Page 299 - 20dynamics of cancer

284 CHAPTER 13

Figure 13.7 Symmetric stem cell division and regulation of the stem pool to
a constant size by random selection of daughter cells. The three patterns in
each generation—polymorphism, fixation for the light type, or fixation for the
dark type—are shown in Figure 13.6. Here those three patterns are combined
over two generations to form nine patterns. The probability for each pattern
can be obtained by using the hypergeometric distribution. In general, if the
stem pool size remains at N, and symmetric daughter cells migrate randomly
to either the stem or transit pool, then starting with n black stem cells that
double to 2n, and m gray stem cells that double to 2m, with n + m = N, the
probability of retaining 0 ≤ x ≤ α n = min(2n, N) black stem cells in the next

2n 2m 2N
pool of N is given by P(x, n, N) = . Over two generations,
x N−x N

α n
P 2 (x, n, N) = i P(x, i, N)P(i, n, N). From this formula, the probability of re-
taining polymorphism after two generations starting with n = 1 black cell and
N = 2 stem cells is 16/36; the probability of ending with two black cells is
10/36; and the probability of ending with two white cells is 10/36.
With asymmetric division, the stem pool maintains N independent
cell lineages. Any heritable change remains confined to the particular
lineage in which it arose. The N distinct lineages form N parallel lines
of evolution.
With symmetric division, the random selection process causes each
heritable change eventually to disappear or to become fixed in the stem
pool. In effect, only one lineage survives over many generation.
Figure 13.6 introduces a rough guide to the sorting of lineages under
symmetric division. That figure shows a stem pool with N = 2, and the