entropic forces - university of cagliari• biological physics (updated 1st ed.), philip nelson,...

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!

Entropic Forces

1

Molecular crowding Osmotic flows

Molecular recognition in ionic solvents


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.

2


References •  Books and other sources

•  Biological Physics (updated 1st ed.), Philip Nelson, Chaps. 6, 7

•  Movies

•  Exercise

3


Entropic forces Free energy is the currency for all kinds of transactions in real world

•  Macroscopic system at fixed temperature T characterized by Helmoltz free energy:

•  For microscopic subsystem a replace macroscopic quantity with ensemble

average over all allowed microstates:

•  Free energy related to partition function (Boltzmann statistics):

4

F = F(E,S) = E −TS

Fa = Ea −TSa Ea = EjPjj∑ Sa = −kB Pj lnPj

j∑with

Fa = −kBT lnZ Z = e−Ej kBT

j∑


Entropic forces •  Physical interpretation of Helmoltz free energy:

•  Bringing a small system a into thermal contact with a big system B in

equilibrium at temperature T will shift the state of a into a new equilibrium that minimizes Fa (B is virtually unaffected by a).

•  Physical interpretation of Gibbs free energy:

•  Bringing a small system a into thermal and mechanical contact with a big system B in equilibrium at temperature T and pressure p will shift the state of a

into a new equilibrium that minimizes Ga (B is virtually unaffected by a).

5

Fa = Ea −TSa

Ga = Ea + pVa −TSa Ha = Ea + pVa enthalpy


Entropic forces •  Statistical Postulate: A big enough, isolated system, subject to some

macroscopic constraints, evolves to an equilibrium if left alone long enough. Equilibrium is not a microstate, but rather a probability distribution, namely that with the greatest disorder (entropy), that is, the one acknowledging the

greatest ignorance of the detailed microstate subject to any given constraints.

•  Second Law of Thermodynamics: Any sudden relaxation of internal constraints (for example, opening an internal door) will lead to a new

distribution, corresponding to the maximum disorder among a bigger class of possibilities. Hence the new equilibrium state will have entropy at least as

great as the old one.

•  Thermodynamic work: If a system is found in a state with greater than minimum (equilibrium) free energy, the maximum work it can do on external

world is:

6

ΔFa = Fa −Fa,min


Entropic forces

•  In the case of ideal gas (system a) in thermal equilibrium at temperature T with bath B within a cube with sides of length L:

1.  If volume is fixed, Z depends on V only: Z=Z(V) F=F(V)

The pressure is obtained as thermodynamic derivative of F with respect to V=L3:

7

p = − dFdV

=kBNTV

Pressure and volume are thermodynamically conjugates variables

2.  If the pressure is fixed, Z is function of p only: Z=Z(p) F=F(p) The average volume is the thermodynamic derivative of F with respect to p:

V =dFdp

=kBNTp


Entropic forces and osmosis

8

Osmosis through semipermeable membrane is driven by solute’s Δc

•  At equilibrium, hydrostatic pressure is higher in the chamber containing the solute.

•  At low concentration of solute, this can be approximated by an ideal gas.

•  The value of the equilibrium pressure difference is thus:

V

van’t Hoff relation Δp = Ns

VkBT = cskBT


Osmotic pressure in cellular world

9

•  If cell is suspended in water, pressure needed to stop inward flow of water is:

•  Work done by pressure to increase spherical cell area by dA=8πRdR is equal to product of surface tension Σ by dA:

Δpequil = cskBT ≈ 300 Pa

•  To get estimate of osmotic pressure in cellular world, approximate cells’ interior as water solution with volume fraction φ=0.3 of globular proteins

(spheres with radii ~ 10 nm):

φ = csVs → cs = φ Vs≈ 7 ⋅1022 m−3 ≈ 0.1 mM

L = p dV = p dVdR

dR = p4πR2dR = Σ dA = Σ 8πRdR→ Σ = pR 2Laplace’s formula


Osmotic pressure in cellular world

10

•  Substituting value of Δpequil gives for Σ a value of ~ 10-3 N m-1.

