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Branching processes in biology. (English) Zbl 0994.92001

Interdisciplinary Applied Mathematics. 19. New York, NY: Springer. xviii, 230 p. (2002).
This is a book written jointly by a mathematician and a cell biologist, who have collaborated on research in branching processes for more than a decade. In their own words, their monograph is intended for “mathematicians and statisticians who have had an introduction to stochastic processes but have forgotten much of their college biology, and for biologists who wish to collaborate with mathematicians and statisticians.” They have largely succeeded in achieving their goal. There are 7 chapters, the eighth being a comprehensive list of references (16 1/2 pages), three Appendices, and Glossaries of biological terms for mathematicians and mathematical terms for biologists.
Chapter 1, Motivating Examples and other Preliminaries, introduces the branching processes through examples such as the extinction of family names, PCR (Polymerase Chain Reaction), and a genealogical approach. The branching property is described, the probability generating function (pgf) method outlined and branching processes classified.
Chapter 2, Biological Background, presents mathematicians with the main features of genes, cells and cancer. Genomes are discussed, as are changes in DNA and chromosomes. Cell cycle kinetics and cell division are considered, as is cancer drug resistance and chemotherapy.
Chapter 3, The Galton-Watson Process, describes the oldest and simplest of discrete-time branching processes. The functional equation for the pgf of the \(n\)-th generation offspring \(Z(n)\) is derived, and the model is applied to the cell cycle with death and quiescence. The effect of criticality on extinction is examined, as are some asymptotic properties of \(Z(n)\). The process in a random environment, and the bisexual process are also considered.
Chapter 4, The Age-dependent Process: the Markov Case, analyzes time-continuous branching processes with exponential lifetime distributions. While this latter assumption is not realistic, it leads to conveniently computable expressions. The model is applied to the clonal resistance of cancer cells, and to the genealogies of branching processes; the age of the mitochodrial Eve is estimated.
Chapter 5, the Bellman-Harris Process, considers the case where the lifetimes of particles have arbitrary distributions. This is generally a non-Markovian process, but two of its special cases, the Galton-Watson process and the age-dependent branching process with exponential lifetimes are Markovian. Integral equations for the pgf of the process are derived, and its basic properties found. Some asymptotic properties in the supercritical case are given.
Chapter 6, Multitype Processes, extends the previous models to cases where there are several types of particles. The Luria-Delbrück model is examined, as is also the positive regular case of the multitype Galton-Watson process. Applications are provided for two cell populations, the cell cycle with chemotherapy, and cell surface aggregation phenomena, as well as PCR, among others.
Chapter 7, Branching Processes with Infinitely Many Types, examines examples of branching processes with infinite-type spaces. While no systematic theory is presented, several applications are outlined: pgfs and expectations are derived and the loss of telomere sequences modeled. A model of unstable gene amplifiction is presented, and a variety of Galton-Watson and Bellman-Harris processes with denumerably many types and branching random walks are outlined. The chapter concludes with an account of Yule’s evolutionary process.
The book can be strongly recommended to all students of branching processes; all libraries should have a copy of it.

MSC:

92B05 General biology and biomathematics
60J85 Applications of branching processes
92-02 Research exposition (monographs, survey articles) pertaining to biology
92C37 Cell biology
92C40 Biochemistry, molecular biology
92C50 Medical applications (general)
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