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Encyclopedia of Complexity and Systems Science pp 1–22Cite as

Biochemistry, Chaotic Dynamics, Noise, and Fractal Space in

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Definition of the Subject and Its Importance

The description of fractals, noise, and chaotic dynamics is deeply associated with the concepts of scaling and critical phenomena. The discovery of the occurrence of scaling, criticality, and colored noise behavior in mitochondrial networks – at a fundamental level of cellular organization and dynamics as it relates to energetics, life, and death – further emphasizes the role of fractals and chaos in biochemistry.

The concept of fractals introduced by Mandelbrot was initially applied to the geometric description of irregular objects but quickly spread to the field of dynamics where it joined chaos. Initially described by Lorenz in the context of meteorology, chaos extended its influence from fluid dynamics to biology and biochemistry.

Analysis of time series of biological variables with techniques inspired by fractal and chaos theories is providing a more clear understanding of the relationship between the scaling in space and time exhibited...

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Abbreviations

Chaotic :

Is a dynamic, nonperiodic behavior that describes “turbulent” motions in time that exhibit sensitive dependence on initial conditions.

Critical :

Applies to states or phenomena characterized by their extreme sensitivity to small changes or perturbations.

Deterministic chaos :

Is a mechanism which can generate random-looking time series from simple deterministic systems. This complicated behavior can be a consequence of very simple equations that describe the dynamics. Such a system is highly dependent on its initial conditions and is unpredictable in the longer term (i.e., for more than a few cycles), as small divergence becomes progressively amplified.

Deterministic :

Dynamic behavior specified by difference or differential equations.

Dynamic fractals :

Self-similar statistical objects described in time series, i.e., the time-dependent behavior of an observable (e.g., mitochondrial membrane potential, reduced nicotinamide adenine dinucleotide [NADH] concentration). These statistical fractals exhibit scale-free dynamics, i.e., they span a wide range of temporal scales simultaneously (e.g., from milliseconds to several minutes), in such a way that the short-term fluctuations are related to longer-term trends in the dynamic behavior. A fundamental property of dynamic fractals is that affecting a process on one time scale will affect other processes on all time scales.

Emergent behavior :

Novel and sometimes surprising macroscopic properties arising from the self-organizing capacity of the internal structure and dynamics of a biological system. Self-organization appears in nonlinear open systems (cells, organisms, ecosystems) that perform away from thermodynamic equilibrium, i.e., are constantly being driven by exchange of matter, energy, and information with the environment.

Fractal :

Is an object of geometric, statistical, or dynamical nature that obeys power laws of the form \( \mathrm{M}\left(\mathrm{L}\right)\infty {\mathrm{L}}^{\mathrm{D}} \), with D as the non-integer fractal dimension. Mandelbrot introduced “fractal geometry” and defined a fractal “as a shape made of parts similar to the whole in some way.”

Mitochondria :

Subcellular organelles considered to be the main energy producers of the cell as well as the source of important signaling functions, among them apoptosis or cell death.

Network :

The collective organization of an ensemble of objects, or groups of them, in space (structural and topological) and time (dynamics).

Noise :

Refers to random time series originating from stochastic fluctuations in events of different nature (e.g., molecular, electric, energetic, as in the biological realm) that under certain circumstances may represent a signal.

Scale free :

Refers to the geometry or dynamic of structures or processes with no single characteristic scale (space) or frequency (temporal).

Scaling :

A quantity scales if it changes according to a power law whose exponent is insensitive to the details of the processes involved, a feature usually referred as universality.

