@@ -1666,10 +1666,13 @@ \section{Introduction}\label{introduction-1}
intercepts and slopes have changed during the diversification of this
group. Moreover, integration patterns in skull form are quite stable in
both New World (Marroig \& Cheverud, 2001) and Old World Monkeys, with
some exceptions (Oliveira \emph {et al. }, 2009). Several authors have
argued that allometric constraints contribute to integration patterns
some exceptions (Oliveira \emph {et al. }, 2009). The association between
relative size variation and integration has been shown for mammals in
general (Porto \emph {et al. }, 2009, 2013) and several authors have
argued that such association is imposed through allometric constraints
(e.g. Porto \emph {et al. }, 2009; Armbruster \emph {et al. }, 2014), but
this association has never been formalized. Thus, we test the hypothesis
this association has never been formalized, considering a strict
definition of allometric relationships. Thus, we test the hypothesis
that variation in allometric parameters, if present, will contribute to
variation in the strength of association between skull traits.
Considering both the spatial and temporal dynamics of mammalian skull
@@ -1745,7 +1748,18 @@ \subsection{Allometric Slopes and
lengths obtained from the phylogenetic hypothesis we use. This allows
the model to estimate $ a_s$ $ b_s$
ancestral nodes, enabling us to track changes in both parameters along
the phylogeny.
the phylogeny. Furthermore, in order to evaluate whether such
reconstruction is affected by the model we chose, we compared the
estimated posterior distribution of both intercepts and slopes for all
nodes within the phylogeny with point estimates produced under a linear
parcimony model, implemented in Mesquite ({\textbf {??? }}). For
allometric slopes, all parsimony estimates fall within the 95\% credible
intervals obtained from the random regression model. For intercepts,
only in five nodes (Catarrhini, Pitheciidae, Cebidae, \emph {Pithecia },
and the clade composed of \emph {Brachyteles }, \emph {Lagothrix } and
\emph {Ateles }) differences are observed between the two types of
estimates; however, such differences do not hamper the interpretation of
changes in intercepts we elaborate here.
We projected all individuals in our sample along the Common Allometric
Component (CAC; Mitteroecker \emph {et al. }, 2004), which is the pooled
@@ -1759,7 +1773,9 @@ \subsection{Allometric Slopes and
whether the strength of association between size and allometric shape,
represented by projections over this CAC (which we consider the best
representation of the ancestral allometric shape variation) has changed
during the diversification of Anthropoid Primates.
during the diversification of Anthropoid Primates, measuring such
strength of association as increases in CAC values for an unit increase
in logCS standard deviation within each OTU.
We used uniform prior distributions for all $ a_s$ $ b_s$
sample the posterior distribution for our model, we used a MCMC sampler
@@ -1810,16 +1826,21 @@ \subsection{Morphological Integration and
variables and allometric parameters as predictors. We also use a
Bayesian framework to estimate such models. In order to evaluate which
parameters are sufficient to explain variation in MHI values, we
adjusted three different models: one for static intercepts alone
($ a_s$ $ b_s$
the joint effect of both parameters, without interactions ($ a_s$
$ b_s$
Criterion (DIC; Gelman \emph {et al. }, 2004), which increases as a
function of the average posterior likelihood for a particular model and
decreases as a function of the number of parameters considered; in a
equivalent manner to Akaike's (1974) Information Criterion, the model
with the smallest DIC is considered the best fit; models whose
difference in DICs is lower than two are considered equivalent.
adjusted four different models: one for static intercepts alone ($ a_s$
one for static slopes ($ b_s$
effect of both parameters, without interactions ($ a_s$ $ b_s$
fit a model that considers only realtive size variation (measured as
coefficients of variation in log Centroid Size for each OTU) acting on
MHI values, since it has been shown that such variation suffices in
explaining variation in integration patterns, albeit on a larger
phylogenetic scale (Porto \emph {et al. }, 2009, 2013). We compared these
four models using the Deviance Information Criterion (DIC; Gelman
\emph {et al. }, 2004), which increases as a function of the average
posterior likelihood for a particular model and decreases as a function
of the number of parameters considered; in a equivalent manner to
Akaike's (1974) Information Criterion, the model with the smallest DIC
is considered the best fit; models whose difference in DICs is lower
than two are considered equivalent.
\subsection {Software }\label {software-1 }
@@ -1833,13 +1854,15 @@ \subsection{Software}\label{software-1}
\section {Results }\label {results-1 }
We used the Common Allometric Component (CAC; Mitteroecker \emph {et
al. }, 2004), which represents a pooled estimate of skull shape allometry
(represented as local shape variables; Márquez \emph {et al. }, 2012) as a
proxy for the ancestral allometric relationships for Anthropoid
Primates. This axis is associated with positive loadings for Facial
traits and negative loadings for Neurocranial traits, thus representing
a contrast between these regional sets (\autoref {fig:cac_logCS }a).
