# Searching for morphological convergence

#### Silvia Castiglione, Carmela Serio, Pasquale Raia

## Index

1. search.conv basics
2. Morphological convergence between clades
3. Morphological convergence within/between categories
4. Guided examples

## search.conv basics

Dealing with multivariate data, each species at the tree tips is represented by a phenotypic vector, including one entry value for each variable. Naming \(A\) and \(B\) the phenotypic vectors of a given pair of species in the tree, the angle \(θ\) between them is computed as the inverse cosine of the ratio between the dot product of \(A\) and \(B\), and the product of vectors sizes: \[θ = arccos(\frac{A•B}{|A||B|})\] The cosine of angle \(θ\) actually represents the correlation coefficient between the two vectors. As such, it exemplifies a measure of phenotypic resemblance. Possible \(θ\) values span from 0° to 180°. Small angles (i.e. close to 0˚) imply similar phenotypes. At around 90˚ the phenotypes are dissimilar, whereas towards 180˚ the two phenotypic vectors point in opposing directions (i.e. the two phenotypes have different sign for most variables). For a phenotype with \(n\) variables, the two vectors intersect at a vector of \(n\) zeros.

However, it is important to note that with geometric morphometric data (PC scores) the origin coincides with the consensus shape (where all PC scores are 0), so that, for instance, a large \(θ\) indicates the two species diverge from the consensus in opposite directions and the phenotypic vectors can be visualized in the PC space (see the figures below).

Under the Brownian Motion (BM) model of evolution, the phenotypic dissimilarity between any two species in the tree (hence the \(θ\) angle between them) is expected to grow proportionally to their phylogenetic distance. In the figure above, the mean directions of phenotypic change from the consensus shape formed by the species in two distinct clades (in light colors) diverge by a large angle (represented by the blue arc). This angle is expected to be larger than the angle formed by the direction of phenotypic change calculated at the ancestors of the two clades (the red arc).

Under convergence, the expected positive relationship between phylogenetic and phenotypic distances is violated and the mean angle between the species of the two clades will be shallow.

One particular case of convergence applies when species in the two clades start from similar ancestral phenotypes and tend to remain similar, on average, despite the passing of evolutionary time. These parallel trajectories are evident in the figure above, representing two clades evolving towards the same mean phenotype.

The function `search.conv` (Castiglione et al. 2019) is specifically meant to calculate \(θ\) values and to test whether actual \(θ\)s between groups of species are smaller than expected by their phylogenetic distance. The function tests for convergence in either entire clades or species grouped under different evolutionary ‘states’.

## Morphological convergence between clades

When convergence between clades is tested, the user indicates the clade pair supposed to converge by setting the argument `node`. Otherwise, the function automatically scans the phylogeny searching for significant instance of convergent clades. In this case, the minimum distance (meant as either number of nodes or evolutionary time), and the maximum and minimum sizes (in term of number of tips) for the clades to be tested are pre-set within the function or indicated by the user through the arguments `min.dist`, `max.dim`, and `min.dim`, respectively.

Given two monophyletic clades (subtrees) \(C1\) and \(C2\), `search.conv` computes the mean angle \(θ_{real}\) over all possible combinations of pairs of species taking one species per clade. This \(θ_{real}\) is divided by the patristic (i.e. the sum of branch lengths) distance between the most recent common ancestors (mrcas) to \(C1\) and \(C2\), \(mrcaC1\) and \(mrcaC2\), respectively, to account for the fact that the mean angle (hence the phenotypic distance) is expected to increase, on average, with phylogenetic distance. To assess significance, `search.conv` randomly takes a pair of tips from the tree (\(t1\) and \(t2\)), computes the angle \(θ_{random}\) between their phenotypes and divides \(θ_{random}\) by the distance between \(t1\) and \(t2\) respective immediate ancestors (i.e. the distance between the first node \(N1\) above \(t1\), and the first node \(N2\) above \(t2\)). This procedure is repeated 1,000 times generating \(θ_{random}\) per unit time values, directly from the tree and data. The \(θ_{random}\) per unit time distribution is used to test whether \(θ_{real}\) divided by the distance between \(mrcaC1\) and \(mrcaC2\) is statistically significant, meaning if it is smaller than 5% of \(θ_{random}\) values the two clades are said to converge.