Bibliography

Primary Literature

  • Akar FG, Aon MA, Tomaselli GF, O’Rourke B (2005) The mitochondrial origin of postischemic arrhythmias. J Clin Invest 115:3527–3535

    Article  Google Scholar 

  • Alon U (2007) An introduction to systems biology. Design principles of biological circuits. Chapman & Hall/CRC, London

    MATH  Google Scholar 

  • Alon U, Surette MG, Barkai N, Leibler S (1999) Robustness in bacterial chemotaxis. Nature 397:168–171

    Article  ADS  Google Scholar 

  • Aon MA (2010) From isolated to networked: a paradigmatic shift in mitochondrial physiology. Front Physiol 1. doi: 10.3389/fphys.2010.00020

    Google Scholar 

  • Aon MA (2013) Complex systems biology of networks: the riddle and the challenge. In: Aon MA, Saks V, Schlattner U (eds) Systems biology of metabolic and signaling networks. Energy, mass and information transfer. Springer, Berlin/Heidelberg/New York/Dordrecht/London, pp 19–35

    Google Scholar 

  • Aon MA, Cortassa S (1993) An allometric interpretation of the spatio-temporal organization of molecular and cellular processes. Mol Cell Biochem 120:1–13

    Article  Google Scholar 

  • Aon MA, Cortassa S (1997) Dynamic biological organization. Fundamentals as applied to cellular systems. Chapman & Hall, London

    Book  Google Scholar 

  • Aon MA, Cortassa S (2006) Metabolic dynamics in cells viewed as multilayered, distributed, mass-energy-information networks. In: Encyclopedia of genetics, genomics, proteomics and bioinformatics. Wiley Interscience, New York

    Google Scholar 

  • Aon MA, Cortassa S (2009) Chaotic dynamics, noise and fractal space in biochemistry. In: Meyers R (ed) Encyclopedia of complexity and systems science. Springer, New York

    Google Scholar 

  • Aon MA, Cortassa S (2012) Mitochondrial network energetics in the heart. Wiley Interdiscip Rev Syst Biol Med 4:599–613

    Article  Google Scholar 

  • Aon MA, Gomez-Casati DF, Iglesias AA, Cortassa S (2001) Ultrasensitivity in (supra)molecularly organized and crowded environments. Cell Biol Int 25:1091–1099

    Article  Google Scholar 

  • Aon MA, Cortassa S, Marban E, O’Rourke B (2003) Synchronized whole cell oscillations in mitochondrial metabolism triggered by a local release of reactive oxygen species in cardiac myocytes. J Biol Chem 278:44735–44744

    Article  Google Scholar 

  • Aon MA, Cortassa S, O’Rourke B (2004a) Percolation and criticality in a mitochondrial network. Proc Natl Acad Sci U S A 101:4447–4452

    Article  ADS  Google Scholar 

  • Aon MA, O’Rourke B, Cortassa S (2004b) The fractal architecture of cytoplasmic organization: scaling, kinetics and emergence in metabolic networks. Mol Cell Biochem 256–257:169–184

    Article  Google Scholar 

  • Aon MA, Cortassa S, Akar FG, O’Rourke B (2006a) Mitochondrial criticality: a new concept at the turning point of life or death. Biochim Biophys Acta 1762:232–240

    Article  Google Scholar 

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

    Article  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

    Article  ADS  Google Scholar 

  • Aon MA, Cortassa S, Maack C, O’Rourke B (2007b) Sequential opening of mitochondrial ion channels as a function of glutathione redox thiol status. J Biol Chem 282:21889–21900

    Article  Google Scholar 

  • Aon MA, Cortassa S, O’Rourke B (2007c) On the network properties of mitochondria, In: Saks V, (ed.) Molecular system bioenergetics: energy for life, Wiley-VCH Verlag GmbH, Weinheim, pp 111–135

    Google Scholar 

  • Aon MA, Cortassa S, O’Rourke B (2008a) Is there a mitochondrial clock? In: Lloyd D, Rossi EL (eds) Ultradian rhythms from molecules to mind. A new vision of life. Springer, New York, pp 129–144

    Chapter  Google Scholar 

  • Aon MA, Cortassa S, O’Rourke B (2008b) Mitochondrial oscillations in physiology and pathophysiology. Adv Exp Med Biol 641:98–117

