interplay of energy, diffusion and friction indispense del corso di biofisica, dipartimento di...

Dispense del Corso di Biofisica, Dipartimento di Fisica, Università di Cagliari. A.A.: 2014/2015. Docente: Dott. Attilio Vittorio Vargiu NB: Queste dispense non sostituiscono il materiale didattico suggerito a piè del programma!

Dynamics in nanoworlds

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Interplay of energy, diffusion and friction in

(sub)cellular world


NB Queste diapositive sono state

preparate per il corso di Biofisica tenuto dal Dr. Attilio V. Vargiu presso il

Dipartimento di Fisica nell’A.A. 2014/2015

Non sostituiscono il materiale didattico consigliato a piè del programma.

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References •  Books and other sources

•  Biological Physics (updated 1st ed.), Philip Nelson, Chap. 5

•  Biochemistry (5th ed.), Berg et al., Chap. 4

•  Movies

•  Exercise

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Sorting molecules by mass/charge •  A way of separating molecules by their mass is to centrifuge them in order to

accelerate sedimentation.

•  Recalling Archimedes’ principle, force driving sedimentation of an object of mass m under the action of a “acceleration gravity” as directed along z is:

•  Flux j of molecules under force f and diffusion D can be calculated as (analogous of Nerst-Planck formula):

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f = −dU dz = − m−Vlρl( )as ≡ −mnetas U =masΔz− Vlρlas( )Δz$% &'

j z( ) =mnetasc z( )

ς−D∇c z( ) ςD=kBT

# →## j z( ) = D mnet

kBTasc z( )−∇c z( )

%

&'

(

)*


Sorting molecules by mass/charge •  Centrifuges are needed because at equilibrium (j=0) thermal agitation often

keeps a relevant fraction of particles in suspension:

•  Example as=g and m typical values of a globular protein (e.g. myoglobin): m ≈ 17Kg/mol (mnet ≈ m/4) gives for z* a value of ~60m.

•  In a tube of 10 cm the concentration at the top is equal to ~ 99.8 % of that at the bottom:

•  The suspension never settles out equilibrium colloid or colloidal suspension.

•  Settling occurs when:

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c 10 cm( )∝ c 0 cm( )e−0.1m 60m ≈ 99.8 c 0 cm( )

c z( )∝ e−mnetasz kBT = e−z z* z*= kBT mnetas z* ° scale height

mnetash kBT >1 h = ztop − zbottom"# $%


sedimentation distribution in centrifuge

Sorting molecules by mass/charge •  For a given molecule, one can act on h, T, and on as, as done in centrifuges,

where as can be as high as 106 m/s2.

•  Equilibrium found again by imposing compensation between drift flux, due to viscosity of the medium where particles are suspended:

and diffusion flux due to Fick’s law:

giving the equilibrium concentration as a function of radial position:

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jdrift r( ) = vdriftc r( ) =fςc r( ) = mnetωs

2rς

c r( ) = mnetωs2rD

kBTc r( )

jD r( ) = −Ddc r( )dr

c r( )∝ e−mnetωs2r2 2kBT


Sorting molecules by mass/charge •  A sedimentation coefficient s can be defined which is an intrinsic property of

the particle:

•  s is the time needed for a particle to reach a terminal velocity, and it is measured in svedbergs (10-13 s).

•  s depends on friction ζ, in turn depending on viscosity η of the fluid where particles are suspended, as well as on their shape.

•  For a spherical particle of radius R in a medium with viscosity η, friction is given by Stokes’ formula:

•  η for water at room temperature is 10-3 Pa⋅s.

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s = vdrift g =mnet ς

ς = 6πηR


Sorting molecules by mass/charge

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s = vdrift g =mnet ς

ς = 6πηR


Sorting molecules by mass/charge •  In electrophoresis sedimentation is facilitated by means of applied electric

field to a ionic solution.

•  Drift velocity will be proportional to charge and electric field strength:

•  As for the centrifuge, an intrinsic coefficient can be defined, the electrophoretic mobility µe:

•  Stokes approximation often valid also for non-spherical particles due to solvation shell of charged molecules in polar ionic solutions.

