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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 .
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