Branching Processes Applied to Cell Surface Aggregation by Catherine A. Macken, Alan S. Perelson

By Catherine A. Macken, Alan S. Perelson

Aggregation procedures are studied inside of a few assorted fields--c- loid chemistry, atmospheric physics, astrophysics, polymer technological know-how, and biology, to call just a couple of. Aggregation professional 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 was once first formulated in 1916 by means of Smoluchowski who proposed that the time e- lution of the mixture dimension distribution is ruled by way of the endless procedure of differential equations: (1) okay . . c. c. - c ok = 1, 2, . . . ok 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 by means of ir okay reversible condensation reactions [i-mer ] j-mer -+ (i+j)-mer]. whilst the kernel okay . . will be represented through A + B(i+j) + Cij, with A, B, and C consistent; and the in- 1. J itial is selected to correspond to a monodisperse answer (i. e., c (0) = 1 zero, ok > 1), then the Smoluchowski equation may 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 ok, the answer ij isn't recognized and in a few ca ses would possibly not even exist.

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Results we obtain are the weight fractions The principle for a two-type process wi th general 29 branching probabilities, from which we derive Goldberg's (1952) result on the size distribution of antigen-bivalent antibody aggregates in solution. Goldberg's (1953) generalization for the size distribution of aggregates formed between multivalent antibody and multivalent antigen is derived in Appendix B. B. 2 We first show how antigen-antibody aggregates can be described by family trees containing more than one particle type.

N)i = (ni -- I1) 1. 13) n The genera1ization of a binomial coefficient to a multinomia1 coefficient is apparent if we write i = (i 1 , i 2 , .... , i f ) and define 2: f k 1, 2, ... 14) 34 When f = 2, the multinomial coefficient reduces C. 1 Let w~~) = ~J to the binomial coefficient. P(a rooted tree contains i particles of type A and j particles of type G, given the root is of type A) w~~) = ~J P(a rooted tree contains i particles of type A and particles of type G, given the root is of type G) and p = P(randomly chosen root is of type A) .

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. 22) p The substitution of these into straightforward algebra, DeLis i To make this comparison easier, (1980). bound antigens in a chain. a relations result which agrees Eq. with Eq.

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