We obtained local shape variables (Márquez \emph {et al. }, 2012) to
represent skull shape variation, and projected them over the Common
Allometric Component (CAC; Mitteroecker \emph {et al. }, 2004), which
represents the pooled within-species allometric shape variation. We used
such component as a proxy for the ancestral allometric relationships for
Anthropoid Primates (see `Methods' for details). This axis is associated
with positive loadings for Facial traits and negative loadings for
Neurocranial traits, thus representing a contrast between these regional
sets (\autoref {fig:cac_logCS }a).
\begin {figure }[htbp]
\centering
@@ -1857,17 +1880,17 @@ \section{Results}\label{results-1}
over the CAC and log Centroid Size (logCS) within each species
(\autoref {fig:cac_logCS }b) are codependent upon their phylogenetic
relationships. The posterior distribution of deviations from mean for
both parameters estimated in this manner (\autoref {fig:phylo_W })
intercepts and slopes estimated in this manner (\autoref {fig:phylo_W })
indicates that intercepts deviate from the mean in at least seven
lineages: increases occur in \emph {Pithecia } and \emph {Callicebus }
within Pitheciids, in the clade composed of Callithrichinae and Aotinae,
and in \emph {Alouatta } within Atelids, while lower intercepts are found
in the clade composed of \emph {Homo } and \emph {Pan } within Hominidae, in
\emph {Hylobates } within Hylobatids, and also in \emph {Ateles } within
Atelids (\autoref {fig:phylo_W }a). Allometric slopes for terminals, on
the other hand, deviate from the mean in only three species: \emph {Homo
sapiens } and both representatives of \emph {Gorilla }, all with slopes
shallower than the mean (\autoref {fig:phylo_W }b).
Atelids (\autoref {fig:phylo_W }a). Allometric slopes on the other hand
deviate from the mean in only three species: \emph {Homo sapiens } and
both representatives of \emph {Gorilla }, all with slopes shallower than
the mean (\autoref {fig:phylo_W }b).
\begin {figure }[htbp]
\centering
@@ -1882,27 +1905,28 @@ \section{Results}\label{results-1}
Porto \emph {et al. }, 2013) for two regions (Face and Neurocranium) and
three sub-regions within each region (Oral, Nasal and Zygomatic in the
Face; Orbit, Vault and Basicranium in the Neurocranium) on the estimated
allometric parameters. Model selection using Deviance Information
Criteria (DIC; Gelman \emph {et al. }, 2004) shows that models considering
only the intercept always have the worst fit. In most cases, models
including only the slope are better than or equal ($ \Delta DIC < 2 $
models with both parameters (intercept and slope), suggesting a
negligible effect of the intercept. For the Neurocranium region and the
Vault sub-region, the model including intercept and slope shows a better
fit than the slope-only model; in both cases, however, the effects of
the intercept are not significant when one takes into account the
posterior distribution of the regression coefficients (results not
shown). We thus consider that the models including only allometric
slopes provide the best representation of the association between
allometric parameters and modularity indexes
(\autoref {tab:dic_allo_im }). It should be noted that none of the models
tested for the Orbit and Basicranium sub-regions show any association
between allometric parameters and Modularity Indexes, therefore these
sub-regions were not included in the model selection. Allometric slopes
show opposite effects on MHI values for Face and Neurocranium
(\autoref {fig:MI_vs_slopeW_main }), and, with the exception of the Orbit
and Basicranium, sub-regions follow the same pattern as the more
inclusive partition in which they are contained
allometric parameters, and also on size variation alone, measured as the
coefficient of variation in log Centroid Size. Model selection using
Deviance Information Criteria (DIC; Gelman \emph {et al. }, 2004) shows
that models considering either intercept or size variation only always
have the worst fit. In most cases, models including only the slope are
better than or equal ($ \Delta DIC < 2 $
(intercept and slope), suggesting a negligible effect of the intercept.
For the Neurocranium region and the Vault sub-region, the model
including intercept and slope shows a better fit than the slope-only
model; in both cases, however, the effects of the intercept are not
significant when one takes into account the posterior distribution of
the regression coefficients (results not shown). We thus consider that
the models including only allometric slopes provide the best
representation of the association between allometric parameters and
modularity indexes (\autoref {tab:dic_allo_im }). It should be noted that
none of the models tested for the Orbit and Basicranium sub-regions show
any association between allometric parameters and Modularity Indexes,
therefore these sub-regions were not included in the model selection.
Allometric slopes show opposite effects on MHI values for Face and
Neurocranium (\autoref {fig:MI_vs_slopeW_main }), and, with the exception
of the Orbit and Basicranium, sub-regions follow the same pattern as the
more inclusive partition in which they are contained
(\autoref {fig:MI_vs_slopeW_si }).
\input {Tables/dic_allo_im.tex }
@@ -1933,17 +1957,17 @@ \section{Discussion}\label{discussion-1}
along a size-trait gradient (e.g. Gould, 1974) has historically been
confronted with the opposing view that allometry may be in fact much
more dynamic and evolvable (e.g. Kodric-Brown \emph {et al. }, 2006;
Bonduriansky, 2007; Frankino \emph {et al. }, 2009). As recently stated by
some authors (Voje \emph { et al. }, 2013; Pélabon \emph {et al. }, 2014) ,
this seeming paradox may in large part owe its existence to different
meanings for allometry: one related to any monotonic relationship
between variables , while the other specifically deals with power-law
relationships between variables as defined by Huxley (1932). This
imprecision of definition greatly hampers evidence in favor of a dynamic
interpretation of allometry; considering the few studies estimating
evolvability of narrow-sense allometry, only two demonstrate changes in
allometric slopes (Tobler \& Nijhout, 2010; Bolstad \emph { et al. },
2015).
