Electrostatics in Macromolecular Solutions - Food Colloids: Self-Assembly and Material Science - page 141

interaction between their average charges, also will contain terms originating from
induced charges. These interactions can be formalized in a statistical mechanical
perturbation approach, 20,21 so that a protein is characterized not only by its
average net charge, but also by its capacitance. The induction interaction is
important for the interaction of an approximately neutral protein with another
charged macromolecule. The protein capacitance is a function of the number of
titrating residues, and it will display maxima close to the pK a of the titrating
amino acids. In what follows, we derive a formal expression for the capacitance.
Consider the macromolecules A and B, described by two sets of charges [r i ,z i ]
and [r j ,z j ], respectively. Their centres of mass are separated by R, which means
that the distance between two charges i and j is given by r ij ¼ |R + r j r i |. The
average net charge of the distributions need not be zero. The free energy of
interaction can be written as

A ð R Þ= kT ¼ ln exp ð U ð R Þ= kT Þ h i 0 U ð R Þ= kT h i 0

D E

1
2

ð U ð R Þ= kT Þ 2

ð 11 Þ

0

2

D E

þ 1
2

i 0 þ 1
2

Þ 2 h U ð R Þ= kT ð U ð R Þ= kT ;

0

where U(R) is the interaction between the two charge distributions and h ... i 0
denotes an average over the unperturbed system, which in the present case is
the single isolated protein in solution. The interaction energy is simply the
direct Coulomb interaction between the two charge distributions, i.e.,
U ð R Þ= kT ¼ X
i

X

l B z i z j
r ij

: ð 12 Þ

j

We can make a Taylor series expansion of U, assuming that R c r i . This
expansion includes ion-ion interaction, ion-dipole interaction, dipole-dipole
interaction, etc. It also includes charge induced-charge and induced-
charge induced-charge interactions. Thus, we can write an approximation to
the free energy including all terms of order up to 1/R 2 . Note that the ion dipole
interaction disappears at first order and that the first non-vanishing dipole
term, l B Z 2 m 2 /6R 4 , is of order 1/R 4 :

A ð R Þ= kT l B Z hi Z hi
R
l B
2R 2

D E

Z A Z h 2

Z B Z h 2

ð 13 Þ

h Z h 2

Z A Z h 2 l B

2R 2

Z h 2 i :

þ Z B Z h 2

The first term in Equation (13) is the direct Coulomb term, and the following
term is the induced-charge induced-charge term, and the final terms are the