Fluid Dynamics
and
the Evolution of
Biological Complexity
R E Goldstein
DAMTP
The Size-Complexity Relation
Amoebas, Ciliates, Brown Seaweeds
Green Algae and Plants
Red Seaweeds
Fungi
Animals
?
Bell & Mooers (1997)
Bonner (2004)
Multicellularity
Uni- to multicellular organisms
Park, et al. (2003)
D. Mandoli (U. Washington)
Populations of individuals
Vibrio harveyi
Signaling, adaptation,
chemotaxis, quorum sensing
(e.g., Berg & Purcell)
Acetabularia crenulata
Advantages of size, complexity,
differentiation, transport
(e.g., Bonner, Niklas)
Mendelson (1976)
Some Terminology*
Homocytic – assembly of prokaryotic
or eukaryotic cells that are
structurally and functionally
equivalent
Heterocytic – differentiation of
structure/function
Lee, Cox, G (1996)
Colonial – physical association without cytoplasmic connections
Multicellular – … with cytoplasmic connections
Colonialism need not precede
heterocytic lifestyle
*(D.L. Kirk)
D. discoideum
I. Bacteria
III. Plants
Case Studies
II. Algae
Real-time Imaging of Fluorescent Flagella
1-4 μm
20 nm
10-20 μm
B. subtilis
C. Dombrowski
Turner, Ryu, and Berg (2000)
Macnab and Ornstein (1977)
Bacterial Swimming
Nanostructure of Flagella
11 protofilaments
•Nanoscale bistability of monomer packing produces microscale polymorphism
Asakura, Adv. Biophys. 1, 99 (1970).
Calladine, J. Mol. Biol. 118, 457 (1996).
Yamashita, et al., Nature Struct. Biol. 5, 92 (1998).
Samatey, et al., Nature 410, 331 (2001).
Competing Chirality in Other Systems
Supercoiled B. subtilis [Mendelson]
(plectonemic)
Supercoiled
B. subtilis
[Tilby]
(solenoidal)
Tendril Perversion
[Goldstein & Goriely]
Hotani’s Remarkable Experiment
Fluid flow, microns/sec
H. Hotani, J. Mol. Bio. 156, 791 (1982).
Microscope slide
Flagellum, pinned at hook end
Persistent oscillations at
low Reynolds number (!)
normal (LH)
Flow
curly (RH)
Time
Viscous Nonlinear Dynamics of Twist and Writhe
Ωt = ωs + ( −Ωt + t × t s ) ⋅ [rt ]s
Twist:
differential rotation
stretching
writhing
For the simplest (twist) elastic energy, torque balance
Ωt =
Bend:
C
ςr
Ω ss +
1
ς⊥
t × t s ⋅ fs
twist diffuses (generally fast),
forced by out-of-plane bending
ς ||tt ⋅ rt + ς ⊥ (1 − tt ) ⋅ rt = f s
ς ⊥ ≈ 2ς || ≈ 4πη /[ln( L / 2a ) + c ]
ς r = 4πηa 2
⇒
[ς rω = CΩ s ] gives
ε2 ≡
ςr
ς ⊥ L2
Goldstein, Powers & Wiggins, PRL 80, 5232 (1998)
The natural small
parameter associated
with a separation of
time scales:
twist = fast
bend = slow
Twirling and Whirling, Twisting and Writhing
E=
1
2
2
2
ds
(
A
κ
+
C
Ω
)
∫
~Ea4
Equating twisting and buckling torques:
2
A
⎛a⎞ E
ωc ≈ 2 ≈ ⎜ ⎟
≈ (10−3 ) 2109 ≈ 103 s −1
ςrL ⎝ L ⎠ η
Slender-body hydrodynamics + elasticity theory yield relevant scaling
Wolgemuth, Powers, and Goldstein, PRL 84, 1623 (2000)
Reconstituted Salmonella Flagella
Hotani
The Model (Statics)
To accommodate bistability:
Generalize conventional
elastic energy that is quadratic
in curvature and twist
E =
1
2
2
2
ds
(
A
κ
+
C
Ω
)
∫
to a Landau theory in the twist
with spontaneous curvature.
