Elastic deformation energy for beams and thin filamentsakosmrlj/MAE545_F2015/lecture... · 2014. 10. 28. · 1 = 34 A corresponding to a helical repeat every 10.5 base pairs (bp)

MAE 545: Lecture 13 (10/29)

Elastic deformation energy for beams and thin filaments

2

Deformations of macroscopic beams

Lt

undeformed beam beamcross-sectionbeam made of material with

Young’s modulusE0

L(1 + ✏)

stretching

✏strain

EsL

=k✏2

2

k / E0t2

twisting

�

angle of twist� = ⌦L

EtL

=C⌦2

2C / E0t4

Bending and twisting is much easier than stretching for long and narrow beams!

R R

radius of curvature

bending

A / E0t4

EbL

=A

2R2

3

Energy cost of deformationsrotation rate ofmaterial frame

ẑx̂

ŷ

E =Z

ds

2⇥A1⌦21 + A2⌦

22 + C⌦

23

⇤

R1 = ⌦�11 R2 = ⌦�12

p = 2⇡⌦�13

bending around twisting aroundbending around e2e1 e3

Bending and twisting represented as rotations of material frame

undeformed beam deformed beam

s

d~eids

= ~⌦⇥ ~ei~⌦ = ⌦1~e1 + ⌦2~e2 + ⌦3~e3

~e1 ~e1~e2 ~e2

~e3~e3

4

Energy cost of deformationsrotation rate ofmaterial frame

ẑx̂

ŷ

E =Z

ds

2⇥A1⌦21 + A2⌦

22 + C⌦

23

⇤

Bending and twisting represented as rotations of material frame

undeformed beam deformed beam

s

d~eids

= ~⌦⇥ ~ei~⌦ = ⌦1~e1 + ⌦2~e2 + ⌦3~e3

~e1 ~e1~e2 ~e2

~e3~e3

Bending and twisting modes are coupled,because successive rotations do not commute!

Ry⇣⇡

2

⌘Rz

⇣⇡2

⌘ẑ = Ry

⇣⇡2

⌘ẑ = x̂

Rz⇣⇡

2

⌘Ry

⇣⇡2

⌘ẑ = Rz

⇣⇡2

⌘x̂ = ŷ

plectonemes

5

Elastic energy of deformations in the general form

Energy density for a deformed filament can be Taylor expanded around the minimum energy ground state

E =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23 + 2A12⌦1⌦2 +2A13⌦1⌦3 + 2A23⌦2⌦3

twist-bend coupling

+k✏2 +2D1✏⌦1 + 2D2✏⌦2 + 2D3✏⌦3

�

twist-stretchcoupling

bend-stretchcoupling

Energy density is positive definitive functional!A11, A22, A33, k > 0

A2ij < AiiAjjD2i < kAii

In principle 10 elastic constants, but symmetries of filament shape determine how many independent

elastic constants are allowed!

6

Beams with uniform cross-section along the long axis

~e1~e2

~e3

beamcross-section

Beam has mirror symmetry through a plane perpendicular to . ~e3

Two beam deformations that are mirror images of each other must have the same energy cost!

7

Beams with uniform cross-section along the long axis

~e1~e2

~e3

beamcross-section

E =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23+2A12⌦1⌦2 �

Twist is decoupled from bending and stretching!

+k✏2 + 2D1✏⌦1 + 2D2✏⌦2


A13 = A23 = D3 = 0

8

Blades of propellers and turbines are chiral, therefore there is coupling between twist and bend deformations!

Twist-bend coupling in propellers and turbineswind turbine airplane propeller

ship propeller

9

Beams with isosceles triangle cross-section

~e1~e2

~e3

beamcross-section

~e1

~e2

~e3�

Beam has 2-fold rotational symmetry around axis . ~e2

Note: n-fold rotational symmetry is symmetry

due to rotation by angle .2⇡/n



10

bending around ~e1 bending around ~e2 twisting around

How mirroring around affects bending and twisting?

⌦1 ⌦2 ⌦3

⌦1 �⌦3Note: mirroring doesn’t

affect stretching

mirrorimage mirrorimage

mirrorimage

~e1~e2

~e3

beamcross-section

~e1

~e2

~e3�

~e1

�⌦2

~e3

A12 = A13 = D2 = D3 = 0

⌦3

11


How rotation by around affects bending and twisting?

⌦1 ⌦2 ⌦3

⌦1

Note: rotation doesn’t affect stretching

rotation

~e1~e2

~e3

beamcross-section

~e1

~e2

~e3�

�⌦2

~e3

⇡ ~e2

rotation rotation

A12 = A23 = D2 = 0

12

Elastic energy for beams of various cross-sectionsbeam

cross-sectionE =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23+2A12⌦1⌦2 �

E =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23

+k✏2 + 2D1✏⌦1

�

+k✏2 + 2D1✏⌦1 + 2D2✏⌦2

13

Beams with rectangular cross-section

~e1~e2

~e3

beamcross-section

~e1

~e2

~e3�






⌦2

14


How mirroring around affects bending and twisting?

⌦1 ⌦2 ⌦3

�⌦3Note: mirroring doesn’t

affect stretching

mirrorimage mirrorimage

mirrorimage

~e1

~e2

~e3�

~e3

~e2

~e1~e2

~e3

beamcross-section

�⌦1

A12 = A23 = D1 = D3 = 0

⌦3�⌦1

15



⌦2


rotation

�⌦2

~e3

⇡

rotation rotation

~e3

~e1

~e2

~e3�

~e1~e2

~e3

beamcross-section

⌦1 ⌦3

A13 = A23 = D1 = D2 = 0

16


cross-sectionE =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23+2A12⌦1⌦2�

E =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23

+k✏2 + 2D1✏⌦1

�

+k✏2 + 2D1✏⌦1 + 2D2✏⌦2

E =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23 + k✏

2

�

17

Beams with square cross-section

~e1~e2

~e3

beamcross-section

~e1

~e2

~e3�






⌦3�⌦1

18



⌦2


rotation

~e3

rotation rotation

~e3

~e1

~e2

~e3�

beamcross-section

⌦1 ⌦3

⇡/2

~e1~e2

~e3

⌦2

A11 = A22, A12 = D1 = D2 = 0

19


cross-section

E =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23+2A12⌦1⌦2�

E =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23

+k✏2 + 2D1✏⌦1

�

+k✏2 + 2D1✏⌦1 + 2D2✏⌦2

E =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23 + k✏

2

�

E =

Z L

0

ds

2

A�⌦21 + ⌦

22

�+ C⌦23 + k✏

2

�

+2A12(s)⌦1(s)⌦2(s) + 2A13(s)⌦1(s)⌦3(s) + 2A23(s)⌦2(s)⌦3(s)

E =

Z L

0

ds

2

A11(s)⌦

21(s) +A22(s)⌦

22(s) + C(s)⌦

23(s)

+k(s)✏(s)2 + 2D1(s)✏(s)⌦1(s) + 2D2(s)✏(s)⌦2(s) + 2D3(s)✏(s)⌦3(s)

�

20

DNA

982

Mar

ko a

nd

Sig

gia

Mac

rom

olec

ules

, V

ol.

27, N

o. 4, 1

994

V

Fig

ure

1.

Sch

emat

ic d

iagr

am o

f th

e B

-DN

A m

olec

ule.

T

he

mol

ecul

ar d

iam

eter

is d

= 2

0 A

, th

e he

lical

rep

eat l

engt

h is

1 =

27

r/w

0 =

34 A

, cor

resp

ondi

ng to

a s

tack

of a

bout 1

0.5

nuc

leic

aci

d bas

es.

The

nucl

eotide

s ar

e bo

und

betw

een

the

suga

r-ph

osph

ate

back

bone

hel

ices

: w

e no

te t

he

arro

ws

on t

he

side

vie

w (

upper

p

ort

ion

of

figure

), w

hich

indic

ate

the

oppo

site

dir

ecte

dnes

s of

the

two

helic

es.

The

wid

e m

ajor

gro

ove

is m

arke

d “M”,

whi

le th

e na

rrow

er m

inor

gro

ove

is m

arke

d “m

”. T

he

low

er p

art

of t

he

fig

ure

sho

ws t

he

end

view

of

B-D

NA

, wit

h ta

ngen

t t d

irec

ted o

ut

of t

he

pag

e.

11.

Ela

stic

Fre

e E

ne

rgy

of DNA

A.

Sy

mm

etr

y A

naly

sis.

B-D

NA

mo

lecu

les2

are

rig

ht-

h

and

ed c

hir

al r

od

s of

cro

ss-s

ecti

on

al d

iam

eter

d =

21

A.

As sc

hem

atic

ally

sho

wn

in F

igu

re 1

, th

e p

airs

of n

ucl

eoti

des

(o

ccu

py

ing

the m

ajo

r gro

ove

reg

ion

, den

ote

d M

in

Fig

ure

1) a

re a

rran

ged

in

a h

elix

wit

h a

pit

ch o

f ab

ou

t 1

= 3

4 A

co

rres

po

nd

ing

to

a h

elic

al r

epea

t ev

ery

10.5

bas

e p

airs

(b

p).

