UC Riverside Undergraduate Research Journal

Computational Prediction of Association Free Energies for the C3d-CR2 Complex and Comparison to Experimental Data
Alexander S. Cheung
Association in vacuum is described simply by Coulomb’s
law, using the same dielectric coefficient for protein and
solvent in the absence of ions (ε p
= ε s
= 2; κ = 0). Association
in solution is described by the Poisson-Boltzmann equation
with different dielectric coefficients for the protein and
solvent in the presence of ions (ε p
= 2; ε s
= 78.5; κ ≠ 0).
Solvation describes the transfer of the protein or protein
complex from vacuum into solution.
The electrostatic free energy of association is given by
(5)
where
, with tip and base referring to
the direction of the arrows in the thermodynamic cycle. We
solve Eq. (5) for
Table 1 lists the mutants (also shown graphically
in Fig. 1), solvation free energy differences (Eq. 5),
calculated electrostatic free energies of association in
solution without solvation effects (Eq. 6), experimental
binding ability data for C3d mutants [9] and CR2
mutants [6], previously published calculated ionization
free energy differences [5], and calculated distances
of each mutated amino acid from the association site
contact residues. The distances were calculated using
the C3d-CR2 structure (Fig. 1) and Glu116 (in C3d) and
Arg390 (in CR2) as the association site contact residues.
Distances were measured between the central atoms of
the ionization sites: C γ for Asp, C δ for Glu, N ζ for Lys, C ζ
and for Arg. Experimental binding abilities are relative
to the parent proteins [6,9]. The key for the experimental
binding abilities is as follows: +++++, 2-fold increase;
(6)
which represents the solvation free energy difference
upon C3d-CR2 association between solution and vacuum
environments.
In order to perform this calculation, we must
know , , and , which are
determined by calculating the 6 electrostatic free energies
for C3d, CR2, and C3d-CR2 in vacuum and in solution and
by taking their differences according to the thermodynamic
cycle of Fig. 3 and Eqs. (5) and (6). As can be seen in Eq.
(6) the solvation free energy difference, ΔΔG solvation ,
is equal to the electrostatic free energy of association in
solution,
minus the Coulombic free energy
of association in vacuum, (also seen in the
thermodynamic cycle of Fig. 3).
To assess the effect of solvation in association, we
also calculated the electrostatic free energy of association
in solution alone (bottom horizontal step only in Fig. 3)
according to
(7)
The values of and as
described by Eqs. (6) and (7) were calculated 24 times:
once for the parent proteins and once for each of the 23
sets of mutant protein.
Figure 4. (A) ΔΔG solvation (calculated using the complete
thermodynamic cycle of Fig. 3) versus relative association ability
with a linear fit drawn in red. (B)
(calculated using
the bottom horizontal reaction only of the thermodynamic cycle
of Fig. 3) versus relative association ability.
18 UCR Un d e r g r a d u a t e Re s e a r c h Jo u r n a l