Locomotion
Low Re hydrodynamics
Low Re hydrodynamics
Motion of an object completely immersed in fluid
An object can accelerate only if the fluid exerts a net force on it
We understand how an organism moves, we must also solve for the motion of all the fluid around it.
Force balance for a volume of fluid (Newton’s law per volume):
ma
F
=
= ρa
V
V
ρ is fluid density
12.2. HYDRODYNAMICS OF WATER AND OTHER FLUIDS
Forces can come from many places, but most importantly
(A)
635
(B)
p(z+Dz)
1. Pressure gradients
vz(x)
p(x)
Fx = −∆(pA)x →
F
∂p
Fx
= −∇p
=−
→
V
V
∂x
12.2. HYDRODYNAMICS OF WATER AND OTHER FLUIDS
2. Shear forces
(A)
Dx
z
Dy
z
p(x+Dx)
Dx
Dy
y
x
y
x
p(z+Dz)
vz(x)
p(x)
∂ 2 vz
Fz ∝
∂x2
p(y)
635
p(z)
(B)
Fluid equivalent of frictional forces
Dz
p(y+Dy)
Dz
→
F
= η∇2 v
V
Dz
p(y+Dy)
Dz
p(y)
Dx
z
Dy
z
p(x+Dx)
Dx
Dy
Figure 12.5: Pressure and viscous forces acting on a small volume element of
fluid. (A) Pressure acts on the faces of the volume element perpendicular to
each face, with the force pointing into the fluid element. (B) Spatial gradients
of the fluid velocity lead to shear forces on the faces of the fluid element. If the
fluid velocity points in the z-direction, and it spatially varies along the x-axis,
shear forces will be acting in the z-direction on the faces of the fluid element
that are perpendicular to the x-axis.
p(z)
η is viscosity; typically ηwater = 1 cPoise = 0.01 g / cm s
y
x
y
x
Now that we have the geometry of fluid motion in hand, we turn to the
Figure 12.5: Pressure and viscous forces acting on a small volume element of
of the forces acting on the fluid element. For simplicity we take
η is sometimes written μ; you often also see the ratiofluid.
η/ρ(A)
defined
“kinematic
viscosity”
ν. calculation
Pressure as
actsthe
on the
faces of the volume
element perpendicular
to element to be a box of dimensions ∆x × ∆y × ∆z and whose faces
the fluid
3. We usually ignore gravity
F
= −ρgẑ
V
each face, with the force pointing into the fluid element. (B) Spatial are
gradients
perpendicular to the x, y or z axis. For a fluid in flow the force due to
of the fluid velocity lead to shear forces on the faces of the fluid element.
If the with other fluid elements can be split up into a pressure force and
interaction
fluid velocity points in the z-direction, and it spatially varies along the
x-axis,
a viscous stress force. The first is a consequence of pressure variations in the
shear forces will be acting in the z-direction on the faces of the fluid element
fluid, while the second is the frictional force that arises due to gradients in the
that are perpendicular to the x-axis.
flow velocity. Both contributions are illustrated in fig. 12.5.
The pressure field p(x, y, z, t) exerts opposing forces on parallel faces of the
fluid element as shown in fig. 12.5(A). We examine the force on the two faces of
Now that we have the geometry of fluid motion in hand, we turn
theelement which are perpendicular to the x-axis. The face at position x
thetofluid
calculation of the forces acting on the fluid element. For simplicityexperiences
we take a force p(x, y, z, t)∆y∆z in the positive x direction, while the force
the fluid element to be a box of dimensions ∆x × ∆y × ∆z and whose
faces
on the
face at x + ∆x is p(x + ∆x, y, z, t)∆y∆z in the negative x direction. The
are perpendicular to the x, y or z axis. For a fluid in flow the force
duepressure
to
total
force in the positive x-direction can be obtained by adding these
interaction with other fluid elements can be split up into a pressure two
forceforces
and as
a viscous stress force. The first is a consequence of pressure variations in the
fluid, while the second is the frictional force that arises due to gradients in the
δFxp = p(x, y, z, t)∆y∆z − p(x + ∆x, y, z, t)∆y∆z.
(12.6)
flow velocity. Both contributions are illustrated in fig. 12.5.
and electrostatic forces
q
F
= E
V
V
Low Re hydrodynamics
What about ρa?
