INTERACTIONS OF HAPTENS WITH ANTIBODIES 277 



various components, the values of J(pdtsp may be calculated and compared 

 with the experimental values derived from the combining constants Kq. 



The equilibrium distance d^ may be estimated by adding the group van 

 der Waals' radius to an average protein group radius. The latter was as- 

 signed a value of 2.0 A by Pauling and Pressman (1945) but it is felt that 

 this figure is a little large. Perusal of Table 6-9 for groups common in pro- 

 teins, such as amino, hydroxyl, phenyl, hydrocarbon, and peptide, and 

 considering that these groups will often be flexible and adjust to near- 

 minimal distance, leads one to conclude that the average protein group 

 radius is nearer 1.8 A. Next, one must assume some reasonable figure for 

 the number of water molecules adjacent to the hapten groups and antibody 

 site. Pauling and Pressman assumed n = 8 but it is likely, since these groups 

 usually will not disrupt the water structure and pack closely about the 

 group, that this value is high. If one assumes that the hapten groups are 

 not greatly different in size from a water molecule, it will be seen that 

 n — 3 will be reasonable (the fourth water molecule that would surround 

 the isolated group is replaced by the bond). Finally, the molar refraction 

 of the antibody, which refers to the integrated refractions of the protein 

 groups over the surface in contact with the hapten group, must be esti- 

 mated. Pauling and Pressman calculated a value of 7?^ = 5.90 ml for a 

 portion of the protein equivalent to one water molecule, using the index 

 of refraction of squash seed globulin. From the summary of the indices 

 of refraction of eight wa' !r-free proteins by Doty and Geiduschek (1953), 

 the average value would be 1.604, from which -R4 = 5.28 ml, and this va- 

 lue, as probably more accurate, will be used here. It must be noted that 

 R^ will differ in different regions of the protein and that the figure assumed 

 above is approximate only. 



Equation 6-116 may now be rewritten using n = 3 and reasonable values 

 for the electron number Z: 



18 7 

 ^a.sr, = - -y^ [2.837?^ - 27?^] [3/2^ - 2>R^] (6-117) 



for all the groups considered except NO2 and C00~, for which the numerical 

 factor of i?Y is 3.46. Calculations based on this equation are presented in 

 Table 6-19 and compared with the experimental values for the smaller 

 hapten groups. The combining constants for the two antibodies often differ 

 due to the different patterns presented by the regions adjacent to the groups; 

 it is hoped that by averaging the energy differences, forces others than 

 dispersion, such as hydrogen bonding, may be partly eliminated. The 

 agreement between the experimental over-all free energy changes and the 

 calculated dispersion energy is quite good considering that the equilibrium 

 distances can only be approximately estimated. It would appear that dis- 

 psrsion forces are mainly responsible for the interaction of these groups 