W

e d

efin

e th

e m

ole

cula

r ax

is (

the c

ente

r of

th

e

mo

lecu

le) to

be

des

crib

ed b

y t

he s

pac

e cu

rve

r(s),

wit

h s

b

ein

g a

rcle

ng

th.

Th

e t

ang

ent

t 1 d

r/d

s th

us

has

un

it

len

gth

.

At

any

s,

con

sid

er t

he

pla

ne

per

pen

dic

ula

r to

t.

Th

e

two s

ug

ar-p

ho

sph

ate

bac

kb

on

es (

the tw

o h

elic

es, d

raw

n

wit

h o

pp

osi

ng

arr

ow

s in

Fig

ure

1) i

nte

rsec

t th

is p

lan

e a

t tw

o p

oin

ts R

an

d S

. W

e d

efin

e u t

o b

e th

e u

nit

vec

tor

in

this

pla

ne

tha

t p

oin

ts f

rom

th

e m

ole

cula

r ax

is t

o t

he

mid

po

int

of E

. A

fin

al u

nit

vec

tor

v i

s d

efin

ed b

y v =

t

X u s

o t

ha

t th

e s

et (u, v,

t) f

orm

s a

rig

ht-

han

ded

co

ord

inat

e sy

stem

at

each

po

int

s.

It w

ill

be

hel

pfu

l to

te

mp

ora

rily

use

in

dex

ed v

ecto

rs e

(l) u, e

(2) =

v, a

nd

e(3

)

= t.

A g

ener

al d

efo

rmat

ion

of

the m

ole

cule

th

at

mai

nta

ins

t2 = 1

may

be

des

crib

ed b

y i

nfi

nit

esim

al r

ota

tio

ns

Q(s

) of

th

e c

oo

rdin

ate

axe^

:^^^

de“

’

ds

-- -

+

ill

x e

“’

wh

ere 00 =

2*/

1=

0.1

85 A

-l d

eter

min

es t

he h

elic

al r

epea

t le

ng

th i

n t

he a

bse

nce

of

def

orm

atio

ns.

W

e m

ay t

hin

k o

f th

e c

om

po

nen

ts O

i =

il.e

ci)

as “

stra

ins”

whic

h lo

call

y

gen

erat

e ro

tati

on

s of

th

e c

oo

rdin

ates

aro

un

d e

(’).

If Q =

0

, th

e m

ole

cule

tak

es i

ts u

nd

isto

rted

co

nfi

gu

rati

on

sh

ow

n

in F

igure

1.

Th

e m

ole

cula

r ax

is r

(s) i

s o

bta

ined

for g

ener

al

Q b

y i

nte

gra

tin

g t

he t

ang

ent

equ

atio

n d

r/d

s =

e(3

).

Th

e in

teg

ral

Tw

= L

/1 +

J d

s Q

3/(

2r

) is d

efin

ed t

o b

e th

e d

ou

ble

hel

ix “

twis

t”3

wh

ere

L i

s th

e m

ole

cule

len

gth

, an

d w

her

e th

e i

nte

gra

l is

fro

m s

= 0

to

s =

L.

Fo

r an

u

nd

isto

rted

mo

lecu

le, T

w =

L/1, an

d T

w ju

st c

ou

nts

th

e

nu

mb

er o

f h

elic

al t

urn

s of

len

gth

1 a

lon

g t

he c

hai

n.

Fo

r a

dis

tort

ed c

hai

n, t

he e

xces

s tw

ist

per

hel

ix r

epea

t is

(T

w

- L/l)/(L/l)

= (

Q3

)/

~0

,

wh

ere

we

use

th

e n

ota

tio

n (

Q3

) =

L-’J

; d

s Q

3(s) to

den

ote

an

av

erag

e al

on

g t

he c

hai

n o

f le

ng

th L

>>

1.

Sin

ce w

e as

sum

e th

at th

e Q

= 0

sta

te is

eq

uil

ibri

um

, we

may

wri

te t

he f

ree

ener

gy

fo

r sm

all

stra

ins

as a

Tay

lor

exp

ansi

on

in

a a

nd

its

s d

eriv

ativ

es.1

0 T

he lo

wes

t o

rder

term

s ar

e

(2)

wh

ere

we

hav

e in

tro

du

ced

th

e m

atri

ces

Aij

an

d A

ijk, w

hic

h

are

sym

met

ric

un

der

all

per

mu

tati

on

s of

th

eir

ind

ices

, an

d w

her

e th

e i

nte

gra

l ru

ns

over

th

e m

ole

cula

r ax

is o

f le

ng

th L

. If

we

ign

ore

nu

cleo

tid

e-se

qu

ence

dep

end

ence

of

th

e e

last

ic p

rop

erti

es o

f th

e m

ole

cule

(o

r if

we

rest

rict

o

ur

atte

nti

on

to

sy

mm

etri

c re

pea

ts s

uch

as ($

i)N

th

en

th

e

A m

atri

ces

hav

e n

o s

dep

end

ence

, si

nce

in

th

ese

coo

rdi-

n

ates

, ev

ery

po

int

alo

ng

th

e m

ole

cule

in

th

e u

nd

isto

rted

st

ate

is e

qu

ival

ent.

W

e w

ill r

efer

to

thes

e m

atri

ces

as th

e

“ela

stic

co

nst

ants

”: t

hey

may

dep

end

on

en

vir

on

men

tal

fact

ors

(te

mp

erat

ure

, io

nic

str

eng

th,

pH

, et

c.).

W

e w

ill

ign

ore

th

e c

on

stan

t fr

ee e

ner

gy

A0 fo

r th

e r

emai

nd

er o

f th

is p

aper

.

Th

e s

ec

on

d-o

rde

r m

atr

ix h

as

six

in

de

pe

nd

en

t co

mp

on

ents

: A

ll,

A22

, A

33,

A12,

A13

9 an

d A

23.

We

now

sh

ow

how

sy

mm

etri

es m

ake

som

e of

th

ese

com

po

nen

ts

van

ish

. N

ote

th

at

rota

tio

n b

y 1

80’

aro

un

d t

he v

ecto

r u

is a

sy

mm

etry

of

the u

nd

isto

rted

mo

lecu

le (

see

Fig

ure

1).

N

ow c

on

sid

er a

n in

fin

ites

imal

seg

men

t of

len

gth

ds

fro

m

s =

-d

s/2

to

ds/

2,

wit

h u

nif

orm

str

ain

Q =

(01, 02, O

S).

R

ota

tio

n o

f th

is s

egm

ent

by

180

’ ar

ou

nd

u(s

= 0

) y

ield

s p

reci

sely

th

e s

egm

ent

con

fig

ura

tio

n w

ith

un

ifo

rm s

trai

n

il’ =

(4

1,

Qz, O

3).

T

her

efo

re c

on

fig

ura

tio

ns D a

nd

D’ h

ave

the s

ame

free

en

erg

y,

ind

icat

ing

th

at

A12

=

A13 =

0.l

’

We

no

te t

ha

t ro

tati

on

s b

y 1

80’

aro

un

d e

ith

er t

(0)

or

v(0

) d

o n

ot

tak

e o

ur

infi

nit

esim

al d

isto

rted

seg

men

t to

a

con

fig

ura

tio

n w

ith

tran

sfo

rmed

D b

ecau

se th

ese

op

erat

ion

s are

no

t sy

mm

etri

es o

f th

e u

nd

isto

rted

mole

cule

. T

he la

ck

of s

ym

met

ry u

nd

er t

hes

e o

per

atio

ns

is d

ue to

the e

xis

ten

ce

of t

wo

dis

tin

ct reg

ion

s of

th

e D

NA

su

rfac

e b

ou

nd

ed b

y th

e

two

hel

ices

, mar

ked

“M

” an

d “

m”

in F

igu

re 1

. T

hes

e tw

o

regio

ns ar

e re

ferr

ed to

as t

he ”

maj

or g

roove”

an

d th

e “

min

or

gro

ov

e”, r

esp

ecti

vel

y.

Th

e m

ino

r gro

ove

is n

arro

w, w

hil

e th

e m

ajo

r gro

ove

is w

ider

, fi

lled

up

by t

he n

ucl

eoti

des

w

hic

h a

re b

ou

nd

bet

wee

n th

e b

ack

bo

nes

. T

he tw

o g

rooves

ar

e al

so d

isti

ng

uis

hed

by

th

e o

pp

osi

te d

irec

ted

nes

s of

th

e

sug

ar-p

ho

sph

ate

bac

kb

on

es o

n t

hei

r b

ou

nd

arie

s.

A r

o-

tati

on

of

the u

nd

isto

rted

mo

lecu

le a

rou

nd

eit

her

t o

r v b

y

180”

exch

ang

es t

he m

ajo

r an

d m

ino

r gro

oves

.

