Motility, Mixing, and Multicellularity

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Motility, Mixing, and Multicellularity
Raymond E. Goldstein Department of Physics
Program in Applied Mathematics BIO5
Institute University of Arizona
The Zooming Bio- Nematic, a non- equilibrium
phase with turbulent dynamics
Large-scale coherent flows from chemotaxis, with
diffusion dom- Inated by advection
Physical driving forces underlying evolutionary
transitions to multicellularity in Volvox
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Bacterial Swimming and Chemotaxis
Macnab and Ornstein, J. Mol. Biol. (1977)
Real-time Imaging of Fluorescent Flagella
1-4 mm
10-20 mm
Turner, Ryu, and Berg, J. Bacteriol. (2000)
20 nm
normal LH helix curly RH helix straight
straight
Swimming speed 10 mm/s Propulsive force 1 pN
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Advection, Dissipation Diffusion Reynolds and
Peclet Numbers
Navier-Stokes equations
Passive scalar dynamics
Reynolds number
Peclet number
If U10 mm/s, L10 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
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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)
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Mechanism of Self-Concentration
Dombrowski, et al. (2004)
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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).
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Side Views Depletion and Flow
2 mm
Video 100x actual speed
Dombrowski, et al. (2004)
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Diffusion and Chemotaxis
Oxygen diffusion/advection
Chemotaxis
Navier-Stokes/Boussinesq
depletion layer D/v
n(z)
C(z)
z
z
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Experiment vs. Theory
Tuval, et al. PNAS (2005)
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Moffat Vortex
1 mm
Experiment (PIV)
Numerics (FEM)
Tuval, et al. (2005)
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Chemotactic Singularities Mixing
Stirring re-oxygenates the entire drop
Tuval, et al. (2005)
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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)
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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)
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Velocity Correlation Functions in Space Time
space
oscillations due to multiple vortices (individual
images)
sequence average
time
oscillations due to recurring vortices (individual
images)
spatial average
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Advection of Microspheres
contact line
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Historical Ideas

• Flocking models (Toner and Tu, 1995, traffic
  flow)

A Landau theory in the velocity field clever
but not at all faithful to the physics of Stokes
flow

• Sedimentation (interacting Stokeslets)

as few as three particles exhibit
chaotic trajectories (Janosi, et al., 1997)

• Conventional chemotaxis picture (e.g.
  Keller-Segel) - MISSES ADVECTION

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
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(consider the Volvocalean green algae)
Part III. Driving Forces for Multicellularity
Chlamydomonas
V. carteri
Discovered by van Leeuwenhoek (1700), name means
fierce roller
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The Diffusional Bottleneck
Smoluchowski result diffusion to an
absorbing sphere
Number of peripheral cells, and hence
their requirements, scale as R2
Fluxes
Organism radius R
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Volvox On A Stick
S. Ganguly
Solari, Ganguly, Kessler, Michod Goldstein,
Multicellularity and the Functional
Interdependence Of Motility and Molecular
Transport, preprint (2005).
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Stirring by Volvox carteri
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A Closer View
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Even Closer (Flagellar Motions Visible)
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Locally Chaotic Advection
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High-Speed Movie (125 fps) of Volvox Flagella
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Flow Field Viewed On Axis
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Fluid Velocities During Life Cycle

• Hatch
• Division
• Daughter
• Pre-Hatch

Solari, et al. (2005)
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Peclet Number During Life Cycle (Large!)
Solari, et al. (2005)
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Flagellar-Driven Flows and Scaling Laws
Specified shear stress t at surface
Detailed calculation (Gegenbauer polynomials,
etc.) yields
This implies that the Peclet number scales as
Finally, large Pe scaling (FluxRPe1/3) yields
This almost eliminates the bottleneck!
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Velocity Profile
Solari, et al. (2005a)
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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