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

I. FREELY-JOINTED CHAIN

&

CONSTRAINTS

POLYMER PHYSICS & DNA

Troisième Cycle de la physique en Suisse romande

École Polytechnique Fédérale de Lausanne

12 March 2009

Rigid Rod Anchored at a Wall

F

θ

ϕ

0

b

y

z

x

M

FIG. 1 – Spherical coordinates of a rigid rod anchored at thexOy plane. Constraint : 0

�

z , 0

� θ � π

�

2.

Protein & Rigid Rod Anchored at a Wall

F

R0

θ

ϕ

0

b

y

z

x

M

FIG. 2 – Rigid rod anchored above the xOy plane wall, with abound protein of radius R0. Constraint : R0

�

z , 0

� θ � θ0 ,cosθ0

� R0

�

b.

Partition Functions

Z α

�

u

�

�

θ0

0sinθdθexp

�

βFbcosθ�

� expu � expαuu

��

α � cosθ0

� R0

�

b

�

� expu � 1u

�

�α � 0 � θ0

� π

�

2

�

� 2sinhu

u�

�

α � � 1 � θ0

� π

�

as a function of the dimensionless variable u � βFb.

Freely-Jointed Chain in a Half-Space

FIG. 3 – A freely-jointed chain of n segments of length b, subjec-ted to a force F, anchored at O and constrained to stay abovethe xOy plane : z

�0. [Q. by A. Stasiak, 2007]

Partition Function of a Half-Space FJC

Zn

�

u

�

�

1

� 1

n

∏i �1

dzi ϑi

∑j �1

z j exp un

∑j �1

z j

zi

� cosθi � dzi

� � sinθi dθi �

�

u � βFb

�

Exact Partition Function of a Half-Space FJC

Zn

�

u

�

� 1un J n

�

u

� �

u � βFb�

J n

�

u

�

�

�

1 � Tn � 1

�

K n�

u�

K n

�

u

�

�

�

eu � e

� u �

J n � 1

�

u

��

� � J 0

�

u

�

� 1

Tn � 1K n

�

u

�

�

n � 1

∑p �0

1p!

K�

p�

n

�

0

�

up

B.D., 2007

Exact Partition Functions of Half-Space FJCs

Zn

�

u

�

� 1un J n

�

u

� �

u � βFb

�

J 0

�

u

�

� 1

J 1

�

u

�

� eu � 1

J 2

�

u

�

� e2u � eu � u

J 3

�

u

�

� e3u � e2u � eu � ueu ��

u2 � u � 1

�

J 4

�

u

�

� e4u � e3u � 2e2u � ue2u 2eu ueu � u2eu

u32

u2 �

43

u3�

Zn

�

u

�

� 1un J n

�

u

�

J n �5

�

u

�

� � 2

�

2eu �

3e2u � 3e3u � e4u �

e5u

� 2u

�

2euu

�

e2uu � e3uu

� u2

2

� 3euu2

2� e2uu2 � 7u3

3

�

4euu3

3

�

23u4

12

J n �5

�

u � 0

�

� 63u5

256��� � � � 1

5!945u5 ��� � �

Zn �5

�

0

�

� 15!

945

J n �10

�

u

�

� � 42eu �

42e2u �

48e3u � 48e4u � 27e5u �

27e6u �

8e7u � 8e8u � e9u �

e10u

� 14u � 28euu

�

28e2uu

�

20e3uu � 20e4uu � 7e5uu�

7e6uu

�

e7uu � e8uu

� 21u2 �

7euu2 �

21e2uu2 � 11e3uu2 � 9e4uu2 � 112

e5uu2 � 32

e6uu2 � e7uu2

�

23u3

6

� 121euu3

6

�

376

e2uu3 �

232

e3uu3 � 112

e4uu3 � 73

e5uu3 �

43

e6uu3

� 65u4

4

� euu4

24

�

1158

e2uu4 � 214

e3uu4 � 154

e4uu4 �

2312

e5uu4

� 317u5

40

�

253euu5

15

� 24760

e2uu5 � 37160

e3uu5 �

4415

e4uu5

�

1013u6

60

� 22euu6

45� 937

90e2uu6 �

841180

e3uu6

�

799u7

90

� 7489euu7

420

�

2416315

e2uu7 � 208139u8

6720

�

259723euu8

20160

�

62476u9

2835

Zn

�

u

�

� 1un J n

�

u

�

J n �10

�

u � 0

�

� 46189u10

262144

��� � � � 110!