Ch

iral

po

lym

ers

wit

ho

ut

this

par

ticu

lar

bro

ken

sym

- m

etry

(e.

g. a

do

ub

le-h

elix

po

lym

er w

ith

in

dis

tin

gu

ish

able

b

ack

bo

nes

, an

d th

eref

ore

inv

aria

nt u

nd

er r

ota

tio

n b

y 18

0’

abo

ut t)

wil

l h

ave

A23 =

0.

Fin

ally

, o

ne

sho

uld

no

te t

ha

t re

flec

tio

ns

(use

ful

in d

eriv

atio

n o

f th

e e

last

ic e

ner

gy

of

thin

ro

ds

wit

h r

efle

ctio

n s

ym

met

ries

lO can

no

t b

e u

sed

to

an

aly

ze a

ch

iral

ro

d s

uch

as

DN

A.

Th

us,

the n

on

zero

ele

men

ts o

f th

en

= 2

ela

stic

co

nst

ant

mat

rix

are

All

, A22, A

33, an

d A

23.

Ph

ysi

call

y, A

11 an

d A

22

are

(dis

tin

ct)

ben

din

g c

on

stan

ts a

sso

ciat

ed w

ith

ben

ds

loca

lly

in th

e p

lan

es p

erp

end

icu

lar

to u a

nd

v, r

esp

ecti

vel

y.

We

exp

ect

thes

e co

nst

ants

to

be

app

rox

imat

ely

eq

ual

to

th

e b

end

per

sist

ence

len

gth

~5

00

A

.4 T

he c

on

stan

t A33 i

s ju

st t

he

tw

ist

rig

idit

y,

an

d i

s ro

ug

hly

eq

ual

to

th

e t

wis

t p

ersi

sten

ce l

eng

th 2

50

0 A

.4

In a

dd

itio

n t

o th

e b

end

ing

rig

idit

ies

an

d tw

ist

rig

idit

y,

we

hav

e a

cou

pli

ng

A23 o

f b

end

s ab

ou

t th

e lo

cal v

ax

is a

nd

th

e tw

ist in

the q

uad

rati

c (O

(Q2

))

elas

tic

theo

ry.

Its

ov

eral

l m

agn

itu

de

sho

uld

als

o b

e co

ntr

oll

ed b

y th

e d

egre

e by w

hic

h

rota

tio

ns

of

the m

ole

cule

by 180°

abo

ut

v a

nd

t a

re n

ot

sym

met

ries

. F

or

DN

A,

thes

e o

per

atio

ns

esse

nti

ally

ex-

chan

ge

the m

ajo

r an

d m

ino

r gro

oves

of t

he

mo

lecu

le, w

hic

h

are

rath

er d

iffe

ren

t in

str

uct

ure

: w

e th

us

exp

ect A

23 =

Ai,

.

~e3~e1

~e2

982 Marko and Siggia Macromolecules, Vol. 27, No. 4, 1994

V

Figure 1. Schematic diagram of the B-DNA molecule. The molecular diameter is d = 20 A, the helical repeat length is 1 = 27r/w0 = 34 A, corresponding to a stack of about 10.5 nucleic acid bases. The nucleotides are bound between the sugar-phosphate backbone helices: we note the arrows on the side view (upper portion of figure), which indicate the opposite directedness of the two helices. The wide major groove is marked “M”, while the narrower minor groove is marked “m”. The lower part of the figure shows the end view of B-DNA, with tangent t directed out of the page.

11. Elastic Free Energy of DNA

A. Symmetry Analysis. B-DNA molecules2 are right- handed chiral rods of cross-sectional diameter d = 21 A. As schematically shown in Figure 1, the pairs of nucleotides (occupying the major groove region, denoted M in Figure 1) are arranged in a helix with a pitch of about 1 = 34 A corresponding to a helical repeat every 10.5 base pairs (bp). We define the molecular axis (the center of the molecule) to be described by the space curve r(s), with s being arclength. The tangent t 1 dr/ds thus has unit length.

At any s, consider the plane perpendicular to t. The two sugar-phosphate backbones (the two helices, drawn with opposing arrows in Figure 1) intersect this plane at two points R and S. We define u to be the unit vector in this plane that points from the molecular axis to the

midpoint of E. A final unit vector v is defined by v = t X u so that the set (u, v, t) forms a right-handed coordinate system at each point s. It will be helpful to temporarily use indexed vectors e(l) u, e(2) = v, and e(3) = t.

A general deformation of the molecule that maintains t2 = 1 may be described by infinitesimal rotations Q(s) of the coordinate axe^:^^^

de“’

ds -- - + ill x e“’

where 00 = 2* /1= 0.185 A-l determines the helical repeat length in the absence of deformations. We may think of the components Oi = il.eci) as “strains” which locally generate rotations of the coordinates around e(’). If Q = 0, the molecule takes its undistorted configuration shown in Figure 1. The molecular axis r(s) is obtained for general Q by integrating the tangent equation dr/ds = e(3).

The integral T w = L/1 + J ds Q 3 / ( 2 r ) is defined to be the double helix “twist”3 where L is the molecule length, and where the integral is from s = 0 to s = L. For an undistorted molecule, T w = L/1, and Tw just counts the number of helical turns of length 1 along the chain. For a distorted chain, the excess twist per helix repeat is (Tw - L/l)/(L/l) = ( Q 3 ) / ~ 0 , where we use the notation ( Q 3 ) = L-’J; ds Q3(s) to denote an average along the chain of length L >> 1.

Since we assume that the Q = 0 state is equilibrium, we may write the free energy for small strains as a Taylor expansion in a and its s derivatives.10 The lowest order

terms are

(2)

where we have introduced the matrices Aij and Aijk, which are symmetric under all permutations of their indices, and where the integral runs over the molecular axis of length L. If we ignore nucleotide-sequence dependence of the elastic properties of the molecule (or if we restrict our attention to symmetric repeats such as ($i)N then the A matrices have no s dependence, since in these coordinates, every point along the molecule in the undistorted state is equivalent. We will refer to these matrices as the “elastic constants”: they may depend on environmental factors (temperature, ionic strength, pH, etc.). We will ignore the constant free energy A0 for the remainder of this paper.

The second-order matrix has six independent components: All , A22, A33, A12, A139 and A23. We now show how symmetries make some of these components vanish. Note that rotation by 180’ around the vector u is a symmetry of the undistorted molecule (see Figure 1). Now consider an infinitesimal segment of length ds from s = -ds/2 to ds/2, with uniform strain Q = (01, 02, OS). Rotation of this segment by 180’ around u(s = 0) yields precisely the segment configuration with uniform strain il’ = ( 4 1 , Qz, O3). Therefore configurations D and D’ have the same free energy, indicating that A12 = A13 = 0.l’

We note that rotations by 180’ around either t(0) or v(0) do not take our infinitesimal distorted segment to a configuration with transformed D because these operations are not symmetries of the undistorted molecule. The lack of symmetry under these operations is due to the existence of two distinct regions of the DNA surface bounded by the two helices, marked “M” and “m” in Figure 1. These two regions are referred to as the ”major groove” and the “minor groove”, respectively. The minor groove is narrow, while the major groove is wider, filled up by the nucleotides which are bound between the backbones. The two grooves are also distinguished by the opposite directedness of the sugar-phosphate backbones on their boundaries. A rotation of the undistorted molecule around either t or v by 180” exchanges the major and minor grooves.

Chiral polymers without this particular broken symmetry (e.g. a double-helix polymer with indistinguishable backbones, and therefore invariant under rotation by 180’ about t) will have A23 = 0. Finally, one should note that reflections (useful in derivation of the elastic energy of thin rods with reflection symmetrieslO cannot be used to analyze a chiral rod such as DNA.

Thus, the nonzero elements of then = 2 elastic constant matrix are All , A22, A33, and A23. Physically, A11 and A22 are (distinct) bending constants associated with bends locally in the planes perpendicular to u and v, respectively. We expect these constants to be approximately equal to the bend persistence length ~ 5 0 0 A.4 The constant A33 is just the twist rigidity, and is roughly equal to the twist persistence length 2500 A.4

In addition to the bending rigidities and twist rigidity, we have a coupling A23 of bends about the local v axis and the twist in the quadratic (O(Q2)) elastic theory. Its overall magnitude should also be controlled by the degree by which rotations of the molecule by 180° about v and t are not symmetries. For DNA, these operations essentially ex- change the major and minor grooves of the molecule, which are rather different in structure: we thus expect A23 =Ai,.

~e1

~e2

~e3

p⇡

3.4n

m

For simplicity we ignore DNA sequence dependence

of elastic constants!

DNA is chiral and has right-handed helical

structure cross section

In the undeformed state DNA has spontaneous twist

!0 = 2⇡/p ⇡ 1.8 nm�1

Twist strain is measured relative to the

spontaneous twist

⌦3

d~eids

=⇣~⌦+ !0~e3

⌘⇥ ~ei

21

DNA982

Mar

ko a

nd

Sig

gia

Mac

rom

olec

ules

, V

ol.

27, N

o. 4, 1

994

V

Fig

ure

1.

Sch

emat

ic d

iagr

am o

f th

e B

-DN

A m

olec

ule.

