Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / C5 / c

In the Yule process, each of $k$ branches gives birth independently at rate $\lambda$ per generation. The next birth therefore has total rate $k\lambda$. Its waiting time has an exponential distribution, so the mean phylogenetic epoch duration isFor the ensuing calculation use $E_2=1/(2\lambda )$, $E_3=1/(3\lambda )$ and $E_4=1/(4\lambda )$. This is the inter-event waiting-time convention requested in the question, including an observation interval with four branches. Stopping the forward process immediately when the fourth branch is born would instead leave a zero terminal interval and describe a different sampling convention.

For a haploid population with $N$ possible parents, each pair of ancestral lineages merges backward at rate $1/N$ in the Kingman's coalescent approximation. With $k$ lineages there are $\left(\genfrac{}{}{0ex}{}{k}{2}\right)$ pairs. Hence the mean phylogenetic epoch duration isThus $E_2=N$, $E_3=N/3$ and $E_4=N/6$. For an autosomal sample from $N$ diploid individuals, replace the number of gene copies by $2N$, giving $4N/[k(k-1)]$. A mitochondrial sample naturally uses a haploid maternal effective population size; the common factor cancels in the ratios below.