Page 289 - 20dynamics of cancer

274 CHAPTER 13

8/14 4/14 2/14

Figure 13.2 Probability of the number of mutated cells for a single mutational
event. Each of the three sequences starts with a single cell that then proceeds
3
through three generations of cell division, yielding N = 2 = 8 descendants.
In each sequence, there are 2(N − 1) = 14 branches, each branch representing
an independent DNA copying process. I assume one mutational event with
equal probability of occurring on any branch. On the left, there are 8 third-
level branches, so the probability that the sequence yields one mutated cell is
3
2 /[2(N − 1)] = 8/14. In the middle, there are 4 second-level branches, so
2
the probability that the sequence yields two mutated cells is 2 /[2(N − 1)] =
4/14. On the right, there are 2 first-level branches, so the probability that the
1
sequence yields four mutated cells is 2 /[2(N − 1)] = 2/14. Early mutations
in the sequence occur relatively rarely because there are fewer branches. When
early mutations do occur, they carry forward to a large number of descendant
cells; for this reason, the Luria-Delbrück distribution is sometimes called the
jackpot distribution.

may occur. If one mutational event occurs among those 14 replications,
then how many of the final 8 cells carry the mutation?
Figure 13.2 enumerates the possible outcomes for the simple example
in which there is exactly one mutational event and a single cell divides
regularly to produce 8 descendants. We can gain an intuitive under-
standing of the problem by generalizing the example in Figure 13.2.
Suppose we begin with one precursor cell, which then divides n times
n
to yield N = 2 descendants. Assume that exactly one mutational event
occurs, and that the mutational event happens with equal probability on
any of the 2(N − 1) branches. If the mutation occurs on one branch in
the first division, then 2 −1 = 1/2 of the descendants carry the mutation;
if the mutation occurs on one branch in the second division, then 2 −2 =
1/4 of the descendants carry the mutation. In general, a fraction 2 −i of
i
the descendants carries the mutation with probability 2 /[2(N − 1)] for
i = 1,...,n (Frank 2003b).
My simple calculations in the previous paragraph do not provide a
full description of the Luria-Delbrück distribution, because I assumed
exactly one mutational event over the entire population growth period.
In reality, mutational events arise stochastically, so a full analysis must