•  Tension sufficient to cause rupture of eukariotic cell membrane.

  osmotic pressure is significant for cells!

•  Situation more drastic with ionic solutions, to which cell membranes are almost impermeable.

•  1M solution of NaCl contains ~1027 ions/m3, ~ 104 more particles than proteins, so that the osmotic pressure in that case will literally cause the cell (e.g.

red blood cells) to burst.

•  This is confirmed by experiments, showing that is not possible to dilute red blood cells with pure water or low concentrated solutions.


Molecular crowding

11

It’s crowded inside the cell!


Molecular crowding

12

It’s crowded inside the cell, and everything is jiggling around!


Molecular crowding

13

•  Hierarchy of objects of different sizes lead to entropic effect that

facilitates specific molecular recognition events.

•  depletion interaction or

molecular crowding.


Molecular crowding

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•  A large object in bath containing suspended particles approximated by spheres of radius a will reduce the volume accessible to the spheres by ~ Vr=A⋅ a, where A is the

surface of the large object. This region is called depletion zone.

•  If two large objects have matching surfaces, they can merge, thus reducing the size of the total depletion zone with respect to when unbound.

•  Depletion interaction leading to binding is short range, arising only when intermolecular distance between large objects less than 2a (radius of small objects!)


Molecular crowding

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•  Depletion force is entropic in nature, driving assembly by increasing the entropy of small objects.

•  Can be understood also in terms of osmotic pressure, as the concentration of small objects will drop to 0 at the interface between two large ones.

•  This will sucks water molecules out of the interface, thus reducing the hydrostatic pressure and favoring molecular association.


Molecular crowding

16

•  Pressure is equal to change in free energy per change in volume ΔV ~ 2Vr = 2A⋅ a.

•  This must equal the drop in osmotic pressure:

Δp = − ΔFΔV

=ΔF2Aa

= cskBT→ΔFA

= 2a cskBT

ΔFkBT

= 2acsA


Molecular crowding

17

•  Despite involving a thin layer, effect can be considerable!

•  If encounter between two macromolecular objects occurs with contact area of ~ 1µm2, assuming they in bath of proteins with radii 10 nm gives a free energy

change of several hundreds kBT:

ΔFkBT

= 2acsA ≈ 2 10−9102310−12 = 2 102


Molecular crowding

18

•  Effect of depletion interactions verified experimentally by means of experiment with vesicles.

•  In absence of small particles, a large one of size ~0.25 µm keeps diffusing within the vesicle (b), while in presence of smaller particles of radii ~0.04 µm, sticks to the

internal membrane for most of the time (c).

•  Ratio between times spent at vesicle’s walls in (b) vs. (c) confirmed relation between change in free energy and osmotic pressure.


Molecular crowding

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•  Crowding favors macromolecular recognition of specific sites.

•  Crowding can speedup reactions by orders of magnitude by increasing the entropy at the transition state, thus lowering the activation free energy:

-  E.g. DNA replication system of E. coli does not work in vitro without some crowding agent added to the broth.

-  Crowding crucial for chromosome compaction.


Osmotic forces & Brownian motion

20

•  Osmotic pressure pushes piston through action of a force.

•  Force must originate in semipermeable membrane, only element fixed to cylinder.

•  Membrane bow as it pushes the fluid (reaction force).



21

•  At equilibrium, external force acting on generic element of fluid must compensate force due to hydrostatic pressure.

•  Defining a force density F(r) acting along +z, equilibrium condition gives:

•  Thus pressure varies so as to compensate F(z)dz to realize hydrostatic equilibrium.

•  When F(z)=F =-ρg one recovers Stevin law:

p z+ 12 dz( )− p z− 1

2 dz( )"# $%dxdy =F r( )dxdydz dz→0' →'' F z( ) =dp z( )dz

dp z( ) = −ρmdz



22

•  In a colloidal suspension of concentration c(z), with force f(z) acting on each particle at low Reynolds numbers, inertial effects negligible, only viscous drag counterbalances f.

