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Random Walks

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Mathematics as a Laboratory Tool

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Abstract

We began this book with the notion that the mechanisms that underlie biological processes are deterministic. This hypothesis underscores the use of models based on ordinary and delay differential equations. Then, in the last two chapters, we admitted the possibility that there is likely to be a stochastic (random) element as well. In particular, the stochastic differential equations discussed in the previous chapter assert that biological dynamics reflect the interplay between deterministic and stochastic processes. Here we consider the final possibility, that some biological processes are dominated by random processes.

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Notes

  1. 1.

    Derived in 1855 by Adolf Eugen Fick (1829–1901), German physician and physiologist.

  2. 2.

    The reason for 2D rather than D is explained in [36]. Briefly, including the factor 2 simplifies the calculations.

  3. 3.

    Didier Sornette (b. 1957), French physicist and economist.

  4. 4.

    John W. Van Ness (PhD 1964), American mathematician and artist.

  5. 5.

    Benoît Mandelbrot (1924–2010), Polish-born French and American mathematician noted for developing the field of fractal geometry.

  6. 6.

    Christian W. Eurich (b. 1964), German neurophysicist and educator.

  7. 7.

    Jeffrey M. Hausdorff (PhD 1995), American biomechanist.

  8. 8.

    Paul Ehrenfest (1880–1933), Austrian and Dutch theoretical physicist who did pioneering work on random walks and phase transitions. Tatyana Ehrenfest (1905–1984) was a Dutch mathematician and daughter of Paul Ehrenfest.

  9. 9.

    Howard Curtis Berg (b. 1934), American biophysicist; Edward Mills Purcell (1912–1997), American physicist.

  10. 10.

    Marian Smoluchowski (1872–1917), Polish physicist.

  11. 11.

    Nicolaas Godfried van Kampen (1921–2013), Dutch theoretical physicist.

  12. 12.

    We can consider the master equation for a continuous state variable. In that case, the sum is replaced by an integral. If we further expand this continuous-time and continuous-state master equation with small changes of state in short time intervals, it leads to the Fokker–Planck equation.

  13. 13.

    Daniel Thomas Gillespie (b. 1938), American physicist.

  14. 14.

    Siméon Denis Poisson (1781–1840), French mathematician and physicist.

References

  1. H. C. Berg. Random walks in biology. Princeton University Press, Princeton, New Jersey, 1993.

    Google Scholar 

  2. H. C. Berg and E. M. Purcell. Physics of chemoreception. Biophys. J., 20:193–219, 1977.

    Article  Google Scholar 

  3. H. S. Bhat and N. Kumar. Spectral solution of delayed random walks. Phys. Rev. E, 86:045701, 2012.

    Article  Google Scholar 

  4. J. Boeckh, K-E. Kaissling, and D. A. Schneider. Insect olfactory receptors. Cold Spring Harbor Symp. Quant. Biol., 30:1263–1280, 1965.

    Google Scholar 

  5. D. Brockman and T. Giesel. The ecology of gaze shifts. Neurocomputing, 32–33:643–650, 2000.

    Article  Google Scholar 

  6. L. M. Browning, K. J. Lee, T. Huang, P. D. Nallathamby, J. E. Lowman, and X.-H. N. Xu. Random walk of single gold nanoparticles in zebrafish embryos leading to stochastic toxic effects on embryonic developments. Nanoscale, 1:138–152, 2009.

    Article  Google Scholar 

  7. J. L. Cabrera and J. G. Milton. Human stick balancing: Tuning Lévy flights to improve balance control. CHAOS, 14:691–698, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  8. Z. Chen, P. C. Ivanov, K. Hu, and H. E. Stanley. Effect of nonstationarities on detrended fluctuation analysis. Phys. Rev. E, 65:041107, 2002.

    Article  Google Scholar 

  9. C. C. Chow and J. J. Collins. A pinned polymer model of postural control. Phys. Rev. E, 52:907–912, 1995.

    Article  Google Scholar 

  10. T. Cluff and R. Balasubramanian. Motor learning characterized by changing Lévy distributions. PLoS ONE, 4:e5998, 2009.

    Article  Google Scholar 

  11. J. J. Collins and C. J. de Luca. Random walking during quiet standing. Phys. Rev. Lett., 73:764–767, 1994.

    Article  Google Scholar 

  12. M. C. DeSantis, J.-L. Li, and Y. M. Wang. Protein sliding and hopping kinetics on DNA. Phys. Rev. E, 83:021907, 2011.

    Article  Google Scholar 

  13. A. M. Edwards, R. A. Phillips, N. W. Watkins, M. P. Freeman, E. J. Murphy, V. Afanasyev, S. V. Buldyrev, M. G. E. da Luz, E. P. Raposo, and H. E. Stanley. Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer. Nature, 449:1044–1049, 2007.