With `seach.conv`, it is also possible to test for the initiation of convergence. In fact, given a pair of candidate clades under testing, the phenotypes at \(mrcaC1\) and \(mrcaC2\) are estimated by `RRphylo`, and the angle between the ancestral states (\(θ_{ace}\)) is calculated. Then, \(θ_{ace}\) is added to \(θ_{real}\) and the resulting sum divided by the distance between \(mrcaC1\) and \(mrcaC2\). The sum \(θ_{ace} + θ_{real}\) should be small for clades evolving from similar ancestors towards similar daughter phenotypes. Importantly, a small \(θ_{ace}\) means similar phenotypes at the mrcas of the two clades, whereas a small \(θ_{real}\) implies similar phenotypes between their descendants. It does not mean, though, that the mrcas have to be similar to their own descendants. Two clades might, in principle, start with certain phenotypes and both evolve towards a similar phenotype which is different from the initial shape. This means that the two clades literally evolve along parallel trajectories. Under `search.conv`, simple convergence is distinguished by such instances of convergence with parallel evolution. The former is tested by looking at the significance of \(θ_{real}\). The latter is assessed by testing whether the quantity \(θ_{ace} + θ_{real}\) is small (at alpha = 0.05) compared to the distribution of the same quantity generated by summing the \(θ_{random}\) calculated for each randomly selected pair of species \(t1\) and \(t2\) plus the angle between the phenotypic estimates at their respective ancestors \(N1\) and \(N2\) divided by their distance.

clade18 | clade24 | angle
--- | --- | ---
t11 | t6 | 2.079
t3 | t6 | 35.642
t4 | t6 | 46.413
t10 | t6 | 27.257
t11 | t7 | 38.774
t3 | t7 | 4.18
t4 | t7 | 59.379
t10 | t7 | 39.497
t11 | t13 | 42.862
t3 | t13 | 52.011
t4 | t13 | 4.571
t10 | t13 | 17.719
mrcaC-18 | mrca-24 | 24.86
\(\theta_{real}\) | = | 30.865
\(\theta_{real}\)+\(\theta_{ace}\) | = | 55.725
\(distance_{mrcas}\) | = | 1.786

\[\frac{\theta_{real}}{dist_{mrcas}} = 17.286 ; \frac{\theta_{real}+\theta_{ace}}{dist_{mrcas}} = 31.242\]

Regardless of whether clades are indicated (by the argument `node`) or not (i.e. the function automatically locates convergent clades), `search.conv` returns the metrics (i.e. \(θ_{real}\), \(θ_{ace}\) and so on) and the relative significance level for each clade pair under testing (`$node pairs`).

    search.conv(RR=RR,y=y,min.dim=3,max.dim=4,nsim=100,rsim=100,clus=2/parallel::detectCores())->SC

| distance | p-value | Clade size
--- | --- | --- | ---
node.pair | ang.bydist.tip | ang.conv | ang.ace | ang.tip | node | time | ang.bydist | ang.conv | n1 | n2
18-24 | 17.286 | 31.209 | 24.860 | 30.865 | 7 | 1.786 | 0.11 | 0.01 | 4 | 3
23-18 | 25.476 | 46.883 | 32.173 | 38.288 | 6 | 1.503 | 0.17 | 0.01 | 4 | 4

Here, ang.bydist.tip and ang.conv correspond to \(\frac{\theta_{real}}{dist_{mrcas}}\) and \(\frac{\theta_{real}+\theta_{ace}}{dist_{mrcas}}\), respectively; ang.tip and ang.ace are \(θ_{real}\) and \(θ_{ace}\); the distance between the clades is computed both in terms of number of nodes (node) and time (time; N.B. this is \(dist_{mrcas}\)); p-values for ang.bydist and ang.conv are the significance levels for such metrics; clade size indicates the number of tips within the clades under testing.

The function also returns the `$average distance from group centroids`, that is the average phenotypic distance of each single species within the paired clades to the centroid of each pair (i.e. the mean phenotype for the pair as a whole) in multivariate space. Such distances are compared between significantly convergent pairs to identify the pair with the most similar phenotypes (`$node pairs comparison`).

node pairs comparison | average distance from group centroids
--- | ---
| diff | lwr | upr | p adj | 18/24 | 23/18
23/18-18/24 | 0.021 | -0.022 | 0.064 | 0.324 | 0.085 | 0.106

As for the example above, `search.conv` found two clade pairs under “convergence and parallelism” (which is also printed out in the console when the function ends running). In both cases, \(\theta_{real}\) by time (ang.bydist.tip) is not significant (p.ang.bydist > 0.05) while \(\theta_{real}+\theta_{ace}\) by time (ang.conv) is significantly different from random (p.ang.conv < 0.05). This means the clades within each pair started with similar phenotypes and evolved along parallel trajectories. Although not significantly different (p adj), the average distance from group centroid for the pair 18/24 is smaller than for 23/18, which means the former has less phenotypic variance.

