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