52
4 The Smoluchowski Model
A B
A + C D
(4.13)
A + C E + F
The rates of the reactions can be written, as
d[A]
dt
= −k 1 [A] + k 2 [B]
d[A]
dt
= −k 3 [A][C] + k 4 [D]
(4.14)
d[A]
dt
= −k 5 [A][C] + k 6 [E][F ]
The brackets indicate concentrations in the classical version of the theory, although
the problem may be recast into activities. The k s are called rate parameters or rate
constants; that is, they do not depend explicitly on the concentrations, and have the
dimensions of (time) −1 . If in the second of Eqs. 3.13, we make A − ≡ A, H + ≡ C,
D ≡ [H.A] ≡ A 0 then we get
A
−
+ H
+
A
0
(4.15)
The former Eq. 4.15 resembles nucleotide hydrolisis which is one of the most
common energy supply to drive molecular motors. If we center only in the amino
acid (nucleotide), we see it exists in two states: charged (A − ) and neutral (H.A ≡
A 0 ). Then Eq. 4.15 from the view point of the amino acid is simply
k ∗
f
A
−
A
0
(4.16)
k r
Accordingly the second Eq. 4.15 with k 3 = k f and k 4 = k r1 , “f ” of forward and
“r” of reverse.
d[A − ]
dt
= −k f [H
+
][A
−
] + k r [A
0
] =
− k
∗
f [A
−
] + k r [A
0
]
(4.17)
where k ∗
f = k f [H + ] is called a pseudo-first order rate constant.
Returning to the rate constants, k’s, in general they are functions of the temperature and other environmental variables. Such purely experimental variables usually
have an exponential dependence on temperature. Arrhenius in 1889 formulated the
following equation (empirical generalization).