POLYMER PHYSICS & DNA

Bertrand Duplantier

Institut de Physique Théorique, Saclay, France

Troisième Cycle de la physique en Suisse romande

École Polytechnique Fédérale de Lausanne

February-March 2009

STATISTICAL MECHANICS

OF POLYMER SOLUTIONS

POLYMER PHYSICS & DNA

Troisième Cycle de la physique en Suisse romande

École Polytechnique Fédérale de Lausanne

26 February 2009

MATHEMATICAL

BROWNIAN MOTION

Gaussian DistributionIn d 1 dimension :

∂P∂t

D ∂2P∂x2

x2

t

∞

∞x2P

x t

dx 2Dt

4Dt

2xe

4 Dtπ

-4 -2 2 4

0.1

0.2

0.3

0.4

0.5

x

8Dt

In d dimensions :

P

r t

14πDt

d

2exp r2

4Dt

r2

t

2dDt

N. WIENER, 1923MARK KAC (1964) : “Only Paul Lévy in France, who had himself been

thinking along similar lines, fully appreciated their significance.”

Wx0

dw

n

∏i 1

δd

w

ti

ri

n

∏i 1

1

4πD

ti ti 1

d

2exp

ri

ri 1

2

4D

ti ti 1

This is the Wiener integral’s fundamental property. Take ti

ti 1

t

n,multiply by ∏n

i 1 ddri, and take the formal limit n ∞ to get the Wienermeasure as a functional integral measure :

Wx0

dw

“ limn ∞

n

∏i 1

ddri

n

∏i 1

1

4πDt

n

d

2exp

ri

ri 1

2

4Dt

n

”

Dw

x0exp 1

4D

t

0

d

w

t

dt

2

dt

Dw “ limn ∞

n

∏i 1

ddri

4πDt

n

d

2

”

BROWNIAN MOTION

&

FIELD THEORY

Free Field Propagator in d

P

r t

1

4πDt

d

2e

r24Dt ud

r

D

∞0

em2 DtP

r t

dt

∆

m2

ud

r

δd

r

dddr ei

k

rud

r

1k2

m2

FEYNMAN-SCHWINGER amplitude

dddr

∞0

3

∏i 1

dti e

m2 D ∑3i 1 ti

3

∏i 1

P

r ti

K. Symanzyk, 1969 ; P.-G. de Gennes, 1972 ; J. des Cloizeaux, 1975 ;

M. Aizenman, 1981 (d 4) ; D. Brydges, J. Fröhlich & T. Spencer, 1982

BROWNIAN MOTION

&

POLYMER THEORY

EDWARDS POLYMER MEASURE, 1965Minimal model for self-avoiding polymers :

P

dw

W

dw

exp

b

2

t

0dt1

t

0dt2 δd

w

t1

w

t2

Dw exp

14D

t

0

d

w

t

dt

2

dt

b

2

t

0dt1

t

0dt2 δd

w

t1

w

t2

(Dimensional) UV regularization, perturbation theory and

renormalization theory for d

4, either via the equivalence to an N

component

Φ2 2 field theory, in the limit N 0 (K. Symanzyk, 1969 ;

P.-G. de Gennes, 1982 ; M. Aizenman, 1981 ; D. Brydges, J. Fröhlich & T.

Spencer, 1982), or by a direct method (J. des Cloizeaux, 1982).

Polymerized Membrane

0

Dt

w t( )

A polymerized membrane of internal dimension D embedded in

d .

Polymerized Membrane MeasurePolymerized self-avoiding membranesa (Kardar & Nelson ; Aronowitz &

Lubensky, B.D., 1987) :

M

dw

Dw exp

1

2 M

∇t

w

t

2 dDt b2 M

dDt1M

dDt2 δd

w

t1

w

t2

w d t D

Renormalizable theory for d

4D2 D

0 D 2, but no longer equivalence

to a local field theory. The validity of the direct renormalization method is

proven to all orders (David, B.D. & Guitter, 1992-97).

aRobin Ball, 1981

BROWNIAN MOTION,

THE BUILDING BLOCK OF PHYSICS ?

