Applied

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

By Catherine A. Macken, Alan S. Perelson

Aggregation strategies are studied inside a few various fields--c- loid chemistry, atmospheric physics, astrophysics, polymer technology, 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 perhaps stars) that subscribe to jointly to shape polymers or aggregates. A quantitative thought of aggre- tion was once first formulated in 1916 via Smoluchowski who proposed that the time e- lution of the combination measurement distribution is ruled by way of the countless 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 via ir ok reversible condensation reactions [i-mer ] j-mer -+ (i+j)-mer]. whilst the kernel ok . . might 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, okay > 1), then the Smoluchowski equation might 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 identified and in a few ca ses won't even exist.

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44) derived below for f > 2. C. 5 Here we present the major new result of the paper, the weight fraction dis- tribut ion for aggregates formed when f-valent antigens bind and cross-link bivalent cell surface receptors. 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.

18) as one would expect. 10 ... , The simplest assumption to make about the generating functions F 2 (8), F3 (8), is that r = 2, 3, .. 19) This is equivalent to assuming that the probability that an individual in generation r has k offspring is equal for all generations except the zeroth. In the following, we assume that Eq. 19) holds. E. 11 As formulated in Section B of this chapter, the maj or problem in applying branching processes to the computation of the number or weight fraction distribution of aggregates is to determine the probability distribution PCY = n).

Assuming that within a given generation parents of either type act independently of each other and of their past, the aggregation process may be modeled as a multitype branching process. Again, gene rating functions will prove to be ef- fective tools for deriving weight fractions. The root (zeroth generation) of a tree may be an individual of either type. We will assume forthwith that fertility of either type of "parent" is independent of generation after the zeroth. ] [The same assumption was made in Chapter 2, Eq.

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