Transcript Mar05lc - University of Arizona

PIV Studies of the Zooming Bionematic Phase
Luis Cisneros
Department of Physics
University of Arizona
NSF: MCB (NER)
Chris Dombrowski
John O. Kessler
Raymond E. Goldstein
Earlier work: Dombrowski, et al., PRL 93, 098103 (2004)
Advection, Dissipation & Diffusion:
Reynolds and Peclet Numbers
Navier-Stokes equations:

  
2
 (ut  u  u )  p   u  nf
Passive scalar dynamics:

2
ct  u  c  D c
Reynolds number:
 
 u  u U 2 / L UL


 Re
2
2
 u
U / L

Peclet number:
 
u  c UC / L UL


 Pe
2
2
D c DC / L
D
If U=10 mm/s, L=10 mm, Re ~ 10-4, Pe ~ 10-1
At the scale of an individual bacterium, dissipation dominates
inertia, and diffusion dominates.
With multicellularity, Pe > or >> 1.
Self-Concentration and the
Chemotactic Boycott Effect
2 mm
Video ~100x actual speed
Dombrowski, et al. (2004); Tuval, et al. (2005)
Experimental Details
Bacterial protocols using B. subtilis strain 1085 (and various mutants)
Simple: Overnight growth in Terrific Broth in a still petri dish
More controlled: Start with -20o C stock, prepared from spores stored on
sand. [Add to TB at RT, 24h of growth, 1 ml + 50 ml TB, incubated for 18 h.
Then 1 ml + 50 ml TB, incubated for 5 hrs. 0.75 ml + 0.25 ml glycerol].
1 ml of -20o stock + 50 ml TB, incubate for 18 h (shaker bath, 37o, 100
rpm), then 1 ml + 50 ml TB (5 hr), then into chamber
Fluorescent microspheres (Molecular Probes, Nile Red, 0.1-2.0 mm)
The ZBN in Brightfield and Fluorescence
210 mm
Velocity Field from Cinemagraphic PIV
Peclet number ~10-100 (vs. 0.01-0.1 for individual bacterium)
35 mm
Dombrowski, et al. (2004). See also Wu and Libchaber (2000)
The ZBN in Brightfield and Fluorescence
210 mm
PIV Velocity Field
210 mm
Streamlines (Note intermittency)
210 mm
Velocity-Velocity Correlation Function (spatial)
I (r) 
v(x  r, t )  v(x, t ) x  v
v
2
x
I(r)
r (mm)
 v
2
x
2
x
Velocity-Velocity Correlation Function (temporal)
J (t ) 
v(x, s  t )  v(x, s) s  v
v
2
J(t)
t (s)
s
 v
2
s
2
s
Vorticity (homage a Miró)
210 mm
Summary: Peclet Number Revisited
In the Zooming Bionematic (ZBN) phase, there are large
coherent regions of high-speed swimming, whose
internal fluid velocities and scale generate an effective
diffusion constant DZBN =L2/T~10-4 cm2/s which is an
order of magnitude larger than the molecular oxygen
diffusion constant. Alternatively, the (chaotic) Peclet
number is >> 1.
In the ZBN, the bacterial concentration is so high that
dissolved oxygen is used up in the time T~1 s, matching
the time scale of the coherent structures.
Side Views of Sessile Drops
drop
Tuval, et al. PNAS 102, 227 (2005)
Bacterial Swimming and Chemotaxis
(Macnab and Ornstein, 1977)
1-4 mm
10-20 mm
20 nm
Swimming speed ~10 mm/s
Propulsive force ~1 pN
Real-time Imaging of Fluorescent Flagella
t 
Turner, Ryu, and Berg,
J. Bacteriol. 182, 2793 (2000)
“normal = LH helix
“curly” = RH helix
“straight” = straight
Swimming Near the Contact Line
Bacterial Bioconvection
J.O. Kessler
The Chemotactic Boycott Effect
1 cm
Dombrowski, Cisneros, Chatkaew, Goldstein, and Kessler, PRL 93, 098103 (2004)
Mechanism of Self-Concentration
Dombrowski, et al. (2004)
Historical Ideas
•Flocking models (Toner and Tu, 1995, …; traffic flow…)
v t  ( v  ) v  v   | v |2 v  p  D12 v    
t    ( v)  0
A Landau theory in the velocity field – clever but
not relevant to the physics of Stokes flow
•Sedimentation (interacting Stokeslets)
n
ri  v 0  av 0  U(ri  r j )
j i
U (r ) 
as few as three particles exhibit chaotic
trajectories (Janosi, et al., 1997)
3a  e (e  r )r 
 

3
4 r
r 
•Conventional chemotaxis picture (e.g. Keller-Segel) - MISSES ADVECTION
ct  Dc2c  f (c,  )

 t  D 2     ( rc)
ct  (u  )c

 t  (u  ) 
Velocity field must be
determined self-consistently
with density field
•A synthesis is emerging from coarse-grained models of sedimentation
(Bruinsma, et al.) and self-propelled objects (Ramaswamy, et al. 2002, 2004)…
IMPLICATIONS FOR QUORUM SENSING…
Side Views of Sessile Drops
Tuval, Cisneros, Dombrowski, Wolgemuth, Kessler & Goldstein, preprint (2004)
Side Views: Depletion and Flow
2 mm
Dombrowski, et al. (2004)
Circulation Near the “Nose”
Self-trapping in the corner
Diffusion and Chemotaxis
Oxygen diffusion/advection

2
ct  u  c  Dc c  nf (c)
nt  u  n  Dn n    ( rn c) Chemotaxis

2
 (u t  (u  )u)   p   u  ngzˆ
2
Navier-Stokes/Boussinesq
C(z)
n(z)
z
depletion layer: D/v
z
Experiment vs. Theory
Tuval, Cisneros, Dombrowski, Wolgemuth, Kessler & Goldstein, preprint (2004)
Numerics (FEM)
Experiment (PIV)
Moffat Vortex
Tuval, et al. (2004)
Depletion Layers
Geometry of the Contact Line Region

c( r,  )  cs   am r
m / 2
cos( m / 2 )
m 1
nt  u  n  Dn n    ( rn c)
2
Tuval, Cisneros, Dombrowski, Wolgemuth, Kessler & Goldstein, preprint (2004)
Chemotactic Singularities & Mixing
Tuval, et al. (2004)
Supported Drops
Tuval, et al. (2004)