E = ∫ ds [ A (κ − Q ) + V ( Ω ) +
1
2
2
1
2
γ Ωs ]
2
2
Key result for static properties:
•Kinks, or domain walls, between
the two twist states correspond to
coexisting helices meeting at a
block angle α – consistent with
“Hotani’s Law”
Goldstein, Goriely, Huber & Wolgemuth, PRL 84, 1631 (2000)
Mechanism of Pulse Generation
τ
U>Uc
τc
nucleation
U<Uc
τ
τ
c
Ω
L z
torque
cancellation
L z
τ
τc
Ω+
Ω‐
z
Ω+
Ω‐
z
Ω+
resetting
L z
Ω‐
z
Coupled PDEs for Twist and Bend
(extremely “stiff” differential equations)
Bend dynamics
with flow
Relaxation to preferred curvature
iθ
ς p (ξt + Uξ z ) = − A(ξ zz − Qe ) zz
iθ
ς rθ t = −γ θ zzzz + V ' ' (θ z )θ zz + A Im(Qe ξ zz )
2
Hyperdiffusion from
twist-gradient energy
Multistable diffusion
*
Twist-bend coupling
Separation of time scales from the two distinct drag coefficients:
rotations are fast, bends relax slowly
Coombs, Huber, Kessler & Goldstein, PRL 89, 118102 (2002)
Dynamics of Flagellar Polymorphism
Key Physics: Torque Cancellation (Hotani, Purcell, …)
Filament dynamics
Twist dynamics
Coombs, Huber, Kessler & Goldstein, PRL 89, 118102 (2002)
Predictions (experimental tests underway)
numerical results for PDEs
scaling law
critical velocity
> 3 fronts
2-3 fronts
1-2 fronts
Filament growth rate
Extensions: Growth & Dynamic Bifurcations
Intrinsic twist
Wolgemuth, Goldstein, Powers, Physica D 190, 266 (2004)
Microscopic origin of
twist? Errington, et al.
Advection, Dissipation & Diffusion:
Reynolds and Peclet Numbers
Navier-Stokes equations:
r
r r r
2r
ρ ( u t + u ⋅ ∇u ) = − ∇ p + η ∇ u
Passive scalar dynamics:
r
2
c t + u ⋅ ∇c = D ∇ c
Reynolds number:
r rr
ρ u ⋅ ∇u ρU 2 / L UL
≈
≈
≡ Re
2r
2
η∇ u
ηU / L
ν
Peclet number:
r r
u ⋅ ∇c UC / L UL
≈
≈
≡ Pe
2
2
D∇ c DC / L
D
If U=10 μm/s, L=10 μm, Re ~ 10-4, Pe ~ 10-1
At the scale of an individual bacterium, dissipation dominates
inertia, and diffusion dominates advection.
The second relation breaks down with multicellularity…
Self-Concentration via Bioconvection
1 cm
Dombrowski, et al., PRL (2004)
Tuval, et al., PNAS (2005)
Diffusion, Chemotaxis, and Flow
Oxygen diffusion/advection
r
2
ct + u ⋅ ∇c = Dc ∇ c − βnf ( c )
Pedley & Kessler (1992)
nt + u ⋅ ∇n = Dn ∇2n − ∇ ⋅ ( nr ∇c ) Chemotaxis
r
2
ρ (u t + (u ⋅ ∇)u ) = −∇ p + η∇ u − αngzˆ
Navier-Stokes/Boussinesq
C(z)
n(z)
z
depletion layer: D/v
z
Side Views Reveal Large-Scale Flows
drop
laser (532 nm)
ring
light
Keplerian
telescope
shutter
dichroic
longpass filter
ccd camera
Tuval, et al. (2005)
Side Views: Depletion and Flow
2 mm
Video ~100x actual speed
1 mm
Numerics (FEM)
Experiment (PIV)
Persistent Vortex & Large Peclet Number
Vortex in wedge: Moffatt (1964).
r
ct + u ⋅ ∇c = D∇2c
r r
u ⋅ ∇c UL
≈
≡ Pe
2
D∇ c D
If U=10 μm/s, L=10 μm,
Re ~ 10-4, Pe ~ 10-1
At the scale of individual
bacteria, advection is
unimportant.
Collective dynamics can
lead to very large Pe.
Here, L~1000 μm,
U~100 μm/s so
Pe ~ 100.
The Zooming Bio-Nematic (ZBN)
Petri dish
contact line
brightfield
epi-fluorescence
300 μm
A “bacterial bath” with enormously enhanced diffusion (even superdiffusion)
[Kessler (‘98); Wu & Libchaber (‘00); Soni, et al., (‘03/4)]
Microfluidic applications [Kim & Breuer (‘04), Darnton, Turner, Breuer & Berg (’04)]
Velocity Field from PIV (pendant drop)
Peclet number ~10-100 (vs. 0.01-0.1 for individual bacterium)
35 μm
Like a van Kármán vortex street, but the Reynolds number is 10-2 (!)
Phenomenology like that of recurring jets and swirls in sedimentation.
