Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-19T05:04:34.125Z Has data issue: false hasContentIssue false
  • English
  • Français

Induced diffusion of tracers in a bacterial suspension: theory and experiments

Published online by Cambridge University Press:  24 July 2013

G. L. Miño ,
J. Dunstan ,
A. Rousselet ,
E. Clément  and
R. Soto
G. L. Miño
Affiliation:
PMMH-ESPCI, UMR 7636, CNRS–ESPCI–Université Paris 6 and Paris 7, 10 rue Vauquelin, 75005 Paris, France
J. Dunstan
Affiliation:
Departamento de Física, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
A. Rousselet
Affiliation:
PMMH-ESPCI, UMR 7636, CNRS–ESPCI–Université Paris 6 and Paris 7, 10 rue Vauquelin, 75005 Paris, France
E. Clément
Affiliation:
PMMH-ESPCI, UMR 7636, CNRS–ESPCI–Université Paris 6 and Paris 7, 10 rue Vauquelin, 75005 Paris, France
R. Soto*
Affiliation:
Departamento de Física, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile
*
†Email address for correspondence: rsoto@dfi.uchile.cl
Get access Rights & Permissions [Opens in a new window]

Abstract

The induced diffusion of tracers in a bacterial suspension is studied theoretically and experimentally at low bacterial concentrations. Considering the swimmer–tracer hydrodynamic interactions at low Reynolds number and using a kinetic theory approach, it is shown that the induced diffusion coefficient is proportional to the swimmer concentration, their mean velocity and a coefficient $\beta $, as observed experimentally. This paper shows that $\beta $ increases as a result of the interaction with solid surfaces. The coefficient $\beta $ scales as the tracer–swimmer cross-section times the mean square displacement produced by single scattering events, which depends on the swimmer propulsion forces. Considering simple swimmer models (acting on the fluid as two monopoles or as a force dipole), it is shown that $\beta $ increases for decreasing swimming efficiencies. Close to solid surfaces, the swimming efficiency degrades and, consequently, the induced diffusion increases. Experiments on wild-type Escherichia coli in a Hele-Shaw cell, under buoyant conditions, are performed to measure the induced diffusion on tracers near surfaces. The modification of the suspension pH varies the swimmers’ velocity over a wide range, allowing the $\beta $ coefficient to be extracted with precision. It is found that solid surfaces modify the induced diffusion: decreasing the confinement height of the cell, $\beta $ increases by a factor of 4. The theoretical model reproduces this increase, although there are quantitative differences, probably attributed to the simplicity of the swimmer models and to the estimates for the parameters that model E. coli.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Low-Reynolds-number flows Biological Fluid Dynamics: Micro-organism dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 729 , 25 August 2013 , pp. 423 - 444
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adler, J. 1973 A method for measuring chemotaxis and use of the method to determine optimum conditions for chemotaxis by Escherichia coli . J. Gen. Microbiol. 74, 77–91.Google Scholar
Adler, J. & Templeton, B. 1967 The effect of environmental conditions on the motility of Escherichia coli . J. Gen. Microbiol. 46, 175–184.Google Scholar
Amsler, C. D., Cho, M. & Matsumura, P. 1993 Multiple factors underlying the maximum motility of Escherichia coli as cultures enter post-exponential growth. J. Bacteriol. 175, 6238–6244.CrossRefGoogle ScholarPubMed
