By Catherine A. Macken, Alan S. Perelson

Aggregation techniques are studied inside a couple of various fields--c- loid chemistry, atmospheric physics, astrophysics, polymer technological know-how, and biology, to call just a couple of. Aggregation seasoned ces ses contain monomer devices (e. g., organic cells, liquid or colloidal droplets, latex beads, molecules, or maybe stars) that sign up for jointly to shape polymers or aggregates. A quantitative idea of aggre- tion used to be first formulated in 1916 by way of Smoluchowski who proposed that the time e- lution of the mixture measurement distribution is ruled via the limitless approach of differential equations: (1) okay . . c. c. - c okay = 1, 2, . . . okay 1. J 1. J L i+j=k j=l the place c is the focus of k-mers, and aggregates are assumed to shape through ir ok reversible condensation reactions [i-mer ] j-mer -+ (i+j)-mer]. whilst the kernel okay . . may be represented by way of A + B(i+j) + Cij, with A, B, and C consistent; and the in- 1. J itial is selected to correspond to a monodisperse resolution (i. e., c (0) = 1 zero, ok > 1), then the Smoluchowski equation will be co' a continuing; and ck(O) solved precisely (Trubnikov, 1971; Drake, 1972; Ernst, Hendriks, and Ziff, 1982; Dongen and Ernst, 1983; Spouge, 1983; Ziff, 1984). For arbitrary okay, the answer ij isn't really identified and in a few ca ses would possibly not even exist.

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This result is directly applicable to the production of antibody by antigen-stimulated B lymphocytes, and to the release of histamine by basophils and mast cells. As for the case of bivalent antigens, we shall use a specific binding model to evaluate PAk and PGQ' and hence wij . Here we shall rely upon the model of Perelson (1981) shown schematically in Fig. 2. In this model antigen molecules, each with a total of v binding sites, are placed in the medium at concentration Co at time t = o. Subsequently, antigen free in solution at 51 concentration C reversibly binds to the surface to become singly bound antigen with concentration Cl' In this binding reaction any of the v antigen sites can bind with rate constant k 1 to a free receptor site present at concentration S, Once antigen is bound at a single site, an antigen-receptor complex is formed : : :) c ( ~ cell surfoce Figure 4,2 A model for multivalent antigens binding to and cross-linking bivalent cell surface receptors, taken from Perelson (1981), Antigen, at concen- tration C in the solution surrounding a population of cells, can bind at any one of v (v=6) sites to a free cell surface receptor site, with rate constant k l , Singly bound antigen, at concentration Cl' can dissociate with rate constant k_ l , or bind -- using any of f-l (f=3) free determinants -- a second receptor, with rate constant k2 , Doubly bound antigen, at concentration C2 , can return to the singly bound state by having either of the two receptor-antigen bonds break with rate constant k_ 2 , or can become triply bound with rate constant k 3 , Triply bound antigen, at concentration C3 , can form no additional bonds wi th surface receptors, but any of its three bound sites can dissociate with rate constant 52 whieh ean dissoeiate with rate eonstant k_ 1 .

6 Remark In what follows we call w.. the weight fraction of aggregates con- taining i antibodies and j antigens. ~J However, in analogy with the arguments made in Section B of Chapter 2, one can see that this is rigorously true only if antibody and antigen each have unit weight. e. p = MWAA/(MWAA + MWGG)) will allow proper weight fractions to be computed with Eq. 15). 12. To compute weight fractions we must determine ties can be expressed in terms of {PAk; k = 0, w(~) 1J and 1, 2} and w~~). 1J {PG~; ~ These quanti- = 0, 1, ...

The concentration of antigen bound to the surface at exactly one site is denoted by C1 (t). permits a second The fluidity of the membrane singly bound antigen to react with a receptor. free receptor site on a The concentration of antigen that cross-links two receptors is denoted by C2 (t). The total concentration of antigen in the system Co = C(t) + C1 (t) + C2 (t). 17) and hence from Eqs. 18) In the model, receptor sites are assumed to act independently. 19) is used to denote the probability that a receptor site is bound, then the probability that k sites on a single receptor are bound is given by k [cL Eq.