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Probability distributions for the run-and-tumble bacterial dynamics: An analogy to the Lorentz model

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Abstract

In this paper, we exploit an analogy of the run-and-tumble process for bacterial motility with the Lorentz model of electron conduction in order to obtain analytical results for the intermediate scattering function. This allows to obtain an analytical result for the van Hove function in real space for two-dimensional systems. We furthermore consider the 2D circling motion of bacteria close to solid boundaries with tumbling, and show that the analogy to electron conduction in a magnetic field allows to predict the effective diffusion coefficient of the bacteria. The latter is shown to be reduced by the circling motion of the bacteria.

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References

  1. H.C. Berg, E. Coli In Motion (Springer, New York, 2004).

  2. H.C. Berg, D.A. Brown, Nature (London) 239, 500 (1972).

    Article  ADS  Google Scholar 

  3. H.G. Othmer, S.R. Dunbar, W. Alt, J. Math. Biol. 26, 263 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  4. M.J. Schnitzer, Phys. Rev. E 48, 2553 (1993).

    Article  MathSciNet  ADS  Google Scholar 

  5. J. Tailleur, M.E. Cates, Phys. Rev. Lett. 100, 218103 (2008).

    Article  ADS  Google Scholar 

  6. J. Tailleur, M.E. Cates, EPL 86, 60002 (2009).

    Article  ADS  Google Scholar 

  7. R.W. Nash, R. Adhikari, J. Tailleur, M.E. Cates, Phys. Rev. Lett. 104, 258101 (2010).

    Article  ADS  Google Scholar 

  8. M.E. Cates, Rep. Prog. Phys. 75, 042601 (2012).

    Article  ADS  Google Scholar 

  9. J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguina, P. Silberzan, Proc. Natl. Acad. Sci. U.S.A. 108, 16235 (2011).

    Article  ADS  Google Scholar 

  10. L. Angelani, A. Costanzo, R. Di Leonardo, EPL 96, 68002 (2011).

    Article  ADS  Google Scholar 

  11. L.G. Wilson, V.A. Martinez, J. Schwarz-Linek, J. Tailleur, G. Bryant, P.N. Pusey, W.C.K. Poon, Phys. Rev. Lett. 106, 018101 (2011).

    Article  ADS  Google Scholar 

  12. Vincent A. Martinez, Rut Besseling, Ottavio A. Croze, Julien Tailleur, Mathias Reufer, Jana Schwarz-Linek, Laurence G. Wilson, Martin A. Bees, Wilson C. K. Poon, arXiv:1202.1702v1.

  13. H.A. Lorentz, Arch. Néerl. 10, 336 (1905).

    Google Scholar 

  14. E. Hauge, Phys. Fluids 13, 1201 (1970).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. A.D. Kolesnik, J. Stat. Phys. 131, 1039 (2008).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Pavel L. Krapivsky, Sidney Redner, Eli Ben-Naim (Editors), A Kinetic View of Statistical Physics (Cambridge University Press, 2010).

  17. We use relation 6.575, in I.S Gradshteyn, I.M. Ryzhik, Table of integrals, series and products, 6th ed. (Academic Press, San Diego, 2000). The term n = 0 needs a specific treatment, see 6.512 (8).

  18. W. Stadje, J. Stat. Phys. 46, 207 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  19. Allison P. Berke, Linda Turner, Howard C. Berg, Eric Lauga, Phys. Rev. Lett. 106 101, 038102 (2008).

  20. E. Lauga, W.R. DiLuzio, G.M. Whitesides, H.A. Stone, Biophys. J. 90, 400 (2006).

    Article  ADS  Google Scholar 

  21. P.D. Frymier, R.M. Ford, H.C. Berg, P.T. Cummings, Proc. Natl. Acad. Sci. U.S.A. 92, 6195 (1995).

    Article  ADS  Google Scholar 

  22. R. Di Leonardo, D. Dell’Arciprete, L. Angelani, V. Iebba, Phys. Rev. Lett. 106, 038101 (2011).

    Article  ADS  Google Scholar 

  23. K.M. Ottemann, J.F. Miller, Mol. Microbiol. 24, 1109 (1997).

    Article  Google Scholar 

  24. F. Cornu, J. Piasecki, Physica A 370, 591 (2006).

    Article  ADS  Google Scholar 

  25. A.J. Spakowitz, Z.-G. Wang, Macromolecules 37, 5814 (2004).

    Article  ADS  Google Scholar 

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Correspondence to L. Bocquet.

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Martens, K., Angelani, L., Di Leonardo, R. et al. Probability distributions for the run-and-tumble bacterial dynamics: An analogy to the Lorentz model. Eur. Phys. J. E 35, 84 (2012). https://doi.org/10.1140/epje/i2012-12084-y

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  • DOI: https://doi.org/10.1140/epje/i2012-12084-y

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