Branching processes

A branching process starts with one individual in generation 0. This individual produces a random number of individuals for generation 1, the number distributed according to a probability law G. Each individual in generation 1 and all subsequent generations produces offspring independently according to the law G.

The process can be studied generation by generation focussing on the (random) numbers in each generation and on issues of population growth and extinction. If generation times are assumed to be variable, however, the generations soon overlap in time and then properties such as "total numbers at time t" are of interest. Such a process is usually referred to as the "age-dependent branching process". (There are also the more general models of Moyal and Jagers, where many of the independence assumptions are dispensed with.)

Papers #13 , #23, #35 and #63 use the age-dependent branching process, and variants, as a model for cell division. The generation times of proliferating cells are variable. When cells are all dividing, each cell produces 2 daughter cells for the next generation (so G is very simple, with all probability mass on the integer 2). #13 and #23 deal with experiments where drugs are administered to a cell culture, affecting DNA replication in a manner which gives useful information on the variability of cell generation times and some components of this time.

During the period when cellular differentiation occurs, one or both of the cells might be non-proliferating (living almost indefinitely relative to the generation times of those cells which continue to proliferate). So there are 0, 1 or 2 proliferating cells produced for the next generation. Using the basic construct of the age-dependent branching process, Valerie Morris and I investigate in #35 and #63 the gradual cessation of cell division and lengthening of cell generation times during the development of a mature tissue (eg. the chick retina). Paper #38 is a dual to #35, analysing the same issues with a semi-deterministic model.

Paper #32 provides a fundamental contribution, explained by the title, to the mathematical theory of age-dependent branching processes.

13. Morris, V.B., Cowan, R. and Culpin, D. The variability of cell cycle times measured in vivo  in the embryonic chick retina by continuous labelling with 5-bromodeoxyuridine. Nature 280 68-70 (1979).

23. Cowan, R. , Culpin, D. and Morris, V. B. A method for the measurement of variability in cell lifetimes. Math Biosciences 54 249-263 (1981).

38. Cowan, R. and Morris, V. B. Determination of proliferative parameters from growth curves. Cell &Tissue Kinetics 20 153-159 (1987).

31. Staudte, R. G., and Cowan, R. Models for dependent cell populations. Proc. Imacs Conference, Oslo (1985).

32. Cowan, R. Branching process results in terms of moments of the generation time distribution. Biometrics 41 681-689 (1985).

33. Cowan, R. and Staudte, R. G. The bifurcating autoregression model in cell lineage studies. Biometrics 42 769-783 (1986) .

35. Cowan, R. and Morris, V. B. Cell population dynamics during the differentiative phase of Tissue Development. J. Theoretical Biology 122 , 205-224 (1985).

40. Cowan, R. and Morris, V. B. Division Rules for Polygonal Cells. J. Theoretical Biology 131 33-42 (1988).

43. Cowan, R. The division of space and the Poisson distribution. Adv. Appl. Prob. 21 233-234 (1989) .

63. Morris, V. B. and Cowan, R. An analysis of the growth of the retinal cell population in embryonic chicks yielding proliferative ratios and cell-cycle times for successive generations of cell cycles. Cell Proliferation 28, 373-391 (1995).

65. Cowan, R. and Chen, S. The random division of faces in a planar graph. Adv. Appl. Prob. 28, 377-383 (1996). Download postscript version (with figures).

66. Staudte, R. G., Zhang, J., Huggins, R. M. and Cowan, R. A re-examination of the cell-lineage data of E.O.Powell. Biometrics 52 1214-1222 (1996)

The works with Robert Staudte, papers #31, #33 and #66, do not conform to the branching-process models since cell generation-times, as observed in a finite tree of cells, are assumed to be dependent -- mother with daughter and sister with sister. The papers are about the statistical estimation of such correlations from cell lineage data.

Papers #40, #43 and #65 are ostensibly about the random geometry, or more precisely, the random topology of polygons which are continually divided into two by various random rules. The mathematics is, however, heavily dependent on the theory of "multitype" branching processes. Nice examples for any teacher attempting to illustrate his/her lectures on the multitype process!

Papers #40 and #43 have a very fundamental, and somewhat surprising, result about the genesis of the Poisson distribution. Link to a description of these "random topology" papers.


Here is a personal branching process,   working backward in time

Link to a related page on cellular biology.