Modeling of Biofilaments: Elasticity and Fluctuations Combined

D. Kessler, Y. Kats, S. Rappaport (Bar-Ilan)

S. Panyukov (Lebedev)

Mathematics of Materials and MacromoleculesIMA, Minneapolis, October 3, 2004

Stretching of helical springs

Overview

1. Motivation

2. Ribbons: geometry, elasticity, fluctuations

3. Computer simulations: Frenet algorithm

Stretching of filaments

Twisting dsDNA

CyclizationDistribution functions

Polymers – objects with atomic thickness (1 A) and arbitrary length

Atomic resolution

Quantum mechanics

RIS models

Coarse grained description

Statistical mechanics

Random walks

What sort of objects are described by this model?

N

n

nnn l

P1

2

21exp)(

RRR

This is the probability distribution of a random walk!

Beads connected by entropic springs

The standard model of polymers:

nn-1

nRspring constant= kT/l2

Random walks are not lines!

s

R(s)

0

L

1|/| dsdR

Continuous curve:

1d f

Inextensible line

Random walk:

2d f

arbitrarydsd |/| R

Extensible fractal

What about nano-filaments: thickness 10-100 A?

1 Intrinsic shape 2 Resistance to change of shape (bending, twist)

Biofilaments: DNA, actin and tubulin fibers, flagella, viruses …

Synthetic filaments: organic microtubules, carbon nanotubes, …

But: thermal fluctuations are still important!

Theory of elasticity of fluctuating filaments with arbitrary intrinsic shape

New elements:

Bending elasticity of inextensible lines

Modeling dsDNA at large deformations

Bustamante et al., Science 265, 1599 (1994)

The first step:

dsDNA under stretching and torque

1. Cannot twist lines 2. Lines have no chirality

degree of over/unwinding

Strick et al., Science 271, 1835 (1996)

Geometry of space curves:

s

s’t

tn

n

b

b

,nb

ds

d,bt

n ds

dn

t ds

d

Frenet eqs: generate curve by rotation of the triad bnt ,,

- curvature, - torsion This is not a physical twist !

Helix

p

2r22

22

2

r

p

Straight line

0

0

/1

r

Circle

r

Ribbons (stripes)

t2(s)

t1(s)

Physical triad: t1, t2, t3

n(s)

b(s))(s

3

2

1

12

13

23

3

2

1

0

0

0

t

t

t

t

t

t

ds

d

Generalized Frenet eqs. – rotation of physical axes

Ribbon - principal axes ; tangent 21, tt 3t

cos1 sin2 ds

d 3

Configuration of the ribbon – uniquely defined by )(sior by )(),(),( sss

rate of twist

Mechanics: Linear Elasticity

Deviations from stress-free state: kkk 0

Elastic Energy k

L

kkel dsbU0

2

2

1

kb - rigidity with respect to bending and twist

Small local but arbitrarily large global deviations from equilibrium configuration!

k0Equilibrium shape defined by spontaneous curvatures

Stretching a helical spring

pitch > radius, bending rigidity > twist rigidity4 turns,

minimize ))(,,()(2

100

2

0sFRsdsbE ii

L

ii

pitch < radius, bending rigidity < twist rigidity

Phys. Rev. Lett. 90, 024301 (2003)

The energy landscape E(R) has multiple minima with depths and locations that vary with F

Stretching helical ribbons of cholesterol:

Smith, Zastavker and Benedek, Phys. Rev. Lett. 87, 278101 (2001)

Mechanical noise-induced transitions?

Stretching transitions and hysteresis in chromatin ?

Y. Cui and C. Bustamente, PNAS 97, 127 (2000).

Correlation functions )'()( ss ji tt for ribbons with arbitrary spontaneous

shape and rigidity!