EINSTEIN, 1910 : “If to conclude we ask once more the question :

“Are the observable physical facts correlated one to another in an

entirely causal way ?”, we must surely answer this question in the

negative. The positions of a particle engaged in a Brownian motion at two

instants separated by one second must always appear, even to the most

conscientious observer, as independent from each other, and the greatest

mathematician would never succeed in any determined case to compute in

advance, even approximately, the path covered in a second by such a

particle. According to the theory, to be able to do so one should know the

position and speed of each molecule exactly, which appears in principle

excluded. However, the laws of mean values, which proved themselves

everywhere, as well as the statistical laws of fluctuations, valid in these

domains of finest effects, lead us to the conviction that in theory we must

firmly hold onto the hypothesis of a complete causal connection of events,

even if we should not hope to ever obtain by improved observations of

Nature the direct confirmation of such a concept.

Institut Henri Poincaré · Amphi Hermite11, rue Pierre et Marie Curie · 75005 Paris

Crédit photos :

Julie Mehretu, excerpt (suprematist evasion), 2003

ink and acrylic on canvas, 32"x54"

Archives Lotte Jacobi, Université du New Hampshire

www.lpthe.jussieu.fr/poincare

O. Darrigol :C. M. Will :

B. Duplantier :Ph. Grangier :

T. Damour :

La genèse de la Relativité · 10h00Tests of Special Relativity · 11h00Le mouvement brownien · 14h00Expériences à un seul photon · 15h00Einstein épistémologue · 16h00

Séminaire Poincaré

Samedi 9 avril 2005

Einstein,1905-2005

FondationIagolnitzer

Einstein 09.03.2005 9:07 Uhr Seite 1

BIRKHÄUSER, 2006(www.bourbaphy.fr , , arXiv :0705.1951)

POLYMER THEORY

A Brief HistoryNOBEL PRIZES IN CHEMISTRY & PHYSICS

(www.nobel.se)

H. Staudinger 1920 1953χ

P. J. Flory 1934 1974χ

P.-G. de Gennes 1970 1991ϕ(S. F. Edwards, J. des Cloizeaux)

FIELDS MEDAL IN MATHEMATICS

W. Werner 1993 2006M

Polymers & Biology

Information, functions : DNA, RNA

Functions : Proteins (folding)

Mechanical : Actin, collagen, polysaccharides

POLYMER CHAIN

FLEXIBILITY

Polymer Chain FlexibilityMacromolecules locally rigid : Intrachain hydrogen bonds,steric effects, and chemical bonds limit conformationalchanges.

At large scale : persistence length ξ

p, at which the chainbecomes flexible.

(Depends on possible conformational states and their relativeenergies.)

Flexibility & Persistence Length

PSfrag replacements

cϕθ

c

c

c θ

θ fixed, ϕ variableθ 71o

E∆E 3 kcal/mole

δE 0 5 kcal/mole

ϕ120o

gauche +0

trans240o

gauche -

Polyethylene molecule : « trans » conformations withminimal energy Et and « gauche » with Eg

Et

δE Eg

Et kBT : equiprobable conformations withrandom orientations : Flexible chain.

δE Eg

Et kBT : Rigid « trans » chain.

Persistence LengthPersistence length

p (Boltzmann’s law) :

p ∝ a expδEkBT

(a typical monomer size.)

(Transition time τ τ0 exp ∆EkBT

10

12s.)

Actin & DNA Filaments

Electronic microscopy of actin filaments with strong bending.

(“Semi-flexible” chains)

IDEAL POLYMER CHAIN

0

a

R

Ideal chain with N links on the square lattice with mesh sizea : random walk (with probability 1

4

along each direction :

R2

N

Na2

A long ideal chain will look from far away like a Gaussianchain (cf. Brownian motion).

Gaussian Distribution

4Dt

2xe

4 Dtπ

-4 -2 2 4

0.1

0.2

0.3

0.4

0.5

x

8Dt

r2

t

R2

N

2dDt Na2 f ixed

N ∞ a 0

P

R N

P

r t

14πDt

d

2exp r2

4Dt d

2πN a2

d

2

exp d2

R2

N a2

Left : successive links of a ideal polymer.