    Article  MathSciNet  Google Scholar 

  • Aon MA, Roussel MR, Cortassa S, O’Rourke B, Murray DB, Beckmann M, Lloyd D (2008c) The scale-free dynamics of eukaryotic cells. PLoS One 3, e3624

    Article  ADS  Google Scholar 

  • Aon MA, Cortassa S, Lloyd D (2011) Chaos in biochemistry and physiology. In: Encyclopedia of molecular cell biology and molecular medicine. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

    Google Scholar 

  • Arkin AP (2000) Signal processing by biochemical reaction networks. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Barabasi AL (2009) Scale-free networks: a decade and beyond. Science (New York, NY) 325:412–413

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Barabasi AL, Oltvai ZN (2004) Network biology: understanding the cell’s functional organization. Nat Rev Genet 5:101–113

    Article  Google Scholar 

  • Barkai N, Leibler S (1997) Robustness in simple biochemical networks. Nature 387:913–917

    Article  ADS  Google Scholar 

  • Bassingthwaighte JB, Liebovitch LS, West BJ (1994) Fractal physiology. Oxford University Press for the American Physiological Society, New York

    Book  Google Scholar 

  • Becskei A, Serrano L (2000) Engineering stability in gene networks by autoregulation. Nature 405:590–593

    Article  ADS  Google Scholar 

  • Becskei A, Seraphin B, Serrano L (2001) Positive feedback in eukaryotic gene networks: cell differentiation by graded to binary response conversion. EMBO J 20:2528–2535

    Article  Google Scholar 

  • Berry H (2002) Monte carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation. Biophys J 83:1891–1901

    Article  Google Scholar 

  • Boccara N (2004) Modeling complex systems. Springer, New York

    MATH  Google Scholar 

  • Brown JH, West GB, Enquist BJ (2000) Scaling in biology. Oxford University Press, New York

    MATH  Google Scholar 

  • Brown DA, Aon MA, Frasier CR, Sloan RC, Maloney AH, Anderson EJ, O’Rourke B (2010) Cardiac arrhythmias induced by glutathione oxidation can be inhibited by preventing mitochondrial depolarization. J Mol Cell Cardiol 48:673–679

    Article  Google Scholar 

  • Casolo G, Balli E, Taddei T, Amuhasi J, Gori C (1989) Decreased spontaneous heart rate variability in congestive heart failure. Am J Cardiol 64:1162–1167

    Article  Google Scholar 

  • Cortassa S, Aon MA (2013) Dynamics of mitochondrial redox and energy networks: Insights from an experimental-computational synergy. In: Aon MA, Saks V, Schlattner U (eds) Systems biology of metabolic and signaling networks. Energy, mass and information transfer. Springer, Berlin/Heidelberg/New York/Dordrecht/London, pp 115–144

    Google Scholar 

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

    Article  Google Scholar 

  • Cortassa SC, Aon MA, O’Rourke B, Jacques R, Tseng HJ, Marban E, Winslow RL (2006) A computational model integrating electrophysiology, contraction and mitochondrial bioenergetics in the ventricular myocyte. Biophys J 91:1564–1589

    Article  Google Scholar 

  • Cortassa S, Aon MA, Iglesias AA, Aon JC, Lloyd D (2012) An introduction to metabolic and cellular engineering, 2nd edn. World Scientific, Singapore

    Google Scholar 

  • Cvitanovic P (1989) Universality in chaos. Adam Hilger, Bristol/New York

    MATH  Google Scholar 

  • Enquist BJ, West GB, Brown JH (2000) Quarter-power allometric scaling in vascular plants: functional basis and ecological consequences. Oxford University Press, New York

    MATH  Google Scholar 

  • Ewing DJ (1991) Heart rate variability: an important new risk factor in patients following myocardial infarction. Clin Cardiol 14:683–685

    Article  MathSciNet  Google Scholar 

  • Fakhri N, Wessel AD, Willms C, Pasquali M, Klopfenstein DR, MacKintosh FC, Schmidt CF (2014) High-resolution mapping of intracellular fluctuations using carbon nanotubes. Science (New York, NY) 344:1031–1035