ρa = ρ
Dv
Dt
D/Dt is the “convective derivative”:
The velocity of a volume at a fixed point in space can change in two ways:
1. the velocity is “legitimately” changing in time
∂
D
=
+ (v · ∇)
Dt
∂t
2. new fluid entering the space has a different velocity from old fluid leaving the space
To get this simple form for the convective derivative we assumed incompressibility:
∇·v =0
All together, these give the Navier-Stokes equation
−∇p + η∇2 v = ρ
∂v
+ ρ(v · ∇)v
∂t
Low Re hydrodynamics
Simplifying Navier-Stokes
The N-S equation is nonlinear (bad)
Estimating terms in the N-S equation for a small, slow object gives, for a bacterium in water:
η ~ 10-2 g / cm s
v ~ 10-3 cm / s
ρ~1g/
cm3
L ~ 10-4 cm
−∇p + η∇2 v = ρ
∆p
L
ηv
L2
?
103
∂v
+ ρ(v · ∇)v
∂t
0
ρv 2
L
10−2
The ratio of inertial to viscous terms is the Reynolds number Re. For our bacterium,
Re =
ρv 2 /L
ρvL
∼ 10−5
=
2
ηv/L
η
“Inertial forces” are totally negligible compared to viscous ones,
Viscous forces must counterbalance pressure gradients (to an accuracy of 1 in 105, anyway)
We might as well just solve
−∇p + η∇2 v = 0
This is the Navier-Stokes equation for Re ≪ 1.
This equation governs the motion of almost all cells.
Only in the case of extremely large cells moving extremely quickly do you need to include the inertial terms.
This is a linear equation: much, much more mathematically tractable.
Low Re hydrodynamics
Weirdness of the low Re world
Mass doesn’t matter
ρ has dropped out of the equations: no inertia
No coasting: a bacterium that stops propelling itself will stop within 1 Å.
History is irrelevant: motion is entirely dictated by instantaneous application of forces
Reversing applied forces perfectly reverses motion:
This is called as kinematic reversibility
Mixing is difficult at low Re
Low Re hydrodynamics
Since there is no explicit time dependence, it doesn’t matter whether you execute a motion quickly or slowly
In physics terms, there is no characteristic time to define the difference between “fast” and “slow”
The scallop theorem:
Since reversing applied forces reverses motion, any symmetric / reciprocal / oscillatory motion produces no net motion.
A Re = 0 scallop cannot move.
(Real scallops operate at Re > 1)
A low-Re organism with a single degree of freedom in its motion (for example, a single hinge) cannot propel itself
Many high-Re methods for locomotion fail at low Re:
A simple scissor kick is symmetric, so at low Re it will produce no net thrust over a complete cycle
Breaststroke kick (frog kick) should work because it’s not symmetric
Low Re hydrodynamics
Purcell’s article “Life at Low Reynolds Number” suggested three ways to break the trap of
kinematic reversibility:
1. Add another degree of freedom
Peko Hosoi (MIT)
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Examples of Re=0 flow
Examples of Re=0 flow
Simple geometries can be solved analytically
Shear flow: flow in a long, wide rectangular pipe due to a pressure head p0
Starting from the assumption
v = v(x, y)x̂
p = p(x, y)
no-slip condition: v=0
And applying PDEs
+y
The solution is
!
v(y) = v0 1 −
" y #2 $
h
x̂
p(x) = p0
x"
1−
L
!
0
max. v at center
with
v0 =
h2 p0
2η L
In terms of total flux Φ, this is Φ = ρ
length L and width w
2h3 w p0
3η L
+x
lower pressure
∇·v =0
higher pressure
h
∂2v
∂p
=η 2
∂x
∂y
Examples of Re=0 flow
Shear flow: flow in a long circular pipe of radius a due to a pressure head p0
By a sequence of arguments similar to previous slide,
!
!
" r #2 $
z"
p = p0 1 −
v = v0 1 −
ẑ
L
a
with
v0 =
a2 p0
4η L
from which we can calculate the flux
Φ=ρ
πa4 p0
8η L
It is not a coincidence that Φ ∝ p0.
The equations that govern low Re flow are linear, so we always end up with Φ ∝ v ∝ p0, though the constants of
proportionality depend on the geometry.
