Simulation of Single Molecular Bond Rupture in Dynamic Force Spectroscopy

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