Up right : asymptotic continuum limit. Inset : scaleinvariance “ad infinitum” of the Gaussian chain.

ENTROPIC FORCE

& IDEAL POLYMER CHAIN

Number of Configurations of an Ideal Chain

P

R N

d2πN a2

d

2

exp d2

R2

N a2

Ω

R N

2d

N

Entropy & Free Energy

S

R N

kB lnΩ

R N

Boltzmann

F

R N

U 0

T S

R N

kBTd2

R2

N a2

F

0 N

Entropic Elastic Force

f ∂F

R N

∂

R

kBT d

RN a2

kBT

R2Dt

CONFORMATIONS OF A

SINGLE CHAIN IN A SOLUTION

Good Poor Solvents

a b

Excluded volume effects : Left, the “poor solvent” case : the

mutual interactions of the chain monomers are energetically more

favorable than interactions with solvent molecules and the chain folds

onto itself ; Right, the “good solvent” case : the chain monomers prefer to

be in contact with solvent molecules, there results an effective repulsion

between momomers.

Effective Interaction between Monomers

0.5 1 1.5 2 2.5 3r-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

VHrL

Effective potential V

r

between monomers in presence of thesolvent.

Interacting Gas Theory

Cloud of monomers, local concentration :

c

R

N

Rd

∝ N1 d

2 Interaction energy (/unit volume) :

u 12

kBT v

T

c2 O

c3

(kBT # of contacts between monomers per unit volume)

Second virial coefficient :

v

T

dddr 1 exp

V

r

kBT

Second Virial Coefficient

2 4 6 8 T

-10

-8

-6

-4

-2

2

vHTL

v

T

d ddr

1 exp V

r

kBT

.

Effect of TemperatureSecond virial coefficient :

v

T

dddr 1 exp

V

r

kBT

T Θ v

T

0 : attraction, poor solvent

Θ-point, v

T Θ

0 : “ideal” chain

T Θ v

T

0 : repulsion, good solvent*

THE FLORY METHOD

Excluded Volume Effects in a Good SolventSAW in plane - 1,000,000 steps

An ideal chain crossing over to a self-avoiding configurationfor v

T Θ

0 Complete change of the statisticalproperties (here in two dimensions).

Chain Internal Energy

Within the chain of radius R, local concentration

c c

R

∝ NRd in volume V ∝ Rd :

U

R N

uV 12

kBT v

T

V c2 12

kBT v

T

N2

Rd

Elastic Entropy of the Chain

T S

R N

kBTd2

R2

N a2

Free Energy

F

R N

U

R N

T S

R N

Flory Free Energy

F

R N

kBTd2

R2

N a2

12

kBT v

T N2

Rd

Minimization

∂∂R

F

R N

0 RF a

v

T

a3 N31

d

2

Swelling Critical Exponent ν

RF

aNν

νFlory

d

3d 2

DHausdorff

1ν

d 23

Resultsv

T

0, Chain with Excluded Volume

R aNν

νFlory

d

3d 2

d 1 νFlory

d

1

exact

d 2 νFlory

d

3

4

exact

d 3 νFlory

d

3

5

0 5880 0 0015

d 4 νFlory

d

1

2

exact

Rd 4

aN1

2

lnN

1

8

THE THETA POINT

At the Θ-pointv

T Θ

0, Gaussian chain

d 3 ν 1

2

exact

In a poor solventv

T Θ

0, collapsed chain, crumpled globule

Rd

aN1

d

Θ-Point Exact Resultsv

T Θ

0, Θ-point

Rd 3

aN1

2 1

372 363lnN

d 3 ν 1

2

exact

d 2 ν 4

7

exact

v

T Θ

0, collapsed chain, crumpled globule

Rd

aN1

d

LATTICE CHAINS

0

a

R

0

a

R

Ideal and self-avoiding polymer chains of N steps on thesquare lattice of mesh size a :

R2

N

Na2;

R2

N

Nνa2

A very long ideal chain will look from far away as aGaussian chain ; a self-avoiding chain, like a polymer chainwith excluded volume.

