relations of maize.  Plant, Cell and Environment. 21 1143-1151
    CrossrefPubMedSearch in Google Scholar
  • 70 Theißen,  G., and Saedler,  H.. (1997);  Molecular architects of plant body plans.  Prog. Bot.. 59 227-256
    PubMedSearch in Google Scholar
  • 71 Walcroft,  A. S.,, Whitehead,  D.,, Silvester,  W. B.,, and Kelliher,  F. M.. (1997);  The response of photosynthetic model parameters to temperature and nitrogen concentration in Pinus radiata D. Don.  Plant, Cell and Environment. 20 1338-1348
    CrossrefPubMedSearch in Google Scholar
  • 72 Williams,  M.,, Malhi,  Y.,, Nobre,  A. D.,, Rastetter,  E. B.,, Grace,  J.,, and Pereira,  M. G. P.. (1998);  Seasonal variation in net carbon exchange and evapotranspiration in a Brazilian rain forest: a modelling analysis.  Plant, Cell and Environment. 21 953-968
    CrossrefPubMedSearch in Google Scholar
  • 73 Winfree,  A. T.. (1972);  Spiral waves of chemical activity.  Science. 175 634-637
    PubMedSearch in Google Scholar
  • 74 Winfree,  A. T.. (1974);  Rotating chemical reactions.  Scientific American. 230 82-89
    PubMedSearch in Google Scholar
  • 75 Winfree,  A. T.. (1980) The geometry of biological time. New York; Springer
    Search in Google Scholar
  • 76 Woodward,  F. J.. (1987) Climate and plant distribution. Cambridge; Cambridge University Press
    Search in Google Scholar

1 Dedicated to Professor Dr. Freder Beck on the occasion of his 75th birthday.

M.-Th. Hütt

Institute of Botany
Darmstadt University of Technology

Schnittspahnstr. 3 - 5
64287 Darmstadt
Germany

Email: huett@bio.tu-darmstadt.de

Section Editor: H. Rennenberg