    Google Scholar 

  • Feigenbaum MJ (1980) Universal behavior in nonlinear systems. Los Alamos Sci 1:4–27

    MathSciNet  Google Scholar 

  • Garfinkel A, Spano ML, Ditto WL, Weiss JN (1992) Controlling cardiac chaos. Science (New York, NY) 257:1230–1235

    Google Scholar 

  • Garfinkel A, Chen PS, Walter DO, Karagueuzian HS, Kogan B, Evans SJ, Karpoukhin M, Hwang C, Uchida T, Gotoh M, Nwasokwa O, Sager P, Weiss JN (1997) Quasiperiodicity and chaos in cardiac fibrillation. J Clin Invest 99:305–314

    Article  Google Scholar 

  • Glass L (2009) Introduction to controversial topics in nonlinear science: is the normal heart rate chaotic? Chaos (Woodbury, NY) 19:028501

    Google Scholar 

  • Gleick J (1988) Chaos: making a new science. Penguin, New York

    MATH  Google Scholar 

  • Goldberger AL, West BJ (1987) Applications of nonlinear dynamics to clinical cardiology. Ann N Y Acad Sci 504:195–213

    Article  ADS  Google Scholar 

  • Goldberger AL, Bhargava V, West BJ, Mandell AJ (1985) On a mechanism of cardiac electrical stability. The fractal hypothesis. Biophys J 48:525–528

    Article  Google Scholar 

  • Goldberger AL, Amaral LA, Hausdorff JM, Ivanov P, Peng CK, Stanley HE (2002) Fractal dynamics in physiology: alterations with disease and aging. Proc Natl Acad Sci U S A 99(Suppl 1):2466–2472

    Article  ADS  Google Scholar 

  • Goldbeter A, Koshland DE Jr (1982) Sensitivity amplification in biochemical systems. Q Rev Biophys 15:555–591

    Article  Google Scholar 

  • Gomez-Casati DF, Aon MA, Iglesias AA (1999) Ultrasensitive glycogen synthesis in Cyanobacteria. FEBS Lett 446:117–121

    Article  Google Scholar 

  • Gomez-Casati DF, Cortassa S, Aon MA, Iglesias AA (2003) Ultrasensitive behavior in the synthesis of storage polysaccharides in Cyanobacteria. Planta 216:969–975

    Google Scholar 

  • Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9:189–208

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Helms V (2008) Principles of computational cell biology. Wiley-VCH, Weinheim

    Google Scholar 

  • Hiroi N, Lu J, Iba K, Tabira A, Yamashita S, Okada Y, Flamm C, Oka K, Kohler G, Funahashi A (2011) Physiological environment induces quick response – slow exhaustion reactions. Front Physiol 2:50

    Article  Google Scholar 

  • Hood L, Tian Q (2012) Systems approaches to biology and disease enable translational systems medicine. Genomics Proteomics Bioinformatics 10:181–185

    Article  Google Scholar 

  • Ivanov PC, Amaral LA, Goldberger AL, Havlin S, Rosenblum MG, Struzik ZR, Stanley HE (1999) Multifractality in human heartbeat dynamics. Nature 399:461–465

    Article  ADS  Google Scholar 

  • Ivanov PC, Nunes Amaral LA, Goldberger AL, Havlin S, Rosenblum MG, Stanley HE, Struzik ZR (2001) From 1/f noise to multifractal cascades in heartbeat dynamics. Chaos (Woodbury, NY) 11:641–652

    Article  MATH  ADS  Google Scholar 

  • Jin H, Nass RD, Joudrey PJ, Lyon AR, Chemaly ER, Rapti K, Akar FG (2010) Altered spatiotemporal dynamics of the mitochondrial membrane potential in the hypertrophied heart. Biophys J 98:2063–2071

