modeling the phase transformation which controls the mechanical behavior of a protein filament
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Modeling the phase transformation which controls the mechanical behavior of a protein filament. Peter Fratzl Matthew Harrington Dieter Fischer. Potsdam, Germany. 108th STATISTICAL MECHANICS CONFERENCE December 2012. mussel byssus. whelk egg capsule. i mportant yield. - PowerPoint PPT PresentationTRANSCRIPT
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Modeling the phase transformation which controls the mechanical behavior of a protein filament
Peter FratzlMatthew
HarringtonDieter Fischer
Potsdam, Germany
108th STATISTICAL MECHANICS CONFERENCEDecember 2012
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musselbyssus
whelk eggcapsule
Relatively high initial stiffness
400 MPa 100 MPa 1) Stiffness
important yieldimportant yield
2) Extensibility
slow
immediate recovery
3) Recovery
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Mussel byssal threads
Self-healing fibres
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yield
relaxation
„healing“ ~ 24h
Mechanical function of Zn – Histidine bonds
M. Harrington et al, 2008
elastic
1h
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Egg capsules of marine whelk
Busycotypus canaliculatus
Harrington et al. 2012J Roy Soc Interface
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α-helix
extended β*
αβ*
Raman
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X-ray (small-angle) diffraction
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Ramanintensity
XRD intensity
stress strain
αβ*
Phase coexistenceyield
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Co-existence of two phases during yield
Elastic behaviour
W(s) = (k/2) (s – s0)2
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Force f
actuallength
s
extended(contour)
lengthL
21 1
14 4
p
kT s sf
l L L
persistencelengthlp
kinknumber
ν
lengthat rest
s0
Worm-like chain(Kratky/Porod 1949)
Molecule with kinks(Misof et al. 1998)
(s > s0)
extendedphase
β*
0 0
0
s s skT
fL L s L s
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21
( ) 24
B
p
k T s s LW s W
l L L L s
0 0
0
( ) 1 log 1
B s sk T s s
W s WL L s L L L
21 1
14 4
B
p
k T s sf
l L L
0 0
0
B s s sk T
fL L s L s
f k s s
21( )
2 W s W k s s
Relation between force and potential energy:
W
fs
β* phase (entropic)α phase (elastic)
Low strainHigh strain
WLC
kinkmodel
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All molecular segments in the fiber see the same force
fa
mechanical equilibrium: *
a
WWf
s s
Complete analogy to thermodynamic equilibrium:
*
a
WW
c c
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( ) aW s f s s
D-period(nm)
100 120 140 160
ela
stic ene
rgy density (M
Jm-3
)
0
1
2
3
4
Total energy
D-period(nm)
100 120 140 160-1
0
1
2
3
100 120
W(x) -
(x - s)
-1
0
1
2
3
100 120-1
0
1
2
3
WLC and kink model nearlyidentical on this scale
internalenergy
work ofapplied force
α stable stability limit α + β*
sclow
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𝜎= ρ 𝑓
Relation to experiment
What can be measured(by in-situ synchrotron
x-ray diffraction):
Force as a function of mean elongation
The critical force at yield (α-β* coexistence)
The yield point (start of α-β* coexistence)
( )af s
Yaf
clows
Number of moleculesper cross-sectional area
Reconstruct W(s)
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* * aW W f s s
Based on: R. Abeyaratne, J.K. Knowles, Evolution of Phase Transitions – A Continuum Theory (Cambridge University Press, Cambridge, 2006)
Phase transformation kineticsin analogy to pseudoelasticity in NiTi
thermodynamic driving force
d
dt
kinetic equation
fraction of β* segments in the fiber
Hypothesis: load at contant stress rate, (loading) and (unloading)af
af
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Slow or fast stretching
Blue: Red:
Green:
WLC
Equilibriumline
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musselbyssus
whelk eggcapsule
Cooperativity of many weak bonds phase transition