By Catherine A. Macken, Alan S. Perelson

Aggregation tactics are studied inside of a couple of 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 perhaps stars) that sign up for jointly to shape polymers or aggregates. A quantitative concept of aggre- tion was once first formulated in 1916 by way of Smoluchowski who proposed that the time e- lution of the combination measurement distribution is ruled by means of the countless method of differential equations: (1) ok . . 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 way of ir okay reversible condensation reactions [i-mer ] j-mer -+ (i+j)-mer]. whilst the kernel okay . . might be represented by means 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 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 ok, the answer ij isn't identified and in a few ca ses won't even exist.

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**Example text**

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). , 2. 20) 23 =1 By Eq. 22) Under the assumption of equal generating functions for all generations after the zeroth, cf. Eq. 23) = 2, 3, '" where u r _ 1 (6) is 6F 1 composed with itself r-1 times. Recall that an n-mer is represented by a set of n family trees all of which have gone extinct before the nth generation.

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). , 2. 20) 23 =1 By Eq. 22) Under the assumption of equal generating functions for all generations after the zeroth, cf. Eq. 23) = 2, 3, '" where u r _ 1 (6) is 6F 1 composed with itself r-1 times. Recall that an n-mer is represented by a set of n family trees all of which have gone extinct before the nth generation.

9) and provided i > 1, and j become (A) w.. 15) w .. 1J i > From the properties of the binomial coefficient [Eq. 12)), w.. 1J j > I =0 if li-jl ~ 1. Thus if there are i bivalent antibodies in an aggregate, there can only be i-I, i, or i +1 bivalent antigens. This is to be expected since aggregates of bivalent antigen and bivalent antibody must be linear chains with antigens and antibodies alternating along the chain. 4 Linear aggregates formed between bivalent receptors (cell surface immunoglo- bulin) and bivalent ligand have been weIl studied.