Geometría y Física del Movimiento
de Microorganismos
Jair Koiller, FGV/RJ, GMC y AGIMB
UniAndes, Deciembre 11 2008
El bobo, Doctor Universalis
Outline
I.
Rowers and Squirmers (Taylor, Lighthill, Purcell, 1950  …)
Aristotelian mechanics, gauge theory of micro-swimming
Purcell
a short course: www.impa.br/~jair
• Geometry: the hydro-dynamical connection
• Optimizing Efficiency: the sub-riemannian control problem
II. Singers/shakers (joint work with Kurt Ehlers, 2008, unpublished)
• Acoustic streaming
• AS in micro-engineering devices
• AS in locomotion strategies: diatoms and cyanobacteria
Collaboration with Kurt Ehlers
I.
Rowers and Squirmers
Aristotelian mechanics, gauge theory of micro-swimming
Geometry: the hydro-dynamical connection
Optimizing Efficiency: sub-riemannian control problem
a short course (JK): www.impa.br/~jair
seminar
A survey in bacterial motility: Howard Berg (Harvard)
Purcell
II. Singers/shakers
(joint work with Kurt Ehlers, 2008, unpublished)
• Acoustic streaming
• AS in micro-engineering devices
• AS in locomotion strategies:
diatoms and cyanobacteria
Acoustic streaming
Acoustics timeline
• Lord Rayleigh (Theory of Sound, 1896)
• Nyborg, Westervelt
(RNW streaming, 1953)
• Lighthill (1978):
“not only can a jet generate sound,
but also sound can generate a jet”
AS holds for all Reynolds numbers
focus here in low Reynolds regime
MEMS devices
NEMS devices
MIT Gallery
Berkeley
Nanophysics
Roukes
New stuff AS in MEMS
Monash Research Tanetal
Near future: NEMS Review
Nanotech
Old stuff:
Moroney Hashimoto
Two main types of acoustic streaming:
1. The quartz wind effect. Here the attenuation takes place
in the bulk of the fluid. Streaming is normal to the source.
(When piezoelectrically excited, the faces of a quartz
crystal vibrate, creating an ultrasonic beam. AS
generates a turbulent jet with velocities reaching 10’s of
cm/s.)
2. Boundary induced streaming. Here the attenuation takes
place near a solid surface. The induced streaming is
tangential to the surface.
Quartz wind effect
Attenuation of the sound wave occurs in the bulk of the fluid
Video1
Video2
800kHz ultrasonic wave in glycerol
W. Dridi, V. Button, X. Escriva, H. BenHadid, D. Henry
Boundary induced streaming
• Let U be an irrotational oscillatory vector field in a fluid
representing an acoustic wave.
• Owing to the no slip boundary condition, U must vanish
at a solid boundary. There is a thin layer (the Stokes
boundary layer) where U is rotational. The thickness of
the Stokes boundary layer is 5√(n/w) where n is the
kinematic viscosity and w is the frequency of U.
• Within the Stokes boundary layer shear stresses cause
strong attenuation of U leading to streaming.
Rayleigh’s Law
• In the late 19th century Lord Rayleigh showed that the
streaming velocity at the edge of the boundary layer due
to an oscillatory vector field U=U(x) is
-3/(4w) U(x)U’(x)
and that the streaming is in the direction of the nodes:
Kundt’s Tube
A resonant standing acoustic wave is established
using a sound transducer at one end of the tube.
(Quartz wind)
Boundary induced streaming blows dust into piles at
the nodes.
We propose two possible mechanisms for self-propulsion via
acoustic streaming:
1. The Quartz Wind (QW) model.
2. The Surface Acoustic Wave (SAW) model based on
boundary induced streaming
The Quartz Wind model
• In this model the spicules in a small region vibrate at a high
frequency buckling the crystalline shell in a manner similar to an
electric door buzzer. Our inspiration for this mechanism came from
the cuica: aBrazilian samba instrument
• A flow, normal to the cell, is generated by attenuation in the bulk of
the fluid.
• Problem: Low efficiency. (Quartz wind swimmers are Hummers!)
Power/Efficiency estimate for Synechococcus
employing the QWmechanism:
Lighthill defines the efficiency to be the ratio of the power required to push
the cell through the water to the power required by the mechanism (P):
η = viscosity, for water η = 0.01 g / cm sec
a = radius, for Synechococcus a = 10^(-4) cm
V = velocity, for Synechococcus V = 2.5×10^(-3) cm / sec
The force (F) exerted on the fluid by the QW effect and acoustic power (P)
are related by P=Fc where c is the speed of sound.
The force required to drive the cell with velocity V is
F = 6πμaV
making the power output for Synechococcus P=7×10^(-10) watts.
The efficiency of the QW mechanism for Synechococus is then
η=1.7×10^(-6) %
The squirming and boundary induced streaming mechanisms have
efficiencies between 0.1-1%.
We have not ruled out the QW mechanism completely. There are
possible power enhancement mechanisms.
