Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology (1995)
Chapter: Top-down
Page 124
Forwards and Backwards in the Tree
Hudson (1991) describes many situations in which simulation of genealogical trees is useful. In its simplest form, the idea is to construct (a simulation of) a coalescent tree, with times and branching order, and then superimpose the effects of mutation on this tree using the Poisson nature of the mutation process. In this section we make use of two equivalent descriptions of the effects of mutation in the coalescent tree in which the mutation and coalescence events evolve simultaneously.
Top-down
The first of these methods is a very useful "top-down" scheme exploited by Ethier and Griffiths (1987) in the context of the infinitely-many-sites model. We start at the common ancestor of the sample and think of the genetic process running down to the sample. Just after the first split, we have a sample of two individuals, each of identical genetic type. Attach to each individual a pair of independent exponential alarm clocksone of rate q / 2, the second of rate 1/2and suppose the clocks are independent for different individuals. The q clocks will determine mutations, the other clocks split times. Now watch the clocks until the first one rings: if a q-clock rings, a mutation occurs in that gene, whereas if one of the other clocks rings, a split occurs in which that gene is copied, now making a sample of three individuals. Using the standard "competing exponentials" argument, the probability that a mutation occurs first is
whereas a split occurs first with probability 1/ (q + 1). Furthermore, given that a mutation occurs first, the gene in which it occurs is chosen uniformly and at random, and given that a split occurs first, the gene that is copied is chosen uniformly and at random.
Once an event occurs, the process repeats itself in a similar way. Suppose, then, that there are currently m genes in the sample. Attach independent mutation clocks of rate q / 2 and independent split clocks of