    Article  Google Scholar 

  14. P. Ehrenfest and T. Ehrenfest. Über zwei bekannte Einwände gegen der Boltzmannschen H-Theorem. Phys. Zeit., 8:311–314, 1907.

    MATH  Google Scholar 

  15. A. Einstein. On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat. Ann. Physik, 17:549–560, 1905.

    Article  MATH  Google Scholar 

  16. C. W. Eurich and J. G. Milton. Noise–induced transitions in human postural sway. Phys. Rev. E, 54:6681–6684, 1996.

    Article  Google Scholar 

  17. M. Evans, N. Hastings, and B. Peacock. Statistical distributions, 2nd ed. Wiley-Interscience, New York, 1993.

    MATH  Google Scholar 

  18. J. Fort, D. Jana, and J. Humet. Multidelayed random walk: Theory and applications to the neolithic transition in Europe. Phys. Rev. E, 70:031913, 2004.

    Article  Google Scholar 

  19. R. P. Futrelle. How molecules get to their detectors: The physics of diffusion of insect pheromones. Trends Neurosci., 7:116–120, 1984.

    Article  Google Scholar 

  20. C. W. Gardiner. Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer-Verlag, New York, 1990.

    MATH  Google Scholar 

  21. D. T. Gillespie. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81:2340–2361, 1977.

    Article  Google Scholar 

  22. P. S. Grassia. Delay, feedback and quenching in financial markets. Eur. Phys. J. B, 17:347–362, 2000.

    Article  Google Scholar 

  23. Y. Y. Grinberg, J. G. Milton, and R. P. Kraig. Spreading depression sends microglia on Lévy flights. PLoS ONE, 6:e19294, 2011.

    Article  Google Scholar 

  24. S. E. Halford and J. F. Marko. How do site-specific DNA-binding proteins find their targets? Nucleic Acids Res., 32:3040–3052, 2004.

    Article  Google Scholar 

  25. J. M. Hausdorff. Gait dynamics, fractals and falls: Finding meaning in the stride-to-stride fluctuations of human walking. Hum. Mov. Sci., 26: 555–589, 2007.

    Article  Google Scholar 

  26. J. M. Hausdorff, M. E. Cudkowicz, R. Firtion, J. Y. Wei, and A. L. Goldberger. Gait variability and basal ganglia disorders: stride-to-stride variation of gait cycle timing in Parkinson’s disease and Huntington’s disease. Mov. Disord., 13:428–437, 1998.

    Article  Google Scholar 

  27. J. M. Hausdorff, S. L. Mitchell, R. Firtion, C. K. Peng, M. E. Cudkowicz, J. Y. Wei, and A. L. Goldenberg. Altered fractal dynamics of gait: reduced stride interval correlations with aging and Huntington’s disease. J. Appl. Physiol., 82:262–269, 1996.

    Google Scholar 

  28. J. M. Hausdorff, C-K. Peng, Z. Ladin, J. Y. Wei, and A. L. Goldberger. Is walking a random walk? evidence for long-range correlations in stride interval of human gait. J. Appl. Physiol., 78:349–358, 1995.

    Google Scholar 

  29. T. Huang, L. M. Browning, and X.-H. N. Xu. Far–field photostable optical nanoscopy (photon) for real-time super-resolution single-molecular imaging of signaling pathways of single live cells. Nanoscale, 4:2797–2812, 2012.

    Article  Google Scholar 

  30. N. E. Humphries, N. Queiroz, J. R. M. Dyer, N. G. Pade, M. K. Musyl, K. M. Schaefer, D. W. Fuller, J. M. Brunnschweiler, T. K. Doyle, J. D. R. Houghton, G. C. Hays, C. S. Jones, L. R. Noble, V. J. Wearmouth, E. J. Southall1, and D. W. Sims. Environmental context explains Lévy and Brownian movement patterns of marine predators. Nature, 465:1066–1069, 2010.

    Google Scholar 

  31. N. G. Van Kampen. Stochastic processes in physics and chemistry, 3rd ed. Elsevier, New York, 2007.

    Google Scholar 

  32. S. Kar, W. T. Baumann, W. R. Paul, and J. J. Tyson. Exploring the roles of noise in the eukaryotic cell cycle. Proc. Natl. Acad. Sci. USA, 106:6471–6476, 2009.

    Article  Google Scholar 

  33. J. Kempe. Quantum random walks: An introductory overview. Cont. Phys., 44:307–327, 2003.

    Article  Google Scholar 

  34. A. Kusumi, Y. Sako, and M. Yamamoto. Confined lateral diffusion of membrane receptors as studied by single particle tracking (nanovid microscopy). Effects of calcium-induced differentiation in cultured epithelial cells. Biophys. J., 65:2021–2040, 1993.