Archer, C. T., Kim, J. F., Jeong, H., Park, J. H., Vickers, C. E., Lee, S. Y. & Nielsen, L. K. 2011 The genome sequence of E. coli W (ATCC 9637): comparative genome analysis and an improved genome-scale reconstruction of E. coli . BMC Genomics 12, 9.Google Scholar
Atsumi, T., Maekawa, Y., Yamada, T., Kawagishi, I., Imae, Y. & Homma, M. 1996 Effect of viscosity on swimming by the lateral and polar flagella of Vibrio alginolyticus . J. Bacteriol. 178, 5024–5026.Google Scholar
Baskaran, A. & Marchetti, M. C. 2009 Statistical mechanics and hydrodynamics of bacterial suspensions. Proc. Natl Acad. Sci. USA 106, 15 567–15 572.Google Scholar
Becker, L. E., Koehler, S. A. & Stone, H. A. 2003 On self-propulsion of micro-machines at low Reynolds number: Purcell’s three-link swimmer. J. Fluid Mech. 490, 15–35.Google Scholar
Berg, H. C. 2000a Constraints on models for the flagellar rotary motor. Phil. Trans. R. Soc. Lond. B 355, 491–501.Google Scholar
Berg, H. C. 2000b Motile behavior of bacteria. Phys. Today 53, 24–29.Google Scholar
Berg, H. C. 2004 E. coli in Motion. Springer.Google Scholar
Berg, H. C. & Brown, D. A. 1972 Chemotaxis in Escherichia coli analysed by three-dimensional tracking. Nature 239, 500–504.Google Scholar
Berke, A. P., Turner, L., Berg, H. C. & Lauga, E. 2008 Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101, 038102.Google Scholar
Bhattacharya, S. & Bławzdziewicz, J. 2002 Image system for Stokes-flow singularity between two parallel planar walls. J. Math. Phys. 43, 5720–5731.CrossRefGoogle Scholar
Blake, J. R. & Chwang, A. T. 1974 Fundamental singularities of viscous flow. J. Engng Maths 8, 23–29.Google Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242–251.Google Scholar
Chattopadhyay, S., Molddovan, R., Yeung, C. & Wu, X. L. 2006 Swimming efficiency of bacterium Escherichia coli . Proc. Natl Acad. Sci. USA 103, 13 712–13 717.CrossRefGoogle ScholarPubMed
Darwin, C. 1953 Note on hydrodynamics. Proc. Camb. Phil. Soc. 49, 342–354.Google Scholar
Drescher, K., Dunkel, J., Cisneros, L. H., Ganguly, S. & Goldstein, R. E. 2011 Fluid dynamics and noise in bacterial cell–cell and cell–surface scattering. Proc. Natl Acad. Sci. USA 108, 10940–10945.Google Scholar
Drescher, K., Goldstein, R. E., Michel, N., Polin, M. & Tuval, I. 2010 Direct measurement of the flow field around swimming microorganisms. Phys. Rev. Lett. 105, 168101.CrossRefGoogle ScholarPubMed
Dunkel, J., Putz, V. B., Zaid, I. M. & Yeomans, J. M. 2010 Swimmer–tracer scattering at low Reynolds number. Soft Matt. 6, 4268–4276.Google Scholar
Dunstan, J., Miño, G., Clément, E. & Soto, R. 2012 A two-sphere model for bacteria swimming near solid surfaces. Phys. Fluids 24, 011901.Google Scholar
Frymier, P. D., Ford, R. M., Berg, H. C. & Cummings, P. T. 1995 Three-dimensional tracking of motile bacteria near a solid planar surface. Proc. Natl Acad. Sci. USA 20, 6195–6199.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall – I. Motion through a quiescent fluid. Chem. Engng Sci. 22, 637–651.Google Scholar
Golestanian, R. & Ajdari, A. 2008 Analytic results for the three-sphere swimmer at low Reynolds number. Phys. Rev. E 77, 036308.CrossRefGoogle ScholarPubMed
Guasto, J. S., Johnson, K. A. & Gollub, J. P. 2010 Oscillatory flows induced by microorganisms swimming in two dimensions. Phys. Rev. Lett. 105, 168102.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. Prentice-Hall.Google Scholar
Hernandez-Ortiz, J. P., Stoltz, C. G. & Graham, M. D. 2005 Transport and collective dynamics in suspensions of confined swimming particles. Phys. Rev. Lett. 95, 204501.Google Scholar
Holmqvist, P., Dhont, J. K. G. & Lang, P. R. 2006 Anisotropy of Brownian motion caused only by hydrodynamic interaction with a wall. Phys. Rev. E 74, 021402.Google Scholar
