56
4 The Smoluchowski Model
d[A 0 ]
dt
= −
d[A − ]
dt
= net flow over the energy barrier = J ξ = k
∗
f [A
−
] − k r [A
0
]
(4.32)
where we have used Eq. 4.17. The former Eq. 4.32 in the vector form:
d
dt
P = J ξ = K.P,
P =
p −
p 0
,
K =
k ∗
f −k r
−k ∗
f k r
(4.33)
where p − and p 0 are the probabilities to have a negatively charged and neutral
amino acid, respectively.
4.3 A Mechanochemical Model
For a Brownian motor the potential V (x, t) in Eq. 4.11 has to be broken into two
parts:
V (x, t) =
V I (x, t)
Internally generated forces
+
V L (x, t)
External load forces
(4.34)
Each chemical state is characterized by its own probability distribution p k (x, t),
where k ranges over all the chemical states, and each chemical state is characterized
by a separate driving potential, V k (x, t) Thus there will be a Smoluchowski equation 4.11 for each chemical state, and these equations must be solved simultaneously
to obtain the motor’s motion. For the amino acid hydrolisis Eq. 4.16, the total change
in probability, p(x, ξ, t), is given by
∂
∂t
p 1
p 2
= Net flow in x + Net flow in ξ
= −
(∂/∂x 1 ) J x 1
(∂/∂x 2 ) J x 2
+
J ξ 1
J ξ 2
= D
∂/∂x 1 [p 1 ∂ (V 1 /k B T )/∂x 1 + ∂p 1 /∂x 1 ]
∂/∂x 2 [p 2 ∂ (V 2 /k B T )/∂x 2 + ∂p 2 /∂x 2 ]
+
−k 12 p 1 +k 21 p 2
−k 21 p 2 +k 12 p 1
(4.35)
where p − ≡ p 1 and p 0 ≡ p 2 . Also k 12 ≡ k ∗
f and k 21 ≡ k ∗
r .
Observe that J ξ 1 = −J ξ 2 . We can visualize the mechanochemical coupling
in Fig. 4.3