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 contrasting values for each variable). 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’.
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
).
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
).
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.
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.
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 | 47.341 | 2.126 |
t4 | t7 | 13.187 | 3.438 |
t4 | t12 | 54.947 | 3.051 |
t4 | t9 | 13.24 | 3.031 |
t13 | t3 | 71.28 | 3.517 |
t13 | t7 | 19.532 | 0.261 |
t13 | t12 | 49.591 | 2.57 |
t13 | t9 | 32.685 | 2.55 |
t14 | t3 | 54.447 | 2.532 |
t14 | t7 | 2.214 | 1.865 |
t14 | t12 | 43.587 | 1.584 |
t14 | t9 | 24.69 | 1.564 |
mean \(\theta_{real}\) = 35.562 | |||
mean \(\frac{\theta_{real}}{distance}\) = 20.131 | |||
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).
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.
# 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
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