Simulation of Single Molecular Bond Rupture
in Dynamic Force Spectroscopy
Prepared for MatSE385 by
Fang Li(TAM)
Samson Odunuga(MatSE)
Phenomenological description of bonds rupture
dS1 t 
 k  t S1 t   k  t 1  S1 t 
dt
S1 t 
Probability of being in state 1 at time t
Pt   
dS1 t 
Probability distribution of lifetime
dt
Probability of lifetime within [t, t+dt]
 Pf  
Pf  
dS1 f  1
   k  f S1 f   k  f 1  S1 f 
df
kv
1
 Pt 
kv
f  kvt
Dissociation rate
Bell’s Expression
1
k  t   k 0 Exp[
f x ]
k BT
k 0 Intrinsic dissociation rate
Recent Explanation
L ts
Eb
Eb
D
1
k0 
Exp[]  Exp[]
L c L ts
k BT
tD
k BT
Eb
1
1
Eb  f t x ]
k  t   Exp[tD
k BT
? x  : xmin , x ts ,
xts  xmin 
Lc
Rupture forces for a non-reversible bond
Probability distribution of rupture forces




k0
 f x
Pf  
Exp   f x  
e
1 
kv
kvx


k 0
 kvx
x  f  ln
 k0
*
 k0
x  f  Exp
 kvx




  k0
 E1 
  kvx

 
E1 x    e y dy
0
y
1

1
k BT
High loading rate




 kvx
0.5772 


x  f  ln
e
 k0

Low loading rate
x  f 
kvx
k0
Modeling the Pulling Experiment
V
Lennard-Jones potential
Z0
E0
Zmin
ZFmax
z 0 12
z0 6
U LJ ( z )  4 E 0 [( )  ( ) ]
z
z
dU LJ ( z )
 0; z
 6 2z0
min
dz
zz
min
d 2U LJ ( z )
26
6

0
;
z

z0
2
F
max
7
dz
zz
Fmax
405 E 0
Fmax  U LJ ( z
)
Fmax
169 z 0
Nanoscopic description of the pulling experiment
dz
D

[ F  f ] Overdamped Langevin Equation
dt k B T
DF
z 
t  random( z )
k BT
U ( z )  4E0 [(
F ( z) 
z 0 12
z
1
)  ( 0 ) 6 ]  k (vt  z  z min ) 2
z
z
2
E z
z
dU
 24 0 [( 0 ) 7  2( 0 )13 ]  k vt  z  z min 
dz
z0 z
z
random( z )
Brownian displacement
E random( z )  0
 2 random( z )  2 Dt
Simulate the Pulling Experiment
Initial Position
t=0, Z=Z min,
Compute F(z)
Move cantilever end
Move the particle
Measure force
No
Detached
yes
E0 z 0 7
z 0 13
F ( z )  24 [( )  2( ) ]  k vt  z n  z min 
z0 z
z
z c n 1  z c n   vt
z n 1  z n 
DF
t  random( z )
k BT
F  k c vt  z n 1  z min 
Forced in spring is
the rupture force
Dimensionless description
Dimensionless distance and time
z

z0
Dt
 2
z0
Dimensionless displacement of the particle

1
 U  1
   random   48 0  13  7
2

 k BT  
Dimensionless loading rate
 random
vz0

D
  k c z 02 vz0 k c z 02
  


  k BT D k BT

z min  

  
 
z 0  


Scaled Units
E 0 : 1K B T z 0 : 1nm
Brownian displacement
E0
F:
 4pN
z0
 random(  )  2
E0
k : 2  4pN/nm
z0
Erandom(  )  0
2
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) = (22)-1/2 exp-[(x-<x>)2/22]
• x = <x> + x’
Single Molecular Bond Rupture
Detachment under low loading rate
2
1.2
1.4
1.6
1.8
2
-2
-4
-6
E0  5K B T ; z 0  0.1nm; T  300K; D  10 18 m 2  s 1
k m  3N  m 1 ; k c  0.03N  m 1
2.2
2.4
Detachment under high loading rate
1.2
1.4
1.6
1.8
2
2.2
-5
-10
-15
-20
-25
E0  5K B T ; z 0  0.1nm; T  300K; D  10 18 m 2  s 1
k m  3N  m 1 ; k c  0.03N  m 1
2.4
Mean rupture force V.S loading rates
Mean rupture force V.S loading rates
E0  20K B T
E0  10K B T
E0  5K B T
E0  2.5K B T
z0  0.1nm


D  10 18 m2 s 1 ; T  300K


km  3N  m 1 ; kc  0.03N  m 1
Rupture of Multiple Parallel Molecular Bonds
under Dynamic Loading
k m kc
Fm t   k m xm t  
vt
N t k m  k c
Bell’s Expression
1
k  t   k 0 Exp[
f t x  ]
k BT
Time dependent decrease of the bonds number
1
 t N   N t k 0 Exp[
Fm t x  ]
k BT
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