•  With charged biomolecules mobility will depend on viscosity of medium, and on shape and charge of particles.

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vdrift = f ς = qE ς

µe = q ς spherical particles! →!!!!! µe = q 6πηR


Sorting molecules by mass/charge •  Gel electrophoresis one of preferred techniques to separate and analyze

molecular components of biological assemblies.

•  Put macromolecule (DNA, RNA, protein) in a gel and switch on electric field.

•  Add denaturing agent (e.g. SDS for proteins) to reduce effects of different folding. One SDS anion binds about two aa in proteins charge proportional

to protein mass.

•  Gel used as molecular sieve enhancing filtration of molecules by their size.

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© Biochemistry, J. Berg et al., 5th ed., 2001


Sorting molecules by mass/charge •  Separation is reflected by occurrence of different migration bands (e.g.

staining with dye).

•  Small proteins move rapidly through the gel, whereas large ones not.

•  µe of most polypeptide chains under these conditions ~ log M (some carbohydrate-rich proteins and membrane proteins escape this rule).

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© Biochemistry, J. Berg et al., 5th ed., 2001


Mixing and viscosity •  From Einstein and Stokes relations, inverse proportionality between diffusion

coefficient of spherical object and viscosity of medium:

•  Greater the viscosity, slower will be diffusion

•  Explains why a small blob of thick fluid can apparently mix and reappear if a slow mixing movement is followed by its reverse.

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D = kBT ς ∝η−1

© Biological Physics, P. Nelson, 1st ed. updated, 2008


•  In laminar flow conditions (low Reynolds number) with thick fluids, motion of stirring rod or one cylinder relatively to a second concentric one only cause mixture of peripheral layers of molecules, due to sliding of fluid layers one

over another.

•  Diffusion randomize molecules over relatively long times, thus fluid layers can slide back if reverse movement is applied, reassembling blob.

•  The blob has never fully mixed due to the viscosity of the fluid.

•  The opposite occurs with turbulent flows (e.g. milk on coffee).

Mixing and viscosity

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•  In laminar flow viscous force due to shear motions with (low) v0 among fluid planes of area A and separated by distance dx given by:

•  Though no intrinsic definition of large and small viscosity, for isotropic Newtonian fluids viscous regime takes place when ratio ffrict/fcrit is small, the

viscous critical force fcrit being:

Inertia vs. friction

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f frict = −ηAdv dxNewtonian fluid, planar geometry

fcrit =η2 ρm

•  ffrict/fcrit > 1 inertial forces dominates over friction (turbulent flow: large density mass or low viscosity of fluid).

•  ffrict/fcrit < 1 friction quickly damps out inertial effects (laminar flow).



•  Newtonian fluid has not intrinsic scale length! Size of forces involved in the process relative to a critical value discriminates between laminar and turbulent

flows.

•  fcrit of water in range of nN .

•  In molecular world forces have values ~pN friction dominates molecular processes occurring in aqueous solvents! Flows are laminar!

Inertia vs. friction

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•  Reynolds number corresponds to the ratio between inertial and frictional forces in the dynamics of a fluid:

•  L typical linear dimension of object moving relatively to the fluid.

•  R small (< 1000) friction dominates, laminar flow.

•  R large (> 3000) inertial effects dominate, turbulence.

•  Experiments by Reynolds shown that distinction based on R is generally applicable whenever the object can be characterized by a length scale L.

•  R related to the concept of critical force:

Reynolds number

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R= finertial f frict = vLρm η

R ≈ f frict fcrit


•  Relation demonstrated considering a spherical object immersed in a fluid under laminar flow conditions.

•  Newton law for small volume of fluid involves pressure and viscous forces:

•  Small volume of fluid accelerates nearly sphere: direction changes in Δt ≈ R/v.

•  Acceleration comparable to v in size dv/dt ≈ v2/R.