    Google Scholar 

  35. G. W. Lee, A. Ishihara, and K. A. Jacobsen. Direct observation of Brownian motion of lipids in a membrane. Proc. Natl. Acad. Sci. USA, 88:6274–6278, 1991.

    Article  Google Scholar 

  36. S. V. F. Levy, C. Tsallis, A. M. C. Souza, and R. Maynard. Statistical-mechanical foundation of the ubiquity of Lévy distributions in nature. Phys. Rev. Lett., 75:3589–3593, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  37. B. G. Malkiel. A random walk down Wall Street. W. W. Norton & Company, New York, 2003.

    Google Scholar 

  38. B. B. Mandlebrot and J. W. Van Ness. Fractional Brownian motions, fractional noises, and applications. SIAM Rev., 10:422–437, 1968.

    Article  MathSciNet  Google Scholar 

  39. R. M. Mazo. Brownian motion: Fluctuations, dynamics and applications. Oxford Science Publications, New York, 2002.

    Google Scholar 

  40. K. Mergenthaler and R. Engbert. Modeling the control of fixational eye movements with neurophysiological delays. Phys. Rev. Lett., 98:138104, 2007.

    Article  Google Scholar 

  41. M. J. Miller, S. H. Wei, M. D. Chandler, and I. Parker. Autonomous T cell trafficking examined in vivo with intravital two-photon microscopy. Proc. Natl. Acad. Sci. USA, 100:2604–2609, 2003.

    Article  Google Scholar 

  42. M. J. Miller, S. H. Wei, I. Parker, and M. D. Cahalan. Two-photon imaging of lymphocyte motility and antigen response in intact lymph node. Science, 296:1869–1873, 2002.

    Article  Google Scholar 

  43. J. Milton, J. Gyorffy, J. L. Cabrera, and T. Ohira. Amplitude control of human postural sway using Achilles tendon vibration. USNCTAN2010, 2010:791, 2010.

    Google Scholar 

  44. J. G. Milton, T. Ohira T, J. L. Cabrera, R. M. Fraiser, J. B. Gyorffy, F. K. Ruiz, M. A. Strauss, E. C. Balch, P. J. Marin, and J. L. Alexander. Balancing with vibration: A prelude for “drift and act” balance control. PLoS ONE, 4:e7427, 2009.

    Google Scholar 

  45. J. G. Milton, J. L. Townsend, M. A. King, and T. Ohira. Balancing with positive feedback: the case for discontinuous control. Phil. Trans. R. Soc. A, 367:1181–1193, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  46. F. Moss and J. G. Milton. Balancing the unbalanced. Nature, 425:911–912, 2003.

    Article  Google Scholar 

  47. P. D. Nallathamby, K. J. Lee, and X.-H. N. Xu. Design of stable and uniform single photonics for in vivo dynamics imaging of nanoenvironments of zebrafish embryonic fluids. ACS Nano, 2:1371–1380, 2008.

    Article  Google Scholar 

  48. S. R. Nelson, M. Y. Ali, K. M. Trybus, and D. M. Warshaw. Random walk of processive, quantum dot-labeled myosin Va molecules within the actin cortex of COS-7 cells. Biophys. J., 97:509–518, 2009.

    Article  Google Scholar 

  49. A. Nimmerjahn, F. Kirchhoff, and F. Helmchen. Resting microglial cells are highly dynamic surveillants of brain parenchyma in vivo. Science, 308:1314–1318, 2005.

    Google Scholar 

  50. T. Ohira and J. Milton. Delayed random walks: investigating the interplay between noise and delay. In B. Balachandran, T. Kalmár-Nagy, and D. E. Gilsinn, editors, Delay differential equations: Recent advances and new directions, pp. 305–335. New York, Springer, 2009.

    Google Scholar 

  51. T. Ohira and J. G. Milton. Delayed random walks. Phys. Rev. E, 52:3277–3280, 1995.

    Article  Google Scholar 

  52. T. Ohira, N. Sazuka, K. Marumo, T. Shimizu, M. Takayasu, and H. Takayasu. Preditability of currency market exchange. Physica A, 308:368–374, 2002.

    Article  MATH  Google Scholar 

  53. T. Ohira and T. Yamane. Delayed stochastic systems. Phys. Rev. E, 61:1247–1257, 2000.

    Article  Google Scholar 

  54. C-K. Peng, S. V. Buldyrev, A. L. Goldberger, S. Havlin, F. Sciortino, M. Simons, and H. E. Stanley. Long-range correlations in nucleotide sequences. Nature, 356:168–170, 1992.

    Google Scholar 

  55. C-K. Peng, S. V. Buldyrev, A. L. Goldberger, S. Havlin, M. Simons, and H. E. Stanley. Finite-size effects on long-range correlations: Implications for analyzing DNA sequences. Phys. Rev. E, 47:3730–3733, 1993.

