ENERGY OF ANTIBODY-ANTIGEN REACTIONS 137 



the value of —7.4 kcal. per bond found by Epstein, Doty, and Boyd 

 (1956) for the reaction of anti-benzenearsonic acid antibodies with 

 the divalent hapten T (terephthalanilide-/',/''-diarsonic acid) (Fig. 

 10-1). 



H203As<' p>NHOC<r ^CONH<^ ^AsOaH, 



Fig. 10-1. Divalent hapten used by Epstein, Doty, and Boyd (1956). 



It would have been expected that the benzenearsonic acid groups 

 in the coupled bovine serum albumin, being coupled through the azo 

 linkage with tyrosine and histidine residues just as in the coupled 

 protein used for immunization, would correspond to the combining 

 sites of the antibody better than the amide-coupled benzenearsonic 

 acid groups of the hapten T. Epstein, Doty, and Boyd suggested 

 that the decreased bond strength was due to some unfavorable feature 

 in the orientation of the groups in the coupled protein. 



In dealing with multivalent antigens which may combine simul- 

 taneously with a number of molecules of antibody, the mathematical 

 problems of formulating the reaction become formidable unless we 

 introduce simplifying assumptions. The simplest assumption is that 

 the free energy of combination of an antibody molecule with a com- 

 bining site of the antigen is the same for all such sites and is not 

 afifected by the number of antibody molecules which have already com- 

 bined with the antigen. With the aid of this assumption, which can 

 hardly be strictly true but which is certainly adequate as a first ap- 

 proximation, we can easily solve the problem, as shown by Linder- 

 str0m-Lang (1924), von Muralt (1930), Eowler (1936), Wyman 

 (1943), and Klotz (1946). If we let the association constant for 

 the formation of a single antibody-antigen bond be K, and the num- 

 ber of combining sites on the antigen molecule (or cell) be ;//, we 

 find the ratio r of antibody molecules combined with an antigen mole- 

 cule (or cell) to be 



r = mKiA)/l\ + A'(A)] (6) 



where (A) is the concentration of free antibody. 



A summary of the principal thermodynamic studies on the antibody- 

 antigen or antibody-hapten reaction is given in Table 10-1. 