T

he

mol

ecul

ar d

iam

eter

is d

= 2

0 A

, th

e he

lical

rep

eat l

engt

h is

1 =

27

r/w

0 =

34 A

, cor

resp

ondi

ng to

a s

tack

of a

bout 1

0.5

nuc

leic

aci

d bas

es.

The

nucl

eotide

s ar

e bo

und

betw

een

the

suga

r-ph

osph

ate

back

bone

hel

ices

: w

e no

te t

he

arro

ws

on t

he

side

vie

w (

upper

p

ort

ion

of

figure

), w

hich

indic

ate

the

oppo

site

dir

ecte

dnes

s of

the

two

helic

es.

The

wid

e m

ajor

gro

ove

is m

arke

d “M”,

whi

le th

e na

rrow

er m

inor

gro

ove

is m

arke

d “m

”. T

he

low

er p

art

of t

he

fig

ure

sho

ws t

he

end

view

of

B-D

NA

, wit

h ta

ngen

t t d

irec

ted o

ut

of t

he

pag

e.

11.

Ela

stic

Fre

e E

ne

rgy

of DNA

A.

Sy

mm

etr

y A

naly

sis.

B-D

NA

mo

lecu

les2

are

rig

ht-

h

and

ed c

hir

al r

od

s of

cro

ss-s

ecti

on

al d

iam

eter

d =

21

A.

As sc

hem

atic

ally

sho

wn

in F

igu

re 1

, th

e p

airs

of n

ucl

eoti

des

(o

ccu

py

ing

the m

ajo

r gro

ove

reg

ion

, den

ote

d M

in

Fig

ure

1) a

re a

rran

ged

in

a h

elix

wit

h a

pit

ch o

f ab

ou

t 1

= 3

4 A

co

rres

po

nd

ing

to

a h

elic

al r

epea

t ev

ery

10.5

bas

e p

airs

(b

p).

W

e d

efin

e th

e m

ole

cula

r ax

is (

the c

ente

r of

th

e

mo

lecu

le) to

be

des

crib

ed b

y t

he s

pac

e cu

rve

r(s),

wit

h s

b

ein

g a

rcle

ng

th.

Th

e t

ang

ent

t 1 d

r/d

s th

us

has

un

it

len

gth

.

At

any

s,

con

sid

er t

he

pla

ne

per

pen

dic

ula

r to

t.

Th

e

two s

ug

ar-p

ho

sph

ate

bac

kb

on

es (

the tw

o h

elic

es, d

raw

n

wit

h o

pp

osi

ng

arr

ow

s in

Fig

ure

1) i

nte

rsec

t th

is p

lan

e a

t tw

o p

oin

ts R

an

d S

. W

e d

efin

e u t

o b

e th

e u

nit

vec

tor

in

this

pla

ne

tha

t p

oin

ts f

rom

th

e m

ole

cula

r ax

is t

o t

he

mid

po

int

of E

. A

fin

al u

nit

vec

tor

v i

s d

efin

ed b

y v =

t

X u s

o t

ha

t th

e s

et (u, v,

t) f

orm

s a

rig

ht-

han

ded

co

ord

inat

e sy

stem

at

each

po

int

s.

It w

ill

be

hel

pfu

l to

te

mp

ora

rily

use

in

dex

ed v

ecto

rs e

(l) u, e

(2) =

v, a

nd

e(3

)

= t.

A g

ener

al d

efo

rmat

ion

of

the m

ole

cule

th

at

mai

nta

ins

t2 = 1

may

be

des

crib

ed b

y i

nfi

nit

esim

al r

ota

tio

ns

Q(s

) of

th

e c

oo

rdin

ate

axe^

:^^^

de“

’

ds

-- -

+

ill

x e

“’

wh

ere 00 =

2*/

1=

0.1

85 A

-l d

eter

min

es t

he h

elic

al r

epea

t le

ng

th i

n t

he a

bse

nce

of

def

orm

atio

ns.

W

e m

ay t

hin

k o

f th

e c

om

po

nen

ts O

i =

il.e

ci)

as “

stra

ins”

whic

h lo

call

y

gen

erat

e ro

tati

on

s of

th

e c

oo

rdin

ates

aro

un

d e

(’).

If Q =

0

, th

e m

ole

cule

tak

es i

ts u

nd

isto

rted

co

nfi

gu

rati

on

sh

ow

n

in F

igure

1.

Th

e m

ole

cula

r ax

is r

(s) i

s o

bta

ined

for g

ener

al

Q b

y i

nte

gra

tin

g t

he t

ang

ent

equ

atio

n d

r/d

s =

e(3

).

Th

e in

teg

ral

Tw

= L

/1 +

J d

s Q

3/(

2r

) is d

efin

ed t

o b

e th

e d

ou

ble

hel

ix “

twis

t”3

wh

ere

L i

s th

e m

ole

cule

len

gth

, an

d w

her

e th

e i

nte

gra

l is

fro

m s

= 0

to

s =

L.

Fo

r an

u

nd

isto

rted

mo

lecu

le, T

w =

L/1, an

d T

w ju

st c

ou

nts

th

e

nu

mb

er o

f h

elic

al t

urn

s of

len

gth

1 a

lon

g t

he c

hai

n.

Fo

r a

dis

tort

ed c

hai

n, t

he e

xces

s tw

ist

per

hel

ix r

epea

t is

(T

w

- L/l)/(L/l)

= (

Q3

)/

~0

,

wh

ere

we

use

th

e n

ota

tio

n (

Q3

) =

L-’J

; d

s Q

3(s) to

den

ote

an

av

erag

e al

on

g t

he c

hai

n o

f le

ng

th L

>>

1.

Sin

ce w

e as

sum

e th

at th

e Q

= 0

sta

te is

eq

uil

ibri

um

, we

may

wri

te t

he f

ree

ener

gy

fo

r sm

all

stra

ins

as a

Tay

lor

exp

ansi

on

in

a a

nd

its

s d

eriv

ativ

es.1

0 T

he lo

wes

t o

rder

term

s ar

e

(2)

wh

ere

we

hav

e in

tro

du

ced

th

e m

atri

ces

Aij

an

d A

ijk, w

hic

h

are

sym

met

ric

un

der

all

per

mu

tati

on

s of

th

eir

ind

ices

, an

d w

her

e th

e i

nte

gra

l ru

ns

over

th

e m

ole

cula

r ax

is o

f le

ng

th L

. If

we

ign

ore

nu

cleo

tid

e-se

qu

ence

dep

end

ence

of

th

e e

last

ic p

rop

erti

es o

f th

e m

ole

cule

(o

r if

we

rest

rict

o

ur

atte

nti

on

to

sy

mm

etri

c re

pea

ts s

uch

as ($

i)N

th

en

th

e

A m

atri

ces

hav

e n

o s

dep

end

ence

, si

nce

in

th

ese

coo

rdi-

n

ates

, ev

ery

po

int

alo

ng

th

e m

ole

cule

in

th

e u

nd

isto

rted

st

ate

is e

qu

ival

ent.

W

e w

ill r

efer

to

thes

e m

atri

ces

as th

e

“ela

stic

co

nst

ants

”: t

hey

may

dep

end

on

en

vir

on

men

tal

fact

ors

(te

mp

erat

ure

, io

nic

str

eng

th,

pH

, et

c.).

W

e w

ill

ign

ore

th

e c

on

stan

t fr

ee e

ner

gy

A0 fo

r th

e r

emai

nd

er o

f th

is p

aper

.

Th

e s

ec

on

d-o

rde

r m

atr

ix h

as

six

in

de

pe

nd

en

t co

mp

on

ents

: A

ll,

A22

, A

33,

A12,

A13

9 an

d A

23.

We

now

sh

ow

how

sy

mm

etri

es m

ake

som

e of

th

ese

com

po

nen

ts

van

ish

. N

ote

th

at

rota

tio

n b

y 1

80’

aro

un

d t

he v

ecto

r u

is a

sy

mm

etry

of

the u

nd

isto

rted

mo

lecu

le (

see

Fig

ure

1).

N

ow c

on

sid

er a

n in

fin

ites

imal

seg

men

t of

len

gth

ds

fro

m

s =

-d

s/2

to

ds/

2,

wit

h u

nif

orm

str

ain

Q =

(01, 02, O

S).

R

ota

tio

n o

f th

is s

egm

ent

by

180

’ ar

ou

nd

u(s

= 0

) y

ield

s p

reci

sely

th

e s

egm

ent

con

fig

ura

tio

n w

ith

un

ifo

rm s

trai

n

il’ =

(4

1,

Qz, O

3).