0)( si )'()'()( 1 ssass ijiji

i - random Gaussian variables

L

iiiel ads

kTU

0

2

2Fluctuation energy:

Thermal Fluctuations

ia - persistence lengths

Phys. Rev. Lett. 85, 2404 (2000) Phys. Rev. E 62, 7135 (2000)

01

31

21

1 aaa

Weak fluctuations of a helix:

skijk

sjiij

sjiji

RR esesetst

)sin()cos()0()( 0

0

002

0

00

20

00 1

e1

t3(s )t2(s )

t1(s )

s

e3 ( )

e2 ( )

Persistence lengths > helical period

,001 ,002 003 20

200 frequency

Ribbon with spontaneous twist – model for dsDNA?

20 10 Lk

,10Lk0 10,100,1 321 aaa

)(1000);(350);(50/ cbakTFlf Europhys. Lett. 57, 512 (2002)

Buckling under torsion: stability diagram

Frenet-Based Computer Simulations

)'()'()( ssb

kTss ij

iji

1. Generate random numbers i

2. Integrate Frenet eqs. to generate configurations

3. Excluded volume, attractive interactions – Boltzmann weights

Direct simulation of fluctuating lines!

Phys. Rev. E 65 020801 (2002)

Rectilinear rod 12321 bbb

L=2

Does twist affect conformation?

is independent of twist !2R

Exact result: if there is no spontaneous curvature -

WLC model ok ?

321 105.7,75.0 aa

Rectilinear ribbon

Twist affects conformation!

J. Chem. Phys. 118, 897 (2003)

L=2

What about objects with spontaneous curvature?

Consider small deformations of a planar ring

y

x

2/0

rss /)(0

00

2r

Twist and bending fluctuations – always decouple, but:

for curved filaments – twist is not simply rotation of cross-section!

Example: small fluctuations of a planar ring

andTwist rigidity - coupling between (rotation) (conformation)

0ta zero-energy modesrds

d

Out-of-plane fluctuations diverge!

(vanishes for )r

222

rds

da

ds

d

rds

dadsE tb

Euler Angles

)0()( s )0()( s

)0()( s )0()( s

s/r

Open Ring

1

4321 10,1 bbb

510

310

110

10

Pro

babi

lity

T=

Fluctuation-induced shape transitions – at fixed local curvature!

elastic moduli

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35Y

Axi

s T

itle

X Axis Title

k0=0

k0=1

k0=2

k0=3

k0=4

Length L=1.5

Effect of spontaneous curvature on cyclization

Pro

babi

lity

of

R

End-to-end distance Rcyclization

0.1 0.15 0.2 0.25 0.3

1E-3

0.01

log

(P(r

|r<

R)

log(R)

r3

Fundamental Exponent

2 4 6 8 10 12 14 160.00000

0.00001

0.00002

0.00003

0.00004

k0=0

Yamakawa

P0

L

1 22 3 44

1E-4

1E-3

0.01

k0=0

k0=1

k0=2

k0=3

k0=4

log

(P0)

L

Effect of constant spontaneous curvature

0 1 2 3 4 5 6 7 8

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

Y A

xis

Titl

e

X Axis Title

sequence 1,2<3

k0=0

Effect of random spontaneous crvature

0 1 2 3 4 5 6 7 8

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007Y

Axi

s T

itle

X Axis Title

sequence 1 sequence 2 k

0=1.5

1 2 3 4

0.005

0.010

0.015

0.020

0.025

P0

L

b3=0.01 b3=100 b3=1

40

Effect of twist rigidity on cyclization of curved filaments

Stretching fluctuating filaments

Unbiased sampling of configurations – works only for small f

f

How are fluctuations affected by the force?

Large-scale fluctuations are suppressed by stretching

MS approximation breaks down for short filaments with L<a (neglect orientational effects)!

L=6.28

All orientations are equally probable

Flexible chain Rigid filament

No Wall

f=1 f=10f=2 f=3

y

xEnd fluctuations of stretched filaments:simulation results

Experiments: short dsDNA segments (ca 1000 bp)actin filaments

Take home message:

Bending rigidity is not enough!

New generation of models of biofilaments that account for :

• intrinsic shape (spontaneous curvature and twist)

• twist rigidity