Lecture 12. Life in the low

Reynolds-number world

Zhanchun Tu (涂展春 )

Department of Physics, BNU

Email: tuzc@bnu.edu.cn

Homepage: www.tuzc.org

Main contents

● Friction in fluids

● Low Reynolds-number world

● Biological applications

§12.1 Friction in fluids

Suspension ( 悬浮液 )

Suspension of globular protein

0

z

mg

w g V

Gravity: mg Buoyant force: w g V

Net force: m−w V g≡mnet g

Profile of particle density

c z =c0 e−

mnet g z

k B T r

[sedimentation ( 沉降 ) equilibrium in gravity]

Myoglobin (肌红蛋白 ): mnet

≈4 kg/mol

z∗≡k B T r /mnet g≈60 m

In a 4cm test tube c z /c0=e−0.04 /60=0.999

Thus the suspension never settles out (沉降 ). Called colloid

Homework: Please estimate the thickness of PM2.5 in our atmosphere

Centrifuge (离心机 )

r

ω

m

Centripetal (向心 ) acceleration: 2 r

Equivalence principle

Centrifugal (离心 ) force f =m2 r

Centrifugal potential

U r =−∫ f dr=−12

m2 r2

Boltzmann law==> c r =c0em

2 r2/2 k BT r

(sedimentation equilibrium in centrifuge)

Problem: derive the above equation from Nernst-Planck formula

Sedimentation time scale(沉降系数 )● Sedimentation velocity

v drift=mnet g

Depends on g

NOT intrinsic property of particles

● Sedimentation time scale

s=vdrift / g=mnet /

unit: Svedbergs (10-13s)

Problem: Find the relation between s and the mass of an ideal polymer

mnet=m−w V

2Rg

s=mnet /

V ∝N : number of monomers⇒mnet∝m

=6w Rg

Rg2=⟨ 1

N ∑n=1

N

Rn−Rc2⟩=⟨ 1

N ∑n=1

N

Rn−1N ∑

m=1

N

Rm2

⟩=

1

2 N 2∑n ,m

N

⟨Rn−Rm2⟩

Gaussian chain: ⟨Rn−Rm2⟩=∣n−m∣b2

⇒ Rg2=Nb2

/6 for large N

⇒ Rg∝N 1 /2∝m1 /2

⇒ s∝m0.5[Data: Meyerhoff & Schultz(1952)]

s∝m0.5

s∝m0.44

Hard to mix a viscous liquid● Experiment: ink in corn syrup (玉米糊 )

slowly

● Question:

2nd law==>mix. Is 2nd law violated in the above experiment?

Einstein relation

Stokes relation

D=k B T

∝corn

D∝k BT

corn

∞⇒ D0Thus the blob of ink cannot change much in a short time due to diffusion, and no evident mixing phenomenon

Slowly Stirring (搅拌 ) causes an organized motion, in which successive layers of fluid simply slide over each other. Such a fluid motion is called laminar flow (层流 ).

§12.2 Low Reynolds-number world

Newtonian fluid

Uniform flow along x

x

y

zv y y

y+dy

τfdxdz v

vdv

Force applied on y+dy layer by y layer

f =−dvdy

(Newtonian viscous formula)

Coefficient of viscosity

Discussion: guess the above formula from simple analysis.

Viscous critical force● Dimensional analysis for Newtonian fluid

[η]: Pa s=kg m-1 s-1 [ρ]: kg m-3

Hence no intrinsic distinction between “thick” and “thin” fluids.

We cannot construct a dimensionless quantity from η and ρ.

We cannot construct a dimension of length from η and ρ.

Newtonian fluid “has no intrinsic length scale”, or is “scale invariant”.

Macroscopic experiments can tell us something relevant to a microscopic organism.

● Viscous critical force

[ 2

]: kg m−1 s−12

kg m−3 (dimension of force)

f crit=2/

If an external force f < fcrit

, the fluid can be

called “thick”. Friction will quickly damp out inertial effects. Flow is dominated by friction.

Typical scale of forces inside cells is in pN range.

Friction rules the world of the inner cell!