    Article  Google Scholar 

  • Kembro JM, Aon MA, Winslow RL, O’Rourke B, Cortassa S (2013) Integrating mitochondrial energetics, redox and ROS metabolic networks: a two-compartment model. Biophys J 104:332–343

    Article  Google Scholar 

  • Kembro JM, Cortassa S, Aon MA (2014) Complex oscillatory redox dynamics with signaling potential at the edge between normal and pathological mitochondrial function. Front Physiol 5:257

    Article  Google Scholar 

  • Keulers M, Suzuki T, Satroutdinov AD, Kuriyama H (1996) Autonomous metabolic oscillation in continuous culture of Saccharomyces cerevisiae grown on ethanol. FEMS Microbiol Lett 142:253–258

    Article  Google Scholar 

  • Klevecz RR, Bolen J, Forrest G, Murray DB (2004) A genomewide oscillation in transcription gates DNA replication and cell cycle. Proc Natl Acad Sci U S A 101:1200–1205

    Article  ADS  Google Scholar 

  • Kopelman R (1988) Fractal reaction kinetics. Science (New York, NY) 241:1620–1626

    Google Scholar 

  • Koshland DE Jr, Goldbeter A, Stock JB (1982) Amplification and adaptation in regulatory and sensory systems. Science (New York, NY) 217:220–225

    Google Scholar 

  • Kosko B (2006) Noise. Viking, New York

    Google Scholar 

  • Kurz FT, Aon MA, O’Rourke B, Armoundas AA (2010) Spatio-temporal oscillations of individual mitochondria in cardiac myocytes reveal modulation of synchronized mitochondrial clusters. Proc Natl Acad Sci U S A 107:14315–14320

    Article  ADS  Google Scholar 

  • Kurz FT, Aon MA, O’Rourke B, Armoundas AA (2014) Cardiac mitochondria exhibit dynamic functional clustering. Front Physiol 5:329

    Article  Google Scholar 

  • Lemar KM, Passa O, Aon MA, Cortassa S, Muller CT, Plummer S, O’Rourke B, Lloyd D (2005) Allyl alcohol and garlic (Allium sativum) extract produce oxidative stress in Candida albicans. Microbiology (Reading, England) 151:3257–3265

    Article  Google Scholar 

  • Lemar KM, Aon MA, Cortassa S, O’Rourke B, Muller CT, Lloyd D (2007) Diallyl disulphide depletes glutathione in Candida albicans: oxidative stress-mediated cell death studied by two-photon microscopy. Yeast (Chichester, England) 24:695–706

    Article  Google Scholar 

  • Liebovitch LS, Todorov AT (1996) Using fractals and nonlinear dynamics to determine the physical properties of ion channel proteins. Crit Rev Neurobiol 10:169–187

    Article  Google Scholar 

  • Lloyd D (1992) Intracellular time keeping: epigenetic oscillations reveal the functions of an ultradian clock. In: Lloyd D, Rossi E (ed) Ultradian rhythms in life processes. Springer, London, pp 5–22

    Google Scholar 

  • Lloyd D (1998) Circadian and ultradian clock-controlled rhythms in unicellular microorganisms. Adv Microb Physiol 39:291–338

    Article  Google Scholar 

  • Lloyd D (2006) Hydrogen sulfide: clandestine microbial messenger? Trends Microbiol 14:456–462

    Article  Google Scholar 

  • Lloyd D (2008) Advances in experimental medicine and biology 641, 118–140

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Lloyd D, Rossi EL (2008) Epilogue: a new vision of life. In: Lloyd D, Rossi EL (eds) Ultradian rhythms from molecules to mind. Springer Science + Business Media B.V, New York, pp 431–439

    Chapter  Google Scholar 

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

    Article  Google Scholar 

  • Lloyd D, Salgado LE, Turner MP, Suller MT, Murray D (2002) Cycles of mitochondrial energization driven by the ultradian clock in a continuous culture of Saccharomyces cerevisiae. Microbiology (Reading, England) 148:3715–3724