2D Polymer ChainsSAW in plane - 1,000,000 steps

Ideal chain becoming a self-avoiding chain forv

T Θ

0 Complete change of statistical properties(here in two dimensions).

B. Nienhuis (1982) :

R2

I ∝ N3

4a DHausdorff

43

(EXACT, cf. W. Werner, 2006 Fields Medal).

Self-Avoiding Walks in 2DSAW in plane - 1,000,000 steps SAW in plane - 1,000,000 steps

(T. Kennedy)

B. Nienhuis, (1982) : DH

43

R2

I ∝ N3

4a(EXACT, cf. W. Werner, Fields Medal 2006).

Excluded Volume Effects in 2 DimensionsSAW in plane - 1,000,000 steps

Here in 2 dimensions : Gaussian chain crossing over to aself-avoiding chain, for N 106 :

R2

Inter

R2

Gauss

Nν 1

2 N1

4 32

Excluded Volume Effects in 3 Dimensions

In three dimensions : Gaussian chain crossing over to aself-avoiding chain, for N 106 :

R2

Inter

R2

Gauss

Nν 1

2 N0 088 N1

10 3 37

DNA AS A POLYMER

FIG. 1 – AFM tapping mode images of linear λ-phage DNA molecules,strongly adsorbed (48502 pb ; shorter fragments shown here). Scale : 100

nm. Mica surface positively charged with 3-aminopropyltiethoxy silane

(APTES), DNA solution rinsed and dried. (F. Valle, M. Favre, P. De Los

Rios, A. Rosa, and G. Dietler, Phys. Rev. Lett. 95, 158105 (2005), courtesy

of Prof. Giovanni Dietler, EPFL.)

!

"## $# %& '

( ) * ( ) + ( ) , ( ) - ( ) .

/0 1 20 34 56 1 7 2 8 9 1: ;

< < = < = = < = = = < = = = =

FIG. 2 – Mean distance as a function of contour length s along the chain in 3D.s

>

2

?

p

@ 100 nm : rod regime with ν0

1 030

A

0 017 ; 102 nm

>

s

>

104 nm :

fractal regime with exponent ν 0 589

A

0 006. (F. Valle, M. Favre, P. De Los Rios,

A. Rosa, and G. Dietler, Phys. Rev. Lett. 95, 158105 (2005).)

PROBABILITY DISTRIBUTIONS

OF END-TO-END DISTANCES

400

300

200

100

0

Num

ber

of S

ampl

es

706050403020100

End-to-end Distance (nm)

FIG. 3 – Probability distribution 4πr 2P

r s0

for the end-to-end dis-tance r for a contour length s0

71 4 nm and persistence length

p

46 6

nm. (F. Valle, M. Favre, P. De Los Rios, A. Rosa, and G. Dietler, Phys. Rev.

Lett. 95, 158105 (2005).)

!"#$

%'& (*) +, ) -& (./10 +2 & 3 - 4065798 :;<

FIG. 4 – Probability distribution 4πr 2P

r

s0ν

for the normalizedend-to-end distance r

s0ν for two contour lengths s0

548 nm = 12

p

(circles) and s0

748 nm = 17

p (triangles) [ν 0 589

> 006 δ 2 58

>

76]. (F. Valle, M. Favre, P. De Los Rios, A. Rosa, and G. Dietler, Phys.

Rev. Lett. 95, 158105 (2005).)