T

her

efo

re c

on

fig

ura

tio

ns D a

nd

D’ h

ave

the s

ame

free

en

erg

y,

ind

icat

ing

th

at

A12

=

A13 =

0.l

’

We

no

te t

ha

t ro

tati

on

s b

y 1

80’

aro

un

d e

ith

er t

(0)

or

v(0

) d

o n

ot

tak

e o

ur

infi

nit

esim

al d

isto

rted

seg

men

t to

a

con

fig

ura

tio

n w

ith

tran

sfo

rmed

D b

ecau

se th

ese

op

erat

ion

s are

no

t sy

mm

etri

es o

f th

e u

nd

isto

rted

mole

cule

. T

he la

ck

of s

ym

met

ry u

nd

er t

hes

e o

per

atio

ns

is d

ue to

the e

xis

ten

ce

of t

wo

dis

tin

ct reg

ion

s of

th

e D

NA

su

rfac

e b

ou

nd

ed b

y th

e

two

hel

ices

, mar

ked

“M

” an

d “

m”

in F

igu

re 1

. T

hes

e tw

o

regio

ns ar

e re

ferr

ed to

as t

he ”

maj

or g

roove”

an

d th

e “

min

or

gro

ov

e”, r

esp

ecti

vel

y.

Th

e m

ino

r gro

ove

is n

arro

w, w

hil

e th

e m

ajo

r gro

ove

is w

ider

, fi

lled

up

by t

he n

ucl

eoti

des

w

hic

h a

re b

ou

nd

bet

wee

n th

e b

ack

bo

nes

. T

he tw

o g

rooves

ar

e al

so d

isti

ng

uis

hed

by

th

e o

pp

osi

te d

irec

ted

nes

s of

th

e

sug

ar-p

ho

sph

ate

bac

kb

on

es o

n t

hei

r b

ou

nd

arie

s.

A r

o-

tati

on

of

the u

nd

isto

rted

mo

lecu

le a

rou

nd

eit

her

t o

r v b

y

180”

exch

ang

es t

he m

ajo

r an

d m

ino

r gro

oves

.

Ch

iral

po

lym

ers

wit

ho

ut

this

par

ticu

lar

bro

ken

sym

- m

etry

(e.

g. a

do

ub

le-h

elix

po

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aly

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men

ts o

f th

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= 2

ela

stic

co

nst

ant

mat

rix

are

All

, A22, A

33, an

d A

23.

Ph

ysi

call

y, A

11 an

d A

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are

(dis

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ct)

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idit

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cal v

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))

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itu

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ure

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exp

ect A

23 =

Ai,

.

~e3~e1

~e2

982 Marko and Siggia Macromolecules, Vol. 27, No. 4, 1994

V

Figure 1. Schematic diagram of the B-DNA molecule. The molecular diameter is d = 20 A, the helical repeat length is 1 = 27r/w0 = 34 A, corresponding to a stack of about 10.5 nucleic acid bases. The nucleotides are bound between the sugar-phosphate backbone helices: we note the arrows on the side view (upper portion of figure), which indicate the opposite directedness of the two helices. The wide major groove is marked “M”, while the narrower minor groove is marked “m”. The lower part of the figure shows the end view of B-DNA, with tangent t directed out of the page.

11. Elastic Free Energy of DNA

A. Symmetry Analysis. B-DNA molecules2 are right- handed chiral rods of cross-sectional diameter d = 21 A. As schematically shown in Figure 1, the pairs of nucleotides (occupying the major groove region, denoted M in Figure 1) are arranged in a helix with a pitch of about 1 = 34 A corresponding to a helical repeat every 10.5 base pairs (bp). We define the molecular axis (the center of the molecule) to be described by the space curve r(s), with s being arclength. The tangent t 1 dr/ds thus has unit length.

At any s, consider the plane perpendicular to t. The two sugar-phosphate backbones (the two helices, drawn with opposing arrows in Figure 1) intersect this plane at two points R and S. We define u to be the unit vector in this plane that points from the molecular axis to the

midpoint of E. A final unit vector v is defined by v = t X u so that the set (u, v, t) forms a right-handed coordinate system at each point s. It will be helpful to temporarily use indexed vectors e(l) u, e(2) = v, and e(3) = t.

A general deformation of the molecule that maintains t2 = 1 may be described by infinitesimal rotations Q(s) of the coordinate axe^:^^^

de“’

ds -- - + ill x e“’

where 00 = 2* /1= 0.185 A-l determines the helical repeat length in the absence of deformations. We may think of the components Oi = il.eci) as “strains” which locally generate rotations of the coordinates around e(’). If Q = 0, the molecule takes its undistorted configuration shown in Figure 1. The molecular axis r(s) is obtained for general Q by integrating the tangent equation dr/ds = e(3).

The integral T w = L/1 + J ds Q 3 / ( 2 r ) is defined to be the double helix “twist”3 where L is the molecule length, and where the integral is from s = 0 to s = L. For an undistorted molecule, T w = L/1, and Tw just counts the number of helical turns of length 1 along the chain. For a distorted chain, the excess twist per helix repeat is (Tw - L/l)/(L/l) = ( Q 3 ) / ~ 0 , where we use the notation ( Q 3 ) = L-’J; ds Q3(s) to denote an average along the chain of length L >> 1.

Since we assume that the Q = 0 state is equilibrium, we may write the free energy for small strains as a Taylor expansion in a and its s derivatives.10 The lowest order

terms are

(2)

where we have introduced the matrices Aij and Aijk, which are symmetric under all permutations of their indices, and where the integral runs over the molecular axis of length L. If we ignore nucleotide-sequence dependence of the elastic properties of the molecule (or if we restrict our attention to symmetric repeats such as ($i)N then the A matrices have no s dependence, since in these coordinates, every point along the molecule in the undistorted state is equivalent. We will refer to these matrices as the “elastic constants”: they may depend on environmental factors (temperature, ionic strength, pH, etc.). We will ignore the constant free energy A0 for the remainder of this paper.

The second-order matrix has six independent components: All , A22, A33, A12, A139 and A23. We now show how symmetries make some of these components vanish. Note that rotation by 180’ around the vector u is a symmetry of the undistorted molecule (see Figure 1). Now consider an infinitesimal segment of length ds from s = -ds/2 to ds/2, with uniform strain Q = (01, 02, OS). Rotation of this segment by 180’ around u(s = 0) yields precisely the segment configuration with uniform strain il’ = ( 4 1 , Qz, O3). Therefore configurations D and D’ have the same free energy, indicating that A12 = A13 = 0.l’

We note that rotations by 180’ around either t(0) or v(0) do not take our infinitesimal distorted segment to a configuration with transformed D because these operations are not symmetries of the undistorted molecule. The lack of symmetry under these operations is due to the existence of two distinct regions of the DNA surface bounded by the two helices, marked “M” and “m” in Figure 1. These two regions are referred to as the ”major groove” and the “minor groove”, respectively. The minor groove is narrow, while the major groove is wider, filled up by the nucleotides which are bound between the backbones. The two grooves are also distinguished by the opposite directedness of the sugar-phosphate backbones on their boundaries. A rotation of the undistorted molecule around either t or v by 180” exchanges the major and minor grooves.

Chiral polymers without this particular broken symmetry (e.g. a double-helix polymer with indistinguishable backbones, and therefore invariant under rotation by 180’ about t) will have A23 = 0. Finally, one should note that reflections (useful in derivation of the elastic energy of thin rods with reflection symmetrieslO cannot be used to analyze a chiral rod such as DNA.

Thus, the nonzero elements of then = 2 elastic constant matrix are All , A22, A33, and A23. Physically, A11 and A22 are (distinct) bending constants associated with bends locally in the planes perpendicular to u and v, respectively. We expect these constants to be approximately equal to the bend persistence length ~ 5 0 0 A.4 The constant A33 is just the twist rigidity, and is roughly equal to the twist persistence length 2500 A.4

In addition to the bending rigidities and twist rigidity, we have a coupling A23 of bends about the local v axis and the twist in the quadratic (O(Q2)) elastic theory. Its overall magnitude should also be controlled by the degree by which rotations of the molecule by 180° about v and t are not symmetries. For DNA, these operations essentially ex- change the major and minor grooves of the molecule, which are rather different in structure: we thus expect A23 =Ai,.

~e1

~e2

~e3

cross section

A12 = A13 = D1 = 0

Twist strain is measured relative to the

spontaneous twist

⌦3

d~eids

=⇣~⌦+ !0~e3

⌘⇥ ~ei

DNA has 2-fold rotational symmetry around axis . ~e1

E =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23 + 2A23⌦2⌦3

+k✏2 + 2D2✏⌦2 + 2D3✏⌦3

�

Elastic energy for deforming DNA

k ⇠ 1000pNC/kBT ⇠ 100nm

A11/kBT ⇡ A22/kBT = `p ⇠ 50nm

A23, D2 ⇠ 0D3/kBT ⇠ �20

22

Magnetic tweezers

Force on magnetic bead is proportional to the gradient of magnetic field and can be

adjusted by raising or lowering the magnet

Torque on magnetic bead can be produced by rotating the magnet.

23

Twist-stretch coupling

attachment points at both ends suitable for MTs torque mea-surements by annealing two complementary single strands thatcarry multiple biotin or digoxigenin labels at their respective 5′ends (Fig. 1 C and D and Materials and Methods). The function-alized single-stranded constructs were generated by carrying outinitial in vitro transcription reactions that incorporated labelednucleotides and stalled at a missing fourth nucleotide (Fig. 1 C andD). After purification, transcription reactions were restarted andcompleted in the presence of all four unlabeled nucleotides. Thefinal annealed 4.2-kbp dsRNA constructs can be tethered betweenan anti-digoxigenin–coated flow cell surface and streptavidin-coated magnetic beads for manipulation in the MTs (Fig. 1E).

