Motility, Mixing, and Multicellularity
Raymond E. Goldstein
Department of Physics & Program in Applied Mathematics & BIO5 Institute
University of Arizona
The Zooming BioNematic, a nonequilibrium phase
with turbulent
dynamics
Large-scale coherent
flows from chemotaxis,
with diffusion domInated by advection
Physical driving forces
underlying evolutionary
transitions to multicellularity
in Volvox
Bacterial Swimming and Chemotaxis
Macnab and Ornstein, J. Mol. Biol. (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. (2000)
“normal = LH helix
“curly” = RH helix
“straight” = straight
Advection, Dissipation & Diffusion:
Reynolds and Peclet Numbers
Navier-Stokes equations:

  
2
 (ut  u  u )  p   u
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 advection dominates diffusion.
The second relation breaks down with multicellularity…
Part I. Bacterial Self-Concentration
1 cm
Dombrowski, Cisneros, Chatkaew, Goldstein & Kessler, “Self-concentration
and large-scale coherence in bacterial dynamics,” PRL 93, 098103 (2004)
Tuval, Cisneros, Dombrowski, Wolgemuth, Kessler & Goldstein, “Bacterial
swimming and oxygen transport near contact lines,” PNAS 102, 2277 (2005)
Mechanism of Self-Concentration
Dombrowski, et al. (2004)
The Boycott Effect (in Sedimentation)
g
A.E. Boycott, Nature 102, 532 (1920).
A.A. Acrivos and E. Herbolzheimer, J. Fluid Mech. 92, 435 (1979).
Side Views: Depletion and Flow
2 mm
Video ~100x actual speed
Dombrowski, et al. (2004)
Diffusion and Chemotaxis
Oxygen diffusion/advection

2
ct  u  c  Dc c  nf (c)
nt  u  n  Dn n    ( rnc) Chemotaxis

2
 (ut  (u  )u)   p   u  ngzˆ
2
Navier-Stokes/Boussinesq
C(z)
n(z)
z
depletion layer: D/v
z
Experiment vs. Theory
Tuval, et al. PNAS (2005)
1 mm
Numerics (FEM)
Experiment (PIV)
Moffat Vortex
Tuval, et al. (2005)
Chemotactic Singularities & Mixing
Stirring re-oxygenates
the entire drop
Tuval, et al. (2005)
Part II. The Zooming Bio-Nematic Phase
contact line
Petri dish
300 mm
Dombrowski, Cisneros, Chatkaew, Goldstein & Kessler, “Self-concentration
and large-scale coherence in bacterial dynamics,” PRL 93, 098103 (2004)
Velocity Field from PIV (pendant drop)
Peclet number ~10-100 (vs. 0.01-0.1 for individual bacterium)
35 mm
Dombrowski, et al. (2003). See also Wu and Libchaber (2000)
Velocity Correlation Functions in Space & Time
space
oscillations due to
multiple vortices
(individual images)
sequence
average
time
spatial average
oscillations due to
recurring vortices
(individual images)
contact line
Advection of Microspheres
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 at all faithful 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 
 

4 r
r3 
•Conventional chemotaxis picture (e.g. Keller-Segel) - MISSES ADVECTION
ct  Dc 2 c  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 of self-propelled objects (Ramaswamy, et al.)…
IMPLICATIONS FOR QUORUM SENSING…
Part III. Driving Forces for Multicellularity
(consider the Volvocalean green algae)
Chlamydomonas
V. carteri
Discovered by van Leeuwenhoek (1700), name means “fierce roller”
The Diffusional Bottleneck
Fluxes
Smoluchowski result –
diffusion to an
absorbing sphere
Number of peripheral
cells, and hence their
requirements, scale as R2
 R
C  C 1  
r

Flux  4 C DR
Organism radius R
Volvox On A Stick
S. Ganguly
Solari, Ganguly, Kessler, Michod & Goldstein, “Multicellularity and the Functional
Interdependence Of Motility and Molecular Transport,” preprint (2005).
Stirring by Volvox carteri
A Closer View
Even Closer (Flagellar Motions Visible)
Locally Chaotic Advection
High-Speed Movie (125 fps) of Volvox Flagella
Flow Field Viewed On Axis
Fluid Velocities During Life Cycle
I.
II.
III.
IV.
Hatch
Division
Daughter
Pre-Hatch
Solari, et al. (2005)
Peclet Number During Life Cycle (Large!)
Solari, et al. (2005)
Flagellar-Driven Flows and Scaling Laws
Specified shear stress t at surface
Detailed calculation:
(Gegenbauer polynomials, etc.) yields:
tR
u 
f ( )

This implies that the Peclet number scales as:
Finally, large Pe scaling (Flux~RPe1/3) yields:
This almost eliminates the bottleneck!
2 Ru
Pe 
 R2
D
Flux  R
5/ 3
Velocity Profile
Solari, et al. (2005a)
Issues
Transport, mixing, and chemical signaling at high
concentrations – quorum sensing, etc.
(biology, nonequilibrium statistical mechanics, …)
Mixing, metabolism, and evolutionary transitions
to multicellularity – germ-soma differentiation,
vascularization, morphological transformations