    Google Scholar 

  56. A. A. Priplata, J. B. Niemi, J. D. Harry, L. A. Lipsitz, and J. J. Collins. Vibratory insoles and balance control in elderly people. Lancet, 362:1123–1124, 2003.

    Article  Google Scholar 

  57. G. Raivich. Like cops on the beat: the active role of resting microglia. Trends Neurosci., 28:671–573, 2005.

    Article  Google Scholar 

  58. A. M. Reynolds. Bridging the gap between correlated random walks and Lévy walks: autocorrelation as a source Lévy walk movement patterns. J. Roy. Soc. Interface, 7:1753–1758, 2010.

    Article  Google Scholar 

  59. A. M. Reynolds. Can spontaneous cell movements be modeled as Lévy walks? Physica A, 389:273–277, 2010.

    Article  Google Scholar 

  60. A. D. Riggs, S. Bourgeois, and M. Cohn. The lac repressor–operator interaction. III. Kinetic studies. J. Mol. Biol., 53:401–417, 1970.

    Google Scholar 

  61. H. Risken. The Fokker–Planck equation. Springer, New York, 1989.

    MATH  Google Scholar 

  62. J. Rudnick and G. Gaspari. Elements of the random walk: An introduction for advanced students and researchers. Cambridge University Press, New York, 2004.

    Book  Google Scholar 

  63. Y. Sako and A. Kusumi. Compartmentalized structure of the plasma membrane for receptor movements as revealed by a nanometer-level motion analysis. J. Cell Bio., 125:1251–1264, 1994.

    Article  Google Scholar 

  64. M. J. Saxton. Single-particle tracking: The distribution of diffusion coefficients. Biophys. J., 72:1744–1753, 1997.

    Article  Google Scholar 

  65. D. Selmeczi, L. Li, L. I. I. Pedersen, S. F. Nrrelykke, P. H. Hagedorn, S. Mosler, N. B. Larsen, E. C. Cox, and H. Flyvbjerg. Cell motility as random motion: a review. Eur. Phys. J. Special Topics, 157:1–15, 2008.

    Article  Google Scholar 

  66. W. Simon. Mathematical techniques for biology and medicine. Academic Press, New York, 1972.

    Google Scholar 

  67. L. K. Smith, E. L. Weiss, and L. D. Lehmkuhl. Brunnstrom’s clinical kinesiology, 5th ed. F. A. Davis, Philadelphia, 1983.

    Google Scholar 

  68. D. Sornette. Critical phenomena in natural sciences: Chaos, fractals, selforganization and disorder: Concepts and tools. Springer, New York, 2004.

    Google Scholar 

  69. D. Sornette. Dragon-Kings, black swans, and the prediction of crisis. Int. J. Terraspace Sci. Eng., 2:1–18, 2009.

    Google Scholar 

  70. H. E. Stanley. Phase transitions and critical phenomena. Oxford University Press, London, 1971.

    Google Scholar 

  71. G. M. Viswanathan, V. Afanazyev, S. V. Buldyrev, E. J. Murph, and H. E. Stanley. Lévy flight search patterns of wandering albatrosses. Nature, 381:413–415, 1996.

    Article  Google Scholar 

  72. G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. G. E. da Luz, E. P. Raposo, and H. E. Stanley. Optimizing the success of random searches. Nature, 401:911–914, 1999.

    Article  Google Scholar 

  73. Y. M. Wang, R. H. Austin, and E. C. Cox. Single molecule measurements of repressor protein 1D diffusion on DNA. Phys. Rev. Lett., 97:048302, 2006.

    Article  Google Scholar 

  74. G. H. Weiss. Aspects and applications of random walks. North–Holland, New York, 1994.

    Google Scholar 

  75. K. Willson, D. P. Francis, R. Wensel, A. J. S. Coats, and K. H. Parker. Relationship between detrended fluctuation analysis and spectral analysis of heart-rate variability. Physiol. Meas., 23:385–401, 2002.

    Article  Google Scholar 

  76. D. A. Winter. Biomechanics and motor control of human movement, 3rd ed. John Wiley & Sons, Toronto, 2005.

    Google Scholar 

  77. M. S. Zakynthinaki, J. R. Stirling, C. A. Cordent Martinez, A. López Diíaz de Durana, M. S. Quintana, G. R. Romo, and J. S. Molinueve. Modeling the basin of attraction as a two-dimensional manifold from experimental data: Applications to balance in humans. Chaos, 20:013119, 2010.

    Google Scholar 

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Authors and Affiliations

  1. W.M. Keck Science Department, The Claremont Colleges, Claremont, CA, USA

    John Milton

  2. Graduate School of Mathematics, Nagoya University, Nagoya, Japan

    Toru Ohira

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Milton, J., Ohira, T. (2014). Random Walks. In: Mathematics as a Laboratory Tool. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9096-8_14

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