Exploration of landscapes of phylogenetic trees

Thibaut Jombart, Michelle Kendall

2018-02-21

treespace implements new methods for the exploration and analysis of distributions of phylogenetic trees for a given set of taxa.

Installing treespace

To install the development version from github:

library(devtools)
install_github("thibautjombart/treespace")

The stable version can be installed from CRAN using:

install.packages("treespace")

Then, to load the package, use:

library("treespace")

Content overview

The main functions implemented in treespace are:

Other functions are central to the computations of distances between trees:

Distributed datasets include:

Exploring trees with treespace

We first load treespace, and the packages required for graphics:

library("treespace")
library("adegenet")
library("adegraphics")
library("rgl")

The function treespace defines typologies of phylogenetic trees using a two-step approach:

  1. perform pairwise comparisons of trees using various (Euclidean) metrics; by default, the comparison uses the Kendall and Colijn metric (Kendall and Colijn, 2016) which is described in more detail below; other metrics rely on tip distances implemented in adephylo (Jombart et al., 2010) and phangorn (Schliep 2011).

  2. use Metric Multidimensional Scaling (MDS, aka Principal Coordinates Analysis, PCoA) to summarise pairwise distances between the trees as well as possible into a few dimensions; the output of the MDS is typically visualised using scatterplots of the first few Principal Components (PCs); this step relies on the PCoA implemented in ade4 (Dray and Dufour, 2007).

The function treespace performs both tasks, returning both the matrix of pairwise tree comparisons ($D), and the PCoA ($pco). This can be illustrated using randomly generated trees:

# generate list of trees
set.seed(1)
x <- rmtree(10, 20)
names(x) <- paste("tree", 1:10, sep = "")

# use treespace
res <- treespace(x, nf=3)
names(res)
## [1] "D"   "pco"
res
## $D
##        tree1 tree2 tree3 tree4 tree5 tree6 tree7 tree8 tree9
## tree2  26.00                                                
## tree3  31.06 26.74                                          
## tree4  42.85 42.12 44.44                                    
## tree5  30.66 27.71 27.37 44.79                              
## tree6  36.50 31.18 30.18 41.81 31.59                        
## tree7  34.64 28.71 29.48 40.35 31.11 32.37                  
## tree8  28.97 26.29 24.45 43.74 23.47 30.41 29.00            
## tree9  29.63 27.42 27.48 45.61 26.31 30.89 29.77 24.60      
## tree10 34.87 30.00 29.44 44.97 34.06 31.05 34.41 31.54 32.59
## 
## $pco
## Duality diagramm
## class: pco dudi
## $call: dudi.pco(d = D, scannf = is.null(nf), nf = nf)
## 
## $nf: 3 axis-components saved
## $rank: 9
## eigen values: 142.1 76.52 62.69 49.88 41.07 ...
##   vector length mode    content       
## 1 $cw    9      numeric column weights
## 2 $lw    10     numeric row weights   
## 3 $eig   9      numeric eigen values  
## 
##   data.frame nrow ncol content             
## 1 $tab       10   9    modified array      
## 2 $li        10   3    row coordinates     
## 3 $l1        10   3    row normed scores   
## 4 $co        9    3    column coordinates  
## 5 $c1        9    3    column normed scores
## other elements: NULL

Pairwise tree distances can be visualised using adegraphics:

# table.image
table.image(res$D, nclass=30)

# table.value with some customization
table.value(res$D, nclass=5, method="color", 
            symbol="circle", col=redpal(5))

The best representation of these distances in a 2-dimensional space is given by the first 2 PCs of the MDS. These can be visualised using any scatter plotting tool; here we use the treespace function plotGroves, based on the adegraphics function scatter:

plotGroves(res$pco, lab.show=TRUE, lab.cex=1.5)

Alternatively, plotGrovesD3 creates interactive plots based on d3.js:

plotGrovesD3(res$pco, treeNames=1:10)

Tree labels can be dragged into new positions to avoid problems such as overlapping.

The functionality of treespace can be further illustrated using ape’s dataset woodmouse, from which we built the 201 trees supplied in woodmiceTrees using the neighbour-joining and bootstrapping example from the ape documentation.

data(woodmiceTrees)
wm.res <- treespace(woodmiceTrees,nf=3)

# PCs are stored in:
head(wm.res$pco$li)
##         A1     A2      A3
## 1  -0.9949 -1.363 -0.7918
## 2  -0.6137 -1.014 -0.6798
## 3   2.6667  4.219 -2.9293
## 4 -13.6081  1.854  1.0947
## 5   2.1980  4.176 -3.1960
## 6   3.6013  4.865  2.9853
# plot results
plotGrovesD3(wm.res$pco)

Packages such as adegraphics and ggplot2 can be used to make alternative plots, for example visualising the density of points within the space.

The treespace function multiDist simply performs the pairwise comparison of trees and outputs a distance matrix. This function may be preferable for large datasets, and when principal co-ordinate analysis is not required. It includes an option to save memory at the expense of computation time.

Identifying clusters of trees

Once a typology of trees has been derived using the approach described above, one may want to formally identify clusters of similar trees. One simple approach is:

  1. select a few first PCs of the MDS (retaining signal but getting rid of random noise)

  2. derive pairwise Euclidean distances between trees based on these PCs

  3. use hierarchical clustering to obtain a dendrogram of these trees

  4. cut the dendrogram to obtain clusters

In treespace, the function findGroves implements this approach, offering various clustering options (see ?findGroves). Here we supply the function with our treespace output wm.res since we have already calculated it, but it is also possible to skip the steps above and directly supply findGroves with a multiPhylo list of trees.

wm.groves <- findGroves(wm.res, nclust=6)
names(wm.groves)
## [1] "groups"    "treespace"

Note that when the number of clusters (nclust) is not provided, the function will display a dendrogram and ask for a cut-off height.

