Rate Matrix Scaling

Scaling types

There are 3 scaling schemes implemented:

• synonymous is the default scaling. This forces the branch length to be proportional to the expected number of synonymous substitutions per site.
• non-synonymous scaling forces the branch length to be equivalent to the expected number of non-synonymous substitutions per site.
• substitution scaling forces the branch length to be equivalent to the expected number of both types of substitutions per site.
• none can be specified to leave the substitution rates unscaled.

Scaling type is specified as an optional parameter to substitution model constructors, e.g.:

MutationSelection(..., scaling_type = "synonymous")

Implementation

Scaling rate is calculated as:

\[ S(Q, \pi, t) = \sum \sum_{ij}\pi_i Q_{ij}^{(t)} \]

• \(Q\) is the substitution rate matrix
• \(\pi\) is the equilibrium distribution for \(Q\)
• \(t\) is the scaling type (outlined above)

The inner summation is over all the substitutions from each state (codon) \(i\) to each state \(j\). \(Q^{(t)}\) means substitution rate for a substitution of type \(t\) (e.g. synonymous substitution).