# 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, 2020 at 16:45

## 1 Answer

@Michael's comment is correct but possibly a little difficult to understand, so I'm going to rephrase it here with an example (please let me know if I'm misrepresenting):

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: $$Q = \begin{matrix} 0 & 0.37 \\ 0.37 & 0 \\ \end{matrix}$$

You will have to rescale any result anyways to talk about it in terms of "per unit time", so you might as well do it in the matrix itself so that at each point we are dealing with a more or less intuitively comprehensible numbers, given that the scaling doesn't actually change anything because time is dimensionless as you point out.

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.

So it's not that rescaling has any particular effect on the math, but rather that rescaling makes it easier for human brains to think about, because we like to represent time in units.