Force–Extension Response of dsRNA. Using the ability of MTs toexert precisely calibrated stretching forces (18, 19) (Materials andMethods and SI Appendix, Fig. S1), we first probed the force–extension response of dsRNA. The stretching behavior oftorsionally relaxed dsRNA at low forces (F < 5 pN) is well-described by the (inextensible) worm-like chain (WLC) model(20, 21) (SI Appendix, Fig. S2A). From fits of the WLC model, wedetermined the contour length LC = 1.15 ± 0.02 μm and thebending persistence length ARNA = 57 ± 2 nm in the presence of100 mM monovalent salt (SI Appendix, Fig. S2A), in good

agreement with the expected length (1.16 μm, assuming 0.28 nmper bp) (22, 23) and previous single-molecule stretching experi-ments (15, 16). ARNA decreases with increasing ionic strength(16) (SI Appendix, Fig. S1), in a manner well-described by modelsthat partition it into an electrostatic and a salt-independentcomponent (SI Appendix, Fig. S1K). Taking into account the saltdependence, ARNA is consistently ∼20% larger than ADNA at thesame ionic strength (SI Appendix, Fig. S1).Stretching dsRNA at forces >10 pN, we observed elastic

stretching that can be fit by the extensible WLCmodel (21, 24) upto∼40 pN (SI Appendix, Fig. S2B) and an overstretching transitionfor torsionally unconstrainedmolecules (SI Appendix, Fig. S2C), inagreement with previous single-molecule studies (16, 17). Fromfits of the extensibleWLCmodel, we found SRNA = 350 ± 100 pN,about threefold lower than SDNA (SI Appendix, Fig. S1G and TableS1). Our value for the SRNA is in reasonable agreement with, al-though slightly lower than, the value of SRNA∼500 pN determinedin single-molecule optical tweezers measurements (25), possiblydue to subtle differences between magnetic and optical tweezersexperiments. For torsionally unconstrained molecules, the over-stretching transition is marked by a rapid increase in extension to1.8 ± 0.1 times the crystallographic length over a narrow forcerange at F = 54 ± 5 pN (SI Appendix, Fig. S2C). In contrast, usingour torsionally constrained dsRNA, we observed enthalpicstretching beyond the contour length but no sharp overstretchingtransition up to F = 75 pN (SI Appendix, Fig. S2D). The increasedresistance to overstretching for torsionally constrained dsRNAcompared with torsionally unconstrained dsRNA is qualitativelysimilar to the behavior of dsDNA (26–28) (SI Appendix, Fig. S1Hand I). The dependence of the overstretching transition fordsRNA on torsional constraint and on salt (SI Appendix, Fig. S2Cand D) suggests that it might involve melting as well as a tran-sition to a previously unidentified conformation that we name“S-RNA,” in analogy to S-DNA (SI Appendix, Fig. S1).

Twist Response of dsRNA. We used the ability of MTs to controlthe rotation of the magnetic beads (18) to map out the responseof dsRNA upon over- and underwinding at constant stretchingforces. Starting with a torsionally relaxed molecule (correspondingto zero turns in Fig. 2), the tether extension remains initially ap-proximately constant upon overwinding (corresponding to in-creasing linking number) until the molecule reaches a bucklingpoint (Fig. 2A, dashed lines and SI Appendix, Fig. S3). Furtheroverwinding beyond the buckling point leads to a rapid linear de-crease of the tether extension with an increasing number of turns,due to the formation of plectonemes. The critical supercoilingdensity σbuck for buckling increases with stretching force and agreeswithin experimental error with the values found for DNA and witha mechanical model originally developed for supercoiled DNA (9)(Fig. 2B and SI Appendix,Materials and Methods). The decrease inextension per added turn in the plectonemic regime providesa measure for the size of the plectonemes and decreases with in-creasing stretching force (Fig. 2C). The extension vs. turns slopesfor dsRNA are within experimental error of those for dsDNA, andare in approximate agreement with the mechanical model forsupercoiling (Fig. 2C). Underwinding the dsRNA tether atstretching forces F < 1 pN gives rise to a buckling response similarto what is observed upon overwinding and the formation of neg-atively supercoiled plectonemes. In contrast, for F > 1 pN, theover- and underwinding response is asymmetric and the tetherextension remains approximately constant upon underwinding(Fig. 2A), likely due to melting of the double helix, as has beenobserved for DNA (29) (SI Appendix, Fig. S3 K and L).If unwinding at F > 1 pN is continued for several hundred

turns, we eventually observe another structural transition markedby an abrupt change in the extension vs. turns response at asupercoiling density of σ ∼ –1.9 (Fig. 2D). We term this previouslyunidentified highly underwound and left-handed RNA confor-mation with a helicity of –12.6 bp per turn “L-RNA,” in analogy towhat has been observed for highly underwound DNA (11) (SIAppendix, Fig. S3L). We note that the helicity and elongation that

Fig. 1. Construction of a torsionally constrained double-stranded RNA formagnetic tweezers measurements. (A) Comparison of A-form dsRNA [ProteinData Bank (PDB) ID code 1RNA (57)] and B-form dsDNA [PDB ID code 2BNA(58)]. (B) Cartoon of the elastic deformations of dsRNA: bending, stretching, andtwisting. (C) Schematic of the protocol to generate double-stranded RNAmolecules with multiple attachment points at both ends. Initial transcriptionreactions incorporate multiple biotinylated adenosine (green circles) or digoxi-genated uracil (yellow squares) bases and stall at a fourth nucleotide. Afterpurification, transcription reactions are restarted and complete the 4.2-kbptranscripts. In the final step, thepurifiedRNA strands are annealed to yield dsRNAwith chemical modifications at each end. (D) Schematic of the two DNA tem-plates used to generate dsRNA with multiple labels at both ends. (E) Cartoon ofa magnetic tweezers experiment on dsRNA (not to scale). A streptavidin-coatedmagnetic bead is tethered to an anti-digoxigenin–coated surface by a dsRNAmolecule with multiple attachment points at both ends. A surface-attached ref-erence bead is tracked simultaneously for drift correction. Permanent magnetsabove the flow cell are used to exert a stretching force F and to control the ro-tation of themagnetic bead via its preferred axism0. N, north pole; S, south pole.

Lipfert et al. PNAS | October 28, 2014 | vol. 111 | no. 43 | 15409

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J. Lipfert et al., PNAS 111, 15408 (2014)

We determined the effective twist persistence length CRNAfrom the slopes in the linear torque–response regime, wherethe torque after N turns is 2π·N·kBT·CRNA/LC (where kB isBoltzmann’s constant and T is the absolute temperature; Fig. 3C,solid colored lines). CRNA increases with increasing force andis 99 ± 5 nm at F = 6.5 pN. Compared with dsDNA, CRNA issimilar to but slightly lower than CDNA, and both quantitiesexhibit similar force dependence, in qualitative agreementwith a model valid in the high force limit (37) (Fig. 3F and SIAppendix, Materials and Methods). Combining the results fromstretching and torque measurements at different forces, we de-lineate the phase diagram for dsRNA as a function of appliedforce and torque (Fig. 3G).

Twist–Stretch Coupling. The linear elastic rod model has a fourthparameter, D, that describes the coupling between twist andstretch. We measured the twist–stretch coupling for dsRNA bymonitoring changes in the extension upon over- and under-winding while holding the molecule at constant stretching forcesthat are large enough to suppress bending and writhe fluctua-tions (38, 39) (Fig. 4A). We found that for small deformations (inthe range –0.02 < σ < 0.025, which excludes the melting, buck-ling, and A-to-P–form transitions) dsRNA shortens upon over-winding, with a slope of (d∆L/dN)RNA = –0.85 ± 0.04 nm perturn, independent of stretching force in the range F = 4–8 pN(Fig. 4 B and C). This is in stark contrast to dsDNA, whichwe observed to lengthen upon overwinding by (d∆L/dN)DNA =+0.44 ± 0.1 nm per turn (Fig. 4 B and C), in good agreementwith previous measurements (38–41). Our measurements suggestthat dsRNA has a positive twist–stretch coupling equal to DRNA =–SRNA·(d∆L/dN)RNA/(2π·kBT) = +11.5 ± 3.3 (assuming SRNA = 350pN; SI Appendix,Materials and Methods), in contrast to the negativetwist–stretch coupling of dsDNA (38–41), DDNA = –17 ± 5.