≃1 nN

= kg m s-2

Reynolds number

R

v

A ball moves in water

● Dimensional analysis

[η]: Pa s=kg m-1 s-1 [ρ]: kg m-3

[v]: m s-1[R]: m

ℜ≡ R v

[ R v ]: Dimensionless number

● Reynolds number

[Reynolds (1880s)]

● Physical meaning I

Time for a ball moving 2R: t=2 R /v

The same volume of water moves 2R in time t,the mean acceleration is estimated as

12

at2=2 R⇒ a=

4 R

t 2 =v2

R

~ R3v2

/R =R2 v2

R

v

inertial term of water= mass X acceleration

friction between the ball and water: f frict= v=6 R v~ R v

inertial termfriction

=Rv

=ℜ

The inertial term can be safely omitted if ℜ≪1

● Physical meaning II

R

v Friction between the ball and water

f frict= v=6 R v~ R v

Kinetic energy of water

E K~R3 v2

Dissipative work done by the friction in displacement R:

W frict= f frict R~ R2 v

EK

W frict

= Rv

=ℜ

● Physical meaning III

f frict~ R v

f crit=2/

f frict

f crit

~ Rv

=ℜ

ℜ≪1⇒ f frict= f ext (omit the inertial term)

Force applied on water by the ball,also=the external pulling force on the ball

Thus f ext≪ f crit for ℜ≪1

● Problem: Calculate the Reynolds numbers – (1) A 30 m whale, swimming in water at 10m/s

– (2) A 1 μm bacterium, swimming at 30 μm/s

Solutions:

(1) ℜ=Rv

=103

×30×10

10−3=3×108

≫1

(2) ℜ=Rv

=103

×10−6×30×10−6

10−3=3×10−5

≪1

Bacteria live in the world of low Reynolds number where the viscous friction rules their motions!

Time reversal properties

● Time reversal operation (t→-t)

solution Equation of motion

t→-t t→-t

Time reversaldynamical equation

Time reversalEquation of motion

solution

Dynamical equation

Time reversal Invariance

solution

● Example I: falling ball in gravity

mg

v0

z

0

Dynamical equation

d 2 z

dt 2=−g

Equation of motion

z t =v0t−12

g t2solution

Time reversaldynamical equation

d 2 zdt 2

=−g

Time reversalequation of motion

zrt =z −t

=−v0 t−12

g t 2

t→-tt→-t

solution

Time reversal Invariance!

solution

● Example II: diffusion equation

∂ c∂t

=D∂

2 c

∂ x2solution

t→-t

−∂c∂ t

=D∂

2 c

∂ x2

c x , t =1

4 Dte−x2

/4 Dt

solutioncrx ,t =c x ,−t

=1

i4 Dtex2 /4 Dt

t→-t

NOT time-reversal invariant

NOT a solution

● Viscous friction rule is not time-reversal invariant

R

A ball in highly viscous fluid

f

Time reversaldynamical equation −

dxdt= f

Dynamicalequation

dxdt

= f

NOT time-reversal invariant

Something irreversible in the motion ruled by friction!

Discussion: Does this conclusion contradict the experiment of ink drop in corn syrup?

● Newtonian fluid & simple elastic solid

f =−dvdy

=−ddy

dudt

Newtonian fluid

NOT time-reversal invariant!

Simple elastic solid

=−Gdudy

u

x

y

v

x

y

Time-reversal invariant!

(displacement)

(Hooke relation)

§12.3 Biological applications

Swimming of microorganisms● Strictly reciprocating (往复 ) motion

a. the paddles (桨 ) move backward in speed v relative to the body==>the forward motion of the body in speed u relative to water.

b. the paddles move forward in speed v' relative to the body==>==> the backward motion of the body in speed u' relative to water.

c. repeating a.

Time interval t Time interval t'

Friction force of body from water:

f b=−0u

Speed of paddles relative to water:

u p=−vu

Friction force of paddles from water:

f p=−1 u p=−1u−v

Force balance:

f b f p=0⇒u=1 v /01

Displacement of body after time t: x=u t=1 vt /01

Friction force of body from water:

f b'=−0−u ' =0u '

Speed of paddles relative to water:

u p'=v '−u ' =v '−u '

Friction force of paddles from water:

f p'=−1 u p

'=−1v '−u '

Force balance:f b

' f p

'=0⇒u '=1v ' /01

Displacement of body after time t':

x '=−u ' t '=−1 v ' t ' /01

Net displacement of the body in a period t+t':

Constraint of relative speed in a period t+t':

the paddles should return to their original positions on the body

vt−v ' t '=0

x x '=1 vt /01−1 v ' t ' / 01

=1vt−v ' t ' /01=0Scallop theorem: Strictly reciprocating motion won’t work for swimming in the low-Reynolds world [Purcell (1977) Am. J. Phys.]