In geology or hydrology or civil engineering this is called Darcy’s law.
If Φ ∝ p0, we can define an impedance p0/Φ that characterizes the flow through a channel of a certain cross-section and length.
Flow in the circulatory system is mostly low Re (except in the aorta), so it can be modeled as network of tubes, each with calculable
impedance, arranged in series and in parallel.
This maps mathematically to an electric circuit where the “voltage” p0 is provided by the heart (mostly), the “current” is the blood flow
Φ and the circulatory system is replaced by resistors in series and in parallel (arteries to capillaries to veins).
Examples of Re=0 flow
Flow past a sphere at low Re
Stokes’ solution for a moving sphere:
vr
vθ
p
Sphere at rest; moving fluid
!
$
3 a 1 " a #3
+
= v0 cos θ 1 −
2r
2 r
!
$
3 a 1 " a #3
−
= −v0 sin θ 1 −
4r
4 r
3η cos θ v0 a
= p0 −
2
r2
Sphere moving; fluid at rest
Examples of Re=0 flow
The details of the entire flow field are less important than the force-velocity relations:
Since the low Re N-S equations are linear, the force F required to move at speed v0 must obey F ∝ v0.
The constant of proportionality is the translational drag coefficient γT.
Looking at the pressure drop over a long, fat cylinder, γT = 6π η a.
Strictly speaking, this solution is valid only for an isolated sphere in an infinitely large bath of fluid.
If you put a non-slip surface near the sphere it will increase the drag.
If the object is an ellipsoid instead of a sphere, then it matters which way way you drag it through the water
For major radius a and minor radius b, in the limit b ≪ a, γT becomes
γ! =
4πηa
ln(2a/b) − 1/2
γ⊥ =
8πηa
ln(2a/b) + 1/2
Drag parallel and perpendicular to major axis are not the same
In the limit of very long, skinny ellipsoid (i.e. a rod) γ⊥ = 2γ"
You can look up flow field solutions for a rotating sphere too.
For a sphere of radius a rotating at angular velocity ω, τ = 8πηa3 ω
so γR = 8π η a3
As before, there is a correction for nonspherical objects. For ellipsoids with b ≪ a,
rotation about major axis: τ =
rotation about minor axis: τ =
16
πηab2 ω
3
8
3
3 πηa
ln(2a/b) − 1/2
ω
Again, strictly speaking these solutions only apply to an isolated sphere.
Examples of Re=0 flow
Flow past a helix at low Re
Apply parallel and perpendicular drag coefficients to a helical geometry
This assumes that bits of a helix can be treated as bits of an infinitely long, isolated rod (“resistive force theory”). This is
manifestly untrue when the helix is close to a plane, or when the pitch of the helix is small.
1. Take a uniform helix (like a bacterial flagellum) moving with velocity v and
velocity
ω. MATHEMATICS OF WATER
652 rotating with angular
CHAPTER
12. THE
2. Compute the total velocity vnet = v + ω × r
3. Break vnet into components parallel and perpendicular to the helix
4. Apply parallel and perpendicular drag coefficients to the components
v
v||
q
v^
q
F!
F⊥
=
=
−γ! v!
−γ⊥ v⊥
F||
F^
5. Sum all forces to give the total net force on the helix.
6. Calculate torques r × F and sum them to give the net torque on the helix
In general, for a helix translation and rotation get mixed together:
F depends on both v and ω and so does τ.
Z
F
Figure 12.14: Force diagram for a swimming bacterium. The schematic shows
how the velocity at a given point on the flagellum can be resolved into components that are parallel and perpendicular to the flagellum.
This isn’t the case for a simpler object – or, in general, any nonchiral object.
force
2πηLV is
balanced
by involve
the propulsive
force F
p , leading to an expression
Figuring out the explicit expressions for F and τ is a good exercise in vector
algebra,
but
doesn’t
any new
physics.
for V , namely,
(Equations are on a later slide.)
V = v cos θ sin θ = πDf sin θ cos θ.
(12.43)
This mixing is why a rotating propeller can make a bacterium translate. Using the numbers for E. coli given above, we estimate V ≈ 70 µm/s, while the
measured
is roughly
30see
µm/s.