Bonduriansky, 2007; Frankino \emph {et al. }, 2009). This paradox may owe
its existence to different meanings for allometry (Houle \emph {et al. },
2011; Voje \emph { et al. }, 2013; Pélabon \emph { et al. }, 2014): one
related to any monotonic relationship of a given trait with size,
measured on an arbitrary scale , while the other specifically deals with
power-law relationships as defined by Huxley (1932), thus linear on a
log-log scale. This imprecision of definition greatly hampers evidence
in favor of a dynamic interpretation of allometry; considering the few
studies estimating evolvability of narrow-sense allometry, only two
demonstrate changes in allometric slopes (Tobler \& Nijhout, 2010;
Bolstad \emph { et al. }, 2015).
The results we obtained from the phylogenetic random regression model
indicate that substantial changes have occurred on intercepts,
@@ -2018,29 +2042,33 @@ \section{Discussion}\label{discussion-1}
Shifts in allometric slopes throughout Anthropoid diversification imply
changes in the strength of association between Facial and Neurocranial
traits, in opposing directions (\autoref {fig:MI_vs_slopeW_main }). Thus,
although the distinction between these two regions is the by-product of
developmental interactions since the initial steps of skull ontogeny
(Hallgrímsson \& Lieberman, 2008), allometric relationships reinforce
the effect of such interactions due to the induction of facial growth by
muscular activity from weaning to adulthood (Zelditch \emph {et al. },
1992, 2009), since post-natal growth represents a great portion of body
size variation observed in this group. Hence, groups with steeper
slopes, such as \emph {Ateles } and \emph {Alouatta }, exhibit strong Facial
integration and lower Neurocranial integration, while groups with
shallower slopes, such as Callithrichines, \emph {Homo }, and
\emph {Gorilla }, exhibit similar integration values for both regions,
indicating that the restriction over allometric relationships may be
associated with the maintenance of the functional interactions among
skull traits. Although the bulk of this effect over phenotypic variation
may have an environmental origin (Cheverud, 1982b), muscular activity is
necessary for the proper development of osteological elements (Herring,
2011), thus indicating that even though such functional interactions are
dependent on the environment, they are canalized through skull ontogeny
in a predictable manner, since developmental systems themselves are
under selection for robustness and replicability (Wagner \& Altenberg,
1996; Hansen, 2011; Pavlicev \& Hansen, 2011), exhibiting behaviors
similar to machine learning algorithms (Watson \emph {et al. }, 2013).
traits, in opposing directions (\autoref {fig:MI_vs_slopeW_main });
furthermore, allometric slopes explain this effect on integration to a
greater extent than relative size variation alone
(\autoref {tab:dic_allo_im }). Thus, although the distinction between
these two regions is the by-product of developmental interactions since
the initial steps of skull ontogeny (Hallgrímsson \& Lieberman, 2008),
allometric relationships reinforce the effect of such interactions due
to the induction of facial growth by muscular activity from weaning to
adulthood (Zelditch \emph {et al. }, 1992, 2009), since post-natal growth
represents a great portion of body size variation observed in this
group. Hence, groups with steeper slopes, such as \emph {Ateles } and
\emph {Alouatta }, exhibit strong Facial integration and lower
Neurocranial integration, while groups with shallower slopes, such as
Callithrichines, \emph {Homo }, and \emph {Gorilla }, exhibit similar
integration values for both regions, indicating that the restriction
over allometric relationships are associated with the maintenance of the
functional interactions among skull traits. Although the bulk of this
effect over phenotypic variation may have an environmental origin
(Cheverud, 1982b), muscular activity is necessary for the proper
development of osteological elements (Herring, 2011), thus indicating
that even though such functional interactions are dependent on the
environment, they are canalized in a predictable manner given the
existance of epigenetic effects through the course of skull development
(Lieberman, 2011), since developmental systems themselves are under
selection for robustness and replicability (Wagner \& Altenberg, 1996;
Hansen, 2011; Pavlicev \& Hansen, 2011), exhibiting behaviors similar to
machine learning algorithms (Watson \emph {et al. }, 2013).
Here, we demonstrated under a comparative framework that allometric
intercepts are more labile than slopes in a macroevolutionary scale and
@@ -2053,7 +2081,7 @@ \section{Discussion}\label{discussion-1}
in particular situations. The approach we used here to estimate
allometric parameters benefits from properly separating variation within
and among populations, considering the interdependence arising from
phylogenetic relatedness simultaneously; for instance, the same approach
phylogenetic relatedness simultaneously. For instance, the same approach
could be used to evaluate variation in allometric parameters between
individuals in a population whose genealogy is available, enabling us to
quantify the influence of both genetic and environmental factors over