Gaussian & SAW Probability Distributions

2dDt s0

Na2

Gaussian

N ∞ a 0

PGauss

r s0

1

2πs0

d

2exp r2

2s0

PSAW

r s0

Arκ exp B rs0

ν

δ r s0

ν

rθ0 r s0ν

δ 11 ν

2 427

006

d 3

; 4

d 2

κ 1 γ νd d

21 ν

249

011; 5

8

θ0

γ 1

ν 275

002; 11

24

FIG. 5 – AFM images of circular DNA molecules (pbr plasmid, 11860 basepairs), weakly adsorbed on mica surfaces treated with magnesium. The configu-

rations are equilibrium ones of closed polymer chains, with excluded volume and

in two dimensions, with ν 0 74

A

0 01. [G. Witz, K. Rechendorff, J. Adamcik, and

G. Dietler, Phys. Rev. Lett. 101, 148103 (2008).]

102

103

104

105

<RG

2 > [n

m2 ]

6 7 8100

2 3 4 5 6 7 81000

2 3 4 5 6 7

Plasmid length L [nm]

(L/2π)2

ν = 0.75

FIG. 6 – Radius of gyration as a function of plasmid length. Two lines

are drawn : stiff regime with ν 1 and SAW excluded volume regime,

7

p

L

120

p, with ν 0 75. Errors of 1 3% are within point size.

[G. Witz, K. Rechendorff, J. Adamcik, and G. Dietler, Phys. Rev. Lett. 101,

148103 (2008).]

FIG. 7 – AFM tapping mode images of λ-DNA. A) 0 mM NaCl, B) 5

mM NaCl. Decrease of persistence length with salt concentration. (G. Witz,

J. Adamcik, K. Karczewski, K. Rechendorff, T. Eggel, P. De Los Rios, G.

Dietler, 2009.)

FIG. 8 – Experimental internal distance sν vs. contour length s in 2D. A) Rod

region with ν 1 ; Gaussian region with ν 0 5 for

?

p

>

s

>

14

?

p ; SAW frac-

tal region with exponent ν 0 75. B) Average end-to-end distance for various salt

concentrations in the Gaussian region. Inset : Variation of the ratio Nc

Lc

?

p of

the Gaussian-SAW crossover length s Lc to

?

p. (G. Witz, J. Adamcik, K. Karc-

zewski, K. Rechendorff, T. Eggel, P. De Los Rios, G. Dietler, 2009.)

DNA Renaturation Kinetics

Pairing of two single stranded DNAs : kinetic constants fornucleation and pairing for N base pairs : knucl

N

νθ2 etk2

N knucl

N1 νθ2 .

Excluded volume in 3D : 1 νθ2

0 52

Experiments : k2

N 0 5 (Wetmur & Davidson, 1968).

POLYMER CHAINS

IN SEMI-DILUTE SOLUTION

Chain & Monomer ConcentrationsConcentration for the chain number :

V

volume V

Concentration for the monomer number N :

c N

V N

Osmotic PressureΠ

kBT

βΠ cN

(perfect gas of chains)

Virial Expansion 1Excluded volume interactions :

ΠkBT

12

v

T

c2 O

c3

12

v

T

N2 2

Double expansion to all orders in v

T

& , butdivergent coefficients as N ∞.

Virial Expansion 2Osmotic Pressure :

Π

kBT 12 v

T

N2 2

written as :

ΠkBT

112

v

T

N2

112

v

T

ad N2 d

2 Na2d

2

1 z

R2

0d

2

z v

T

a dN2 d

2

2

R2

0

Na2

(excluded volume parameter) (Gaussian size)

Edwards Polymer ModelMinimal model for self-avoiding polymers :

P

dw

Dw exp

1

4D

t

0

d

w

t

dt

2

dt

b

2

t

0dt1

t

0dt2 δd

w

t1

w

t2

Dw

exp

14D

1

0

d

w

t

dt

2

dt

z

1

0dt

1

1

0dt

2 δd

w

t

1

w

t

2

t

t t

ti t t

i i 1 2

w

ti

t1

2

w

t

i

z v

T

a

dN2 d

2 2

R2

0

Na2 2dDt

bt2 d

2

2 ba4 v

T

2dD

2 d

2

(excluded volume parameter) (Gaussian size)

Virial Expansion 3Dimensionless parameters :

z 12

v

T

a

dN2 d

2

y

R2

0d

2

Na2d

2

Double expansion with respect to z and y :

ΠkBT

1 z

R2 0

d

2

1 zy O

z2y

O

y2

∞

∑n 0

bn

z

yn

Virial Expansion 4Single chain swelling :

