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Random walks with barriers

Abstract

Restrictions to molecular motion by barriers (membranes) are ubiquitous in porous media, composite materials and biological tissues. A major challenge is to characterize the microstructure of a material or an organism non-destructively using a bulk transport measurement. Here we demonstrate how the long-range structural correlations introduced by permeable membranes give rise to distinct features of transport. We consider Brownian motion restricted by randomly placed and oriented membranes (dāˆ’1-dimensional planes in d dimensions) and focus on the disorder-averaged diffusion propagator using a scattering approach. The renormalization group solution reveals a scaling behaviour of the diffusion coefficient for large times, with a characteristically slow inverse square root time dependence for any d. Its origin lies in the strong structural fluctuations introduced by the spatially extended random restrictions, representing a new universality class of the structural disorder. Our results agree well with Monte Carlo simulations in two dimensions. They can be used to identify permeable barriers as restrictions to transport, and to quantify their permeability and surface area.

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Figure 1: A fragment of a two-dimensional patch with randomly placed and oriented membranes.
Figure 2: Time-dependent diffusion coefficient D(t) for the two-dimensional random medium of Fig.Ā 1.
Figure 3: Effect of a single membrane.

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References

  1. Haus, J. W. & Kehr, K. W. Diffusion in regular and disordered lattices. Phys. Rep. 150, 263ā€“406 (1987).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  2. Bouchaud, J-P. & Georges, A. Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep. 195, 127ā€“293 (1990).

    ArticleĀ  ADSĀ  MathSciNetĀ  Google ScholarĀ 

  3. Sinai, Y. G. Dynamical systems with elastic reflections. Russ. Math. Survey 25, 137ā€“189 (1970).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  4. Marinari, E., Parisi, G., Ruelle, D. & Windey, P. Random walk in a random environment and 1/f noise. Phys. Rev. Lett. 50, 1223ā€“1225 (1983).

    ArticleĀ  ADSĀ  MathSciNetĀ  Google ScholarĀ 

  5. Fisher, D. S. Random walks in random environments. Phys. Rev. A 30, 960ā€“964 (1984).

    ArticleĀ  ADSĀ  MathSciNetĀ  Google ScholarĀ 

  6. Aronovitz, A. & Nelson, D. R. Anomalous diffusion in steady fluid flow through a porous medium. Phys. Rev. A 30, 1948ā€“1954 (1984).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  7. Fisher, D. S., Friedan, D., Qiu, Z., Shenker, S. J. & Shenker, S. H. Random walks in two-dimensional random environments with constrained drift forces. Phys. Rev. A 31, 3841ā€“3845 (1985).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  8. Kravtsov, V. E., Lerner, I. V. & Yudson, V. I. Random walks in media with constrained disorder. J. Phys. A 18, L703ā€“L707 (1985).

    ArticleĀ  ADSĀ  MathSciNetĀ  Google ScholarĀ 

  9. Lerner, I. V. Distributions of the diffusion coefficient for the quantum and classical diffusion in disordered media. Nucl. Phys. A 560, 274ā€“292 (1993).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  10. Ernst, M. H., Machta, J., Dorfman, J. R. & van Beijeren, H. Long time tails in stationary random media I. Theory. J. Stat. Phys. 34, 477ā€“495 (1984).

    ArticleĀ  ADSĀ  MathSciNetĀ  Google ScholarĀ 

  11. Visscher, P. B. Universality in disordered diffusive systems: Exact fixed points in one, two, and three dimensions. Phys. Rev. B 29, 5472ā€“5485 (1984).

    ArticleĀ  ADSĀ  MathSciNetĀ  Google ScholarĀ 

  12. Sykova, E. & Nicholson, C. Diffusion in brain extracellular space. Physiol. Rev. 88, 1277ā€“1340 (2008).

    ArticleĀ  Google ScholarĀ 

  13. Cory, D. G. & Garroway, A. N. Measurement of translational displacement probabilities by NMR: An indicator of compartmentation. Magn. Reson. Med. 14, 435ā€“444 (1990).

