30
3 Biased Brownian Motion
Fig. 3.4 Simulation performed with Program 3.1, with the parameters included:
V ( ˜
x) =
1
2π
sin (2π ˜
x) −
1
4 sin (4π ˜
x)
,
V ( ˜
x) =
cos (2π ˜
x) −
1
2 cos (4π ˜
x)
. Most of the
stochastics realizations, < x >, are to the left < x > < 0.
D = 5.10 −4 is in accordance of a
3 nm radius monomeric protein, which the Diffusion coefficient in water is D 0 = K B T /(6πηa) =
8.17×10 −11 m 2 /s, given for the bound state diffusion coefficient D = D 0
D = 4.08×10 −14 m 2 /s,
consistent with the value D = 4.4 × 10 −14 m 2 /s for monomeric kinesin
will use the dimensionless variables and constants in the programs and figures and
omit the tilde symbol (Figs. 3.4 and 3.5).
3.4 Dichotomous Markov Noise
We consider a stochastic variable η (t) which switches between two values a and b
with constant transition rates, DMN. The rate of switching from a to b is μ a and
from b to a is μ b . This two-step process can be described by a probability loss-gain
equation or master equation.
d
dt
P (a, t|x, t 0 ) = −μ a P (a, t|x, t 0 ) + μ b P (b, t|x, t 0 )
(3.23)
d
dt
P (b, t|x, t 0 ) = +μ a P (a, t|x, t 0 ) − μ b P (a, t|x, t 0 )
(3.24)