•  Particles push back on fluid transmitting applied force to it, thus an average force density F(z) is created (indirectly) on fluid, given by c(z)⋅ f(z) and corresponding to

pressure gradient:

dp z( )dz

=F z( ) = c z( ) f z( )

z

f(z) fdrag(z)

f(z)

fdrag(z)

f(z) fdrag(z) f(z) fdrag(z)

f(z) fdrag(z)

F(z)=c(z)⋅ f(z)

c(z)



23

•  Suppose f(z) is conservative, related to potential energy U(z). Examples are:

-  Colloidal suspension within electric plates.

-  Two chambers separated by semipermeable membranes containing colloidal suspensions of different

concentrations.

•  In last case U(z) given by presence of membrane: U(0)=∞ at the walls, Uà0 far from walls.

•  cl=0, cr=c0, c(z) follows Boltzmann distribution, thus:

dp z( )dz

= c z( ) f z( ) = −c0e−U z( ) kBT dU z( )

dz= c0kBT d

dz e−U z( ) kBT"

#$%

↓

dp z( )dz

dz∫ = kBTdc z( )dz

dz →∫ ΔpvH = c0kBTvan’t Hoff relation


Darcy’s law

Osmotic flow and permeability

24

Force generated by rectified Brownian motion of particles, due to presence of semipermeable membrane

•  Particles collide with membrane, which transfers them a net momentum to the right (at odd with truly random

Brownian motion).

•  Particles moving away from membrane drag solvent molecules.

•  Depletion zone of low-pressure created aside of the membrane, water flows through semipermeable pores of

the membrane.

Osmotic flow of solvent molecules from left to right:

j = LpΔpvH = LpΔcskBT membrane filtration coefficient or hydraulic permeability Lp


Osmotic & driven flows

25

•  Piston move to the right (mechanical work is done) increasing entropy of chamber containing colloid.

•  Osmotic pressure not merely due to concentration gradient, only arises when a physical object (i.e. the

membrane) can apply force to solute’s particles!

•  Van’t Hoff relation gives not actual pressure, but value ΔpvH that would be needed to stop osmotic flow, i.e. external pressure to apply as to maintain

equilibrium of initial concentrations on both chambers.

•  If no difference in pressure between chambers (no external force on piston), linear pressure drop

through membrane pores arise compensating ΔpvH.

•  Whenever Δpext lower (higher) than ΔpvH, flux of solvent from low (high) to high (low) concentration of

solute particles (osmosis vs. reverse osmosis).


Osmotic & driven flows

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•  Δpext causes hydraulic permeation, independent on presence of solute particles.

•  Total flux through semipermeable membrane given by balance between external and osmotic pressures:

•  When Δpext null, entropic force per area equals the frictional drag per area:

•  Expression of total flux valid not only for model of rigid membrane with fixed-size cylindrical pores, but

also for biological flexible membranes.

•  Lp must be small enough for j not causing sizeable fluctuations in Δc (equilibrium conditions apply).

j = −Lp Δpext −ΔpvH( ) = −Lp Δpext −ΔcskBT( )

j Lp = ΔcskBT


Osmosis in permeable membranes

27

Osmosis arise even if membrane permeable also to solute molecules.

•  If forces involved small enough, linear response combining Darcy’s and Fick’s laws for fluxes of solvent and solute molecules respectively (cross-

effects arise):

•  P11 membrane filtration coefficient Lp

•  P22 solute permeability Ps

•  P12 osmotic flow

•  P21 solvent drag (mechanically pushing solvent drags al drags also solute molecules)

jsolventjsolute

!

"##

$

%&&= −P

ΔpΔcs

!

"##

$

%&&

jsolvent = −P11Δp−P12Δcsjsolute = −P22Δp−P21Δcs


Flows of interacting particles

28

•  Electrostatic interactions biological macromolecules such as some membranes and nucleic acids often negatively charged at cell

standard pH called macroions.

•  Having most macromolecules with the same sign of total charge (thus repelling each other) in cells helps

avoiding precipitation and maintaining colloidal suspensions of macroions.

•  Surrounded by cloud of counterions (strictly speaking the word only refers to charged groups lost by

macromolecule) in equilibrium near macroion surface, and screening its total charge to other macromolecules

called diffuse charge layer.