    ArticleĀ  Google ScholarĀ 

  14. Le Bihan, D. (ed.) Diffusion and Perfusion Magnetic Resonance Imaging (Raven Press, 1995).

  15. Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy (Clarendon, 1991).

    Google ScholarĀ 

  16. Latour, L. L., Svoboda, K., Mitra, P. P. & Sotak, C. H. Time-dependent diffusion of water in a biological model system. Proc. Natl Acad. Sci. USA 91, 1229ā€“1233 (1994).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  17. Yablonskiy, D. A. et al. Quantitative in vivo assessment of lung microstructure at the alveolar level with hyperpolarized 3He diffusion MRI. Proc. Natl Acad. Sci. USA 99, 3111ā€“3116 (2002).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  18. Friedman, M. H. Principles and Models of Biological Transport (Springer, 2008).

    BookĀ  Google ScholarĀ 

  19. Fujiwara, T., Ritchie, K., Murakoshi, H., Jacobson, K. & Kusumi, A. Phospholipids undergo hop diffusion in compartmentalized cell membrane. J. Cell Biol. 157, 1071ā€“1082 (2002).

    ArticleĀ  Google ScholarĀ 

  20. Kusumi, A. et al. Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid: High-speed single-molecule tracking of membrane molecules. Annu. Rev. Biophys. Biomol. Struct. 34, 351ā€“378 (2005).

    ArticleĀ  Google ScholarĀ 

  21. Wawrezinieck, L., Rigneault, H., Marguet, D. & Lenne, P-F. Fluorescence correlation spectroscopy diffusion laws to probe the submicron cell membrane organization. Biophys. J. 89, 4029ā€“4042 (2005).

    ArticleĀ  Google ScholarĀ 

  22. Cotts, R. M. Diffusion and diffraction. Nature 351, 443ā€“444 (1991).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  23. Callaghan, P. T., Coy, A., Macgowan, D., Packer, K. J. & Zelaya, F. O. Diffraction-like effects in NMR diffusion studies of fluids in porous solids. Nature 351, 467ā€“469 (1991).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  24. Mitra, P. P., Sen, P. N., Schwartz, L. M. & Le Doussal, P. Diffusion propagator as a probe of the structure of porous media. Phys. Rev. Lett. 68, 3555ā€“3558 (1992).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  25. Mair, R. W. et al. Probing porous media with gas diffusion NMR. Phys.Ā Rev.Ā Lett. 83, 3324ā€“3327 (1999).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  26. Song, Y-Q., Ryu, S. & Sen, P. N. Determining multiple length scales in rocks. Nature 406, 178ā€“181 (2000).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  27. Sen, P. N. Time-dependent diffusion coefficient as a probe of the permeability of the pore wall. J. Chem. Phys. 119, 9871ā€“9876 (2003).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  28. Altshuler, B. L. & Aronov, A. G. in Electronā€“Electron Interactions in Disordered Systems (eds Efros, A. L. & Pollak, M.) (North-Holland, 1985).

    Google ScholarĀ 

  29. Scher, H. & Lax, M. Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B 7, 4491ā€“4502 (1973).

    ArticleĀ  ADSĀ  MathSciNetĀ  Google ScholarĀ 

  30. Powles, J. G., Mallett, M. J. D., Rickayzen, G. & Evans, W. A. B. Exact analytic solutions for diffusion impeded by an infinite array of partially permeable barriers. Proc. R. Soc. A 436, 391ā€“403 (1992).

    ArticleĀ  ADSĀ  MathSciNetĀ  Google ScholarĀ 

  31. Crick, F. Diffusion in embryogenesis. Nature 225, 420ā€“422 (1970).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  32. Novikov, D. S. & Kiselev, V. G. Effective medium theory of a diffusion-weighted signal. NMR Biomed. 23, 682ā€“697 (2010).