Dynamics at the Buckling Transition. Next, we investigated the dy-namics at the buckling transition. When a dsRNA was twistedclose to the critical supercoiling density, we observed jumps in theextension traces, corresponding to transitions between the pre-and postbuckling states (Fig. 5A). Recording extension traces ata fixed number of applied turns, the population of the post-buckling state increases whereas the population of the prebuck-ling state decreases with an increasing number of applied turns(Fig. 5A). After selecting a threshold to separate the pre- andpostbuckling states (SI Appendix, Fig. S5 A–D), the pre- andpostbuckling populations and dwell time distributions can bequantified. The dependence of the postbuckling population onthe number of applied turns is well-described by a two-statemodel (42) (Fig. 5B and SI Appendix,Materials and Methods) fromwhich we determined the number of turns converted from twist towrithe during the buckling transition ΔNb ∼4 turns (SI Appendix,Fig. S5L). The dwell times in the pre- and postbuckling state areexponentially distributed (SI Appendix, Fig. S5 E–G), and theirmean residence times depend exponentially on the number ofapplied turns (Fig. 5C). We determined the overall characteristicbuckling times τbuck, that is, the dwell times at the point where thepre- and postbuckling states are equally populated, from fits ofthe exponential dependence of the mean residence times on thenumber of applied turns (Fig. 5C and SI Appendix, Materials andMethods). τbuck increases with increasing salt concentration andstretching force (Fig. 5E). The force dependence of τbuck is well-described by an exponential model (solid lines in Fig. 5E), τbuck =τbuck,0·exp(d·F/kBT); from the fit we obtain the buckling time atzero force τbuck,0 = 13 and 52 ms and the distance to the transitionstate along the reaction coordinate d = 5.1 and 5.5 nm for the 100and 320 mM monovalent salt data, respectively.Interestingly, comparing τbuck for dsRNA with dsDNA of

similar length under otherwise identical conditions (Fig. 5 D andE), we found that the buckling dynamics of dsRNA are muchslower than those of dsDNA, with the characteristic bucklingtimes differing by at least two orders of magnitude. For example,

we found τbuck = 10.1 ± 3.7 s for dsRNA compared with ∼0.05 sfor dsDNA at F = 4 pN and 320 mM salt (Fig. 5E).

DiscussionOur experiments are consistent with dsRNA behaving as a linearelastic rod for small deformations from the A-form helix, andallow us to empirically determine all four elastic constants ofthe model: A, S, C, and D (SI Appendix, Table S1). To go beyondthe isotropic rod model, toward a microscopic interpretationof the results, we describe a “knowledge-based” base pair-levelmodel that considers the six base-step parameters slide, shift,rise, twist, roll, and tilt (SI Appendix, Fig. S6 and Materials andMethods; a full description of modeling for a blind predictionchallenge is given in ref. 43). Average values and elastic cou-plings of the base-step parameters for dsRNA and dsDNA froma database of nucleic acid crystal structures are used in a MonteCarlo protocol to simulate stretching and twisting experiments (SIAppendix, Materials and Methods). This base pair-level modelcorrectly predicts the bending persistence length for dsRNA to beslightly larger than for dsDNA, SRNA to be at least a factor of twosmaller than SDNA, and C to be of similar magnitude for dsRNAand dsDNA (SI Appendix, Table S2). The significant difference instretch modulus S between dsRNA and dsDNA can be explainedfrom the “spring-like” path of the RNA base pairs’ center axis,compared with dsDNA (SI Appendix, Fig. S6B). Beyond theagreement with experiment in terms of ratios of dsRNA anddsDNA properties, the absolute values ofA, S, andC all fall withina factor of two of our experimental results for both molecules.Whereas the values for A, S, and C are fairly similar for

dsRNA and dsDNA, our experiments revealed an unexpecteddifference in the sign of the twist–stretch coupling D for dsRNAand dsDNA. The twist–stretch coupling has important biological

Fig. 4. Double-stranded RNA has a positive twist–stretch coupling. (A) Timetraces of the extension of a dsRNA tether held at F = 7 pN and underwound by−6 or overwound by 12 turns. Raw traces (120 Hz) are in red and filtered data(10 Hz) are in gray. The data demonstrate that dsRNA shortens when over-wound. (B) Changes in tether extension upon over- and underwinding at F = 7pN of a 4.2-kbp dsRNA and a 3.4-kbp dsDNA tether. Linear fits to the data(lines) indicate that the dsDNA lengthens by ∼0.5 nm per turn, whereas thedsRNA shortens by ∼0.8 nm per turn upon overwinding. Symbols denote themean and standard deviation for four measurements on the same molecule.(C) Slopes upon overwinding of dsRNA and dsDNA tethers as a function of F(mean and SEM of at least four molecules in TE + 100 mM NaCl buffer). Dataof Lionnet et al. (38) are shown as a black line with the uncertainty indicatedin gray; data from Gore et al. (39) are shown as a black square. The red line isthe average over all dsRNA data. (D) Models of oppositely twisting 50-bpsegments of dsDNA (Left) and dsRNA (Right) under 0 and 40 pN stretchingforces, derived from base pair-level models consistent with experimentalmeasurements (SI Appendix, Table S6 and Materials and Methods). The or-ange bars represent the long axis of the terminal base pair.


BIOPH

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�5 0 5 10

�15

�10

�5

0

5

turns

�L

[nm]

RNA DNA

Twist-stretch coupling has opposite sign for double stranded RNA and DNA!

F = 7pN bead is turned with a magnet

L+�L

24

Response of DNA to external forces and torques

~r(L) ~F

~M

E =

Z L

0

ds

2

A11⌦

21 +A22⌦

22 + C⌦

23 + k✏

2 + 2D3✏⌦3

��~F · ~r(L)

DNA torque

force

work due to external force and torqueDNA end to end distance

~e1~e2

~e3

~r(L) =

Z L

0ds

d~r

ds=

Z L

0ds ~e3(1 + ✏)

rotation of DNA end

~�(L) =

Z L

0ds

d~�

ds=

Z L

0ds ~⌦

� ~M · ~�(L)

25

Response of DNA to external forces and torques

~r(L) ~F

~M

DNA torque

force

~e1~e2

~e3

E =

Z L

0ds

✓1

2

A11⌦

21 +A22⌦

22 + C⌦

23 + k✏

2 + 2D3✏⌦3

�� ~F · ~e3(1 + ✏)� ~M · ~⌦

◆

The configuration of DNA that minimizes energy is described by Euler-Lagrange equations

E =

Z L

0ds g(✏,⌦1,⌦2,⌦3)

0 =d

ds

✓@g

@(d✏/ds)

◆� @g

@✏0 =

d

ds

✓@g

@(d�i/ds)

◆� @g

@�i

⌦i = d�/ds

26

Tension and torque along DNA backbone

~r(L) ~F

~M

DNA torque

force

~e1~e2

~e3

E =

Z L

0ds

✓1

2

A11⌦

21 +A22⌦

22 + C⌦

23 + k✏

2 + 2D3✏⌦3

�� ~F · ~e3(1 + ✏)� ~M · ~⌦

◆

0 =d

ds

✓@g

@(d✏/ds)

◆� @g

@✏

Euler-Lagrange equations

k✏(s) +D3⌦3(s) = ~F · ~e3(s)

tension along DNA

0 =d

ds

✓@g

@(d�i/ds)

◆� @g

@�i

⌦i = d�/ds

A11⌦1(s)~e1(s) +A22⌦2(s)~e2(s)

+ [C⌦3(s) +D3✏(s)]~e3(s) = ~M

torque along DNA

Euler-Lagrange equations thus describe local force and torque balance!

27

Twist-stretch coupling

J. Lipfert et al., PNAS 111, 15408 (2014)

We determined the effective twist persistence length CRNAfrom the slopes in the linear torque–response regime, wherethe torque after N turns is 2π·N·kBT·CRNA/LC (where kB isBoltzmann’s constant and T is the absolute temperature; Fig. 3C,solid colored lines). CRNA increases with increasing force andis 99 ± 5 nm at F = 6.5 pN. Compared with dsDNA, CRNA issimilar to but slightly lower than CDNA, and both quantitiesexhibit similar force dependence, in qualitative agreementwith a model valid in the high force limit (37) (Fig. 3F and SIAppendix, Materials and Methods). Combining the results fromstretching and torque measurements at different forces, we de-lineate the phase diagram for dsRNA as a function of appliedforce and torque (Fig. 3G).

Twist–Stretch Coupling. The linear elastic rod model has a fourthparameter, D, that describes the coupling between twist andstretch. We measured the twist–stretch coupling for dsRNA bymonitoring changes in the extension upon over- and under-winding while holding the molecule at constant stretching forcesthat are large enough to suppress bending and writhe fluctua-tions (38, 39) (Fig. 4A). We found that for small deformations (inthe range –0.02 < σ < 0.025, which excludes the melting, buck-ling, and A-to-P–form transitions) dsRNA shortens upon over-winding, with a slope of (d∆L/dN)RNA = –0.85 ± 0.04 nm perturn, independent of stretching force in the range F = 4–8 pN(Fig. 4 B and C). This is in stark contrast to dsDNA, whichwe observed to lengthen upon overwinding by (d∆L/dN)DNA =+0.44 ± 0.1 nm per turn (Fig. 4 B and C), in good agreementwith previous measurements (38–41). Our measurements suggestthat dsRNA has a positive twist–stretch coupling equal to DRNA =–SRNA·(d∆L/dN)RNA/(2π·kBT) = +11.5 ± 3.3 (assuming SRNA = 350pN; SI Appendix,Materials and Methods), in contrast to the negativetwist–stretch coupling of dsDNA (38–41), DDNA = –17 ± 5.