    Article  Google Scholar 

  • Lloyd D, Murray DB, Klevecz RR, Wolf J, Kuriyama H (2008) The ultradian clock (40min) in yeast. In: Lloyd DaR EL (ed) Ultradian rhythms from molecules to mind. Springer Science + Business Media B.V, New York, pp 11–42

    Chapter  Google Scholar 

  • Lloyd D, Cortassa S, O’Rourke B, Aon MA (2012) What yeast and cardiomyocytes share: ultradian oscillatory redox mechanisms of cellular coherence and survival. Integr Biol (Camb) 4:65–74

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  • Lyon A, Joudrey PJ, Jin D, Nass RD, Aon MA, O’Rourke B, Akar FG (2010) Optical imaging of mitochondrial function uncovers actively propagating waves of mitochondrial membrane potential collapse across intact heart. J Mol Cell Cardiol 49:565–575

    Google Scholar 

  • Mainzer K (2005) Symmetry and complexity. The spirit and beauty of nonlinear science. World Scientific, Singapore

    MATH  Google Scholar 

  • Mandelbrot BB (1983) The fractal geometry of nature. W.H. Freeman, New York

    MATH  Google Scholar 

  • Mandelbrot BB, Hudson RL (2004) The (mis)behavior of markets. A fractal view of risk, ruin, and reward. Basic Books, New York

    MATH  Google Scholar 

  • Meucci R, Labate A, Ciofini C (1998) Experimental techniques for controlling chaos in lasers. Int J Bifurcation Chaos 8, 1759–1768

    Google Scholar 

  • Moss F (1994) Neurobiology: chaos under control. Nature 370:596–597

    Article  ADS  Google Scholar 

  • Moss F, Ward LM, Sannita WG (2004) Stochastic resonance and sensory information processing: a tutorial and review of application. Clin Neurophysiol 115:267–281

    Article  Google Scholar 

  • Murray DB, Engelen FA, Keulers M, Kuriyama H, Lloyd D (1998) NO+, but not NO., inhibits respiratory oscillations in ethanol-grown chemostat cultures of Saccharomyces cerevisiae. FEBS Lett 431:297–299

    Article  Google Scholar 

  • Murray DB, Engelen F, Lloyd D, Kuriyama H (1999) Involvement of glutathione in the regulation of respiratory oscillation during a continuous culture of Saccharomyces cerevisiae. Microbiology (Reading, England) 145(Pt 10):2739–2745

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Murray DB, Amariel C, Sasidharan K, Machne R, Aon MA, Lloyd D (2013) Temporal partitioning of the yeast cellular network. In: Aon MA, Saks V, Schlattner U (eds) Systems biology of metabolic and signaling networks. Energy, mass and information transfer. Springer, Berlin/Heidelberg/New York/Dordrecht/London, pp 323–349

    Google Scholar 

  • Nicolis G, Prigogine I (1977) Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. Wiley, New York

    MATH  Google Scholar 

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

    Google Scholar 

  • O’Rourke B, Cortassa S, Aon MA (2005) Mitochondrial ion channels: gatekeepers of life and death. Physiology (Bethesda) 20:303–315

    Article  Google Scholar 

  • Osborne AR, Provenzale A (1989) Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35:357–381

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Peng CK, Mietus J, Hausdorff JM, Havlin S, Stanley HE, Goldberger AL (1993) Long-range anticorrelations and non-gaussian behavior of the heartbeat. Phys Rev Lett 70:1343–1346

    Article  ADS  Google Scholar 

  • Rabouille C, Cortassa S, Aon MA (1992) Fractal organisation in biological macromolecular lattices. J Biomol Struct Dyn 9:1013–1024

    Article  Google Scholar 

  • Rao CV, Wolf DM, Arkin AP (2002) Control, exploitation and tolerance of intracellular noise. Nature 420:231–237

    Article  ADS  Google Scholar 

  • Rivas G, Ferrone F, Herzfeld J (2004) Life in a crowded world. EMBO Rep 5:23–27

