3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/01/07
Prof. C. Ortiz, MIT-DMSE
I
LECTURE 20: THEORETICAL ASPECTS OF SINGLE
MOLECULE FORCE SPECTROSCOPY 2 :
EXTENSIBILITY AND THE WORM LIKE CHAIN (WLC)
Outline :
REVIEW LECTURE #19 : THEORETICAL FORMULATIONS FOR THE ELASTICITY OF SINGLE
POLYMER CHAINS (FJC) ......................................................................................................................... 2
VARIOUS MATHEMATICAL FORMS FOR THE INEXTENSIBLE FJC- GRAPHICAL COMPARISON .... 3
EFFECT OF a AND n on INEXTENSIBLE FJC ......................................................................................... 4
EXTENSIBLE FJC ..................................................................................................................................... 5
WORM LIKE CHAIN (WLC) MODEL ......................................................................................................... 6
FITS TO EXPERIMENTAL SINGLE MOLECULE FORCE SPECTROSCOPY DATA............................... 7
Objectives:
To understand how extensibility of chain segments affects the FJC elasticity model
and to understand the differences between the FJC and WLC models
Readings: Course Reader Document 31, CR Documents 32-39 are the original theoretical papers
for reference, English translations of CR 33 and 36 are available as handouts.
Multimedia : Podcast : Sacrificial Bonds in Biological Materials; Fantner, et al. Biophys. J. 2006
90, 1411
1
3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/01/07
Prof. C. Ortiz, MIT-DMSE
REVIEW LECTURE 19 : MODELS FOR SINGLE POLYMER CHAIN ELASTICITY
MODEL
F
≈
F
r
f
F
f
⎞ ⎛ 297 ⎛ r ⎞5 ⎞ ⎛ 1539 ⎛ r ⎞ 7
⎞
⎟⎟ + ⎜⎜
⎜ ⎟ ⎟⎟ + ⎜⎜
⎜ ⎟ + ... ⎟⎟
175
na
875
na
⎝
⎠
⎝
⎠
⎠ ⎝
⎠ ⎝
⎠
-1
r ⎞
⎛ k BT ⎞ ⎛
High Stretch Approximation : f(r) = ⎜
⎟ (4)
⎟ ⎜ 1⎝ a ⎠ ⎝ Lcontour ⎠
3
Non - Gaussian :
⎛ f
⎛ r ⎞
⎛k T⎞
f(r) = ⎜ B ⎟ L -1 ⎜
⎟ ; L total = Lcontour + n ⎜⎜
⎝ a ⎠
⎝ Ltotal ⎠
⎝ k segment
(a, n, ksegment)
⎞
⎟⎟
⎠
Exact : Numerical Solution
F
F
≈
f
r
f
(p, n)
F
F
≈
(Odijk, 1995
Wang, et al. 1997)
⎛ 3k BT ⎞
⎛ 3k T ⎞
Gaussian : f(r) = ⎜ B2 ⎟ r = ⎜
⎟ r (1)
⎝ na ⎠
⎝ aL contour ⎠
Non - Gaussian :
⎛ fa ⎞
1⎞
⎛
Exact Formula : r(f ) = na ⎜ coth ( x ) - ⎟ where : x = ⎜
⎟ (2)
x⎠
⎝
⎝ k BT ⎠
⎛k T⎞
Langevin Expansion : f(r) = ⎜ B ⎟ β (3)
⎝ a ⎠
-1 ⎛ r ⎞
β = L ⎜ ⎟ = Inverse Langevin Function
⎝ na ⎠
⎛ r ⎞ ⎛9⎛ r ⎞
= 3⎜ ⎟ + ⎜ ⎜ ⎟
⎝ na ⎠ ⎜⎝ 5 ⎝ na ⎠
(Kratky and Porod, 1943
Fixman and Kovac, 1973
Bouchiat, et al. 1999)
Extensible
Worm-Like
Chain
f
(a, n)
(Smith, et. al, 1996)
Worm-Like
Chain (WLC)
F
r
f
(Kuhn, 1934
Guth and Mark, 1934)
Extensible
Freely-Jointed
Chain
FORMULAS
≈
FreelyJointed
Chain (FJC)
SCHEMATIC
f
r
f
(p, n, ksegment)
⎛
⎞
⎜
⎟
⎛ k BT ⎞ ⎜ r
1
1⎟
Interpolation Formula : f(r) = ⎜
+
− ⎟
⎟⎜
2
⎛
⎝ p ⎠ ⎜ L contour
r ⎞ 4⎟
4 ⎜1 −
⎟
⎜
⎟
⎝ Lcontour ⎠
⎝
⎠
⎛
⎞
⎜
⎟
⎛ k T ⎞⎜ r
1
1⎟
+
− ⎟;
Interpolation Formula : f(r) = ⎜ B ⎟ ⎜
2
⎛
⎝ p ⎠ ⎜ L total
r ⎞ 4⎟
4 ⎜1 −
⎟
⎜
⎟
⎝ L total ⎠
⎝
⎠
⎛ f ⎞
L total = Lcontour + n ⎜
⎜ k segment ⎟⎟
⎝
⎠
2
3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/01/07
Prof. C. Ortiz, MIT-DMSE
COMPARISON OF VARIOUS MATHEMATICAL FORMS FOR THE INEXTENSIBLE
FREELY JOINTED CHAIN (FJC) MODEL
≈
F
F
r
Felastic
0
Felastic
Gaussian (1)
1/3Lcontour
0
Gaussian
-0.01
Force (nN)
high stretch (4)
dashed
-0.2
-0.3
Langevin (2)
coth (3)
-0.4
Force (nN)
-0.1
high stretch (4)
dashed
-0.02
3/4Lcontour
Langevin (2)
-0.03
-0.04
coth (3)
-0.05
-0.5
0
10
20
30
40
50
60
70
Distance (nm)
80
90 100
0
10
20
30
40
50
60
70
Distance (nm)
Surface separation distance, D= r, chain end-to-end distance; sign convention (-) for attractive back force, however some
scientists plot as (+); e.g. Zauscher (podcast)
(1) Gaussian physically unrealistic; force continues to increase forever beyond Lcontour, valid for r,D<1/3 Lcontour