The required motion must be periodic, so that it can be repeated.

Question: What other options does a microorganism have?

It can’t be of the reciprocal type described above.

● Ciliary propulsion (纤毛推进 )

Many cells use cilia to generate net thrust (推力 ). Each cilium contains internal filaments and motors which create an overall bend in the cilium.

Periodic but NOT reciprocal!

有效行程 复位行程

Typical motion of cilia.

Understand intuitively the trick to generate the net motion

v

u

v'

u'time t

Quickly retractpaddles

and then

time t'Quickly extendpaddles

Key: viscous friction coefficient of paddles are proportional to their lengths

net displacement=[1/01−1'/01

']vt

=0 vt 1−1'/0101

'0

(Homework?)

● Bacterial flagella

衬套

推进器

钩

螺栓

55nm

16nm

41nm

Peptidoglycan(肽聚糖 )

Mechanism of flagellar propulsion (鞭毛推进机理 )

v v

dfdf

Key: although v is perpendicular to z, df can have z component. Why?

v

v┴

v║

f┴=-ζ┴v┴

f║ =-ζ║v║

f

Generally ,⊥∥

only if ⊥=∥⇒ f = f ⊥ f ∥ antiparallel v

⇒ f = f ⊥ f ∥ not antiparallel v

Discussion: prove that for a long flagellum

∫ d f ∥z

Discussion: There exists torsion around z although the total force along z. How can bacteria resist rotating themselves around z axis?

Option 1: the bacteria are rough enough (i.e., large surface area), such that

rotation is verylarge⇒bacteria=torsionrotation

0

Option 2: the bacteria have even number of flagella. A pair of flagella have the opposite rotation. The torsion is canceled out while force

along z axis is double.

Option 3: ...... (maybe)

● Dimer (二聚体 ) of two reciprocating motion

Single pair of paddles

Strictly reciprocal

=>No net motion

Dimer of 2 pairs of paddles

Single pair: strictly reciprocal

Dimer: nonreciprocal

=> Net motion is possible

No many-scallop theorem [Lauga and Bartolo (2008) PRE]

Blood flow in capillary (毛细血管 )● Model

blood vessel

ℜ=Rv

=103

×10−5×0.1

10−3=1

≃103 kg/m3 R≃10 m

v≃10 cm /s ≃10−3 Pa s

Reynolds's study suggests

laminar pipe flow for ℜ103

Profile of v(r) Constraints:

v R=0 non−slip

v 0∞

● Force balance equation

Pressure contributes a force: df p=2 r dr p

Viscous force from inner layer fluid drags forward the fluid shell:

df out=[2rdr L]dv r '

dr ' ∣r '= rdr

df in=−[2 r L]dv r

dr

Viscous force from outer layer fluid pulls backward the fluid shell:

= [2rdr L][ dv r dr

d 2v r

dr2dr]

df pdf indf out=0⇒rp L

dvdr

rd 2 v

dr2=0

note: dv /dr0L---length

● Flow profile and flow rate

Problem: prove that the general solution is

v r=AB ln r−r2 p /4 L

Problem: prove that

B=0 and A=pR2/4 L

Flow profile

v r=p R2

−r2

4 L

Flow rate (Hagen–Poiseuille relation)

Q=∫ v dA=2∫ v r rdr= R4 p8 L

∝R4

Blood vessels can efficiently regulate flow with only slight dilations or contractions!

前导链后随链

Viscous force at DNA replication fork

● Question on DNA replication

Since the two single strands cannot

pass through each other, the original

must continually rotate (arrow).

Would frictional force resisting this

rotation be enormous?

Y-shaped junctionω

2R

● Estimate the effect of friction

R

r+dr

r

v(r)

ω

Constraints:

v R=R and v ∞=0

Torsion from the fluid inner r that drags the fluid shell rotating counter-clockwise:

T in=−dv r

dr2 rL r=−2 Lr2 dv r

dr

Note: dvdr

0

Torsion from the fluid inner r+dr that drags the fluid shell rotating clockwise:

T out=2 L rdr 2 dv r '

dr ' ∣r '= rdr

T inT out=0⇒dr2 dvdr =0 ⇒ v r=

R2

rand

dvdr

=−R2

r2

T R=−2 L r2 dv rdr ∣r=R

=2 LR2

Friction torsion between the rod and fluid:

Book: 4π, which is correct?

