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Branching processes applied to cell surface aggregation phenomena. (English) Zbl 0573.92001

Lecture Notes in Biomathematics, 58. Berlin etc.: Springer-Verlag. VIII, 122 p. DM 25.00 (1985).
This is a short monography on how to apply the machinery of the Galton- Watson branching process to aggregation phenomena, written by two researchers who contributed significantly to this application. To understand the essence of the approach, let us suppose that a (bio)chemical particle has n identical sites to each of which one of the n sites of another (identical) particle can attach, with probability p. This process is repeated so that a particle aggregate is formed. The basic question to answer is what will be the expected distribution of sizes of the aggregates formed independently by many primary particles.
The number of ”first generation” particles attached to the primary is a random variable (r.v.) with probability generating function (p.g.f.) \(F_ 0(s)=[ps+(1-p)]^ n.\) Furthermore, the number of the \((i+1)^{st}\) generation particles attached to each of the \(i^{th}\) generation particles, is a r.v. with p.g.f. \(F_ 1(s)=[ps+(1-p)]^{n- 1}\) (one site is already occupied). Thus, the size (Y) of the aggregate is a r.v. equal to the sum of the number of particles in all the generations present in this particular realization of the Galton-Watson process defined by the p.g.f.-s \(F_ 0(s)\) and \(F_ 1(s)\). Now, a standard functional equation can be written for the p.g.f. of the r.v. Y, from which all the relevant information can be extracted.
It is clear that if the Galton-Watson process is critical or supercritical, then the existence of infinite size aggregates is predicted. These ”theoretically infinite” complexes correspond to a special phase termed ”gel”. As another extension, it is possible to invoke multitype Galton-Watson processes, in order to describe complexes built from particles of different types, with different numbers of binding sites. Numerous generalizations are conceivable, the principle remaining essentially the same. The elegant ”symbolic” p.g.f. approach permits one to avoid the involved combinatorics which plagued older derivations.
The book has seven chapters dealing with: (1) Basics of aggregation processes, (2) Applications of the single type Galton-Watson process (Flory-Stockmayer distribution), (3-4) Application of the multi type process (antigen-antibody complexes, and antigen-receptor aggregates on cell surfaces), (5-6) Criticality and gelation, and (7) Conclusions and extensions. Three appendices contain longer proofs and derivations.
The book is mathematically self-contained and accessible even to readers with very little knowledge of probability.
Reviewer: M.Kimmel

MSC:

92B05 General biology and biomathematics
92F05 Other natural sciences (mathematical treatment)
92-02 Research exposition (monographs, survey articles) pertaining to biology
60J85 Applications of branching processes
92Exx Chemistry
92Cxx Physiological, cellular and medical topics