The results can be plotted directly using plotGrovesD3 (see ?plotGrovesD3 for options):

# basic plot
plotGrovesD3(wm.groves)
# alternative with improved legend and tooltip text, giving the tree numbers:
plotGrovesD3(wm.groves, tooltip_text=paste0("Tree ",1:201), legend_width=50, col_lab="Cluster")
# plot axes 2 and 3. This helps to show why, for example, clusters 2 and 4 have been identified as separate, despite them appearing to overlap when viewing axes 1 and 2.
plotGrovesD3(wm.groves, xax=2, yax=3, tooltip_text=paste0("Tree ",1:201), legend_width=50, col_lab="Cluster")

We can also plot in 3D:

# prepare a colour palette:
colours <- fac2col(wm.groves$groups, col.pal=funky)
plot3d(wm.groves$treespace$pco$li[,1],
       wm.groves$treespace$pco$li[,2],
       wm.groves$treespace$pco$li[,3],
       col=colours, type="s", size=1.5,
       xlab="", ylab="", zlab="")

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treespaceServer: a web application for treespace

The functionalities of treespace are also available via a user-friendly web interface, running locally on the default web browser. It can be started by simply typing treespaceServer(). The interface allows you to import trees and run treespace to view and explore the tree space in 2 or 3 dimensions. It is then straightforward to analyse the tree space by varying \(\lambda\), looking for clusters using findGroves and saving results in various formats. Individual trees can be easily viewed, including median trees per cluster (see below). Pairs of trees can be viewed together with their tip-differences highlighted using the function plotTreeDiff, and collections of trees can be seen together using densiTree from the package phangorn. It is fully documented in the help tab.

Finding median trees

When a set of trees have very similar structures, it makes sense to summarize them into a single ‘consensus’ tree. In treespace, this is achieved by finding the median tree for a set of trees according to the Kendall and Colijn metric. That is, we find the tree which is closest to the centre of the set of trees in the tree landscape defined in treespace. This procedure is implemented by the function medTree:

# get first median tree
tre <- medTree(woodmiceTrees)$trees[[1]]

# plot tree
plot(tre,type="cladogram",edge.width=3, cex=0.8)

However, a more complete and accurate summary of the data can be given by finding a summary tree from each cluster. This is achieved using the groups argument of medTree:

# find median trees for the 6 clusters identified earlier:
res <- medTree(woodmiceTrees, wm.groves$groups)

# there is one output per cluster
names(res)
## [1] "1" "2" "3" "4" "5" "6"
# get the first median of each
med.trees <- lapply(res, function(e) ladderize(e$trees[[1]]))

# plot trees
par(mfrow=c(2,3))
for(i in 1:length(med.trees)) plot(med.trees[[i]], main=paste("cluster",i),cex=1.5)

These trees exhibit a number of topological differences, e.g. in the placement of the (1007S,1208S,0909S) clade. To examine the differences between the trees in a pairwise manner, we can use the function plotTreeDiff, for example:

# Compare median trees from clusters 1 and 2:
plotTreeDiff(med.trees[[1]],med.trees[[2]], use.edge.length=FALSE, 
             treesFacing = TRUE, colourMethod = "palette", palette = funky)

# Compare median trees from clusters 1 and 4, and change aesthetics:
plotTreeDiff(med.trees[[1]],med.trees[[4]], type="cladogram", use.edge.length=FALSE, 
             treesFacing = TRUE, edge.width=2, colourMethod="palette", palette=spectral)

Performing this analysis enables the detection of distinct representative trees supported by data.

Note that in this example we supplied the function medTree with the multiPhylo list of trees. A more computationally efficient process (at the expense of using more memory) is to use the option return.tree.vectors in the initial treespace call, and then supply these vectors directly to medTree. In this case, the tree indices are returned by medTree but the trees are not (since they were not supplied).

Emphasising the placement of certain tips or clades

In some analyses it may be informative to emphasise the placement of particular tips or clades within a set of trees. This can be particularly useful in large trees where the study is focused on a smaller clade. Priority can be given to a list of tips using the argument emphasise.tips, whose corresponding values in the vector comparison will be given a weight of emphasise.weight times the others (the default is 2, i.e. twice the weight).

For example, if we wanted to emphasise where the woodmice trees agree and disagree on the placement of the (1007S,1208S,0909S) clade, we can simply emphasise that clade as follows:

wm3.res <- treespace(woodmiceTrees,nf=2,emphasise.tips=c("No1007S","No1208S","No0909S"),emphasise.weight=3)

# plot results
plotGrovesD3(wm3.res$pco)

It can be seen from the scale of the plot and the density of clustering that the trees are now separated into more distinct clusters.

wm3.groves <- findGroves(woodmiceTrees,nf=3,nclust=6,emphasise.tips=c("No1007S","No1208S","No0909S"),emphasise.weight=3)
plotGrovesD3(wm3.groves)

Conversely, where the structure of a particular clade is not of interest (for example, lineages within an outgroup which was only included for rooting purposes), those tips can be given a weight less than 1 so as to give them less emphasis in the comparison. We note that although it is possible to give tips a weighting of 0, we advise caution with this as the underlying function will no longer be guaranteed to be a metric. That is, a distance of 0 between two trees will no longer necessarily imply that the trees are identical. In most cases it would be wiser to assign a very small weighting to tips which are not of interest.

Method: characterising a tree by a vector

Kendall and Colijn proposed a metric for comparing rooted phylogenetic trees (Kendall and COlijn, 2016). Each tree is characterised by a vector which notes the placement of the most recent common ancestor (MRCA) of each pair of tips, as demonstrated in this example: