Colonies Growing without Bound

Abstract

From Theorem 2 we can predict that colonies containing cell types with branching probabilities greater than 0.5 (i.e., colonies having supercritical growth) will, with non-zero probability, never reach completion of growth. When growth is unbounded, calculating the number of cells in each of the three compartments as n → ∞ is a meaningless exercise, for these numbers will be infinite or zero. (For example, in a colony with an S-cell parent, if $\frac{1}{2} < {p_{s}} < 1$ and p _m < 1, then there is a non-zero probability that the S, M and E compartments will each contain an infinite number of cells.) However, we can calculate the proportion of cells in each of the compartments by appealing to the Kesten-Stigum limit theorems for decomposable Galton-Watson processes (cf. Kesten and Stigum, 1967; Mode, 1971). The essence of these theorems is that, under conditions in which a colony can grow without bound, the limiting proportion of each cell type is a constant that can be calculated as a function of the branching probabilities, p _s , p _m , and p _e . The theorems require calculation of the eigenvalues and eigenvectors of the matrix of means of the offspring distribution for parents of each type. We shall consider separately the cases of colonies with an M-cell parent and colonies with an S-cell parent. Since the former case is easier and is in some sense a subset of the latter case, we shall consider it first.