When particles interact with each other, entropic and energetic forces play role


Flows of interacting particles

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Screening in solution reduces the effective range of electrostatic interactions, typically to a nm or so.

•  Helps organization of cellular life:

•  Reducing length of effective interactions allows macroions to

wander around as in a mean-field force, thus not feeling other

macromolecules.

•  When distance is below effective length, molecular specificity becomes

important.

•  Macromolecular recognition will occur if detailed path of (all kinds of)

surface interactions among partners do match, implying several single kBT

interactions add up.


•  Consider a solid surface negatively charged with uniform surface charge density:

•  Distribution of counterions at equilibrium (coions neglected)

difficult to find because of high number of particles involved, due to long range nature of electrostatic interactions between ions.

Poisson-Boltzmann theory

30

Mean field approximation

•  At equilibrium each ion will feel an average effective potential V(x) due to all remaining counterions, with small fluctuations in effective field due to variations in local density of nearest neighbors counterions (thanks to long range nature of electrostatic).

•  Each ion moves approximately independently from detailed locations of others, i.e. under a mean field potential due to average charge density of others, <ρq>.

−σ q = qtot A


•  Since each ion moves in mean field, simple Boltzmann distribution

•  Imposing the conditions c+"0 at large values of x and V(0)=0, gives V(∞)=∞ and

c0=c+(0) respectively.

•  The electrostatic potential V(x) turns out to be given by self-consistent solutions of:

•  Where the reduced potential V and the the Bjerrum length lB (in water) are:


31

c+ x( ) = c0e−eV x( ) kBT

d 2Vdx2

= −4π lBc0e−V

V

V x( ) = eV x( ) kBT lB = e2 4πεkBT

Poisson-Boltzmann equation

Ratio between potential and thermal energy

Distance at which two ions of same charge can be pushed by thermal energy:

~0.7 nm for monovalent ions in water.


•  In the case of a flat negatively charged surface of area A, E=E(x).

•  In particular, at the surface E is constant, and q=σqA:

•  In the bulk, the only flux is also along x, given by the difference in the

value of E at different heights:

To find potential V generated by average charge density <ρq> (equal to counterions

density <c+> multiplied by electric charge e) recall that electric field is E=-∇V and apply

Gauss law:

[derivation]

32

E ⋅dS =∫ qε

Esurf A = −σ qA ε→ Esurf = −σ q ε

E x + 12 dx( )−E x − 1

2 dx( ) = ρq x( )dx ε

⇓

dE dx = ρq ε

Gauss law at flat charged surface

Gauss law in bulk


•  Flux in bulk due to distance-dependent screening by counterions: a probe at

different distances sees the surface with different charge densities or more simply

we can consider two differently (uniformly) charged surfaces.

•  The difference in the surface charge densities of the screened surfaces is related to the average bulk density:

To find potential V generated by average charge density <ρq> (equal to counterions

density <c+> multiplied by electric charge e) recall that electric field is E=-∇V and apply

Gauss law:

[derivation]

33

E ⋅dS =∫ qε

σ qx+12dx =σ q

x−12dx − ρq dx

⇓

Esurfx+12dx −Esurf

x−12dx = −σ qx+12dx +σ q

x−12dx( ) εEsurf

x+12dx −Esurfx−12dx = ρq dx ε


•  Once V is found solving by iteration the Poisson-Boltzmann self-consistent equation, c+ at equilibrium can be determined.

•  PB differential 2nd order equation has a family of solution, the one of interest found by applying boundary conditions:

•  In this case the solution is the Gouy-Chapman layer:

•  Diffuse layer of thickness ~xGC forms. Appropriate solution in neighborhood of flat

charged surface in water (or highly charged surface in dilute solution).


34

V x( ) = 2 kBTeln 1+ x xGC( )( )

Gouy-Chapman length

Esurf = −dV dxsurf

= −σ q ε→−dV dxsurf

= 4π lBσ q e

dV dx∞= 0

V 0( ) = 0

no charge at ∞

by convention

xGC = e 2π lBσ q

Gauss law at surface


•  Gouy-Chapman solution gives V"∞ far away from surface! Due to approximations of infinite surface and absence of coions.