Example: Bubble induced streaming. Here submicro-bubbles adhere
to the CS. Being of characteristic size, the bubbles resonate
enhancing the local streaming.
Boundary induced streaming: the SAW mechanism
In this model, the cell propagates a high frequency traveling wave
along the CS. Attenuation of the wave within the Stokes boundary layer
generates a mean flow just outside this layer creating an effective ‘slip’
velocity.
Longuet-Higgins (1953) derived a generalization of Rayleigh’s law for
streaming due to a traveling wave. The limiting streaming velocity at
the edge of the Stokes boundary layer is the real part of
(* = complex conjugate)
where
is the tangential velocity at the CS and
is the solution to the
linearized NS equations outside the Stokes boundary layer (=0 for us).
We model the cell as a sphere of radius a with spherical coordinates
where is the azimuthal coordinate. The traveling wave is
where ϕm is a material point on the CS. The slip velocity due to
streaming leads to a swimming velocity of
which is 2.5 times that predicted by the squirming mechanism.
Efficiencies for the boundary induced streaming
mechanism
The efficiency compares well with other known strategies.
But … are the required frequencies biologically feasible?
Question: Is singing biologically feasible?
Bacterial flagellar motors are large membrane embedded structures
and have been observed to rotate at 300Hz when unloaded.
From E-Coli in Motion
HC Berg (2004, Springer)
The required frequency for acoustic streaming is biologically feasible.
More details for people who know some fluid mechanics
Streaming flow = what survives after averaging out the fluctuating part
due to some external source or to internal waves
this idea is also used in statistical turbulence
Averaging already present in the very formulation of Navier Stokes equations
Equations of motion in AS: time-averaged Navier Stokes equations.
Reynolds stress tensor = gives the mean momentum flux.
Its gradient is a force, non-zero when an attenuation mechanism is present.
Attenuation is necessary for streaming can occur in the body of the fluid
or in a thin Stokes boundary layer surrounding a surface.
What is Reynolds stress? (following Lighthill, Waves in fluids pg 338)
acceleration frames produce inertial forces (general fact in mechanics)
Eulerian velocities are different at each point in the fluid.
typically, these inertial forces appear when one averages the turbulent
variations in fluid velocity about a mean flow, or when waves produce
fluid motions.
In short:
Acoustic streaming is the result of a gradient in the Reynolds stress
associated with high frequency (acoustic) oscillations in the fluid.
Reynolds stress = r mean value of ui uj
But … attenuation is needed for a net nonzero forcing
Why is attenuation necessary?
Formal argument in Lighthill pag 338-339.
For unattenuated internal waves:
fluid velocity is parallel to surfaces of constant phase, the gradient is
perpendicular to them.
For unattenuated sound waves: force is gradient of a scalar, will be
cancelled by the gradient of a mean pressure
Will see: attenuation works as an asymmetry leading to directed motion
analogy with other phenomena
Summary:
Next : analogies for AS / attenuation as an asymmetry
Analogies for non-fluidmechanicists (like me)
AS is one among many phenomena in which:
small vibrations are rectified to organized
macroscopic motion (linear or rotational)
this always involves an averaging process,
usually second order in the amplitude
An asymetry is required
Escalado
Example:
holonomy in principal bundles
(requires: nonintegrable distributions)
Berry phases (+ topological interpretations)
In Acoustic Streaming:
assymetry = attenuation
Example:
holonomy due to a time periodic potential
A digression, specially for biologists:
(hot topic!)
Feynman’s Ratchet
Eindhoven
Adelaide
this is wrong!
Relation with game theory
Parrondo games
this is correct
(need T2 > T1
By second law of thermodynamics )
molecular motors
MEMS devices
NEMS devices
MIT Gallery
Berkeley
Nanophysics
Roukes
AS in MEMS
Monash Research Tanetal
Old stuff:
Moroney Hashimoto
Biological applications
Artificial cilia
Wixforth droplets
Renaudin nanoquakes
Advalytix
Microchip
AS in biological MEMS devices?
Diatoms
Diatoms Diatoms Diatoms Diatoms Diatoms
Nanotechnology
Startreck
Life in glass houses
Experiment on a diatom?
Sandra Azevedo’s lab, UFRJ
Bengtsson, Martin; Laurell, Thomas, Analytical and Bioanalytical Chemistry,
Volume 378, Number 7, April 2004 , pp. 1716-1721.
Experiment on Synechococcus?
From: kehlers@scsr.nevada.edu [mailto:kehlers@scsr.nevada.edu]
Sent: Mon 12/8/2008 12:46 AM
To: Jair Koiller
Hi Jair,
For the experiment we will need to work out the details of what to
expect. The wave in the fluid would not cause much bulk flow but
boundary induced streaming would move a 'dead' cell. Maybe Berg's
tracking microscope could be used to figure out what the wave does to
a single bacterium on average over a time period. I'll try to think things
out once classes end (this week).
Enjoy Colombia!
(Berg's wife is from Bogota!)
Cheers, Kurt
Nanotech meeting
GRACIAS!