654729075u10 ��� � �

Zn �10

�

u � 0

�

� 110!

654729075

Zn

�

u � 0

�

� 1n!

�

2n � 1

� �

2n � 3

���� �

3� 1

(O. Bernardi, B.D., P. Nadeau, 2009)

II. DNA TOPOLOGY

POLYMER PHYSICS & DNA

Troisième Cycle de la physique en Suisse romande

École Polytechnique Fédérale de Lausanne

12 March 2009

FIG. 4 – Idealized structure of the double helix.

FIG. 5 – Images of supercoiled plasmids : plectonemes (5 kb), obtained

by electronic microscopy. Supercoiling is increasing from left to right.

Shows the compactifying effect of supercoiling or plectonemic structures

(A. Kornberg et T. Baker, DNA replication, Freeman, 1992).

FIG. 6 – Images of plectonemes of supercoiled plasmids (pUC19, 2686 pb

= 913 nm in B form), obtained by atomic force microscopy. [Courtesy by

G. Dietler and G. Witz, EPFL (January 2008).]

FIG. 7 – Tertiary structures generated by DNA supercoiling. Left : Plec-

tonemic structures of supercoiled plasmids. Right : Solenoidal structure,

stabilized by histone-like proteins.

TOPOLOGICAL FORMALISM

Linking Number Lk

FIG. 8 – Schematic representation of a closed DNA molecule, with a to-

pological non trivial linking number Lk � � 4 between the two oriented

strands of the double helix.

FIG. 9 – Linking number Lk as signed sum over oriented local crossings.

a) Two linked curves, Lk � � 1. b) Linking number of the double helix :

Lk � �

4. [Lk � � 4, if one strand orientation is reversed.] c) Unwinding

of the DNA molecule by one turn, Lk � �

3.

Linking Numbers & Local Crossings

The two DNA strands have a natural linking number Lk0, equal to the

number of base pairs N, divided by the double helix pitch h [h � 10 � 4

base pairs for B-DNA (standard form)] :

Lk0

� Nh

Plane projection of both oriented strands, and signed sum of their

apparent crossings :

Lk � 12 ∑

mutual crossings i

εi �

where εi

� �

1, depending on the right or left orientation of the i crossing.

Gauss’ Topological Invariant

iε = +1i

iε = -1i

FIG. 10 – Local rules for calculating the linking number of two curves :

Lk � 12 ∑crossings i εi, where one sums over all apparent mutual crossings i

of both curves seen in projection, with εi

� �

1 for right/left orientation.

A single plane projection is sufficient for evaluating Lk.

The Gauss linking number is an algebraic topologicalinvariant.

It stays invariant as long as no DNA strand break appears. Upon

replication, or during reading of the genetic code by RNA-polymerases,

this topological invariant stays unchanged. When the extremities of a

linear double helix are fixed, as in a chromosome, a Gauss linking number

can also be defined.

DNA can also be supercoiled. One shall characterize itsdegree of supercoiling, together with torsion.

Twist Tw

Description of the geometry (and no longer the topology) of a DNA

double helix by “twist” and “writhe”.

� “Twisting number” Tw : counts the number of turns of the helix in

space. For an unconstrained linear B-DNA, the natural twist Tw0 is equal

to the base pair number N, divided by the helical pitch h (h � 10 � 4) :

Tw0� N

h

�

Identity of linking number and twist for a natural torsion-free DNA :

Lk0

� Tw0 �

Different in presence of supercoiling.

Writhe Wr

FIG. 11 – Example of writhing number�

Wr�

� 3. Counts theapparent self-crossings of the DNA helix backbone.

