Electrostatics in Macromolecular Solutions - Food Colloids: Self-Assembly and Material Science - page 143

approximated by

X

10 pH pK g
1 þ 10 pH pK g

1
ln 10

C ideal

n g

Þ 2 :

ð 21 Þ

ð

g

We have calculated the capacitance for a number of proteins with different
characteristics in terms of the number and type of residues. An MC simulation
has to be performed at each pH for given salt and protein concentrations.
Unless otherwise stated, we have used a salt concentration of 70 mM and a
protein concentration of 0.7 mM. Figure 8(a) shows the capacitance versus
pH for calbindin. The main difference from the ideal capacitance curve is
a strong broadening of the two peaks corresponding to the responses from
acidic and basic residues, respectively. If the protein has a significant net
charge, the true curve will also shift away from the ideal one, as is seen for
calbindin at high pH.
The protein hisactophilin is of the same size as calbindin, but it has a slightly
different capacitance-pH curve [Figure 8(b)]. The protein contains 31 histidine
residues, which is reflected in a large maximum in C at pH

E

5. The downward
shift of the maximum is due to the high positive charge of hisactophilin at low
pH. The net charge is +28 at pH ¼ 3 and +23 at pH ¼ 4. The isoelectric point
found from the simulations is pI ¼ 7.3, which is in good agreement with
experimental estimates.
The electrostatic interaction between two proteins will be dominated by the
direct Coulomb interaction provided that the net charge Z is sufficiently
different from zero. The induced interactions will only play an important role
at pH values close to the isoelectric point of one of the proteins - this can be
seen from Equation (17). Figure 9(a) shows the free energy of interaction
between the two proteins calbindin and lysozyme at pH ¼ 4, which is close to
the isoelectric point for calbindin. At contact there is a significant difference in
interaction energy for a model with fixed charges as compared with a situation
where the proteins are free to adjust their charges. The difference in free energy
between the two models is mainly due to the interaction between the induced
charge in calbindin and the permanent charge in lysozyme. This is a typical
result, and significant effects from charge regulation can be expected when one
of the interacting proteins has a large net charge and the other a large
capacitance. Following Equation (17), we can approximate the difference as

DA ð R Þ= kT ¼ð A reg A fix Þ= kT
¼ l B
2R 2

:

ð 22 Þ

C calb C lys þ C calb Z lys þ C lys Z calb

Figure 10 shows almost perfect agreement between the simulated free energy
difference and the one calculated according to Equation (22).
A noteworthy finding is that, despite calbindin and lysozyme both being
positively charged at pH ¼ 4, there is still an attractive electrostatic interaction
between the two molecules. Such an attraction could, of course, be due