Free energy calculations

As mentioned earlier, the comparison of grid energies from several DelPhi runs with the same grid conditions can yield solvation energies, binding energies, and salt effects. The grid energy is obtained from the product of the potential at each point on the grid and the charge at that point, summed over all points on the grid. The use of grid energy calculations has been successful in measuring the binding energy of the barnase- barstar complex (Rocchia et al., 2001). The binding energy can be calculated using equation 3.6.

∆Gbinding=∆GgridP•Ca−∆GgridP−∆GgridCa Eq. 3.6

The grid conditions were kept constant for every run. These conditions include the grid size and percentage fill of the lattice. The grid size is an odd integer number of point per side of the cubic lattice. A larger grid size gives a better resolution of the molecules on the lattice and results in more accurate potentials. The grid size for the free energy runs was set at 65, which was more than sufficient for the rather small CD2 protein. The percentage fill of the lattice is the percentage of the objects longest linear dimension to the lattice linear dimension. This will affect the scale of the lattice which is measured in grids/angstrom. A large percentage fill will provide a more detailed mapping of the molecular shape onto the lattice and was set at 20% for these calculations. In addition, the x, y, and z coordinates of each protein were centered at the same position on the lattice for all of the calculations. The interior and exterior dielectric constants were set at 4 and 80, respectively. The ∆Ggrid is calculated by DelPhi in units of kT, where k is

Boltzmann’s constant and T is the absolute temperature, and can be converted into kcal/mol using the relationship 1 kT = 0.582 kcal/mol. Furthermore, the free energy was converted to joules using the relationship 1 cal = 4.184 J and the Kd of each variant was

calculated using equation 3.7.

∆G=RTlnKd Eq. 3.7

where the gas constant R was 8.31451 J/K·mol and the temperature was 298 K. The calculated grid energy of Ca2+ alone was 26.35 kT (15.34 kcal/mol). This number was used for the calculations of the ∆Gbinding for all of the variants. Table 3.6 reveals the

calculated ∆Ggrid for the Ca2+ bound and Ca2+ free forms of the variants, the ∆Gbinding, and

the predicted Kd of each variant.

The free energy of binding and binding affinity follows an interesting trend. Additionally, the predicted binding affinities for the variants are high compared to natural extra-cellular Ca2+ binding proteins. For the T75 variants, T75D has a slightly more negative binding energy as compared to 6D79 indicating the protein may bind Ca2+ stronger than 6D79. Further, for the Kd calculation, a 2-fold difference is observed

between the binding of 6D79 and T75D. These results are expected with the introduction of a negative charge around the Ca2+ coordination shell. On the other hand, the T75K variant displays an opposite effect. Both the ∆Gbinding and Kd change indicate decreased

Ca2+ binding compared with 6D79. In fact, the ∆Gbinding is positive (16.86 kcal/mol),

which may indicate binding is very weak. The predicted Kd value for T75K is 17-fold

weaker than 6D79 indicating the introduction of a positive charge around the Ca2+ binding site has a considerable effect on the Ca2+ binding affinity. It is interesting to note that T75K has a more dramatic effect on the Ca2+ binding affinity than T75D. This result is peculiar but may be attributed to the increased side chain length of Lys as compared to Asp. Lys may have a more significant effect because it is closer to the Ca2+ binding site than Asp.

The effect of removal of a positive charge around the Ca2+ binding site can be observed for the K91 and R34 variants. The differences between these two variants are the distance of each residue from the Ca2+ _{binding site, 11.4 Å for K91 and 9.5 Å for}

(Figure 3.9) and therefore if a negative charge is in this position it may act as an anchor to attract the positively charged Ca2+. On the other hand, the positive charged Lys may act to repel the Ca2+ thus a lower binding affinity is observed. Upon the removal of the positive charge of both of these residues varying results are observed for the ∆Gbinding.

For R34I, a 1.5-fold more negative binding energy compared to 6D79 is observed, while for K91I, the binding energy is actually more positive indicating weaker Ca2+ binding. The introduction of a negative charge at these positions has a more considerable effect on the binding energy. Both R34E and K91E have a more negative binding energy (2.3-fold for R34E and 2-fold for K91E). The calculated binding affinities are also stronger compared to 6D79 with R34E having a slightly more significant effect

Both of the D94 mutants, D94K and D94A, display more negative binding energies and stronger binding affinities than 6D79. The binding affinity differences are 2 and 3-fold stronger for D94K and D94A, respectively. This trend is contradictory to the phenomenon observed in calbindin D9k, where the removal of negative surface charges

around the Ca2+ binding site weakened the Ca2+ binding affinity (Linse et al., 1991). The results of our free energy calculations are flawed due to the limitations of continuum models in the case of metal binding. This may be indicated by the high predicted binding affinities of the variants. The extracellular Ca2+ binding protein cadherin has a Ca2+ binding affinity of 0.16 mM (Yang et al., 2000d). The predicted binding affinity of 6D79 is 300-fold weaker (51.02 mM). There are several reasons continuum models are not accurate in measuring metal binding. First, for measuring binding energies, these continuum models rely on large structural changes between the

apo and bound forms of the components. Because of the electrostatic nature of protein- metal interactions only small differences in positions occur, which result in large errors on the estimation of the interaction energies (Schymkowitz et al., 2005). The effect of these small changes is exacerbated in the 6D79 protein because all of the Ca2+ binding ligands are found in β-sheet secondary structure of the protein. The rigidity of β-sheets severely limits local conformational change upon Ca2+ binding. Second, there has been a lag in the advancement of development of force fields that are sensitive enough to detect small metals and metal induced changes. For this reason, the grid energy calculated by DelPhi is not accurate in calculating the binding energy of metal binding. Due to the small size and charge on the Ca2+, a large error is introduced in the calculation when measuring Ca2+ alone. The use of grid energy has been successful in previous analysis of protein-protein interactions (Nicholls A and Honig, 1991) but to date has not been reported for use in metal binding studies. This study would greatly benefit from advances in the development of force fields that can accurately model metals and metal binding. In the case of the charge distribution variants of 6D79 other methods for predicting binding affinity were explored.