Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-13T14:53:43.997Z Has data issue: false hasContentIssue false
  • English
  • Français

Critical bacterial concentration for the onset of collective swimming

Published online by Cambridge University Press:  27 July 2009

GANESH SUBRAMANIAN  and
DONALD L. KOCH
GANESH SUBRAMANIAN*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India
DONALD L. KOCH
Affiliation:
School of chemical and bio-molecular engineering, Cornell University, Ithaca, NY 14853, USA
*
†Email address for correspondence: sganesh@jncasr.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the stability of a suspension of swimming bacteria in a Newtonian medium. The bacteria execute a run-and-tumble motion, runs being periods when a bacterium on average swims in a given direction; runs are interrupted by tumbles, leading to an abrupt, albeit correlated, change in the swimming direction. An instability is predicted to occur in a suspension of ‘pushers’ (e.g. E. Coli, Bacillus subtilis, etc.), and owes its origin to the intrinsic force dipoles of such bacteria. Unlike the dipole induced in an inextensible fibre subject to an axial straining flow, the forces constituting the dipole of a pusher are directed outward along its axis. As a result, the anisotropy in the orientation distribution of bacteria due to an imposed velocity perturbation drives a disturbance velocity field that acts to reinforce the perturbation. For long wavelengths, the resulting destabilizing bacterial stress is Newtonian but with a negative viscosity. The suspension becomes unstable when the total viscosity becomes negative. In the dilute limit (nL3 ≪ 1), a linear stability analysis gives the threshold concentration for instability as (nL3)crit = ((30/Cℱ(r))(DrL/U)(1 + 1/(6τ Dr)))/(1−(15𝒢(r)/Cℱ(r))(DrL/U)(1 + 1/(6τ Dr))) for perfectly random tumbles; here, L and U are the length and swimming velocity of a bacterium, n is the bacterial number density, Dr characterizes the rotary diffusion during a run and τ−1 is the average tumbling frequency. The function ℱ(r) characterizes the rotation of a bacterium of aspect ratio r in an imposed linear flow; ℱ(r) = (r2 −1)/(r2 + 1) for a spheroid, and ℱ(r) ≈ 1 for a slender bacterium (r ≫ 1). The function 𝒢(r) characterizes the stabilizing viscous response arising from the resistance of a bacterium to a deforming ambient flow; 𝒢(r) = 5π/6 for a rigid spherical bacterium, and 𝒢(r)≈ π/45(ln r) for a slender bacterium. Finally, the constant C denotes the dimensionless strength of the bacterial force dipole in units of μU L2; for E. Coli, C ≈ 0.57. The threshold concentration diverges in the limit ((15𝒢(r)/Cℱ(r)) (DrL/U)(1 + 1/(6τ Dr))) → 1. This limit defines a critical swimming speed, Ucrit = (DrL)(15𝒢(r)/Cℱ(r))(1 + 1/(6τ Dr)). For speeds smaller than this critical value, the destabilizing bacterial stress remains subdominant and a dilute suspension of these swimmers therefore responds to long-wavelength perturbations in a manner similar to a suspension of passive rigid particles, that is, with a net enhancement in viscosity proportional to the bacterial concentration.