Reynolds number

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finertial =mdv dt = fp + f frict

finertial =mdv dt = ρml3 v2 R


•  Inertial term must be compared to frictional force:

•  Net force due to pushing by upper layer and pulling to lower one, approximated by derivative times length of volume fluid:

•  Dividing the modulus of this quantity by ffrict gives the Reynolds number definition:

•  A 30m long whale swimming in water at 10 m/s has R~3⋅108.

•  A 1µm thick bacterium swimming at 30 µm/s has R~3⋅10-5!

Reynolds number

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f frict = −ηl2 dv dx

f frict ≈ l df dx = −ηl3 d 2v dx2 ≈ −ηl3 v R2

R= finertial f frict = vRρm η


Reversibility of flow

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At low Reynolds number, flow is reversible


Reversibility means that anything swimming by repeated flapping motions can’t get anywhere. If it moves forward in one stroke, the other stroke will

bring it right back to where it started!

Swimming of bacteria

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Δx = u Δt Δx ' = u 'Δt '

•  At low Reynolds number, body and paddles of bacterium move with friction

coefficients ζ1 and ζ2 respectively.

fbody = fpaddle → u ζ0 +ζ1( ) = vζ1u = vζ1 ζ0 +ζ1( )→Δx = vζ1 ζ0 +ζ1( )( )Δt

•  To estimate u (drift velocity of the body) and Δx due to paddle movement with velocity v, apply reaction law drag forces:

Δx −Δx ' = ?Net movement:


•  u’ and Δx’ due to back movement with velocity v’ are:

•  as the paddles return to their initial position, it must be:

Reversibility means that anything swimming by repeated flapping motions can’t get anywhere. If it moves forward in one stroke, the other stroke will

bring it right back to where it started!


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u ' = v 'ζ1 ζ0 +ζ1( )→Δx ' = − v 'ζ1 ζ0 +ζ1( )( )Δt '

vΔt = v 'Δt '→Δx ' = − vζ1 ζ0 +ζ1( )( )Δt = −Δx

Δx = u Δt Δx ' = u 'Δt '

•  At low Reynolds number, body and paddles of bacterium move with friction

coefficients ζ1 and ζ2 respectively.

Δx −Δx ' = ?Net movement:


Swimming by repeated flapping motions of macroscopic object in water: turbulence due to dominant effects of inertia allows swimming.


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Swimming by repeated flapping motions of macroscopic object in viscous medium: laminar flow regime hinders motion!


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A bacterium must perform a periodic but not reciprocal motion in order to advance within a fluid under laminar flow conditions

Ciliary propulsion (e.g. Paramecium): modulates effective friction coefficient


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Resistive force theory (1955) [N.B.: Challenged for bacterial motion in 2013!]

•  ζ|| for motion of a rod parallel to its axis smaller than ζ⊥ for perpendicular motion.

•  Dragging a rod along or perpendicularly to its axis feels a resisting force proportional to -v, that is, respectively directed along or perpendicular to the axis.

However, the constant of proportionality is larger in the second case.


Ciliary propulsion (e.g. Paramecium)


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A bacterium must perform a periodic but not reciprocal motion in order to advance within a fluid under laminar flow conditions

Flagella (e.g. E. Coli): helical rod rotates to advance


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•  If speed v neither parallel nor perpendicular to rod axis, drag force will not be along v, but closer to perpendicular direction than does v.

•  Thus cranking the flagellum about its axis (done by flagellar motor) generates net force along the same axis.


Flagella (e.g. E. Coli): a corkscrew for a propeller


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Flagella (e.g. E. Coli): a corkscrew for a propeller


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The inner part of the motor assembly develops a torque relative to the outer

part, which is anchored to the peptidoglycan layer, turning the flagellum


Flagella: a corkscrew for a propeller


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Many schemes for flagellation in bacteria, here reported two:

•  Peritrichous flagella (e.g. E. coli): CCW rotation results in formation of helical bundle propelling cell forward in one direction. CW rotation causes unbundling, allowing

bacterium to randomly reorient its direction (a ‘tumble’).

•  In single polar flagellum (monotrichous), CCW rotation propels the cell forward in a run, CW rotation propels the cell backward with concomitant random reorientation.

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