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 FratzlMatthew
HarringtonDieter Fischer
Potsdam, Germany
108th STATISTICAL MECHANICS CONFERENCEDecember 2012
musselbyssus
whelk eggcapsule
Relatively high initial stiffness
400 MPa 100 MPa 1) Stiffness
important yieldimportant yield
2) Extensibility
slow
immediate recovery
3) Recovery
Mussel byssal threads
Self-healing fibres
yield
relaxation
„healing“ ~ 24h
Mechanical function of Zn – Histidine bonds
M. Harrington et al, 2008
elastic
1h
Egg capsules of marine whelk
Busycotypus canaliculatus
Harrington et al. 2012J Roy Soc Interface
α-helix
extended β*
αβ*
Raman
X-ray (small-angle) diffraction
Ramanintensity
XRD intensity
stress strain
αβ*
Phase coexistenceyield
Co-existence of two phases during yield
Elastic behaviour
W(s) = (k/2) (s – s0)2
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
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
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
( ) 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
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)
* * 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
Slow or fast stretching
Blue: Red:
Green:
WLC
Equilibriumline
musselbyssus
whelk eggcapsule
Cooperativity of many weak bonds phase transition
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