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

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.

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

References

• 1 Albino,  A. M.,, Rapp,  P. E.,, Abraham,  N. B.,, and Passamante,  A.. (1996) Measures of spatiotemporal dynamics. Physica D. 96
• 2 Anishchenko,  V.,, Moss,  F.,, Neiman,  A.,, and Schimansky-Geier,  L.. (1999);  Stochastic resonance: noise induced order.  Sov. Phys. Usp.. 42 (1) 7-36
• 3 Bak,  P.,, Tang,  C.,, and Wiesenfeld,  K.. (1988);  Self-organized criticality.  Phys. Rev. . A 38 364-371
• 4 Bar-Yam, Y.. (1997) Dynamics of complex systems. Reading, Mass.; Addison-Wesley
• 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
• 6 Bezrukov,  S. M., and Vodyanoy,  I.. (1995);  Noise-induced enhancement of signal transduction across voltage-dependent ion channels.  Nature. 378 362-364
• 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
• 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
• 9 Blasius,  B.,, Huppert,  A.,, and Stone,  L.. (1999 a);  Complex dynamics and phase synchronization in spatially extended ecological systems.  Nature. 399 354-359
• 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
• 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
• 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
• 13 Cermakian,  N., and Sassone-Corsi,  P.. (2001);  Rhythmes biologiques: les secrets d'une horloge.  La Recherche. 338 38-42
• 14 Cladis,  P. E.,, Palffy-Muhoray,  P., eds.. (1995) Spatio-Temporal Patterns in Nonequilibrium Complex Systems. Reading, Mass.; Addison-Wesley
• 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
• 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
• 17 Ermentrout,  G. B., and Edelstein-Keshet,  L.. (1993);  Cellular automata approaches to biological modelling.  J. Theor. Biol.. 160 97-133
• 18 Gammaitoni,  L.,, Hänggi,  P.,, Jung,  P.,, and Marchesoni,  F.. (1998);  Stochastic resonance.  Rev. Mod. Physics. 70 223-287
• 19 Garcia-Ojalvo,  J., and Sancho,  J. M.. (1999) Noise in spatially extended systems. Berlin; Springer
• 20 Gering,  W. J.. (1992) Entwicklung der Gene. Weinheim; Spektrum der Wissenschaft Verlagsgesellschaft
• 21 Gierer,  A.. (1998) Im Spiegel der Natur erkennen wir uns selbst. Wissenschaft und Menschenbild. Reinbek bei Hamburg; Rowohlt
• 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
• 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
• 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
• 25 Hütt,  M. Th.. (2001) Datenanalyse in der Biologie. Heidelberg; Springer
• 26 Hütt,  M. Th., and Neff,  R.. (2001);  Quantification of spatio-temporal phenomena by means of cellular automata techniques.  Physica A.. 289 498-516
• 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
• 28 Kadar,  S.,, Wang,  J.,, and Showalter,  K.. (1998);  Noise-supported travelling waves in subexcitable media.  Nature. 39 770-772
• 29 Kapitaniak,  T., and Bishop,  S. R.. (1999) The illustrated dictionary of nonlinear dynamics and chaos. Chichester; Wiley
• 30 Kaplan,  D., and Glass,  L.. (1995) Nonlinear Dynamics and Chaos. New York; Springer
• 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
• 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
• 33 Lee,  S.-G.,, Neiman,  A.,, and Kim,  S.. (1998);  Coherence resonance in a Hodgkin-Huxley neuron.  Phys. Rev. E. 57 3292-3297
• 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
• 35 Lorenz,  E.. (1963);  Deterministic nonperiodic flow.  J. Atmos. Sci.. 2 130-141
• 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
• 37 Lüttge,  U., and Beck,  F.. (1992);  Endogenous rhythms and chaos in crassulacean acid metabolism.  Planta. 188 28-38
• 38 Lüttge,  U.. (2000);  The tonoplast functioning as a master switch for circadian regulation of crassulacean acid metabolism.  Planta. 211 761-769
• 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
• 40 Mackey,  M. C., and Glass,  L.. (1977);  Oscillations and chaos in physiological control systems.  Science. 197 287-289
• 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
• 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
• 43 May,  R. M.. (1976);  Simple mathematical models with very complicated dynamics.  Nature. 26 459-467
• 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
• 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
• 46 Meinhardt,  H.. (1995) The alogrithmic beauty of sea shells. Berlin; Springer
• 47 Meinhardt,  H.. (1997);  Biological pattern formation as a complex dynamic phenomenon.  Int. J. Bifurc. Chaos. 7 1-26
• 48 Mosekilde,  E., and Mouristen,  O. G., eds.. (1995) Modelling the dynamics of living systems: nonlinear phenomena and pattern formation. New York; Springer
• 49 Moss,  F.,, Pierson,  D.,, and O'Gorman,  D.. (1994);  Stochastic resonance: tutorial and update.  Int. J. Bifurc. Chaos. 4 1383-1397
• 50 Moss,  F.. (2000) Stochastic resonance: looking forward:. Self-organized biological dynamics and nonlinear control. Wallaczek, J., ed. Cambridge; Cambridge University Press
• 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
• 52 Murray,  J.. (1989) Mathematical Biology. New York; Springer [Chapter 6]: 140-178
• 53 Nature Medicine Editorial. (2000);  The changing face of biomedical research.  Nature Medicine. 6 113
• 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
• 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
• 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
• 57 Nimmo,  H. G.. (2000);  The regulation of phosphoenolpyruvate carboxylase in CAM plants.  Trends in Plant Science. 5 75-80
• 58 Nijhout,  H. F.,, Nadel,  L.,, and Stein,  D. L., eds.. (1997) Pattern Formation in the Physical and Biological Sciences. Reading, Mass.; Addison-Wesley
• 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
• 60 Pikovsky,  A. S., and Kurths,  J.. (1997);  Coherence resonance in a noise driven excitable system.  Phys. Rev. Lett.. 78 775-778
• 61 Poolman,  M. G.,, Fell,  D. A.,, and Thomas,  S.. (2000);  Modelling photosynthesis and its control.  J. Exp. Botany. 51 319-328
• 62 Pyke,  K.. (1994);  Arabidopsis - its use in the genetic and molecular analysis of plant morphogenesis.  New Phytol.. 128 19-37
• 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
• 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
• 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
• 66 Schreiber,  T.. (1999);  Nonlinear time series analysis.  Physics Reports. 308 1-64
• 67 Solé,  R. V.,, Manrubia,  S. C.,, Luque,  B.,, Delgado,  J.,, and Bascompte,  J.. (1996);  Phase transitions and complex systems.  Complexity. 2 13-29
• 68 Strogatz,  S.. (1994) Nonlinear dynamics and chaos with applications to physics, biology, chemistry, and engineering. Reading, Mass.; Addison-Wesley
• 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
• 70 Theißen,  G., and Saedler,  H.. (1997);  Molecular architects of plant body plans.  Prog. Bot.. 59 227-256
• 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
• 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
• 73 Winfree,  A. T.. (1972);  Spiral waves of chemical activity.  Science. 175 634-637
• 74 Winfree,  A. T.. (1974);  Rotating chemical reactions.  Scientific American. 230 82-89
• 75 Winfree,  A. T.. (1980) The geometry of biological time. New York; Springer
• 76 Woodward,  F. J.. (1987) Climate and plant distribution. Cambridge; Cambridge University Press

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

Section Editor: H. Rennenberg