Definition : DNA arbitrarily oriented, followed by projectionon a random plane with a signed

�

1 counting for eachcrossing. The result is averaged over all planar projections in3-space :

Wr ��

∑self-crossings i

εi

�

plane �

Writhe and Projected Crossings

iε = +1i

iε = -1i

FIG. 12 – To calculate the “writhe” of a single curve, one has to sum over

apparent self-crossings, as seen in a planar projection, and to average over

the projection plane : W r ��

∑i εi

�

. The real number W � �

is the mean

“self-linking number” of the curve, and depends on its geometrical shape.

The Calugareanu-Fuller-White Theorem

� Unconstrained linear DNA : Lk and Tw are both equal tothe number of turns of the double helix : Lk0

� Tw0, withWr0

� 0.

� For a double stranded DNA, the sum of the Twisting andWrithing numbers is a topological invariant, the Linking

number of the two DNA helix strands :

Lk � Tw

�

Wr �

Topological conservation law for DNA functions, relating the intrinsic

(secondary) topological double-stranded structure of the helix to the

tertiary geometrical structure (twist & writhe) ; (G. Calugareanu, 1959, J.

Vinograd, 1965, F.B. Fuller, 1971).

The Calugareanu-Fuller-White Theorem

Historical Note : J. Vinograd and collaborators guessed in 1965 that there

existed a decomposition of the integer number Lk into the sum of two real

numbers, that describe the DNA geometry, and no longer only its

topology. The mathematician F. B. Fuller (1971) clarified Vinograd’s

empirical formula, while the differential geometer W. F. Pohl, contacted

by F. H. C. Crick, remarked that the formula Lk � Tw

�

Wr also resulted

from the PhD thesis of his former student J. H. White (1969), who

considered it in any space dimension. Actually, a definition of the writhe

W r and a proof in three dimensions appeared in earlier work by the

Romanian mathematician G. Calugareanu (1959-61).

Supercoiling

Double stranded DNA is supercoiled when Lk�

� Lk0, i.e.,when the molecule has a linking number excess or deficit,with respect to its torsion-free state

�

Wr0� 0

�

.

∆Lk � Lk � Lk0

� Tw � Tw0�

Wr � ∆Tw

�

Wr

σ � Lk � Lk0

Lk0

� ∆Tw

�

WrLk0

�

σ � 0 � supercoiled DNA ; σ � 0, undercoiled or negativelysupercoiled.

Bacterial plasmids have a natural negative supercoiling σ � � 0 � 06. For a

50 kpb DNA, this represents a deficit of 300 helical turns.

FIG. 13 – Representation of a supercoiled circular DNA, for different par-

titions of the linking number ∆Lk between supplementary twist ∆Tw, as

measured from the natural coiling or torsion Lk0

� Tw0, and writhe W r.

In the figure, the plasmid is modelled as a tube with a square section and

one black face, and its extra linking number ∆Lk and twist ∆Tw are

simply denoted here by Lk and Tw.

FIG. 14 – AFM images of DNA of adsorbed pUC19 plasmids. pUC19 with

5 mM MgCl2 buffer. a) Absence of Dau anthracycline, W r � 0. b) 3.6 µM

Dau : W r � 0. c) 37 µM Dau :W r � 0. Scale : 500 nm. [V. Viglasky et al.,

Electrophoresis 24, 1703-1711 (2003).] Courtesy of G. Dietler, EPFL.

TWISTING DNA :EXPERIMENTS

Experimental Scheme

F = k TB lδx2

N S

δx

l

µ

DNA

magneticBead

F

x

zy

oil

Capillary

X 100OBJECTIVE

FIG. 15 – Micromanipulation of a single DNA molecule. The magnetic

bead allows the rotation and writhe of the molecule under tension. The

double stranded DNA is chiral. One can therefore expect differential effects

depending on positive or negative winding.

FIG. 16 – Relative extension� � �

0 of a DNA molecule as a function of

supercoiling σ or number of turns n for different extension forces. At low

force, symmetric behavior under n � � n. Shortening due to formation of

plectonemes by writhing. (Read here σ � Lk

�

Lk0

� 1) [T. R. STRICK et

al., Rep. Prog. Phys. 66, 1-45 (2003)].

