Simulation of Single Molecular Bond Rupturein Dynamic Force Spectroscopy

Prepared for MatSE385 by

Fang Li(TAM)

Samson Odunuga(MatSE)

Phenomenological description of bonds rupture

tStktStkt

tS11

1 1d

d

Probability of being in state 1 at time t

Probability distribution of lifetime

Probability of lifetime within [t, t+dt]

t

tStP

d

d 1

tS1

tPkv

P 1

f

fffff

ff 11

1 11

d

dSkSk

kv

SP

kvtf

Dissociation rate Bell’s Expression

]k

1[

B0 x

Tktk fExp

Intrinsic dissociation rate0k

Recent Explanation

]k

1[-

1

BxtE

Tttk b

D

fExp

]k

[-t

1]

k[-

LL

D

BDBtsc0 T

E

T

Ek bb ExpExp

cL

tsL

bE

x minx tsx minxxts , ,:?

Rupture forces for a non-reversible bond

Probability distribution of rupture forces

100

xekvx

kx

kv

kP ffExpf

kvx

k

kvx

kx 0

10 EExpf

dyyexE y

0

11

TBk

1

0k

kvxx lnf *

0k

kvxx f

57720

0

.lnf ek

kvxx

High loading rate

Low loading rate

V

Modeling the Pulling Experiment

Lennard-Jones potential

0

0

06

2

2

06

601200LJ

169

405max

max

7

26max

0

maxd

d

2min

0

mind

d

4

z

EF

zUF

zF

Fzzz

zU

zzzz

zUz

z

z

zEzU

LJ

LJ

LJ

z

z

)(

;)(

;)(

])()[()(E0

Z0 Zmin ZFmax

Nanoscopic description of the pulling experiment

tDzrandom

zrandomE

2

02 )(

)(

2min

601200 2

14 )(])()[()( zzvtk

z

z

z

zEzU

)( zrandomtTk

DFz

B

min13070

0

0 224d

dzzvtk

z

z

z

z

z

E

z

UzF ])()[()(

][ fFTk

D

dt

dz

B

Brownian displacement

Overdamped Langevin Equation

)( zrandom

Simulate the Pulling Experiment

)( zrandomtTk

DFzz

Bn1n

minn13070

0

0 224 zzvtkz

z

z

z

z

EzF ])()[()(

tvzz nc1nc

min1n zzvtkF c

Forced in spring is the rupture force

Measure force

No Detached yes

Initial Positiont=0, Z=Z min,

Compute F(z)

Move the particle

Move cantilever end

Dimensionless description

0z

z

20z

DtDimensionless distance and time

0

min200

20

7130

2

1148

z

z

Tk

zk

D

vz

Tk

zk

Tk

U

B

c

B

c

Brandom

Dimensionless displacement of the particle

random

2

02 )(

)(

random

randomE

Brownian displacement

Dimensionless loading rate D

vz0

pN/nm4z

pN4z

nm1z TK1

20

0

0

0

0B0

Ek

EF

E

:

:

::

Scaled Units

Brownian displacement: Random number generation

function ran1 (Bayes et Duham NR pp. 270-271)

•I j+1 = I j (mod m)

•generates uniform deviates (0, 1]

•adjusts against low order correlations

function gasdev (Box-Mueller method NR pp. 279-280)

• generates random deviates with standard normal distribution

Transformation p (x) = (22)-1/2 exp-[(x-<x>)2/22]

• x = <x> + x’

Single Molecular Bond Rupture

Detachment under low loading rate

1.2 1.4 1.6 1.8 2 2.2 2.4

-6

-4

-2

2

11

121800

mN030 ;mN3

sm10D 300K;T nm;10z ; TK5

.

.

cm

B

kk

E

Detachment under high loading rate

1.2 1.4 1.6 1.8 2 2.2 2.4

-25

-20

-15

-10

-5

11

121800

mN030 ;mN3

sm10D 300K;T nm;10z ; TK5

.

.

cm

B

kk

E

Mean rupture force V.S loading rates

Mean rupture force V.S loading rates

11

1218

0

mN030 ;mN3

300K T ;sm10D

nm10z

.

.

ckkm

TK200 BE TK100 BE

TK50 BE TK520 BE .

Rupture of Multiple Parallel Molecular Bonds under Dynamic Loading

]k

1[

B0 xt

Tktk fExp

vtkktN

kktxktF

cm

cmmmm

]k

1[

B0 xtF

TktNN mt Exp

Time dependent decrease of the bonds number

Bell’s Expression

Conclusions

• The model predicts, as it is observed experimentally, the rupture force measured is an increasing function of the loading rate.

• At high loading rate, the rupture force equal to the maximum force corresponding to the LJ potential.

• At low loading rate, the thermal fluctuations take an important role in the detachment process.

Acknowledgements

Prof. Duane Johnson

Prof. Deborah Leckband