ENERGY OF ANTIBODY-ANTIGEN REACTIONS 145 



range of values of K, then the "normaHzed" Gaussian function 

 (PauHng, Campbell, and Grossberg, 1944) is 



[1/V(7r)cr]exp[ - \niK/Ko)/aY 



The fraction of total combining sites, n, which have a specified 

 binding constant K will be, for an infinitesimally small area (in, ex- 

 pressed as follows : 



dn/n = [1/ V (7r)a] exp[ - \n(K/K,/a]- d \n{K/K,) 



From this Karush and Sonenberg (1949) found (the derivation is 

 given by Klotz, 1953) that the fraction of antibody sites occupied, 

 7'/n, where n is the number of combining sites per antibody (found 

 by Karush to be two in confirmation of much earlier work), is in 

 terms of the concentration c of free hapten, as follows : 



r/n = 1 - [ 1/V (tt)] I {[1 - exp( - a^-)]/[i + ^V exp(a'(7)]l da 



where a is [ln(A7A'o)] A. Karush (1957, 1958) found that if 

 for his D-Ip anti-D-I,, system he took the heterogeneity index o- of 

 antibody to be 2.3, the above equation enabled him to account satis- 

 factorily for the experimentally formed relation between r/c and r. 

 (See Fig. 10-5, where the circles are the experimental points and the 

 curves are theoretical.) 



Heat of Reaction of Isoagglutinins 



The thermodynamic constants for the reaction of the human iso- 

 hemagglutinins have been estimated by Wurmser and Filitti-Wurmser 

 (Filitti-Wurmser, Jacquot-Armand, and Aubel-Lesure, and Wurmser, 

 1954; Wurmser and Filitti-Wurmser, 1957), who have devoted a 

 great deal of penetrating thought and experimental skill to the prob- 

 lem. The methods used are somewhat different from those involved 

 in the studies just discussed and deserve a little space to themselves. 



Wurmser and co-workers showed that the combination of iso- 

 agglutinins with human erythrocytes is reversible, so that equilibrium 

 considerations apply. We can use equation (25), which gives us a 

 relation between the equilibrium constant K and the concentration of 

 free antibody at equilibrium. Equation (6 ) contains two unknown 

 constants : the number of combining sites on a red cell, in, and the 