DNA TWIST

& THERMAL FLUCTUATIONS

Twist Energy

Tw number of turns about tangent vector

�

t

�

s

�

, or rotationangle

Θ � 2πTw

Relaxed DNA Values :

Lk0

� Tw0

��

0

h; Θ0

� 2πTw0

� 2π

�

0

h

�

h helical pitch (h � 3 � 6 nm, 10 � 4 bp for standard DNA).

Twist Elastic Energy :

βHTw

� 12

C�

0

�Θ � Θ0

� 2 �

C torsional modulus

�

Twist Elastic Energy

Lk � Tw

�

Wr � Θ � 2πTw

σ � ∆LkLk0

� Lk � Lk0

Lk0

� Lk0� Tw0 � Wr0

� 0

Θ � Θ0

� 2π

�

Lk0σ � Wr

�

� 2π

�

0

hσ � Wr

βHTw

� 12

C

�

0

�

Θ � Θ0� 2

� βE�

σ�

� b

�

σ

�

2πWr

�

c0

�

2πW r

� 2 �

βE

�

σ

�

� 12

�2π

� 2C

�

0

h2 σ2 � b

�

σ

�

� 2πCh

σ � c0

� 12

C

�

0

�

Writhe (F.B. Fuller, 1978)

t(s)zuC

A

FIG. 17 – Trajectory C of

�

t

�

s

�

on the unit sphere for periodic boundary

conditions :

�

t

�

0

�

��

t

� �

0

�� Area A : solid angle as seen from O.

Fuller’s formula :2πWr � A�

Auxiliary vector potential at

�

r � 3

�

A

� �

r

�

�

�

uz

� �

rr

�

r

� �

uz

� �

r

� � ∇ �

r

�

�

A

� �

r

�

�

�rr3

�

2πW r � A �

S

�

rr3

� d�

S �

C

�

A

� �

t

� � d

�

t

�

�

0

0ds

�uz

� �

t�

s

�

1� �

uz

��

t

�

s

� �

d

�

t

�

s

�

ds

�

[N. B. : Vector potential

�

A� �

r�

defined for r

� �

uz

��

r � 0, undefined for

z � �� ∞ � 0

�

, where r

� �

uz�

�r � 0 ; corresponds to a Dirac magnetic

monopole. ]

Tension & Twist of a DNA Chain

H � H 0

� HTw

� F

�� βH 0

� 12

ξ�

0

0ds

d

�

t

�

s

�

ds

2

βHTw

� � b

�

σ

�

2πW r

��� � � �

� ��

uz ��

R �

�

0

0dstz

�

s

�

� � ��

0

� 1 �

1

4βFξ � b2�

σ�

��� � �

�

1 �

1

4βFξ

�12

b2 �

σ

�

�

4βFξ

�

3

�

2

��� � �

�

βFξ b2 �

σ

� �

TAB. 1 – Relative Extension & Curvature

F

�

pN

�

x

�

σ � 0

�

�� � �

0

� � � �0 Curvature

�

βFξ Modulus C (nm)

0.328 0.747886 359.705 3.98615 122.623

0.197 0.672163 791.025 2.39412 124.061

0.116 0.566178 1422.56 1.40974 111.833

Curvature & Torsional Modulus

� � � � �

0

� x

�

σ

�

� x

�

0

�

�

�

σ2 ��� � �

� � 116

2πCh

2 1

�

βFξ

�

3

�

2

C � h2π

�

16

� �

βFξ� 3

�

2� 1

�2

�

One first finds ξ � 50 nm. The torsional modulus, which is also the

persistence length for the thermal fluctuations of the torsional angle Θ, is

C � 100 nm. New experiments by the team of D. BENSIMON, V.

CROQUETTE & J.-F. ALLEMAND, (Physique des biomolécules,

LPS-ENS), give the value C � 94

�

7 nm. [F. Mosconi et al., Torque

Determination on DNA with Magnetic Tweezers, arXiv : 0809.0621, Phys.

Rev. Lett., to appear (2009).]

Average Writhe

2π

�

W r

�

�

�

0

2ξb

�

σ

�

21

4βFξ � b2�

σ�

��� � �

�

�

0

2ξb

�

σ

�

4βFξ1

� 12

b2 �

σ

�

4βFξ

��� � � �

βFξ b2 �

σ

� �

Average Torque

β

�

Γ

�

� b

�

σ

�

�

C

�

02π

�

Wr

�

� b

�

σ

�

�

C2ξ

b

�

σ

�

4βFξ1

� 12

b2 �

σ�

4βFξ

� � � � �

To leading order :

β

�

Γ

�

� b

�

σ

��

1 �

C4ξ

1

βFξ�

As b

�

σ

�

� 2πCh σ, one finds a renormalization of the torsional modulus C

into Ceff [J.D. Moroz & P. Nelson, Proc. Nat. Acad. Sci. USA 94, 14418

(1997)]

Ceff� C

�

1 �

C4ξ

1

βFξ

�

�

The torsion rigidity diminishes for weaker force, with plectonemes

induced by thermal fluctuations.

Experiments

FIG. 18 – Top : Relative extension

� � �

0 as a function of the degree of

supercoiling σ � n

�

Lk. Bottom : Variation of the torque in DNA as a func-

tion of σ. [F. Mosconi et al., Torque Determination on DNA with Magnetic

Tweezers, arXiv : 0809.0621, Phys. Rev. Lett., to appear (2009).].

Experiments

FIG. 19 – Variation with force F of the effective torsional stiffness Ceff of

DNA in 100mM NaCl. Continuous curve : best fit to Moroz & Nelson’s

formula with C � 94�

7 nm. [F. Mosconi et al., Torque Determination on

DNA with Magnetic Tweezers, arXiv :0809.0621, Phys. Rev. Lett. (2009).].

Magnetic Bead

DNA l

DNA

Magnetic Bead

Glass surface

Γ

nnb

Γb

bn

l

n

FIG. 20 – Top : Extension�

of an elastic tube elongated by a force F and

under torsional strain of n turns. For n � nb, buckling transition. Bottom :

Torsional torque Γ, saturating for n � nb by plectoneme formation.

Buckling Transition

�

�

FIG. 21 – The total height is shortened to

�� δ �

�� 2πρ by the formation

of a loop or plectoneme of radius ρ.

Semi-Flexible ChainCurvature Hamiltonian in presence of a pulling force F in the Oz direction

1kBT

H � βH � 12

ξ

�

0

0ds

d

�

t

�

s

�

ds

2

� βF�

�

ξ persistence length and

�

chain elongation along Oz.

Kinematics :d

�

t

�

s

�

ds�

�

n�

s�

ρ�

�

n

�

s

�

unit normal vector, ρ curvature radius. For a circular plectoneme of

radius ρ, tangent to the DNA stretched part, a loop contribution to the

curvature energy

δH0

�

ρ

�

� ξ2β

2πρ

�

Contribution

�

Fδ of shortening δ � 2πρ in the force field :

δH

�

ρ

�

� ξπ

�

βρ �

2πρF �

Equilibrium value ρc of ρ such that δH

�

ρ

�

is minimal :

ρc

��

ξ

�

2βF

� 1

�

2 �

Bead displacement under plectoneme formation :

δc

� 2πρc� 2πξ

12βFξ

1

�

2

�

Numerical data. For F � 1 pN, and a typical DNA persistence length

ξADN

� 53 nm � δc

� 65 nm.

FIG. 22 – Slope d

� �

dn of the extension curve

�

as a function of the number

of turns n. The continuous line is a fit to a power law F

� α, with α � 0 � 4,

while the elementary mechanical model d

�

dn ∝ 1�

Fgives α � 0 � 5.

FIG. 23 – Relative extension

� � �

0 versus supercoiling σ (bottom scale), or

number of turns n (top scale).

For weak forces, the curve is symmetric under σ � � σ, with symmetric

formation of plectonemes. The F � 1 � 2 pN curve still illustrates for σ � 0

the buckling instability : Shortening of the extension increases at a critical

number of turns nb� �

140. On the σ � 0 side, one observes a broken

symmetry at F � 1 � 2 pN.

DNA Melting by Torsional Strain

DNA

DNA

DNA

FIG. 24 – Extension curves

�

as a function of the number of turns n for

increasing forces (respectively : 0.2 pN, 1.2 pN, 8 pN). Negative winding

at about 1 pN does not reduce extension because alternative structures,

like denaturation loops, appear in preference to buckling. For large forces,

hypertwisted domains appear.

