Abstract
Escherichia coli is a motile bacterium that moves up a chemoattractant gradient by performing a biased random walk composed of alternating runs and tumbles. This paper presents calculations of the chemotactic drift velocity vd (the mean velocity up the chemoattractant gradient) of an E. coli cell performing chemotaxis in a uniform, steady shear flow, with a weak chemoattractant gradient at right angles to the flow. Extending earlier models, a combined analytic and numerical approach is used to assess the effect of several complications, namely (i) a cell cannot detect a chemoattractant gradient directly but rather makes temporal comparisons of chemoattractant concentration, (ii) the tumbles exhibit persistence of direction, meaning that the swimming directions before and after a tumble are correlated, (iii) the cell suffers random re-orientations due to rotational Brownian motion, and (iv) the non-spherical shape of the cell affects the way that it is rotated by the shear flow. These complications influence the dependence of vd on the shear rate γ. When they are all included, it is found that (a) shear disrupts chemotaxis and shear rates beyond γ≈2 s−1 render chemotaxis ineffective, (b) in terms of maximizing drift velocity, persistence of direction is advantageous in a quiescent fluid but disadvantageous in a shear flow, and (c) a more elongated body shape is advantageous in performing chemotaxis in a shear flow.
Similar content being viewed by others
References
Azam, F., Malfatti, F., 2007. Microbial structuring of marine ecosystems. Nat. Rev. Microbiol. 5, 782–791. doi:10.1038/nrmicro1747.
Barbara, G.M., Mitchell, J.G., 2003. Bacterial tracking of motile algae. FEMS Microbiol. Ecol. 44(1), 79–87. doi:10.1111/j.1574-6941.2003.tb01092.x.
Bearon, R.N., 2001. Run-and-tumble chemotaxis in an ambient fluid flow. PhD thesis, University of Cambridge
Bearon, R.N., 2003. An extension of generalized Taylor dispersion in unbounded homogeneous shear flows to run-and-tumble chemotactic bacteria. Phys. Fluids 15(6), 1552–1563. doi:10.1063/1.1569482.
Bearon, R.N., Pedley, T.J., 2000. Modeling run-and-tumble chemotaxis in a shear flow. Bull. Math. Biol. 62, 775–791. doi:10.1006/bulm.2000.0178.
Berg, H.C., 1983. Random Walks in Biology. Princeton University Press, Princeton.
Berg, H.C., 2007. Personal communication
Berg, H.C., Brown, D.A., 1972. Chemotaxis in Escherichia coli analysed by three-dimensional tracking. Nature 239, 500–504. doi:10.1038/239500a0.
Berg, H.C., Tedesco, P.M., 1975. Transient response to chemotactic stimuli in Escherichia coli. Proc. Natl. Acad. Sci. USA 72, 3235–3239. http://www.pnas.org/content/72/8/3235.
Block, S.M., Segall, J.E., Berg, H.C., 1982. Impulse responses in bacterial chemotaxis. Cell 31, 215–226
Block, S.M., Segall, J.E., Berg, H.C., 1983. Adaptation kinetics in bacterial chemotaxis. J. Bacteriol. 154(1), 312–323. http://jb.asm.org/cgi/content/abstract/154/1/312.
Bowen, J.D., Stolzenbach, K.D., Chisholm, S.W., 1993. Simulating bacterial clustering around phytoplankton cells in a turbulent ocean. Limnol. Oceanogr. 38(1), 36–51. http://www.jstor.org/stable/2837893.
Bretherton, F.P., 1962. The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14, 284–304. doi:10.1017/S002211206200124X.
Clark, D.A., Grant, L.C., 2005. The bacterial chemotactic response reflects a compromise between transient and steady-state behavior. Proc. Natl. Acad. Sci. USA 102(26), 9150–9155. doi:10.1073/pnas.0407659102.
Costerton, J.W., Lewandowski, Z., 1995. Microbial biofilms. Annu. Rev. Microbiol. 49, 711–745. doi:10.1146/annurev.mi.49.100195.003431.
de Gennes, P.G., 2004. Chemotaxis: the role of internal delays. Eur. Biophys. J. 33, 691–693. doi:10.1007/s00249-004-0426-z.
Erban, R., Othmer, H.G., 2004. From individual to collective behaviour in bacterial chemotaxis. SIAM J. Appl. Math. 65(2), 361–391. doi:10.1137/S0036139903433232.
Erban, R., Othmer, H.G., 2005. From signal transduction to spatial pattern formation in E. coli: a paradigm for multiscale modeling in biology. Multiscale Model. Simul. 3(2), 362–394. doi:10.1137/040603565.
Gray, J., Hancock, G.J., 1955. The propulsion of sea-urchin spermatozoa. J. Exp. Biol. 32, 802–814. http://jeb.biologists.org/cgi/content/abstract/32/4/802.
Hahn, T., 2005. Cuba—a library for multidimensional numerical integration. Comput. Phys. Commun. 168, 78–95. doi:10.1016/j.cpc.2007.03.006.