Huang, P. & Breuer, K. S. 2007 Direct measurement of anisotropic near-wall hindered diffusion using total internal reflection velocimetry. Phys. Rev. E 76, 046307.Google Scholar
Kim, S & Karilla, S. J. 2005 Microhydrodynamics: Principles and Selected Applications. Dover.Google Scholar
Lauga, E., DiLuzio, W. R., Whitesides, G. M. & Stone, H. A. 2006 Swimming in circles: motion of bacteria near solid boundaries. Biophys. J. 90, 400–412.Google Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.Google Scholar
Laurent, T. C., Pertoft, H. & Nordli, O. 1980 Physical chemical characterization of Percoll. I. Particle weight of the colloid. J. Colloid Interface Sci. 76, 124–132.Google Scholar
Leptos, K. C., Guasto, J. S., Gollub, J. P., Pesci, A. I. & Goldstein, R. E. 2009 Dynamics of enhanced tracer diffusion in suspensions of swimming eukaryotic microorganisms. Phys. Rev. Lett. 103, 198103.Google Scholar
Li, G., Tam, L.-K. & Tang, J. X. 2008 Amplified effect of Brownian motion in bacterial near-surface swimming. Proc. Natl Acad. Sci. USA 105, 18 355–18 359.Google Scholar
Lighthill, J. 1975 Mathematical Biofluiddynamics, Regional Conference Series in Applied Mathematics, vol. 17. SIAM.Google Scholar
Lin, Z., Thiffeault, J.-L. & Childres, S. 2011 Stirring by squirmers. J. Fluid Mech. 669, 167–177.Google Scholar
Lobry, L. & Ostrowsky, N. 1996 Diffusion of Brownian particles trapped between two walls: theory and dynamic-light-scattering measurements. Phys. Rev. B 53, 12050.Google Scholar
Lowe, G., Meister, M. & Berg, H. C. 1987 Rapid rotation of flagellar bundles in swimming bacteria. Nature 325, 637–640.CrossRefGoogle Scholar
Minamino, T., Imae, Y., Oosawa, F., Kobayashi, Y. & Oosawa, K. 2003 Effect of intracellular pH on rotational speed of bacterial flagellar motors. J Bacteriol. 185, 1190–1194.CrossRefGoogle ScholarPubMed
Miño, G. L., Mallouk, T. E., Darnige, T., Hoyos, M., Dauchet, J., Dunstan, J., Soto, R., Wang, Y., Rousselet, A. & Clément, E. 2011 Enhanced diffusion due to active swimmers at a solid surface. Phys. Rev. Lett. 106, 048102.Google Scholar
Prüß, B. M. & Matsumura, P. 1997 Cell cycle regulation of flagellar genes. J Bacteriol. 179, 5602–5604.CrossRefGoogle ScholarPubMed
Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45, 3–11.Google Scholar
Purcell, E. M. 1997 The efficiency of propulsion by a rotating flagellum. Proc. Natl Acad. Sci. USA 94, 11 307–11 311.CrossRefGoogle ScholarPubMed
Ramia, M., Tullock, D. L. & Phan-Thien, N. 1993 The role of hydrodynamic interaction in the locomotion of microorganisms. Biophys. 65, 755–778.Google Scholar
Saintillan, D. & Shelley, M. J. 2007 Orientational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett. 99, 058102.CrossRefGoogle ScholarPubMed
Schneider, W. R. & Doetsch, R. N. 1974 Effect of viscosity on bacterial motility. J. Bacteriol. 117, 696–701.Google Scholar
Staropoli, J. F. & Alon, U. 2000 Computerized analysis of chemotaxis at different stages of bacterial growth. Biophys J. 78, 513–519.Google Scholar
Thiffeault, J.-L. & Childres, S. 2010 Stirring by swimming bodies. Phys. Lett. A 374, 3487–3490.Google Scholar
Wilson, L. G., Martinez, V. A., Schwarz-Linek, J., Tailleur, J., Bryant, G., Pusey, P. N. & Poon, W. C. K. 2011 Differential dynamic microscopy of bacterial motility. Phys. Rev. Lett. 106, 018101.Google Scholar
Wu, X.-L. & Libchaber, A. 2000 Particle diffusion in a quasi-two-dimensional bacterial bath. Phys. Rev. Lett. 84, 3017–3020.CrossRefGoogle Scholar
Zaid, I. M., Dunkel, J. & Yeomans, J. M. 2011 Lévy fluctuations and mixing in dilute suspensions of algae and bacteria. J. R. Soc. Interface 8, 1314–1331.Google Scholar