    ArticleĀ  Google ScholarĀ 

  33. Warburg, E. Ueber das Verhalten sogenannter unpolarisirbarer Elektroden gegen Wechselstrom. Ann. Phys. (Lpz.) 303, 493ā€“499 (1899).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  34. Kohlrausch, F. Ueber die elektromotorische Kraft Sehr Dunner Gasschichten auf Metallplatteu. Pogg. Ann. 148, 143ā€“154 (1873).

    MATHĀ  Google ScholarĀ 

  35. Wien, M. Ueber die Polarization bei Wechselstrom. Ann. Phys. Chem. 58, 37ā€“72 (1896).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  36. Cole, K. S. Alternating current conductance and direct current excitation of nerve. Science 79, 164ā€“165 (1934).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  37. Cole, K. S. & Cole, R. H. Dispersion and absorption in dielectrics. J. Chem. Phys. 9, 341ā€“351 (1941).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  38. Sen, P. N., Scala, C. & Cohen, M. H. A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads. Geophysics 46, 781ā€“795 (1981).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  39. Norris, A. N., Callegari, A. J. & Sheng, P. A Generalized differential effective medium theory. J. Mech. Phys. Solids 33, 525ā€“543 (1985).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  40. Avellaneda, M. Iterated homogenization, differential effective medium theory and applications. Commun. Pure Appl. Math. XL, 527ā€“554 (1987).

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  41. Abrahams, E., Anderson, P. W., Licciardello, D. C. & Ramakrishnan, T. V. Scaling theory of localization: Absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, 673ā€“676 (1979).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  42. Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181ā€“1203 (1973).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  43. Novikov, D. S. & Kiselev, V. G. Surface-to-volume ratio with oscillating gradients. J. Magn. Reson. 10.1016/j.jmr.2011.02.011 (2011).

  44. Denteneer, P. J. H. & Ernst, M. H. Diffusion in systems with static disorder. Phys. Rev. B 29, 1755ā€“1768 (1984).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  45. Machta, J. Generalized diffusion coefficient in one-dimensional random walks with static disorder. Phys. Rev. B 24, 5260ā€“5269 (1981).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  46. Zwanzig, R. Non-Markoffian diffusion in a one-dimensional disordered lattice. J. Stat. Phys. 28, 127ā€“133 (1982).

    ArticleĀ  ADSĀ  MathSciNetĀ  Google ScholarĀ 

  47. Sukstanskii, A. L., Yablonskiy, D. A. & Ackerman, J. J. H. Effects of permeable boundaries on the diffusion-attenuated MR signal: Insights from a one-dimensional model. J. Magn. Reson. 170, 56ā€“66 (2004).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  48. Dudko, O. K., Berezhkovskii, A. M. & Weiss, G. H. Diffusion in the presence of periodically spaced permeable membranes. J. Chem. Phys. 121, 11283ā€“11288 (2004).

    ArticleĀ  ADSĀ  Google ScholarĀ 

  49. Landau, L. D. & Lifshits, E. M. Quantum Mechanics (Non-relativistic Theory) (Elsevier, 1977).

    Google ScholarĀ 

  50. Fieremans, E., Novikov, D. S., Jensen, J. H. & Helpern, J. A. Monte Carlo study of a two-compartment exchange model of diffusion. NMR Biomed. 23, 711ā€“724 (2010).

    ArticleĀ  Google ScholarĀ 

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Acknowledgements

We thank V. Kiselev and D. Sodickson for discussions. Research was supported by the Litwin Fund for Alzheimerā€™s Research, and the National Institutes of Health Grants 1R01AG027852 and 1R01EB007656.

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D.S.N. carried out analytical calculations and wrote the manuscript. E.F. carried out numerical calculations. All authors discussed the results and implications and commented on the manuscript at all stages.

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Correspondence to Dmitry S. Novikov.

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Novikov, D., Fieremans, E., Jensen, J. et al. Random walks with barriers. Nature Phys 7, 508ā€“514 (2011). https://doi.org/10.1038/nphys1936

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