Chemistry Reference
In-Depth Information
charge-induced-charge interactions. Note also that
h
Z
2
i
h
Z
i
2
. If the mole-
cules are identical, i.e.,
h
Z
A
i¼h
Z
B
i¼h
Z
i
, then the expression simplifies to
a
A
ð
R
Þ=
kT
l
B
hi
2
R
l
B
2R
2
Z
2
hi
2
:
ð
14
Þ
2
þ
2
hi
2
Z
2
hi
2
If pH
¼
pI, then we have
h
Z
i¼
0, and the induced charge-induced charge
interaction becomes the leading term,
A
ð
R
Þ
l
B
Z
2
2
2R
2
:
ð
15
Þ
The above equations show that the fluctuating charge of a protein or macro-
molecule may under certain circumstances contribute significantly to the net
interaction. We can define a 'charge polarizability' or a capacitance, C,as
C
¼
Z
2
hi
2
:
ð
16
Þ
With this definition of the capacitance, Equation (13) can be rewritten in a
more compact form,
:
l
B
2R
2
A
ð
R
Þ=
kT
l
B
Z
hi
Z
hi
R
C
A
C
B
þ
C
A
Z
h
2
þ
C
B
Z
h
2
ð
17
Þ
We can use general electrostatic equations and relate the capacitance to the
charge induced by a potential DF, i.e.,
Z
ind
¼
Ce
D
F
kT
:
ð
18
Þ
The capacitance C can also be derived from the experimental titration curve.
The ionization degree
a
for a single titrating acid can be found in any
elementary physical chemistry textbook:
a
1
a
pH
:
log K
¼
log
ð
19
Þ
Taking the derivative of
a
with respect to pH gives
d
a
d
ð
pH
Þ
¼
a
ð
1
a
Þ¼
C ln10
;
ð
20
Þ
where in the second step we have identified the capacitance defined in Equa-
tion (16). We can obtain an approximate value for the capacitance in a protein
assuming that there is no interaction between the titrating sites. Any pro-
tein contains several titrating groups, like aspartic acid, glutamic acid,
histidine, etc., each with an ideal pK value. Denoting different titrating groups
with subscript
g
and their number with n
g
, the total capacitance can be
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