The converse is also true: pulling a helix through a viscous fluid makes it rotate.
Wespeed
just don’t
often
this.
Reciprocal Deformation of the Swimmer’s Body Does Not Lead to
Net Motion at Low Reynolds Number
Twirling of a helical filament is one of many strategies used by life forms
that live at low Reynolds number to get around. Interestingly, some swimming
strategies that are quite effective for humans and other large swimmers, fail
miserably at low Reynolds number. Take for example a simple strategy that
is similar to that used by a scallop. The “swimmer” is made of two plates
connected by a hinge, and the plates come together and then swing apart at
different speeds. Even though this kind of motion will lead to the propulsion of
the swimmer at large Reynolds numbers, a low Reynolds number swimmer that
adopts this strategy will be rewarded by fatigue without forward progress. To
show this we exploit the property of low Reynolds number flows that the forces
on moving object are proportional to their velocity.
The angle between the plates, θ(t), varies in time in such a way that at the
Swimming with flagella
Swimming with flagella
On any self-propelled object, torques and forces must balance
As the bacterium starts rotating its flagellum, this creates an unbalanced propulsive force.
The bacterium accelerates until frictional drag equals propulsive force.
This is the Re=0 equivalent of terminal velocity for a falling object.
The cell gets up to terminal velocity essentially instantly.
From far away, all propulsion looks like a force dipole
The forward force (e.g. thrust from the flagellum) is cancelled by a backward force (e.g. drag on the head).
Mathematically, this can be represented by a force doublet (a Stokeslet): either pushing (⇨⇦) or pulling (⇦⇨).
From far away, (distance ≫ doublet size), the flow fields from all force doublets look identical
The flow fields of all free-swimming bacteria, with time and distance cues stripped out, look the same.
Swimming with flagella
Swimming E. coli imaged at 30 Hz
Swimming E. coli imaged at 500 Hz
!
"
v
Fbody + Fbundle = 0
Γbody + Γbundle = 0
|Γbody | = |Γbundle | = |Γmotor |
Swimming with flagella
2a
2b
#
p
m
!
"
$
r
v
Cell body
Bundle
Fbody = −Av
Γbody = −BΩ
Fbundle = −αv + γω
Γbundle = +γv − βω
A = A1 sin2 θ + A2 cos2 θ
α = (8π 2 r2 + p2 )C0
B = (D1 + m A1 ) sin θ + D2 cos θ
32πηae3
A1 =
(3e2 − 1)E + 2e
16πηae3
A2 =
(1 + e2 )E − 2e
32πηab2 e3 (2 − e2 )
D1 =
3(1 − e2 )((1 + e2 )E − 2e)
32πηab2 e3
D2 =
3(2e − (1 − e2 )E)
!
e = a2 − b2 /a
2
2
E = ln((1 − e)/(1 − e))
2
β = (4π 2 r2 + 2p2 )r2 C0
γ = 2πr2 pC0
C0 =
2πηL
(ln(ρ/2p) + 1/2)(4π 2 r2 + p2 )
Physics of swimming
We have 6 equations with 7 unknowns.
We can still say something about whether the system is “optimal”
swimming speed (µm/s) per motor rate (Hz)
Flagellar length is pretty well matched to get maximum speed for a given motor rotation rate
0.07
0.06
0.05
0.04
0.03
0.02
0.01
20
40
60
80
100
flagellum
length (µm)
length
of flagellum
(µm)
In addition, the normal form flagellar pitch/radius combination gives maximum speed for a given length and rotation rate
Physics of swimming
Close the system with some information about the motor’s torque-speed curve:
Torque (relative)
Γmotor = |Γbody | = Γ(ω + Ω)
1
0
0
100
200
motor rate (Hz)
300
cell body
bundle
Speed
(µm/s)
Length
(µm)
Width
(µm)
Moment
(µm)
Wobble
(°)
Rotation
(Hz)
Force
(pN)
Torque
(pN nm)
Length
(µm)
Diameter
(nm)
Pitch
(µm)
Radius
(µm)
Rotation
(Hz)
Force
(pN)
Torque
(pN nm)
29
±6
2.5
± 0.6
0.88
± 0.08
0.05
± 0.3
43
± 12
23
±8
0.32
± 0.08
840
± 360
9.5
± 2.3
12
2.22
0.2
131
± 31
0.41
± 0.23
650
± 220
Swimming with flagella
Constant-torque motor is a good design
It has uniform (and presumably high) efficiency
Don’t have to engineer elaborate timing
Allows flagella to phase-lock in the bundle
Kenny Breuer, Brown University
Swimming with flagella
Movie from Keiichi Namba
Swimming with flagella
Even within the simple world of “passive” flagella, there are other ways to swim
Rhodobacter sphaeroides (monoflagellate)
Spirillum volutans (subpolar flagella: not visible)
Rhizobium lupini (polyflagellate)
Swimming with flagella
Purely internal forces
Internal flagella are useful to evade immune system
Internal forces get are hypothesized to couple to medium through flowing outer membrane
Spirochetes: Borrelia burgdorferi (Lyme disease)
(subpolar internal flagella: not visible)
Flavobacterium johnsoniae (mechanism unknown)
Nelson et al. J Bact 190: 2851 (2008)
Cilia – the other flagella
Cilia – the other flagella
Eukaryotic flagella (aka cilia) have dynein motors distributed along their length.