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Saks V, Dzeja P, Schlattner U, Vendelin M, Terzic A, Wallimann T (2006) Cardiac system bioenergetics: metabolic basis of the Frank-Starling law. J Physiol 571:253–273

    Article  Google Scholar 

  • Sasidharan K, Tomita M, Aon M, Lloyd D, Murray DB (2012) Time-structure of the yeast metabolism in vivo. Adv Exp Med Biol 736:359–379

    Article  Google Scholar 

  • Satroutdinov AD, Kuriyama H, Kobayashi H (1992) Oscillatory metabolism of Saccharomyces cerevisiae in continuous culture. FEMS Microbiol Lett 77:261–267

    Article  Google Scholar 

  • Savageau MA (1995) Michaelis-Menten mechanism reconsidered: implications of fractal kinetics. J Theor Biol 176:115–124

    Article  Google Scholar 

  • Schibler U, Naef F (2005) Cellular oscillators: rhythmic gene expression and metabolism. Curr Opin Cell Biol 17:223–229

    Article  Google Scholar 

  • Schneider ED, Sagan D (2005) Into the cool. Energy flow, thermodynamics and life. The University of Chicago Press, Chicago

    Google Scholar 

  • Schnell S, Turner TE (2004) Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws. Prog Biophys Mol Biol 85:235–260

    Article  Google Scholar 

  • Schrodinger E (1944) What is life? The physical aspects of the living cell. Cambridge University Press, Cambridge

    Google Scholar 

  • Schroeder M (1991) Fractals, chaos, power laws. Minutes from an infinite paradise. W.H. Freeman, New York

    MATH  Google Scholar 

  • Shinbrot T, Ditto W, Grebogi C, Ott E, Spano M, Yorke JA (1992) Using the sensitive dependence of chaos (the “butterfly effect”) to direct trajectories in an experimental chaotic system. Phys Rev Lett 68:2863–2866

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Skinner JE, Pratt CM, Vybiral T (1993) A reduction in the correlation dimension of heartbeat intervals precedes imminent ventricular fibrillation in human subjects. Am Heart J 125:731–743

    Article  Google Scholar 

  • Slodzinski MK, Aon MA, O’Rourke B (2008) Glutathione oxidation as a trigger of mitochondrial depolarization and oscillation in intact hearts. J Mol Cell Cardiol 45:650–660

    Article  Google Scholar 

  • Smith TGJ, Lange GD (1996) Fractal studies of neuronal and glial cellular morphology. CRC Press, Boca Raton

    Google Scholar 

  • Sornette D (2000) Critical phenomena in natural sciences. Chaos, fractals, selforganization and disorder: concepts and tools. Springer, Berlin/Heidelberg

    Google Scholar 

  • Thattai M, van Oudenaarden A (2002) Attenuation of noise in ultrasensitive signaling cascades. Biophys J 82:2943–2950

    Article  Google Scholar 

  • Tu BP, Kudlicki A, Rowicka M, McKnight SL (2005) Logic of the yeast metabolic cycle: temporal compartmentalization of cellular processes. Science (New York, NY) 310:1152–1158

    Google Scholar 

  • van Beek JHGM, Roger SA, Bassingthwaighte JB (1989) Regional myocardial flow heterogeneity explained with fractal networks. Am J Physiol 257:H1670–H1680

    Google Scholar 

  • Varela F, Maturana H, Uribe R (1974) Autopoiesis: the organization of living systems, its characterization and a model. Biosystems 5:187–196

    Article  Google Scholar 

  • Verrier RL, Nearing BD, Lovett EG (1997) Complex oscillatory heart rhythm: a dance macabre. J Clin Invest 99:156–157

    Article  Google Scholar 

  • Vicsek T (2001) Fluctuations and scaling in biology. Oxford University Press, New York

    Google Scholar 

  • Vilar JM, Kueh HY, Barkai N, Leibler S (2002) Mechanisms of noise-resistance in genetic oscillators. Proc Natl Acad Sci U S A 99:5988–5992

