Rate Matrix Scaling

PalantiR scales codon substitution rates to assign particular meanings to branches on a phylogeny.

Scaling types

There are four 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.:

use_genetic_code("Standard nuclear")
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).

At equilibrium the identity is exact: with the default synonymous scaling, a branch of length one carries one expected synonymous substitution per codon. On branches where the model changes and the process is temporarily out of equilibrium, the branch-length currency is maintained by the rescaling machinery described in Temporal Heterogeneity Rescaling.