Motors cause local flexing of the 9+2 microtubule structure.
Since the stroke is asymmetric, an individual cilium can cause net fluid movement (or propulsion)
Chlamydomonas (green alga) has 2 flagella operating in (almost) mirror image of each other
model of ciliary stroke
(b)
Cortez and Fauci, Computing in Science and
tubule cilium nearing the end of its power stroke. Asterisks denote fluid markers, which we
Engineering,
March:38
(2004)
e base of the cilium in a rectangular array. The displacement
to the
right is the result
of the net
g cilium. (b) A ciliary waveform showing a single filament at equally spaced time intervals.
Chlamydomonas
Cilia – the other flagella
Usually one finds whole beds of cilia
These may move fluid (lungs) or propel an organism (paramecium)
Ciliar beats are coordinated into metachronal waves through hydrodynamic interaction.
Full understanding of coordination requires 3D computer model of interaction as well as a model for the beat form.
502
ANNALS OF THE NEW YORK ACADEMY OF SCIENCES
Side view
Top view
FIGURE 6. Transporting a mucus layer at selected times.
Dillon et al, “Fluid Dynamic Models of Flagellar
and
Ciliary Beating”, N.Y. Acad. Sci. 1101: 494 (2007).
ciliary engine affects the ciliary beat form, and how this ciliary beat form
influences the properties of mucus transport.
An immersed boundary model for multiciliary transport is described in detail
in Yang et al.43 Here we show preliminary results of a multiciliary simulation
with the addition of a mucus layer modeled as an elastic mesh consisting of
immersed boundary points connected by linear elastic elements.
In FIGURE 7 (upper row) we show simulation results in which the model
parameters are identical to that in FIGURE 6 but without the mucus layer. A
fictitious mucus layer is shown, but the points in the layer are treated in the
simulation as fluid markers, which have no effect on the numerical solution.
In FIGURE 7 (lower row) we show simulation results with the mucus layer at
comparable stages of the ciliary beats. The cilia transporting a mucus layer
UNC Chapel Hill
Conserved function for embryonic nodal c
A similar m e c h a nis m m a y u n d e rlie th e h a n d e d n e ss s e e n in all v ert e brat e b o d y pla n
H – the other flagella
Cilia
h
Xenopus
Chick
Mouse
ow left–right handedness originates conserved expression of Lrd homologues ing the vertebrate L–R axis. Pre
in the body plan of the developing in different vertebrate model organisms ures to identify nodal cilia
vertebrate embryo is a subject of con- indicate that the activity of nodal cilia is mammals may be related to the d
siderable debate1,2. In mice, a left–right bias probably a universal mechanism for specify- we have observed in the relative t
is thought to arise from a directional extraLrdr
Monocilia
cellular flow (nodal flow) that is generated by
Nodal (primary) cilia:
dynein-dependent rotation of monocilia on
b
c
a
the ventral surface of the embryonic node3,4.
showthe
that
the existence
of node
Rotation of nodalHere
cilia we
define
right-left
asymmetry
in
monocilia
and
the
expression
of
a
dynein
developing vertebrate embryo.
