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{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