## Morphological convergence within/between categories

The clade-wise approach we have described so far ignores instances of phenotypic convergence that occur at the level of species rather than clades. `search.conv` is also designed to deal with this case. To do that, the user must specify distinctive ‘states’ (by providing the argument `state` within the function) for the species presumed to converge. The function will test convergence within a single state or between any pair of given states. The species ascribed to a given state may belong anywhere on the tree or be grouped in two separate regions of it, in which case two states are indicated, one for each region. The former design facilitates testing questions such as whether all hypsodont ungulates converge on similar shapes, while latter aids in testing questions such as whether hypsodont artiodactyls converge on hypsodont perissodactyls.

When searching convergence within/between states, `search.conv` first checks for phylogenetic clustering of species within categories and “declusterizes” them when appropriate. This is accomplished by randomly removing one species at time from the “clustered” category until such condition is not met (this feature can be escaped by setting `declust = FALSE`). Then, the function calculates the mean \(θ_{real}\) between all possible species pairs evolving under a given state (or between the species in the two states presumed to converge on each other). The \(θ_{random}\) angles are calculated by shuffling the states 1,000 times across the tree tips. Both \(θ_{real}\) and individual \(θ_{random}\) are divided by the distance between the respective tips.

state a | state b | angle | distance
--- | --- | --- | ---
t4 | t3 | 2.079 | 2.126
t4 | t7 | 35.642 | 3.438
t4 | t12 | 46.413 | 3.051
t4 | t9 | 27.257 | 3.031
t13 | t3 | 38.774 | 3.517
t13 | t7 | 4.18 | 0.261
t13 | t12 | 59.379 | 2.57
t13 | t9 | 39.497 | 2.55
t14 | t3 | 42.862 | 2.532
t14 | t7 | 52.011 | 1.865
t14 | t12 | 4.571 | 1.584
t14 | t9 | 17.719 | 1.564
mean \(\theta_{real}\) = 30.865
mean \(\frac{\theta_{real}}{distance}\) = 13.349

Under the “state” case, `search.conv` returns the mean \(θ_{real}\) within/between states (ang.state) and the same metric divided by time distance (ang.state.time), along with respective significance level (p.ang.state and p.ang.state.time).

    search.conv(tree=tree,y=y,state=state,nsim=100,clus=2/parallel::detectCores())->SC

state1 | state2 | ang.state | ang.state.time | p.ang.state | p.ang.state.time
--- | --- | --- | --- | --- | ---
b | a | 35.562 | 20.131 | 0.01 | 0.01

The example above produced significant results for convergence between states regarding both mean \(θ_{real}\) (p.ang.state < 0.05) and mean \(θ_{real}\) by time (p.ang.state.time). Whether p.ang.state.time or p.ang.state should be inspected to assess significance depends on the study settings. Ideally, p.ang.state.time provides the most appropriate significance metric, however, for badly incomplete tree with clades pertaining to very distant parts of the tree of life (which is commonplace in studies of morphological convergence), the time distance could be highly uninformative and p.ang.state should be preferred.

## Guided examples

    # load the RRphylo example dataset including Felids tree and data
    data("DataFelids")
    DataFelids$PCscoresfel->PCscoresfel # mandible shape data
    DataFelids$treefel->treefel # phylogenetic tree
    DataFelids$statefel->statefel # conical-toothed ("nostate") or saber-toothed condition

    library(ape)
    plot(ladderize(treefel),show.tip.label = F,no.margin = T)
    colo<-rep("gray50",length(treefel$tip.label))
    colo[match(names(which(statefel=="saber")),treefel$tip.label)]<-"firebrick1"
    tiplabels(text=rep("",Ntip(treefel)),bg=colo,frame="circle",cex=.4)
    legend("bottomleft",legend=c("Sabertooths","nostate"),pch=21,pt.cex=1.5,
           pt.bg=c("firebrick1","gray50"))

    # perform RRphylo on Felids tree and data
    RRphylo(tree=treefel,y=PCscoresfel)->RRfel

    ## Example 1: search for morphological convergence between clades (automatic mode)
    ## by setting 9 nodes as minimum distance between the clades to be tested
    search.conv(RR=RRfel, y=PCscoresfel, min.dim=5, min.dist="node9")->SC.clade

    ## Example 2: search for morphological convergence within sabertoothed species
    search.conv(tree=treefel, y=PCscoresfel, state=statefel)->SC.state

## References

Castiglione, S., Serio, C., Tamagnini, D., Melchionna, M., Mondanaro, A., Di Febbraro, M, Profico, A., Piras, P., Barattolo, F., & Raia, P. (2019). A new, fast method to search for morphological convergence with shape data. PloS one 14: e0226949. https://doi.org/10.1371/journal.pone.0226949