    Article  ADS  Google Scholar 

  • Wagner A, Fell DA (2001) The small world inside large metabolic networks. Proc Biol Sci 268:1803–1810

    Article  Google Scholar 

  • West BJ (1990) Fractal physiology and chaos in medicine. World Scientific, Singapore

    Book  MATH  Google Scholar 

  • West BJ (1999) Physiology, promiscuity and prophecy at The Millennium: a tale of tails. World Scientific, Singapore

    Book  MATH  Google Scholar 

  • West BJ, Grigolini P (2011) Complex webs. Anticipating the improbable. Cambridge University Press, Cambridge UK

    MATH  Google Scholar 

  • Williams GP (2003) Chaos theory tamed. Joseph Henry Press, Washington, DC

    MATH  Google Scholar 

  • Wilson KG (1979) Problems in physics with many scales of length. Sci Am 241:158–179

    Article  Google Scholar 

  • Yamamoto Y, Hughson RL (1994) On the fractal nature of heart rate variability in humans: effects of data length and beta-adrenergic blockade. Am J Physiol 266:R40–R49

    Google Scholar 

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

    Article  Google Scholar 

  • Yates FE (1993) Self-organizing systems. In: Boyd CAR, Noble D (eds) The logic of life. The challenge of integrative physiology. Oxford University Press, New York, pp 189–218

    Google Scholar 

  • Zhou L, Cortassa S, Wei AC, Aon MA, Winslow RL, O’Rourke B (2009) Modeling cardiac action potential shortening driven by oxidative stress-induced mitochondrial oscillations in guinea pig cardiomyocytes. Biophys J 97:1843–1852

    Article  Google Scholar 

Books and Reviews

  • Bak P (1996) How nature works: the science of self-organized criticality. Copernicus, New York

    Book  MATH  Google Scholar 

  • Barabasi AL (2003) Linked. Plume, New York

    Google Scholar 

  • Capra F (1996) The web of life. Anchor Books Doubleday, New York

    Google Scholar 

  • Dewey GT (1997) Fractals in molecular biophysics. Oxford University Press, New York

    MATH  Google Scholar 

  • Feder J (1988) Fractals. Plenum Press, New York

    Book  MATH  Google Scholar 

  • Glass L, Mackey MC (1988) From clocks to chaos. The rhythms of life. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Haken H (1978) Synergetics. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Jantsch E (1989) The self-organizing universe. Scientific and human implications of the emerging paradigm of evolution. Pergamon Press, Oxford

    Google Scholar 

  • Kauffman SA (1993) Origins of order: self-organization and selection in evolution. Oxford University Press, New York

    Google Scholar 

  • Lane N (2005) Power, sex, suicide. Mitochondria and the meaning of life. Oxford University Press, Oxford

    Google Scholar 

  • Noble D (2006) The music of life. Oxford University Press, New York

    Google Scholar 

  • Sornette D (2003) Why stock markets crash. Critical events in complex financial systems. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Varela FJ (1989) Autonomie et connaissance. Seuil, Paris

    Google Scholar 

  • West BJ, Deering B (1995) The lure of modern science. Fractal thinking, vol 3. World Scientific, Singapore

    Google Scholar 

  • Yates EF (1987) Self-organizing systems. The emergence of order. Plenum Press, New York

    Book  Google Scholar 

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Authors and Affiliations

  1. School of Medicine, The Johns Hopkins University, Baltimore, MD, USA

    Miguel Antonio Aon & Sonia Cortassa

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  1. Miguel Antonio Aon
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  2. Sonia Cortassa
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Correspondence to Miguel Antonio Aon .

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  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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Aon, M.A., Cortassa, S. (2015). Biochemistry, Chaotic Dynamics, Noise, and Fractal Space in. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_32-2

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  • DOI: https://doi.org/10.1007/978-3-642-27737-5_32-2

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