By Catherine A. Macken, Alan S. Perelson
Aggregation tactics are studied inside of a few various 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 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 means of Smoluchowski who proposed that the time e- lution of the mixture measurement distribution is ruled via the endless procedure of differential equations: (1) okay . . c. c. - c okay = 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 through ir ok reversible condensation reactions [i-mer ] j-mer -+ (i+j)-mer]. while the kernel ok . . should be represented by means of 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 answer (i. e., c (0) = 1 zero, okay > 1), then the Smoluchowski equation should 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 recognized and in a few ca ses won't even exist.
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Extra info for Branching Processes Applied to Cell Surface Aggregation Phenomena
Writing Eq. 23) 50 Substituting Eqs. 22) into Eq. 24) Finally, introducing the relationship between weight fraction and concentration w. 25) SO/2 + Co one obtains Eq. (81) of Perelson and DeLisi (1980): m. 26) Perelson and DeLis i (1980) construct a system of differential equations based on the law of mass-action from which one can derive Cl and C2 as explicit functions of time. Set) is determined by Eq. 16), and the evolution in time of the aggregate size distribution can then be found from Eq.
We now show that W.. 37) To see the equivalence of these formulae, recall from Eqs. 15) that Hence Eq. J .. which is the alternative form, Eq. 37). 13 one of Remark Using Theorem 3 reduces the effort of calculating w.. J fraction generating function. be handled separately since Cases in which i or j are zero would still need to w~~) and w~~) cannot be evaluated by use of Theorem 3. CHAPTER 4 AGGREGATE SIZE DISTRIBUTION ON A CELL SURFACE A. 1 One of the many benefits of using branching processes to study aggregation phenomena is the ability to specify arbitrary functions for PAk and PGQ' the respective probabilities of k sites on an antibody and Q sites on an antigen being bound.
To see this we need a very important result that relates w(~) with w(~). 33) 41 Proof Let N.. be the number of (i,j)-mers in the chemical system. J Then, follow- ing the argument of Section B of Chapter 2, iN .. J rooted trees corresponding to (i,j)-mers. (A) wij By definition, number of (i,j)-trees with an antibody root number of trees with an antibody root iN .. Q. Q. 35) Multiplying Eq. 34) by j and Eq. 35) by i and then rearranging, yields Eq. 33), thus proving the theorem. We now show that W..
Branching Processes Applied to Cell Surface Aggregation Phenomena by Catherine A. Macken, Alan S. Perelson