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 varied fields--c- loid chemistry, atmospheric physics, astrophysics, polymer technology, 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 subscribe to jointly to shape polymers or aggregates. A quantitative thought of aggre- tion was once first formulated in 1916 by way of Smoluchowski who proposed that the time e- lution of the mixture measurement distribution is ruled by means of the limitless method 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 ok . . should be represented via A + B(i+j) + Cij, with A, B, and C consistent; and the in- 1. J itial situation is selected to correspond to a monodisperse resolution (i. e., c (0) = 1 zero, okay > 1), then the Smoluchowski equation could be co' a relentless; 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 really recognized and in a few ca ses would possibly not even exist.

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A theorem that we modify from Good (1960, 1965) exp10its the implied relationship between coefficients. (G) and w.. 20b) 37 (G) w.. I]} i(j-l) C(uA ){F G1 }C(uG ) ~ j > 2 i > 1 0 j > 2 i 0 ::: ~J . 0 . 21b) where MA ::: FAO (I) and MG ::: F~O(I). The proof of this theorem is given in Appendix A. Tutte (1975) gives a purely combinatorial proof of a more general multidimen- sional version of this theorem. 10 Remark The conditions FA1 (Q) 1 0 , FG1 (Q) 1 0 will generally, although not necessarily, be met in systems restricted to finite-sized aggregates.

1 The concepts and results of Chapter 2 are readily extended to aggregates of more than one particle type. Our interest lies in the aggregation behavior of f-valent antigen and bivalent antibody. Processes involving more than two types, [for example, f-valent antigen in solution with either a mixture of bivalent antibody and Fab fragments antibodies 1 are treated (univalent antibody), or a mixture of different affinity similarly, but the labor involved in carrying out the mathematical details increases dramatically.

26) Although we shall not do so here, one can show that the limit of ur must be the smallest positive root of Eq. 26). Combining Eqs. 27) n=O where wn is the weight fraction of n-mer. Following Good (1963), we introduce the notation C(Sn) {WeS) J for the coefficient of Sn in the expansion of W(S). 28) n F. 26). complicated because first u(S) must be found by solving Eq. Rather than approach the problem this way, we can use Lagrange's expan- sion of an inverse function (cf. Jeffreys and Jeffreys, 1972; Bromwich, 1947; Whittaker and Watson, 1935; Goursat, 1904) to construct from S power series in S.

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