# Walking Through phyinformR, Part2: Probabilistic Quantification

Learn how to conduct phylogenetic informativeness analyses in R, Part 2

This is a three part tutorial. Part 1 focuses on the generation of PI profiles

This part focuses on advanced features and calculations of quartet resolution probabilities

Part 3 focuses on visualization

### 4. Resolution probability quantification

Townsend et al. (2012) introduced theory that takes into the account the interplay of site rates, time, and internode length between species divergences to assess the predicted probabilty of data contributing to accurate topological resolution

Use of this method requires three inputs:

site rates, state space, and internode lengths

Site rates are covered above. Now we’ll introduce the internode lengths and state spaces and will continue using the avian tree and site rates from Prum et al. (2015) that are distributed with phyinformR.In case these are not in memory already use

library(ape) tree<-read.tree(system.file("extdata","Prumetal_timetree. phy",package="PhyInformR")) rr<-as.matrix(prumetalrates)

### Getting Started

There are three quantities that phyloInformeR calculates with regard to a specified internode given a set of site rates: Quartet Internode Resolution Probability (QIRP, “Quirp”), Quartet Internode Homoplasy Probability (QIHP, “Quip”), and Quartet Internode Polytomy Probability (QIPP, “Quippy”). Townsend et al.1 introduced two ways to calculate these quantities: An analytical approximation and a Monte Carlo based solution. Both approaches depend on site rates and two user defined internode lengths, T (time from present) and t (internode)

The theory of Townsend et al. defines T and t based on a phylogenetic quartet with even branch lengths. Under this assumption, T is the time from the present to the ancestor of a taxon (red in the example above) and t, the focal internode length (grey in the example above). Later in this section we will discuss how to perform similar analyses allowing for uneven quartets

Lets start by calculating approximate solutions for QIRP, QIHP, and QIPP

Using the above illustration, you should have an idea of what T (time) and t (internode distance) you will want to use for your data.

For state space, a binary morphological matrix could be assessed by setting the state space to 2 or amino acid (20 or ~5)1, or other types of data with differing numbers of characters. If you are using nucleotides, despite having a four character states, Simmons et al. have demonstrated the state space to be better modelled using 3 states, so we will go with that for the remainder of this guide.

Here is the implementation using T= 100 million year (Ma) and t= 0.5 Ma

Approximator(100,0.5, rr, 3)

Alternatively you can use the Monte Carlo simulation approach1. Since the simulation is time consuming dependent on the number of simulations and the number of sites, we recommend that you do this on a cluster or while you are doing something else. The output is automatically recorded to file. The input looks similar to the approximation, though now you specify a file name for the probabilities and an image file name. Note that the two files must have different names!

You can also toggle the screen output on and off by setting image to either “TRUE” or “FALSE”. The simulation can be run in parallel so

Remember to set your cores appropriately using the registerDoParallel(cores=8)

Here is the function using the same T and t and 5000 simulations on a subset of sites to save time

rr2<-as.matrix(prumetalrates[1:20000]) parallel.cluster.signal.noise(100,0.5,rr2,5000,3,filename="test",imagename="testimage",image="FALSE

### 5. Advanced resolution probability quantification

Calculations based on the equations of Townsend et al.(2012) make several assumptions

1) equal base frequency distributions in the alignment

2) equal branch lengths within the phylogenetic quartet

3) a Jukes-Cantor model of sequence evolution

Su et al. (2014) provided extensions to the equations of Townsend et al.(2012) that enable any nucleotide substitution model or distribution of base frequencies and Su and Townsend (2015) provided the theoretical framework for relaxing the assumption of even branch lengths

To use these extension, first specify your model settings based on a model selection program such as Modeltest, or Partitionfinder6. Model settings are defined as follows and require changing for different nucleotide substitution models, base frequencies, and branch lengths

For example:

a=1 b=1 c=1 d=1 e=1 f=1 Pi_T=0.25 Pi_C=0.25 Pi_A=0.25 Pi_G=0.25 internode<-c(62.4937, 62.4937, 62.4937, 62.4937, 8.9939)

Pi_T through Pi_G are the empirical base frequency parameters

IMPORTANT | these must sum up to one | Pi_T + Pi_C + Pi_A + Pi_G = 1

a through f are the relative rate parameters which are defined as follows

rCT = rTC = a; rAT = rTA = b; rTG = rGT = c; rCA = rAC = d; rCG = rGC = e; rAG = rGA = f

A table of relative rate parameter settings for different substitution models taken from Su et al. (2014) is attached for reference internode in the above code is as an object containing the numerical values of branch lengths of T1, T2, T3, T4, and the internode length t0 for the four - taxon tree in question IMPORTANT -T1 and T2 must belong to the same sister clade and T3 and T4 must belong to the other clade in the hypothesized topology of the four - taxon problem in question (see above)

Quartet trees do not need to be rooted. All four subtending branches may vary

You can quantify QIHP, QIPP, and QIRP with

allmodel.signal.noise (a,b,c,d,e,f,internode,Pi_T,Pi_C,Pi_A,Pi_G, rr)

From left to right, the three values represent QIHP, QIPP, and QIRP

### Posterior Distributions

Since branch length are rarely known with certainty, phyinformR can also be used to calculate QIRP, QIPP, and QIHP values across a distribution of trees such as those obtained from Bayesian analyses.

For this example, we will read in use the sample distribution of bichir trees from a study by Near et al. 2015 that is provided with the release

library(ape) trees<-read.tree(system.file("extdata","polypterus_trees.phy",package="PhyInformR")) rate_vector<-as.matrix(rag1)

First you will need to specify the quartet of interest. In the bichir dataset, we will look at the clade containing Polypterus congicus as this species was not placed with high support in the tree:

We define the quartet as follows

quart<- c("Polypterus_congicus","Polypterus_bichir","Polypterus_ansorgii" ,"Polypterus_endlicheri" )

The remaining objects should be familiar, please review the above and preceeding page if the variable names seem enigmatic. To compute over a distribution of trees, run the function. Note that here we are truncating the tree distribution for the purpose of example

final<-su.bayes(a,b,c,d,e,f,Pi_T,Pi_C,Pi_A,Pi_G,rate_vector,quart,trees)

This function returns a matrix of internodes and T values from the trees and their associated QIHP,QIPP, and QIRP values> su.bayes runs in parallel

Remember to set your cores appropriately using registerDoParallel(cores=8)

su.bayes returns a matrix of internodes and T values from the trees and their associated QIHP, QIPP, and QIRP values. This matrix can be summarized using

plotPosterior(final, plot="QIPs") #or plotPosterior(final,plot="violin")