ADN double brinADN simple brin

F + dF→ →

bulle de dénaturation

plectonèmes

F→

FIG. 25 – The negative linking number ∆Lk � n

�

h, imposed by the rotation

of n � 0 turns (h helical pitch), is no longer stored, in Calugareanu-Fuller-

White’s relation ∆Lk � ∆Tw�

Wr, in writhe W r � 0, as when plectonemes

are formed, but in a change of torsion, i.e., of twist ∆Tw � 0 by the opening

of the double helix.

Hypertwisted DNA

FIG. 26 – Structural transition from standard to hypertwisted DNA. Left :

Standard B-DNA form. Right : Possible model for hypertwisted P-DNA,

corresponding to large forces F � 8 pN. Base pairs are broken and turned

outwards. (Lavery et al., 1995).

DNA & TOPOISOMERASES

Knots in DNA

How can two replicated DNAs be separated ? Their mutual linking

number Lk is conserved in replication, thus equal to the linking number of

the two duplex strands before replication, Lk0

� N�

h, where N is the

number of base pairs and h the helical pitch.

Discovery by J.C. Wang a of a family of enzymes, the topoisomerases,

able to modify the topology of DNA molecules by cleaving and resealing

one or two DNA strands (Topo I or Topo II). Topoisomerases are able to

alter DNA supercoiling, the linking number of two molecules, or to

(un)knot a circular DNA. After replication, the daughter DNA chains are

highly linked together. Hence the simultaneous action of multiple

topoisomerases is required during mitosis.

aJ.C. Wang, J. Mol. Biol. 55, 523-533 (1971).

Knots in DNA

FIG. 27 – Duplex DNA knots obtained from the action of Escherichia coli

topoisomerase I on an unknotted nicked plasmid. [F.B. Dean, A. Stasiak,

Th. Koller & N.R. Cozzarelli, J. Biol. Chem. 260 4975-4983 (1985)].

FIG. 28 – AFM images of circular DNA of P4 phage (10346 base pairs,

�

0

� 3 � 5µm). Left column : Strong adsorption, 3D configurations ; Right

column : Weak adsorption on mica, 2D configurations. Top : Simple knots

with essential crossing numbers < 6. Bottom : Complex knots with essen-

tial crossing numbers between 30 and 40. [E. Ercolini et al., Phys. Rev.

Lett. 98, 058102 (2007).] (Courtesy of G. Dietler, EPFL.)

FIG. 29 – AFM images of circular DNA of P4 phage, weakly adsorbed in

two dimensions on mica. Top left : Unknotted configuration. In all other

images the circular DNAs are knotted. For entropic reasons, the knots ap-

pear as localized along the chains. Images from E. Ercolini et al., Phys.

Rev. Lett. 98, 058102 (2007). (Courtesy by G. Dietler, EPFL.)

Action of Type II Topoisomerase

FIG. 30 – Hypothetical scheme for the action of type II topoisomerase.

Each cycle yields a change

�

∆Lk

�

��

∆W r

�

� 2 and requires ATP transfor-

ming into ADP. [J.C. Wang et al., Nature 379, 225-232 (1996).]

Action of Topoisomerases with ATP

F

ATP ADP

F

Surface

Topo II

l

Bead

DNA

s

FIG. 31 – Supercoils relaxation by topoisomerase II. Each double strand

passage (with conversion of ADP into ATP) leads to the removal of two

turns and increases DNA extension, allowing real time observation of en-

zymatic activity.

FIG. 32 – DNA extension

� �

t

�

in presence of Topo II (nM, 1 � µm

� 3) 300

µM ATP. (T. R. Strick et al.) Top : Number of magnet turns. Bottom :

DNA extension. Enzymatic removal of supercoils (

�

) increases extension.

Rotation of the magnet induces supercoiling and DNA contraction (

�

).

The horizontal line is the maximal extension in the absence of supercoils

(W r � 0).

FIG. 33 – Supercoils removal by Topo II in presence of 10 µm ATP. (T.

R. Strick et al.) A) Extension� �

t�

as a function of time. B) Histogram of

distances

� � �

t

�

�� �

t

� � �

for curve (A). Peaks in the histogram correspond

to plateaus in data (A) with 90 nm steps. The mean waiting time between

events is roughly 20 s.

FIG. 34 – Removal of supercoils by Topo II in presence of ATP (10 µM)

at F � 0 � 7N. A) DNA extension and pause times. B) Histogram of step-

sizes and numbers of turns removed. Peak at n � 2 with steps of 86 nm,

corresponding to ∆Lk � � 2. C) Exponential distribution of pause times.

Action of topoisomerases without ATP

FIG. 35 – Traction on plectonemes without Topo II (left) or with Topo II

(right). DNA overwound by n � 60 turns and submitted to a crenel force

from 1pN to 5 pN. In the absence of enzymes, the force removes plecto-

nemes and the extension oscillates between levels (1) (weakly extended)

and (3) (extended). In the presence of enzymes, intermediate extensions

appear (2), due to the protection of plectonemes clamped by Topo II.

Bursting Tractions on Supercoils

l

F = 1 pN

ADN

Bille

F = 5 pN

Surface

Topo II

1. 3.2.

FIG. 36 – Interpretation : When a Topo II clamps a local dsDNA cros-

sing [1.], it momentarily protects a number of supercoils during the next

force increase from 1 to 5 pN [2.]. When the enzymatic clamp relaxes, the

remaining supercoils vanish [3.]

Random Clamping by Topoisomerases

Z0

Z1

cR2 π

Surface

l

F

TopoII

Bead

DNA

F

FIG. 37 – Stabilization of a transient supercoil by a topoisomerase. The

DNA, stretched by 1 pN, is overwound up to the buckling transition. Ther-

mal fluctuations create a few transient supercoils clamped by Topo II

(strong affinity). The stabilization of a supercoil by Topo II shortens the

DNA by δc

� 2πρc, ρc buckling critical radius.

FIG. 38 – Observation of the clamping of a supercoil by Topo II (in the

absence of ATP). Top : A reduced extension marks a clamping of random

duration Tc. (superimposed on Brownian fluctuations.) Bottom : Cumula-

tive probability distribution for the clamping time Tc

� tclamp.

FIG. 39 – A rare event with a double step of twice 40 nm, indicating the

presence of two random supercoils clamped by Topo II.

An important paradox : Topo II, with a size in the range of 5 nm, is

able to RECOGNIZE THE TOPOLOGY of long DNA chains and to relax or

simplify it, hence DISPLACING THERMODYNAMICAL EQUILIBRIUM !

III. RNA UNFOLDING

&

FLUCTUATION RELATIONS

POLYMER PHYSICS & DNA

Troisième Cycle de la physique en Suisse romande

École Polytechnique Fédérale de Lausanne

12 March 2009

Single Molecule Experiments

FIG. 40 – Various configurations for single molecule experiments. (A)

Stretching DNA ; (B) Unzipping DNA ; (C) Mechanical unfolding of

RNA ; (D) Mechanical unfolding of a protein.

Micromanipulating RNA

FIG. 41 – Experimental set-up for pulling on an RNA molecule. Hybrid

RNA/DNA handles prevent the formation of secondary folded structures.

The movable frame monitors the RNA extension under the action of the

restoring force induced by the optical tweezer.

Stochastic Unfolding of RNA & Hysteresis

FIG. 42 – Out-of-equilibrium pulling curves for a P5abc RNA molecule in

a 10mM EDTA buffer (no Mg� �

). A) Reversible trajectory at rate 2-5 pN/s,

irreversible one at 52 pN/s, with hysteresis at refolding (R) as compared to

unfolding (U). B) Out-of-equilibrium curves for 34 pN/s and 52 pN/s, as

compared to near-equilibrium curves at 2-5 pN/s (40 curves superposed).