Dynamics at the Buckling Transition. Next, we investigated the dy-namics at the buckling transition. When a dsRNA was twistedclose to the critical supercoiling density, we observed jumps in theextension traces, corresponding to transitions between the pre-and postbuckling states (Fig. 5A). Recording extension traces ata fixed number of applied turns, the population of the post-buckling state increases whereas the population of the prebuck-ling state decreases with an increasing number of applied turns(Fig. 5A). After selecting a threshold to separate the pre- andpostbuckling states (SI Appendix, Fig. S5 A–D), the pre- andpostbuckling populations and dwell time distributions can bequantified. The dependence of the postbuckling population onthe number of applied turns is well-described by a two-statemodel (42) (Fig. 5B and SI Appendix,Materials and Methods) fromwhich we determined the number of turns converted from twist towrithe during the buckling transition ΔNb ∼4 turns (SI Appendix,Fig. S5L). The dwell times in the pre- and postbuckling state areexponentially distributed (SI Appendix, Fig. S5 E–G), and theirmean residence times depend exponentially on the number ofapplied turns (Fig. 5C). We determined the overall characteristicbuckling times τbuck, that is, the dwell times at the point where thepre- and postbuckling states are equally populated, from fits ofthe exponential dependence of the mean residence times on thenumber of applied turns (Fig. 5C and SI Appendix, Materials andMethods). τbuck increases with increasing salt concentration andstretching force (Fig. 5E). The force dependence of τbuck is well-described by an exponential model (solid lines in Fig. 5E), τbuck =τbuck,0·exp(d·F/kBT); from the fit we obtain the buckling time atzero force τbuck,0 = 13 and 52 ms and the distance to the transitionstate along the reaction coordinate d = 5.1 and 5.5 nm for the 100and 320 mM monovalent salt data, respectively.Interestingly, comparing τbuck for dsRNA with dsDNA of

similar length under otherwise identical conditions (Fig. 5 D andE), we found that the buckling dynamics of dsRNA are muchslower than those of dsDNA, with the characteristic bucklingtimes differing by at least two orders of magnitude. For example,

we found τbuck = 10.1 ± 3.7 s for dsRNA compared with ∼0.05 sfor dsDNA at F = 4 pN and 320 mM salt (Fig. 5E).

DiscussionOur experiments are consistent with dsRNA behaving as a linearelastic rod for small deformations from the A-form helix, andallow us to empirically determine all four elastic constants ofthe model: A, S, C, and D (SI Appendix, Table S1). To go beyondthe isotropic rod model, toward a microscopic interpretationof the results, we describe a “knowledge-based” base pair-levelmodel that considers the six base-step parameters slide, shift,rise, twist, roll, and tilt (SI Appendix, Fig. S6 and Materials andMethods; a full description of modeling for a blind predictionchallenge is given in ref. 43). Average values and elastic cou-plings of the base-step parameters for dsRNA and dsDNA froma database of nucleic acid crystal structures are used in a MonteCarlo protocol to simulate stretching and twisting experiments (SIAppendix, Materials and Methods). This base pair-level modelcorrectly predicts the bending persistence length for dsRNA to beslightly larger than for dsDNA, SRNA to be at least a factor of twosmaller than SDNA, and C to be of similar magnitude for dsRNAand dsDNA (SI Appendix, Table S2). The significant difference instretch modulus S between dsRNA and dsDNA can be explainedfrom the “spring-like” path of the RNA base pairs’ center axis,compared with dsDNA (SI Appendix, Fig. S6B). Beyond theagreement with experiment in terms of ratios of dsRNA anddsDNA properties, the absolute values ofA, S, andC all fall withina factor of two of our experimental results for both molecules.Whereas the values for A, S, and C are fairly similar for

dsRNA and dsDNA, our experiments revealed an unexpecteddifference in the sign of the twist–stretch coupling D for dsRNAand dsDNA. The twist–stretch coupling has important biological

Fig. 4. Double-stranded RNA has a positive twist–stretch coupling. (A) Timetraces of the extension of a dsRNA tether held at F = 7 pN and underwound by−6 or overwound by 12 turns. Raw traces (120 Hz) are in red and filtered data(10 Hz) are in gray. The data demonstrate that dsRNA shortens when over-wound. (B) Changes in tether extension upon over- and underwinding at F = 7pN of a 4.2-kbp dsRNA and a 3.4-kbp dsDNA tether. Linear fits to the data(lines) indicate that the dsDNA lengthens by ∼0.5 nm per turn, whereas thedsRNA shortens by ∼0.8 nm per turn upon overwinding. Symbols denote themean and standard deviation for four measurements on the same molecule.(C) Slopes upon overwinding of dsRNA and dsDNA tethers as a function of F(mean and SEM of at least four molecules in TE + 100 mM NaCl buffer). Dataof Lionnet et al. (38) are shown as a black line with the uncertainty indicatedin gray; data from Gore et al. (39) are shown as a black square. The red line isthe average over all dsRNA data. (D) Models of oppositely twisting 50-bpsegments of dsDNA (Left) and dsRNA (Right) under 0 and 40 pN stretchingforces, derived from base pair-level models consistent with experimentalmeasurements (SI Appendix, Table S6 and Materials and Methods). The or-ange bars represent the long axis of the terminal base pair.


BIOPH

YSICSAND

COMPU

TATIONALBIOLO

GY

�5 0 5 10

�15

�10

�5

0

5

turns

�L

[nm]

Twist-stretch coupling has opposite sign for double stranded RNA and DNA!

F = 7pN bead is turned with a magnet

L+�L

k✏(s) +D3⌦3(s) = ~F · ~e3(s)

⌦3 = 2⇡Nturns/L

�L = L✏ =FL

k� 2⇡D3

kNturns

28

force

F < Fcr

force

F > Fcrẑx̂

ŷ

F F

z(L) z(L)

Assume the very thin beam (filament) limit with ✏ ! 0

Analyze the stability of flat configuration by investigating the energy cost of slightly deformed profile with

~r(s) = (0, y(s), z(s))

~e3 =d~r

ds= (0, y0, z0) z0 =

p1� y02 z(s) =

Z s

0d`p

1� y0(`)2

Bending strain

⌦2 =1

R2=

✓d2~r

ds2

◆2= y00

2+ z00

2=

y002

1� y02

Euler buckling instability

29

force

F < Fcr

force

F > Fcrẑx̂

ŷ

F F

z(L) z(L)

Assume small deformations around the flat configuration

y(s) =X

q

eiqsỹ(q) analyze with Fourier modesBuckled configurations have lover energy for

F > Fcr = Aq2minE ⇡ +FL+

X

q

1

2

�Aq4 � Fq2

�|ỹ(s)|2

E ⇡Z L

0ds

1

2Ay00

2 � 12Fy02 + F

�

E =

Z L

0ds

A

2⌦2 + Fz(L) =

Z L

0ds

A

2

y002

(1� y02) + Fp

1� y02�

Note: buckling direction corresponds to the lower value of bending rigidity A

Euler buckling instability

30

Euler buckling instabilityclamped

boundaries

Fcr =4⇡2A

L2

y(0) = y0(0) = y(L) = y0(L) = 0

qmin = 2⇡/L

hinged boundaries

qmin = ⇡/L

y(0) = y00(0) = y(L) = y00(L) = 0

Fcr =⇡2A

L2

one end clamped the other free

Fcr =⇡2A

4L2

qmin = ⇡/2L

y(0) = y0(0) = y00(L) = y000(L) = 0

The amplitudes of buckled modes are determined by the 4th order terms in

energy functional that we ignored

E ⇡Z L

0ds

A

2y002(1 + y02) + F

✓1� 1

2y02 +

1

8y04

◆�

ỹ(qmin) /q(F �Aq2min)/(Aq4min)

31

Torsional instabilityM < Mcr M > Mcr

plectoneme

torque

torque

critical torque for clamped boundaries

number of turnsthat lead to torsional instability

Mcr ⇡ 9A

LNturns =

L⌦32⇡

=LMcr2⇡C

⇡ 1.4AC

32

Twist, Writhe and Linking numbersLn=Tw+Wr linking number: total number of turns of a particular endTw =

R L0 ds⌦3(s)/2⇡ twist: number of turns due to twisting the beam

Wr writhe: number of crossings when curve is projected on a plane

33

Torsional instability

M < Mcr

M > Mcr

plectoneme

torque

torque

F

F

E =CL[2⇡Ln]2

2E =

CL[2⇡ (Ln�Wr)]2

2+ Ebend + F�L

Torsional instability occurs, because part of the twisting energy can be released by bending and forming

Documents

Elastic deformation energy for beams and thin filamentsakosmrlj/MAE545_F2015/lecture... · 2014. 10. 28. · 1 = 34 A corresponding to a helical repeat every 10.5 base pairs (bp)