gene that is implicated in ciliary function
are conserved across a wide range of vertebrate
indicating
that a inversus
similar ciliary
Ciliary defects lead
to classes,
50% chance
of situs
mechanism may underlie the establishment
(entire body is inverted left-right)
of handedness in all vertebrates.
no phenotype per se,
in general
paralyzed
ciliary
In but
mice,
mutations
in the
genelead
thatto
problems similar
to
cystic
fibrosis
(lung
infections)
encodes the left–right dynein heavy chain
d
e
f
(Lrd ), a component of the ciliary motor,
Essner et al, Nature 418:37 (2002)
result in immotile node cilia and a reversal
Rotation off vertical
breaks
symmetry
of the
left–right
(L–R) axis in roughly half
of mutant offspring5,6. In humans, mutations in the dynein heavy-chain gene
DNAH5 are also associated with immotile
Nodal
cilia in mouse
embryo
cilia syndrome,
an inherited
disorder that
includes mirror-image reversal of the internal organs in half of affected individuals7.
Both Lrd (refs 5, 6) and Dnahc5 (ref. 7) are
expressed in the ventral layer of the mouse
h
i
g
node in cells containing cilia (Fig. 1a–c).
To test the validity and universality of the
nodal-flow model, we cloned Lrd homologues from chick, Xenopus and zebrafish,
analysed their expression, and examined
embryos for the presence of monocilia
(see supplementary information). In chick
embryos, Lrdr (Lrd-related) is expressed
Hensen’s
node
during gastrulation within
Buceta,
Biophys
J 89:2199 (2005)
and in the primitive streak (Fig. 1d).
k
l
j
Although it has been suggested that there is
no equivalent of the ventral mouse node in
chick8, we observed cilia at Hensen’s node
Surface motility
Surface motility
Other forms of motility work in contact with surfaces.
These generally amount to cycles of catch - pull - release.
Actin
The actin cytoskeleton can be used simultaneously to push the leading edge and pull the retracting edge of a cell.
Adhesion foci (transmembrane adhesion patches) are created at the leading edge as actin anchor points, migrate to the trailing
edge, and then release and dissipate.
Surface motility
Crawling: a cell pulls itself across a surface
Pulling seems to be more frequent than pushing
Based on hydrodynamics, at low Reynolds number either should work.
But hin polymers are probably stronger under tension than under compression (buckling), favoring pulling.
mouse fibroblast (3h)
chick fibroblast (2h)
mouse melanoma cell (2m)
trout epidermal keratocyte (4m)
Movies from Small lab (stunning website at http://cellix.imba.oeaw.ac.at)
Surface motility
Crawling uses actin and microtubule networks.
Actin networks (green) converge on adhesion sites
(red), presumably to apply forces to substrate.
Microtubules (green) also converge on adhesion sites
(red), presumably to supply regulatory proteins.
Surface motility
Twitching
Pili can be used to attach to surface for movement
These Psudomonas pull themselves (right to left) against an orienting flow (left to right)
Pseudomonas aeroginosa: type IV pili
Surface motility
Pili can generate very large forces.
The mechanism of force generation during pilus retraction is still under investigation
High Force Generation in N. gonorrhoeae
Biais et al, PLoS Biology 6:0904 (2008)
Neisseria gonorrhoeae: type IV pili
Maier “Single pilus motor forces exceed
100 pN” PNAS 99:16013 (2002)
PLoS Biology | www.plosbiology.org
0909
April 2008 | Volume 6 | Issue 4 | e87
Fig. 2. Retraction kinetics of pilE mutant under load. Electron micrograph of WT MS11 (A) and pilE mutant MS11C9.10 (B) at 0.01 mM IPTG. (C) Frequency of
retraction events and number of pili visible by electron microscopy. Only those cells were counted that retracted!showed pili. (D) Average stall force of WT
compared with pilE mutant at varying IPTG concentrations. (E) Velocity as a function of force for varying expression level of pilin subunits. WT, black triangle;
pilE, 10 mM IPTG, blue diamond; pilE, 0.1 mM IPTG, red square; pilE, 0.01 mM IPTG, yellow circle; averaged over K & 90 retraction events for each IPTG
concentration. Note that many retraction events were terminated by breakage before the stalling force was reached.