Allowing substitution model parameters to differ between branches

Section author: Gavin Huttley

A common task concerns assessing how substitution model exchangeability parameters differ between evolutionary lineages. This is most commonly of interest for the case of testing for natural selection. Here I’ll demonstrate the different ways of scoping parameters across trees for the codon model case and how these can be used for evolutionary modelling.

We start with the standard imports, plus using a canned codon substitution model and then load the sample data set.

We construct the substitution model and likelihood function and set the alignment.

At this point we have a likelihood function with two exchangeability parameters from the substitution model (kappa the transition/transversion ratio; omega the nonsynonymous/synonymous ratio) plus branch lengths for all tree edges. To facilitate subsequent discussion I now display the tree

In order to scope a parameter on a tree (meaning specifying a subset of edges for which the parameter is to be treated differently to the remainder of the tree) requires uniquely identifying the edges. We do this using the following arguments to the likelihood function set_param_rule method:

  • tip_names: the name of two tips

  • outgroup_name: the name of a tip that is not part of the clade of interest

  • clade: if True, all lineages descended from the tree node identified by the tip_names and outgroup_name argument are affected by the other arguments. If False, then the stem argument must apply.

  • stem: Whether the edge leading to the node is included.

The next concepts include exactly what can be scoped and how. In the case of testing for distinctive periods of natural selection it is common to specify distinct values for omega for an edge. I’ll first illustrate some possible uses for the arguments above by setting omega to be distinctive for specific edges. I will set a value for omega so that printing the likelihood function illustrates what edges have been effected, but I won’t actually do any model fitting.

Specifying a clade

I’m going to cause omega to attain a different value for all branches aside from the primate clade and stem (HowlerMon, Human, edge.0).

As you can see omega for the primate edges I listed above have the default parameter value (1.0), while the others have what I’ve assigned. In fact, you could omit the clade argument as this is the default, but I think for readability of scripts it’s best to be explicit.

Specifying a stem

This time I’ll specify the stem leading to the primates as the edge of interest.

Note

I need to reset the lf so all edges have the default value again. I’ll show this only for this example, but rest assured I’m doing it for all others too.

Specifying clade and stem

I’ll specify that both the primates and their stem are to be considered.

Alternate arguments for specifying edges

The likelihood function set_param_rule method also has the arguments of edge and edges. These allow specific naming of the tree edge(s) to be affected by a rule. In general, however, the tip_names + outgroup_name combo is more robust.

Applications of scoped parameters

The general use-cases for which a tree scope can be applied are:

  1. constraining all edges identified by a rule to have a specific value which is constant and not modifiable

lf.set_param_rule(
    "omega",
    tip_names=["Human", "HowlerMon"],
    outgroup_name="Mouse",
    clade=True,
    is_constant=True,
)

  1. all edges identified by a rule have the same but different value to the rest of the tree

lf.set_param_rule(
    "omega", tip_names=["Human", "HowlerMon"], outgroup_name="Mouse", clade=True
)

  1. allowing all edges identified by a rule to have different values of the parameter with the remaining tree edges having the same value

lf.set_param_rule(
    "omega",
    tip_names=["Human", "HowlerMon"],
    outgroup_name="Mouse",
    clade=True,
    is_independent=True,
)

  1. allowing all edges to have a different value

lf.set_param_rule("omega", is_independent=True)

I’ll demonstrate these cases sequentially as they involve gradually increasing the degrees of freedom in the model. First we’ll constrain omega to equal 1 on the primate edges. I’ll then optimise the model.

Note

here I’m specifying a constant value for the parameter and so I must use the argument value to set it. This not to be confused with the argument init that is used for providing initial (starting) values for fitting.

I’ll now free up omega on the primate clade, but making it a single value shared by all primate lineages.

Finally I’ll allow all primate edges to have different values of omega.

We now allow omega to be different on all edges.