# Why do we scale the rate matrix of a substitution model to make the average rate of substitution equal 1?

The matrix of transition probabilities for a substitution model Q over time t is found as follows: $$e^{Qt}$$. Since Q and t only show up as a product, we cannot distinguish time and rate of change. So we multiple Q by a constant so that the average rate of substitution is 1. Time t is then measured by the expected number of substitutions per site.

But why does forcing the average rate of substitution to equal 1 have this effect?

• Anyway because this is assumed to be per alignment position without rescaling you cannot derrive the number of substitutions per site per unit time across an alignment.
– M__
Apr 6 '20 at 16:45

This matrix (pretending for a moment that we care only about purine/pyrimidine transitions and thus only two states) is directly interpretable in terms of the number of transitions per unit time: $$Q = \begin{matrix} 0 & 1.5 \\ 0.5 & 0 \\ \end{matrix}$$
Whereas this matrix is harder to think about because it is talking about the number of transitions per (I think?) $$\frac{1}{0.37}$$ units of time: $$Q = \begin{matrix} 0 & 0.37 \\ 0.37 & 0 \\ \end{matrix}$$
Of course, if you have a time-calibrated phylogeny then time is not dimensionless, so it might make sense to use a different scaling, such that the average rate in $$Q$$ is instead "per year" or "per million years" or whatever works within the allowances of floating-point precision and brains.