J. Liphardt et al., Science 292, 733 (2001) ; 296, 1832 (2002).

FIG. 43 – Pulling experiments on RNA (A, B) or proteins (C, D [RNase

H]). Inset : Unfolding ( �, right) or refolding ( � , left) force distributions

at r � 1 � 5 � 7 � 5 � 20 pN/s. (B) from M. Manosas et al., Phys. Rev. Lett. 96,

218301 (2006) ; (C, D) from C. Cecconi et al., Science 309, 2057 (2005).

JARZYNSKI EQUALITY

Jarzynski Equality, 1997

exp �

WkBT

� exp �∆FkBT

exp �

Wdiss

kBT

� 1

(1993 : Evans, Cohen & Morris ; Gallavotti-Cohen ; Evans &Searles : Entropy production in steady state)

Jarzynski Equality and Dissipated Work

∆F � ∆E � T∆S � W

� ∆Q � T ∆S � W � Wdiss

�� Wrev

�

Wdiss

� T ∆S � ∆Q�

“with” Wdiss

�

0

��

exp

��

WkBT

�

�

∞

� ∞dW P

�

W

�

exp

��

WkBT

�

� exp

��

∆FkBT

�

exp

��

Wdiss

kBT

�

� 1

Distribution of Dissipated Work

FIG. 44 – Characteristic probability distribution for the dissipated work

Wdiss. Because of Jarzynski equality, it necessarily exists a non-vanishing

probability P

�

Wdiss� 0

�

for events with Wdiss

� 0, that represent transient

violations of the Second Principle.

Second Law from Jarzynski Equality

Wdiss

� T ∆S � ∆Q�

“with” Wdiss

�

0

��

∆F � ∆E � T ∆S � W� ∆Q � T∆S � W � Wdiss

� Wrev

exp �

Wdiss

kBT

� 1 �

Convexity : exp�

x� �

expx � 1 � x

�

0 � Wdiss

�

0 �

Experimental Distribution of Dissipated Work

FIG. 45 – Experimental distributions of dissipated work for the unfolding

of the P5abc RNA molecule for various rates [3-5 pN/s, 34 pN/s, 52 pN/s],

as measured for distances 5, 15 and 25 nm. From J. Liphardt et al., Science

296, 1832 (2002).

Comparison to a Two-State Model

-5 -2.5 0 2.5 5 7.5 10W (k T units)

0

0.1

0.2

0.3

0.4

0.5

P (W

)

dis

dis B

F. Ritort, C. Bustamante, I. Tinoco Jr., PNAS 99, 13544 (2002).

Convergence with the Number of Pulls

BJE � � ln exp �

WkBT �

Npulls

�

∆FkBT

� � ln exp �Wdiss

kBT �

Npulls

Bias

10 100 1000 10000

N

0.01

0.1

1

B

(k

T u

nits

)

1 2 3 4 5W /k T

1

10

100

1000

N

pulls

B

JE

dis

pulls

20

34

52

75100

B

FIG. 46 – Bias BJE (in kBT units), as a function of the number of pulls

Npulls and for rates 34, 52, 75, 100 pN/s. Drops as 1

�

Npulls for Npulls

�

1.

Inset : number of pulls necessary to obtain the equilibrium free energy

within 1kBT , as a function of dissipated energy. [F. Ritort, C. Bustamante,

I. Tinoco Jr, PNAS 99, 13544 (2002).]

CROOKS

FLUCTUATION THEOREM

Crooks Fluctuation Theorem, 1999

PU WPR

�W

� expW � ∆F

kBT

U(nfolding) : Direct pathR(efolding) : Reverse path

FIG. 47 – A) CD4 hairpin RNA. B) Force-extension curves (1.7 pN/s). C)

(Quasi-Gaussian) direct and reverse work distributions. D) Test of Crooks

fluctuation theorem for several molecules (7.5 pN/s). Slope : 0 � 95

�

0 � 15,

∆F � 110 � 3

�

0 � 5kBT � [D. Collin et al., Nature 437, 231 (2005).]

Triple Helix Junction

FIG. 48 – A) S15 triple helix. B) Force-extension curves for the wild

type. C) (Quasi non-overlapping) unfolding (–) and refolding (- -) work

distributions for wild and mutant types (G-C � C-G). Inset : Test of Crooks

fluctuation theorem for the mutant. [D. Collin et al., Nature, ibid.]