{l}={l1,l2,...,lN}

The average end to end distance: l2 l1 l2 ... lN l1 l2 ... l N

0

How large is a Polymer Blob?

l i

2

i 1

N

l i l ji j

l 2

i 1

N

N l2

l i

2

i 1

N

l i l ji j

l2 N l

Estimation: Size of a Viral dsDNA with ca 50kbp ?

with l≈3Å => approx. 70nmWith p≈50nm => ca 1,5 µm !

Freely-Jointed-Chain Modell

Random Walk

Gaub/WS 2006 BPM §1.4.2 2

The excluded Volume

• The simple model of a random walk resulted for the end to end distance oft the polymer blob: r2 N l2

• Flory solved the problem with a simple heuristic argument:

If two monomers overlap, they repell each other. The Probability that 2 monomers occupy the same space increases with the concentration squared

W vkBT cm2

Energy Density: cm N

r23

W vkBT N2

r26

EAusschluß W r23

vkBT N 2

r23

The average end to end distance is used as measure for the radius of the polymers.

• Problem: The polymer cannot occupy the same space. Thus the average quadratic end to end distance should be bigger.

Gaub/WS 2006 BPM §1.4.2 3

• The energy for the excluded volume drives the polymer blob apart. This force has to be balanced by an entropic force which wants to keep the blob together:

EAusschluß W r23

vkBT N2

r 23

FAusschluß EAusschluß

r2 vkBT 3

N2

r 24

Fentr 3kT

N l2 r2

3kT

N l2 r2 vkBT 3N2

r 24

!

0

1

l2 r25

v N 3

r2 N3

5 r2 N 0.5

In contrast to the FJC Model

(von FJC Model)

Java-Simulation Self-avoiding Random Walkhttp://polymer.bu.edu/java/java/saw/sawapplet.html

Gaub/WS 2006 BPM §1.4.2 5

s

s

A measure for the stiffness of a polymer is the persistence length Lp, which measures at which length s=Lp the orientation and s are not correlated any more.

f(s) cos (s) (0)

A measure for the correlation of the orientation is the following average value:

cos (s)

oBdA

df sin( )d 1

2cos( )d 2 O(d 4 ) sin( )d

1

2cos( )d 2

sin( ) d 1

2cos( ) d 2

=0

1

2f (s) d 2

df

ds

1

2f(s)

dds

2

ds

The Worm-Like-Chain Model for semiflexible Polymers

M E

RImit

s

R

R s

dds

1

R

Local Bending Radius

Calculation: Energy change of a beam of lengths, if it is bent by the angle

dU M d

dU EI

Rd

1

Rs

EI

Rd

1

R

s

U 0

1

R0

EI

Rd

1

R

s

1

2

EI

R02 s

1

2EI

dds

2

sdds

2

2U

EI s

df

ds f (s)

U

EI

dds

2

2U

EI sdf

ds

1

2f(s)

dds

2

ds

df

ds

1

2f(s)

kT

EIÄquipartition Theorem

in 3-D two angles can fluctuate, each containing the average energy kT/2.

f(s) f (0)e kT

2EIs

in 2-D

f(s) f (0)e kT

EIs

in 3-D

Lp EI

kTf(s) f (0)e

s

Lp Persistence length

Bending is a thermodynamicdegree of freedom

DNA Lp=53 nmAktin Lp = 10 µmMikrotubuli Lp =1 mm

Gaub/WS 2006 BPM §1.4.2 8

Connection between FJC und WLC-Modell

r2 rr

s

t(s)0

L

ds

t( s )0

L

d s

t(s)0

L

t( s )dsd s 0

L

t ds

r

2 t(s)s s

L

t( s )dsd s s0

L

2 cos ( s ) (s) s s

L

dsd s s0

L

2 e

s s

Lp

s s

L

d s dss0

L

2 f ( s s)s s

L

dsd s s0

L

2Lp

2e

L

Lp 1L

Lp

2L Lp

L Lp

Comparison with FJC

r2 N l2 N l l L l l 2Lp

Both models yield the same average end to end distance when the chain of FJC coincides with twice the persistence length l=2Lp

Force Extension Curves: Comparison of Models

Freely Jointed Chain (FJC) Worm-like Chain Model (WLC)

With Stretch Modulus K0 of Monomer(e.g. stretching of DNA)

r N l cothF l

k T

kT

F l

: N l L

F l

k T

F kT

lL 1 r

N l

For negligible

fluctuations

Force Extension Curve of dsDNA