R2

Inter

R2

0 X

z

x

R2

I

d

2

R2

0 X

z

d

2 y

X

z

d

2

Re-expansion of the osmotic pressure with respect

to z & x, from the original one w. r. t. z & y :

ΠkBT

∞

∑n 0

bn

z

yn

∞

∑n 0

gn

z

xn

1 gx∞

∑n 2

gn

z

xn

g gn 1

z

z

Virial Expansion 5Expansion of the osmotic pressure in terms of the

“physical” variable x “fraction” of chains inside the

volume of a single self-interacting one :

x

R2

I

d

2

Rd

ΠkBT

1 g Rd∞

∑n 2

gn xn

Large Chain Limit, N ∞, d 4

z 12

v

T

a

dN2 d

2 ∞ g

z ∞

g

g has a FINITE LIMIT g

Variation of the second virial coefficient g as a function of theexcluded volume parameter z or size N. Asymptotic limitg

z ∞ g

: measurable and universal.

Experiments

Second virial coefficient g as a function of the excluded

volume parameter z or size N. The theoretical prediction isg

0 233

12

2π

d 2

in three dimensions, in good

agreement with experiments.

Universal Osmotic Pressurex Rd f inite N z ∞

g g

gn

z ∞

g

n

g and all gn’s have FINITE LIMITS g

g

n

ΠkBT

1 g

Rd

1 g

x∞

∑n 2

g

n xn

x

x

is UNIVERSAL, independent of temperature and ofthe polymer-solvent chemical nature !

Semi-Dilute Solutions

blob (g monoméres)

ξ

ξ

Semi-dilute solution of chains in strong overlap.

Overlap Concentration c

cN

1 R Nνa

x Rd cadNνd 1 cc

c

a

dN1 νd a

3N 4

5 in d 3

x

1 : exactly one single chain in its own volume

Universal osmotic pressure

ΠkBT

x

cN

cc

Strong Semi-Dilute Solutionscc

x 1, universal function

with power law

x xγ or

cc

cc

γ

and osmotic pressure :

ΠkBT

cN

cc

γ

c

a dN1 νd

Independence w. r. t. polymerization degree N

1

γ

1 νd

0 γ 1

νd 1

Universal power law, J. des Cloizeaux, 1975 :

ΠkBT

a

d cadνd

νd 1 a 3

ca3 9

4

d 3 νFlory

3

5

c2 309 ν 0 5880

Universal Osmotic Pressure Curve

PSfrag replacements

ln ΠkBT

ln

x

ln

R3 lnc

c

lnx

Experimental data for various polymer-solvent systems. Theoretical

prediction : full line. Asymptotic power law : broken line (after J. des

Cloizeaux, B.D., & L. Schäfer).

Correlation Length

blob (g monoméres)

ξ

ξ

Flexible chains in semi-dilute solution. Correlation length ξ and “blob”

made of gblob monomers (after P.-G. de Gennes.)

Scaling Law for ξ

c

ξ

c

R Nνa

ξ is a power law for cc

1 :

ξ

c

Rcc

λ

c

a dN1 νd gives ξ

c

Nν a

cadNνd 1 λ

Independence of ξ from N in the c

c

1 limit

λ ν

νd 1

ξ

c

a cad

ννd 1

a

ca3 3

4

d 3 νFlory

3

5

“Blobs” (P.-G. de Gennes)

Each blob is made of gblob monomers between two successiveinter-chain contacts. Below scale ξ, the polymer chain isswollen by excluded volume :

ξ

gblob

νaNumber of monomers in a blob :

gblob

ξa

1

ν

cad 1

νd 1 c

c

1

hence in d 3 and in Flory’s approximation

gblob

ca3 5

4

d 3 νFlory

3

5

Scaling Law for Π ξ cOsmotic pressure of a strong semi-dilute solution :

ξ

c

a cad

ννd 1

Π a

d cadνd

νd 1

c

c

1

kBT

ξd

whence an energy kBT associated with each blobof size ξ and volume ξd .