On the other hand, the stability analysis predicts that the above threshold concentration reduces to zero in the limit Dr → 0, τ → ∞, and a suspension of non-interacting straight swimmers is therefore always unstable. It is then argued that the dominant effect of hydrodynamic interactions in a dilute suspension of such swimmers is via an interaction-driven orientation decorrelation mechanism. The latter arises from uncorrelated pair interactions in the limit nL3 ≪ 1, and for slender bacteria in particular, it takes the form of a hydrodynamic rotary diffusivity (Dhr); for E. Coli, we find Dhr = 9.4 × 10−5(nUL2). From the above expression for the threshold concentration, it may be shown that even a weakly interacting suspension of slender smooth-swimming bacteria (r ≫ 1, ℱ(r) ≈ 1, τ → ∞) will be stable provided Dhr > (C/30)(nUL2) in the limit nL3 ≪ 1. The hydrodynamic rotary diffusivity of E. Coli is, however, too small to stabilize a dilute suspension of these swimmers, and a weakly interacting suspension of E. Coli remains unstable.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 632 , 10 August 2009 , pp. 359 - 400
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Abramowitz, M. & Stegun, I. A. 1970 Handbook of Mathematical Functions. Dover.Google Scholar
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419–440.CrossRefGoogle Scholar
Berg, H. C. 1983 Random Walks in Biology, chapter 6. Princeton University Press.Google Scholar
Berg, H. C. January 2000 Motile behaviour of bacteria. Phys. Today 24–29.CrossRefGoogle Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, vol. 1. Wylie.Google Scholar
Brenner, H. & Edwards, D. A. 1993 Macrotransport Processes. Butterworth Heinemann.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, chapter 1. Dover.Google Scholar
Chapman, S. & Cowling, T. G. 1991 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.Google Scholar
Childress, S., Levandowsky, M. & Spiegel, E. A. 1975 Pattern formation in a suspension of swimming micro-organisms: equations and stability theory. J. Fluid Mech. 69, 591–613.CrossRefGoogle Scholar
Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R. E. & Kessler, J. O. 2004 Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. 93 (9), 098103.CrossRefGoogle ScholarPubMed
Gradshteyn, I. S. & Ryzhik, I. M. 1965 Table of Integrals, Series and Products. Academic Press.Google Scholar
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics. Noordhoff International Publishing.Google Scholar
Hernandez-Ortiz, J. P., Stolz, C. G. & Graham, M. D. 2005 Transport and collective dynamics in suspensions of confined swimming particles. Phys. Rev. Lett. 95, 204501.CrossRefGoogle ScholarPubMed
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83, 695–720.CrossRefGoogle Scholar
Hinch, E. J. & Leal, L. G. 1972 The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles. J. Fluid Mech. 52, 683–712.CrossRefGoogle Scholar
Ishikawa, T., Locsei, J. T. & Pedley, T. J. 2008 Development of coherent structures in concentrated suspensions of swimming model micro-organisms. J. Fluid Mech. 615, 401–431.CrossRefGoogle Scholar
Ishikawa, T., Simmonds, M. P. & Pedley, T. J. 2006 Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech. 568, 119–160.CrossRefGoogle Scholar
Kim, M. J. & Breuer, K. S. 2004 Enhanced diffusion due to motile bacteria. Phys. Fluids 16 (9), L78.CrossRefGoogle Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.Google Scholar
Larson, R. G. 1988 Constitutive Equations for Polymer Melts and Solutions. Butterworth-Heinemann.Google Scholar
Liao, Q., Subramanian, G., DeLisa, M. P., Koch, D. L. & Wu, Mingming 2007 Quadrupole force field dominates the hydrodynamic interactions of swimming Escherichia Coli. Phys. Fluids, 19 (6), 061701–061701-4.CrossRefGoogle Scholar
Mackaplow, M. B. & Shaqfeh, E. S. G. 1996 A numerical study of the rheological properties of suspensions of rigid, non-Brownian fibres. J. Fluid Mech. 329, 155–186.CrossRefGoogle Scholar
Mackaplow, M. B. & Shaqfeh, E. S. G. 1998 A numerical study of the sedimentation of fibre suspensions J. Fluid Mech. 376, 149–182.CrossRefGoogle Scholar
Mackaplow, M. B., Shaqfeh, E. S. G. & Schiek, R. L. 1994 A numerical study of heat and mass transport in fibre suspensions. Proc. R. Soc. Lond. A 447, 77–110.Google Scholar
Mehandia, V. & Nott, P. R. 2008 The collective dynamics of self-propelled particles. J. Fluid Mech. 595, 239–264.CrossRefGoogle Scholar
Mendelson, N. H., Bourque, A., Wilkening, K., Anderson, K. R., & Watkins, J. C. 1999 Organized cell swimming motions in Bacillus subtilis colonies: patterns of short-lived whirls and jets. J. Bacteriol. 181 (2), 600–609.CrossRefGoogle ScholarPubMed
Pedley, T. J., Hill, N. A. & Kessler, J. O. 1988 The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms. J. Fluid Mech. 195, 223–238.CrossRefGoogle Scholar
Rahnama, M., Koch, D. L., Iso, Y. & Cohen, C. 1993 Hydrodynamic translational diffusion in fibre suspensions subject to simple shear flow. Phys. Fluids A 5, 849–862.CrossRefGoogle Scholar
Saintillan, D. & Shelley, M. J. 2007 Orientational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett. 99, 058102.CrossRefGoogle ScholarPubMed
Saintillan, D. & Shelley, M. J. 2008 Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulations. Phys. Rev. Lett. 100, 178103.CrossRefGoogle ScholarPubMed
Shaqfeh, E. S. G. & Koch, D. L. 1988 The effect of hydrodynamic interactions on the orientation of axisymmetric particles flowing through a fixed bed of spheres or fibres. Phys. Fluids 31 (4), 728–743.CrossRefGoogle Scholar
Simha, A. R. & Ramaswamy, S. 2002 Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles. Phys. Rev. Lett. 89, 058101.CrossRefGoogle Scholar
Soni, G. V., Jaffar Ali, B. M., Hatwalne, Y. & Shivashankar, G. G. 2003 Single particle tracking of correlated bacterial dynamics. Biophys. J. 84, 2634–2637.CrossRefGoogle ScholarPubMed
Subramanian, G. & Koch, D. L. In press The instability of a suspension of swimming bacteria with and without chemoattractants. J. Fluid Mech.Google Scholar
Underhill, P. T., Hernandez Ortiz, J. P. & Graham, M. D. 2008 Diffusive and spatial correlations in suspensions of swimming particles. Phys Rev. Lett. 100, 248101.CrossRefGoogle ScholarPubMed
Wu, X. & Libchaber, A. 2000 Particle diffusion in a quasi-two-dimensional bacterial bath Phys. Rev. Lett. 84, 3017–3020.CrossRefGoogle Scholar
Wu, Mingming, Roberts, J. W., Kim, S, Koch, D. L. & DeLisa, M. P. 2006 Collective bacterial dynamics revealed using a three-dimensional population-scale defocused particle tracking technique. Appl. Environ. Microbiol. 72, 4987–4994.CrossRefGoogle ScholarPubMed