THE REVERSIBILITY OF ANTIGEN-ANTIBODY UNION 219 



at some point. Modified equations have in fact been evolved that allow for such 

 a saturation Umit. 



With this proviso, the demonstration that there is a close correspondence over 

 most of the range between observed values and values calculated on the basis of 

 Freundlich's isotherm, clearly suggests the possibility that antigen-antibody re- 

 actions are examples of adsorption phenomena. But an equation derived from 

 the law of mass action gives a curve that has a close resemblance to the Freundlich 

 isotherm over a great part of its range. It differs from it in rising rather less steeply 

 at low concentrations of the reagent undergoing adsorption, and asymptoting 

 towards a maximum at high concentrations. Arrhenius and Madsen (see Arrhenius 

 1915) give several examples in which there is a clear correspondence between the 

 observed values and those calculated on the mass-action hypothesis over a con- 

 siderable range of antigen-antibody reactions, though here again there are usually 

 discrepancies at high concentrations of one or other reagent. Biltz (1910) found 

 that the curve of neutralization of tetanolysin by an antitetanic serum would fit 

 the Freundlich isotherm, but would also fit a curve based on the mass-action 

 equation. It does not in fact seem possible, on the basis of the recorded data, to 

 decide between adsorption and mass action by invoking the form of the neutraliza- 

 tion curves obtained. 



The agreement of experimental data with a theoretical equation does not 

 guarantee its validity', whether the equation is derived from the mass law or 

 FreundUch's isotherm. The adsorption isotherm can be used to describe a number 

 of different statistical phenomena such as association between death rates and 

 overcrowding (Brownlee 1925). It describes, in fact, the statistical behaviour 

 of many complex systems of interrelated events, and the agreement of data with 

 it should probably be regarded as evidence of a complex reaction, and not as an 

 indication of the nature of the factors producing the complexity. 



The mass law has been applied by Heidelberger and his colleagues to the exten- 

 sive data they obtained from an analysis of various specific precipitin reactions ; 

 Type III pneumococcal polysaccharide, azoprotein-dye, egg albumin (Heidelberger 

 and Kendall 1935a, b, c, 1937) ; thyroglobins (Stokinger and Heidelberger 1937), 

 and serum albumin (Kabat and Heidelberger 1937). 



The hypothesis assumes thdt antibody may be considered as homogeneous, that antigens 

 and antibody are multivalent with respect to one another, and that before precipitation, 

 there must occur a competing set of bimolecular reactions, whose nature depends on the 

 relative concentrations of antigen and antibody. If antigen and antibody combine in 

 equimolecular proportions an equation apphcable to all cases in which there is excess 

 of antibody is 



y = 2x — x^/A, 



where y is the number of molecules of antibody precipitated when x molecules of antigen 

 are added, and A is the number of antibody molecules precipitated at the equivalence 

 point of a constant-antibody titration. There are a number of objections to this deriva- 

 tion. Among them is the fact that there are not always equimolecular amounts of antigen 

 and antibody at the equivalence point (see Marrack 1938, Heidelberger 1939). Without 

 assuming equimolecular combination at equivalence, the following equation derived from 

 experimental data is found applicable to many systems : 



y = 2Rx - R2xVA , 



where x, y and A are expressed in milligrams, and R is the ratio of antigen to antibody 

 at the equivalence point. This, and other similar equations, are empirically successful 