Velocity Correlation Functions in Space & Time
space
sequence
average
oscillations due to
multiple vortices
(individual images)
What determines
these length and
time scales?
time
spatial average
oscillations due to
recurring vortices
(individual images)
Historical threads
•Conventional chemotaxis picture [e.g. Keller-Segel (1971)]
ct = Dc∇2c + f ( c, ρ )
nt = Dn ∇2n − ∇ ⋅ ( nr∇c )
Rich and diverse behavior, including
singularities (Childress) from
chemical signaling & chemotaxis.
But what about the fluid flow??
•Flocking models [Vicsek, et al. (1995), Toner and Tu, (1995),…]
v t + ( v ⋅ ∇) v = αv − β | v | v − ∇p + D1∇ v + ⋅ ⋅ ⋅
2
2
ρ t + ∇ ⋅ ( ρv ) = 0
A Landau theory:
predicts long-range
swimming order.
Not what we see!!
•“Active medium” models [(Ramaswamy, et al. (’02,’04),
Kruse, et al. (’04)] based on ideas from liquid crystals
predict a finite-wavelength instability from hydrodynamic interactions; long-time behavior unknown…
•Related “swarming models” [Bertozzi, et al. (’05)]
•Thin-film model [Aronson, Sokolov, G (’06)] includes
effects of shear on orientation vector
Challenges
rms velocity
•Direct simulation of a suspension of self-propelled swimmers
[Hernandez-Ortiz, Stoltz & Graham, PRL 95, 204501 (’05)]
0.01
0.1
φ
1
Distinct onset of large velocities at transition
length scale increases with φ
at high φ, fluctuations span box
What is the continuum limit of this system?
Simulations of Self-Propelled Rods
Saintillan & Shelley (2007)
Precursors of Collective Behaviour
Bacterial Crowd Control
Reversal of Bacterial Locomotion
at an Obstacle*
Luis Cisneros1, Christopher Dombrowski1, Raymond E. Goldstein1,2,3, John O. Kessler1
1 Department of Physics, 2 Program in Applied Mathematics, and 3 BIO5 Institute,
University of Arizona, Tucson.
PRE 73, 030901(R)
(2006)
What if the locomotion of a bacterium is impeded by an obstacle?
Like a wall, another cell or….
…group of cells?
To produce Quorum Polarity at
high concentrations of cells, there
might exist some mechanism for
“wrong way” oriented individuals
to join the majority orientation
without “turning around” .
To produce
Reversal of motion, or “backing
up”, is less costly.
Previous work:
1.
Turner & Berg PNAS, 1995 : E. coli swim with either end forward
2.
Magariyama Biophys, 2005 : mono-flagellated organisms
reverse rotation of flagella
Experimental Setup
800 μm
AIR
Glass Sphere
Terrific Broth
Bacterium in Gap
Petri dish bottom
OBJECTIVE
• Under our particular growth conditions B.s. do not display
the usual “run and tumble” behavior
• Petri dish is permeable to O2
Side view
• Bacteria swim along the bottom. Trajectories are long,
slightly curved, runs with essentially no tumbles.
Bacterial Reversal
• Cells occasionally approach to the gap near sphere’s contact point
(excluded region). They generally turn away and keep swimming.
• Close to straight-on “docking” may produce reversal motion, suggesting
that the flagellar bundle flips from one end of the cell to the other.
incoming
outgoing
Gap is less than
1.1mm wide
Top view
Bacterial Reversal in flagrante
Time dependence of average
swimming velocities
35
30
Vin∞
Vout∞
25
20
V (μm/sec)
15
10
5
0
0
1
2
3
4
5
t (s)
tstop
trestart
• Velocities reach an asymptote
on extremes of trajectories far from
reversal point.
• Asymmetry on acceleration time
scales.
Average over 100 runs
Asymptotic velocities
• Data points cluster near the equality line
• Histogram shows a Gaussian distribution
• Statistical symmetry between (V∞)in and (V∞)out
• Swimming ability is independent of the polar location
of the flagellar bundle
Asymmetry between inward and outward motion
Different position of the flagellar bundle with respect to the gap
induces a difference in viscous drag and hence different net
forces over cell body during docking and undocking processes.
Collaborators
Postdocs:
Idan Tuval (HFSP)
Marco Polin
Ph.D. students:
Jan-Willem van de Meent
Knut Drescher
GKB Laboratory
Stuart Dalziel
David Page-Croft
John Milton
Trevor Parkin
Rob Raincock
Tim Pedley
Tobias Locsei
Cristian Solari (Arizona & Buenos Aires)
Sujoy Ganguly (Arizona & Cambridge)
Chris Dombrowski (Arizona->Davis)
Luis Cisneros (Arizona)
Tom Powers (Brown)
John Kessler (Arizona)
Rick Michod (Arizona)
Aurora Nedelcu (New Brunswick)
Schlumberger Chair Fund
NSF, NIH, DOE