Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2025 / Paper 2 / 3 / d

The two descendant branches of an equal-rate Jukes-Cantor model compose to a single branch of duration $2T$. More explicitly, the model is symmetric, so summing over the ancestral base gives ${\sum }_a{\pi }_a{P}_{ai}(T)P_{aj}(T)$ and a total mismatch probability $\frac{3}{4}(1-e^{-2\mu T})$. Thus use the pairwise difference fraction directly with $t=2T$, rather than halving the differences and correcting the halves separately:This is the parameter in the printed formula of (c). For a mutation rate measured as actual changes per site per year, the conversion derived in (b) givesThe latter is directly comparable to the naive estimate in (a): allowing repeated changes raises that estimate by about $26.4\%$. The exponent parameter and actual event rate are two conventions for the same fitted process, not two different biological estimates.