The rod rotates 2π for opening each helical turn. The work of friction

W frict=T R×2=42 LR2

DNA polymerase synthesizes new DNA in E. coli at a rate ~1000bp/s

=2turn

×1000 bp/s

10.5 bp/ turn≈600 rad /s

⇒W frict=42 LR2

≈2.4×10−22 J=0.06 k B T r

10-3 Pa s <~10μm 1nm

Problem: please estimate R.

● DNA helicase (解旋酶 )

Helicase walks along the DNA in front of the polymerase,

unzipping the double helix as it goes along.

1 ATP ≡ 20 kBT

r

Energy source: ATP etc.

W frict=0.06 k B T r≪20 k B T r

The energy lost of viscous friction can be negligible!

(in physiological condition)

Cytoplasmic streaming (胞浆流动 )

Chara corallina珊瑚轮藻

leaf

node

internodalcell

an internodal cell

close-up of indifferent zone

chloroplasts

chloroplasts

close-up of indifferent (中性 ) zone (3D view)

endoplasm

indifferent zone

indifferent zone

vacuole

[PNAS 105(2008) 3663]

● Intracellular (细胞内 ) material transport: Flow dominate or diffusion dominate?

Characteristic length: L Characteristic velocity of flow: U

Diffusion constant: D

⟨ x2⟩=2 D t⇒Time for molecules diffusing L: tD=

L2

2 D≃

L2

D

Time for molecules reach L due to flow: t f=LU

Peclet number: Pe=t D

t f

=ULD

If Pe >>1, molecules reach L more quickly due to flow than diffusion.

Thus, flow dominates the transport.

If Pe <<1, then diffusion dominates the transport.

In an internodal cell: L=1mm, U=30 μm/s, D=10-5 cm2/s.

Pe=ULD

=30≫1⇒ flow dominates the transport.

● Physical description

Flow of streaming: ∇2u=∇ p , ∇⋅u=0

Fluid velocity

The change of concentration of materials:

c tu⋅∇ c=D∇2 c (advection-diffusion equation)

Note: if you stand a frame S' with the velocity of u, you will observe

c t '=D ∇ ' 2c

For large Peclet number, diffusion can be neglected, thus

c tu⋅∇ c=0

(stokes equation)

(advection equation)

If only we have solved the stokes equation and obtain u(x,y,z,t),

then we can obtain c(x,y,z,t).

● Spiral flow

u=v r ,HJ r ,

Parallel to sprial

In radial direction

Flow stems from surface, then passes the inner (vacuole), and reaches the other points in surface again. Helpful to material transport across the vacuole.

pitch/R3

This ratio corresponds to max nutrient (营养 ) uptake rate from environment

Max at pitch/R=3

§ Summary & further reading

Summary● Suspension

– “Suspension never settles out (沉降 )”

– Sedimentation time scale: s=mnet /

● Viscous liquid– Hard to mix a viscous liquid

– Friction rules the world of the cell!

– Critical viscous force:

● Reynolds number:– inertial term / friction

– kinetic energy/dissipative work of friction

– external force/critical viscous force

ℜ= Rv/

f crit=2/

● Time reversal properties– 2nd Newtonian law: time-reversal invariant

– Viscous friction rule: NOT time-reversal invariant

– Simple elastic solid: time-reversal invariant

– Viscous fluid: NOT time-reversal invariant

● Swimming of microorganisms: – Strictly reciprocating motion won’t work

– Periodic but NOT reciprocal motion can work: Ciliary propulsion & flagellar propulsion

● Blood flow in capillary– Flow rate:

– Blood vessels can efficiently regulate flow with only slight dilations or contractions

Q= R4 p8 L

● DNA replication fork– Dissipative work of friction:

– Energy lost of viscous friction can be negligible:

W frict=42 LR2

≈0.06 k BT r

W frict≪EATP≈20 k B T r

● Cytoplasmic streaming– Peclet number:

– Physical description:

Pe=ULD

∇2u=∇ p , ∇⋅u=0

c tu⋅∇ c=D∇2 c

Further reading

● E. M. Purcell, Life at low Reynolds number, Am. J. Phys. 45, 3 (1977)

● H. C. Berg, Motile Behavior of Bacteria, Physics Today, January (2000)

● R. E. Goldstein, I. Tuval, and J. van de Meent, Microfluidics of cytoplasmic streaming and its implications for intracellular transport, PNAS 105, 3663 (2008)