Theoretical Models in Force Spectroscopy - Dynamic Force Spectroscopy and Biomolecular Recognition - page 100

J = kn

ρ (x)

x

β

x 2

x 1

x u

x 0

FIGURE 3.3 The prescription for deriving Kramers rate of escape, k . The distribution of
particles in the well at x o provide a source, while the location outside the barrier x u acts as
a sink. Therefore, a steady flow of particles, or flux J , is passing over the barrier. The flux is
defined as the number of particles available n times rate of escape. Thus, estimating the flux
and integrating over the distribution of particles in the well provides the transition rate (see
discussion in text).

in the well provide a constant flux of particles over the barrier (steady-state condi-
tions). To the far left, the steeply rising energy is essentially a reflecting boundary.
To the right, after particles pass over the barrier and reach x u , they are completely
unbound and carried away from the potential indefinitely. Therefore, we define x u as
an absorbing boundary with ρ 0. To calculate the flux over the barrier, we look

to integrate over x o to x u . Multiplying both sides of Equation 3.9 by an integrating
factor exp

( x u ) ≡

( U ( x ) / k B T ) ,wehave

U ( ∂ρ ( x ) ρ ( x ) x ) η J

k B T e U ( x ) / k B T

e U ( x ) / k B T e U ( x ) / k B T

+ = − (3.10)

∂ x k B T

ρ

e U ( x ) / k B T

d
d x

η J
k B T e U ( x ) / k B T

( x ) = − (3.11)

Where we have exploited the fact that the left side of Equation 3.10 is just the chain
rule derivative of ρ ( x ) exp ( U ( x ) / k B T ) . Upon integration, and noting that ρ ( x u )= 0,

we have

e U ( x 0 ) / k B T

( )

k B T
η

ρ x 0

J = (3.12)

x u
x 0

e U

( x ) / k B T d x

Now the flux J is the steady flow of particles moving from a source, the well, over
the barrier and into a sink (or carried away). To determine the actual rate k of par-
ticles escaping per unit time, we recognize that the flux is the product of the rate
with the number of particles available to escape, J

kn . Therefore, we need to find
how the particle concentration changes along x .Define n as the number of particles
in the well. Near the bottom of the well, around x 0 ,theflux is nearly zero and we

=