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 Singers/shakers (joint work with Kurt Ehlers, 2008, unpublished)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 Purcell 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)

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(joint work with Kurt Ehlers, 2008, unpublished)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

800kHz ultrasonic wave in glycerol

W. Dridi, V. Button, X. Escriva, H. BenHadid, D. Henry

Video1

Video2

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√(where is the kinematic viscosity and 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/(4U(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 = 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

Next : analogies for AS / attenuation as an asymmetry

Summary:

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 Escalado

An asymetry is required

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: molecular motors

(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 )

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

Bengtsson, Martin; Laurell, Thomas, Analytical and Bioanalytical Chemistry, Volume 378, Number 7, April 2004 , pp. 1716-1721.

Experiment on a diatom? Sandra Azevedo’s lab, UFRJ

Experiment on Synechococcus?

From: kehlers@scsr.nevada.edu [mailto:kehlers@scsr.nevada.edu]Sent: Mon 12/8/2008 12:46 AMTo: 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!