(3) Langevin Series Expansion; finite force beyond Lcontour (physically unrealistic); valid for r,D<3/4 Lcontour
(4) High stretch approximation underestimates force for r,D<3/4Lcontour, valid for r,D>3/4Lcontour
3
3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/01/07
Prof. C. Ortiz, MIT-DMSE
EFFECT OF a and n on INEXTENSIBLE FJC
0
0
-0.1
-0.1
-0.2
Felastic (nN)
Felastic (nN)
⎛ 3k T ⎞
Gaussian : f(r) = ⎜ B2 ⎟ r
⎝ na ⎠
a = 0.1 nm
a = 0.2 nm
a = 0.3 nm
a = 0.6 nm
a = 1.2 nm
a = 3.0 nm
-0.3
-0.4
-0.5
-0.2
n=100
-0.3
n=200
n=300
n=400
n=500
-0.4
-0.5
0
50
100
r (nm)
150
200
0
100
200
r (nm)
300
(left) Elastic force versus displacement as a function of the statistical segment length, a, for the non-Gaussian FJC model
(Lcontour = 200 nm) and (right) elastic force versus displacement as a function of the number of chain segments, n, for the
non-Gaussian FJC model (a = 0.6 nm)
4
3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/01/07
Prof. C. Ortiz, MIT-DMSE
EXTENSIBLE FREELY JOINTED CHAIN (FJC) MODEL
≈
F
F
r
Felastic
Felastic
Add displacement term to Lcontour:
L total = L contour +
123
=na
⎛ f ⎞
n⎜
⎟⎟
⎜k
segment ⎠
⎝
14243
extension beyond Lcontour
due to enthalpic stretching of
chain segments
n= number of statistical segments
⎛ r ⎞
⎛k T⎞
f(r) = ⎜ B ⎟ L -1 ⎜
⎟
⎝ a ⎠
⎝ Ltotal ⎠
Now we have three physical (fitting) parameters;
a, n, ksegment
felastic (nN)
- Take into account a small amount of longitudinal (along chain axis) enthalpic deformability (monomer/bond stretching) of
each statistical segment, approximate each statistical segment as a linear elastic entropic spring (valid for small
deformations) with stiffness, ksegment →springs is series, forces are equal, strain additive;
ksegment; fsegment= ksegment δsegment
0
solve for: δsegment =fsegment/ksegment
-0.1
extensible
nonGaussian
FJC
-0.2
nonGaussian
FJC
-0.3
-0.4
-0.5
0
100
200
300
400
r (nm)
Schematic of the stretching of an extensible freely jointed chain and (b) the elastic
force versus displacement for the extensible compared to non-extensible nonGaussian FJC (a = 0.6 nm, n = 100, ksegment = 1 N/m) - note units
5
3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/01/07
Prof. C. Ortiz, MIT-DMSE
WORM LIKE CHAIN (WLC) MODEL
≈
(*Kratky-Porod Model)
"Directed random walk"- segments are correlated, polymer
chains intermediate between a rigid rod and a flexible coil
(e.g. DNA)
- takes into account both local stiffness and long range
flexibility
-chain is treated as an isotropic, homogeneous elastic rod
whose trajectory varies continuously and smoothly through
space as opposed to the jagged contours of the FJC
γ
(lw)1
p= persistence length, length over which statistical
segments remain directionally correlated in space
(lw)n
r
Exact : Numerical Solution
-WLC stiffer at higher extensions, force rises sooner than
FJC since statistical segments are constrained, can also
make an extensible form of WLC→ replace Lcontour by Ltotal as
before for FJC
0
felastic (nN)
⎛
⎞
⎜
⎟
⎛ k BT ⎞ ⎜ r
1
1⎟
Interpolation Formula : f(r) = ⎜
+
− ⎟
⎟⎜
2
⎛
⎝ p ⎠ ⎜ L contour
r ⎞ 4⎟
4 ⎜1 −
⎟
⎜
⎟
⎝ L contour ⎠
⎝
⎠
-0.2
non-Gaussian
FJC
-0.4
-0.6
WLC
-0.8
-1
-In reality the FJC and WLC are very similar and just produce
slightly different values of the local chain stiffness
0
20
40
60
r (nm)
80
100
6
3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/01/07
Prof. C. Ortiz, MIT-DMSE
FITS TO EXPERIMENTAL SINGLE MOLECULE FORCE SPECTROSCOPY DATA
AFM retract data on single polymer chain of polystyrene in toluene; comparison to Freely-Jointed Chain Model (a = 0.68
nm) - right plot aata normalized by Lcontour to create a master plot
0
-0.05
-0.05
-0.1
76
146
290
400
680
n
-0.15
Fchain (nN)
Fchain (nN)
0
-0.1
Lchain/Lcontour=0.93
-0.15
0
100
200
300
Distance (nm)
400
500
0
0.2
0.4
0.6
0.8
1
D / Lcontour
7