Hill, J., Kalkanci, O., McMurry, J.L., Koser, H., 2007. Hydrodynamic surface interactions enable Escherichia coli to seek efficient routes to swim upstream. Phys. Rev. Lett. 98(6), 068101, doi:10.1103/PhysRevLett.98.068101.
Jeffery, G.B., 1922. The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179. doi:10.1098/rspa.1922.0078.
Johansen, J.E., Pinhassi, J., Blackburn, N., Zweifel, U.L., Hagström, Å., 2002. Variability in motility characteristics among marine bacteria. Aquat. Microb. Ecol. 28, 229–237. doi:10.3354/ame028229.
Kim, S., Karrila, S.J., 1992. Microhydrodynamics: Principles and Selected Applications. Butterworth/Heinemann, Stoneham/London.
Lapidus, I.R., 1981. Analysis of bacterial chemotaxis in flowing water. Math. Biosci. 54, 79–90. doi:10.1016/0025-5564(81)90078-X.
Lighthill, J., 1976. Flagellar hydrodynamics. SIAM Rev. 18(2), 161–230. doi:10.1137/1018040.
Locsei, J.T., 2007. Persistence of direction increases the drift velocity of run and tumble chemotaxis. J. Math. Biol. 55(1), 41–60. doi:10.1007/s00285-007-0080-z.
Luchsinger, R.H., Bergerson, B., Mitchell, J.G., 1999. Bacterial swimming strategies and turbulence. Biophys. J. 77, 2377–2386. http://www.biophysj.org/cgi/content/abstract/77/5/2377.
Mesibov, R., Ordal, G.W., Adler, J., 1973. The range of attractant concentrations for bacterial chemotaxis and the threshold and size of response over this range. J. Gen. Physiol. 62, 203–223. http://www.jgp.org/cgi/content/abstract/62/2/203.
Mitchell, J.G., Kogure, K., 2006. Bacterial motility: links to the environment and a driving force for microbial physics. FEMS Microbiol. Ecol. 55(1), 3–16. doi:10.1111/j.1574-6941.2005.00003.x.
Mitchell, J.G., Martinez-Alonso, M., Lalucat, J., Esteve, I., Brown, S., 1991. Velocity changes, long runs, and reversals in the Chromatium minus swimming response. J. Bacteriol. 173(3), 997–1003. http://jb.asm.org/cgi/content/abstract/173/3/997.
Mitchell, J.G., Pearson, L., Dillon, S., 1996. Clustering of marine bacteria in seawater enrichments. Appl. Environ. Microbiol. 62(10), 3716–3521. http://aem.asm.org/cgi/content/abstract/62/10/3716.
Neidhardt, F.C. (Ed.), 1987. Escherichia coli and Salmonella typhimurium: Cellular and Molecular Biology, vol. 1. Am. Soc. Microbiol., Washington.
Schnitzer, M.J., 1993. Theory of continuum random walks and applications to chemotaxis. Phys. Rev. E 48(4), 2553–2568. doi:10.1103/PhysRevE.48.2553.
Segall, J.E., Manson, M.D., Berg, H.C., 1982. Signal processing times in bacterial chemotaxis. Nature 296, 855–857. doi:10.1038/296855a0.
Segall, J.E., Block, S.M., Berg, H.C., 1986. Temporal comparisons in bacterial chemotaxis. Proc. Natl. Acad. Sci. USA 83(23), 8987–8991. http://www.pnas.org/cgi/content/abstract/83/23/8987.
Stocker, R., Seymour, J.R., Samadani, A., Hunt, D.E., Polz, M.F., 2008. Rapid chemotactic response enables marine bacteria to exploit ephemeral microscale nutrient patches. Proc. Natl. Acad. Sci. USA 105(11), 4209–4214. doi:10.1073/pnas.0709765105.
Thar, R., Kühl, M., 2003. Bacteria are not too small for spatial sensing of chemical gradients: An experimental evidence. Proc. Natl. Acad. Sci. USA 100(10), 5748–5753. doi:10.1073/pnas.1030795100.
Turner, L., Ryu, W.S., Berg, H.C., 2000. Real-time imaging of fluorescent flagellar filaments. J. Bacteriol. 182(10), 2793–2801. http://jb.asm.org/cgi/content/abstract/182/10/2793.
Walsh, F., Mitchell, R., 1978. Bacterial chemotactic responses in flowing water. Microb. Ecol. 4, 165–168. doi:10.1007/BF02014286.
Young, K.D., 2006. The selective value of bacterial shape. Microbiol. Mol. Biol. Rev. 70(3), 660–703. doi:10.1128/MMBR.00001-06.
Author information
Authors and Affiliations
Corresponding author
Additional information
J.T. Locsei is supported by an Oliver Gatty Studentship from the University of Cambridge.
Rights and permissions
About this article
Cite this article
Locsei, J.T., Pedley, T.J. Run and Tumble Chemotaxis in a Shear Flow: The Effect of Temporal Comparisons, Persistence, Rotational Diffusion, and Cell Shape. Bull. Math. Biol. 71, 1089–1116 (2009). https://doi.org/10.1007/s11538-009-9395-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-009-9395-9