A complete solution is obtained for the equilibrium between bivalent antigen and bivalent antibody in a system restricted to the formation of linear chains. Some typical results of antigen titrations are tabulated, showing the effects of variation in bond-strength, steric interaction, relative proportion of reactants, and volume, on the maximal degree of aggregation possible. The principal conclusion is that surprisingly little aggregation can occur by specific means unless the reactions are of irreversible character. This conclusion, logically extended to systems of higher valence, emphasizes the importance of the structural effects of ring closure to the application of the lattice hypothesis.
The problem of ring closure and its effect on the equilibrium is discussed, and a general method of approach is indicated. No explicit treatment is attempted at this time, except for the limiting case of perfectly flexible chains.
A general treatment leading from equilibrium to statistical expressions for the structure of linear polymers is appended. Certain inferences, applicable to real precipitates, concerning the relation between the structure of aggregates and the properties of individual bonds, are drawn from the study of the simpler system. In immune precipitates, structural effects are probably far more important in determining the properties of the aggregate than is the intrinsic strength of the bond itself. It is for this reason that the composition of the precipitate, which is intimately connected with the initial equilibrium between specific bond-forming groups, is easier to predict than the amount of precipitate, which is largely determined by structural factors. It follows also that immune reactions may differ considerably depending on the physical character of the antigen, independently of the nature of the combining groups.