•  Real membranes are not infinite, moreover real solutions contain not only counterions but also coions: c∞ is never 0, even away from surface!

•  More realistic to take V"0 far from surface " Boltzmann distributions are:

•  Poisson-Boltzmann equation becomes:

Which has solution (x* found applying Gauss law at surface):


36

Debye screening length λD =1 8πlBc∞ = εkBT 2e2c∞

c− x( ) = c∞e− −e( )V x( ) kBTc+ x( ) = c∞e

−eV x( ) kBT

d 2Vdx2

= −12λD−2 e−V − eV"#

$%

V x( ) = −2 ln1+ e− x+x*( ) λD

1− e− x+x*( ) λDex* λD =

e2π lBλDσ q

1+ 1+ 2π lBλDσ q e( )2!

"#

$%&


•  At fixed salt concentration, for x>>λD, recalling that ln(1+x) " x for small x, the solution of PB equation simplifies to:

•  The electric fields far outside a charged surface in an electrolyte are exponentially screened at distances much greater than the Debye length λD.

•  For 0.1 M solution of table salt, λD ≈ 1nm.

•  λD decreases at low values of dielectric constant ε or temperature T, or by increasing c∞ or the charge of particles.

•  These effects shrink the diffuse layer, shortening the effective range of the electrostatic interactions.


37

V x( )→− 4e−x* λD( )e−x λD = −ke−x λD


•  If the surface is weakly charged (small σq) one get further simplification:

•  The ratio between prefactors for x>>λD, and that for x>>λD and small σq, is called charge renormalization:

•  At large distances any surface will look the same as a weakly charged surface, but

with the “renormalized” charge density:

•  True surface charge only becomes apparent to ions within the strong-field region.


38

V x( )→−σ qλDε

e−x λD−x λD

e−x* λD → π lBλDσ q λD

RC = 4e−x* λD( ) σ qλD ε( )

σ q,R = 4ε λD( )e−x* λD


Repulsion and attraction within PB

39

•  Repulsion between like-charged surfaces in solution of counterions largely due to entropic forces

(if T"0 layer will collapse on surface).

•  Imposing the condition Vmidplane=V(0)=0 the solution to the PB equation and the related concentration of counterions are:

•  β found by applying Gauss law at the surface:

•  When surfaces get closer than ~2xGC, counterions cloud gets squeezed (ρ increases), generating an osmotic pressure responsible for repulsion of like-charged surfaces.

•  As for depletion interaction, Brownian motion of fluid is rectified by osmotic pressure. Force acting on counterions transferred to charged surface by the fluid, creating a pressure drop equal to kBT times Δc between force-free regions x=0 and |x|>D.

V x( ) = 2 lncos βx( ) c+ x( ) = c0 cosβx( )2

4π lBσ q e = 2β tan Dβ( )



40

•  Maximum concentration jump c0 function of β and of lB:

•  β=β(D, σq) solution of:

c+ x( ) = c0 cosβx( )2 → c+ x( ) = c0, c+ x > D( ) = 0⇓

f A = ΔckBT = c0kBT

Repulsive force per area between like-charged surfaces

c0 = β2 4π lB




41

•  Solved numerically or (approximately) graphically.

•  Smaller values of the surface separation correspond to larger values

of β.

•  These corresponds to larger values of c0 and of the repulsive pressure f/A.


•  β=β(T) f not simply proportional to T, thus not due entirely to entropic effect: counterions layer reflect balance between attractive interactions and

entropy.

•  Adding coions (salt) shift the balance towards energy, shrinking the diffuse layer and shortening the range of the interaction.



42

Relation between f/A and D experimentally verified for biological membranes if D>xGC

f A = c0kBT

c0 = β2 4π lB




43

•  Attraction works by the same principle.

•  Ions gets squeezed out (entropy increases) when the two surfaces approaches.

•  Eventually surfaces get in tight contact (energy decreases).

entropic forces - university of cagliari• biological physics (updated 1st ed.), philip nelson,...

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