Supplementary materials
Biomechanical profiling
Biting performance is quantified using a specifically defined biomechanical
performance measure, the mechanical advantage (MA). MA is the ratio between the
moment arm of the muscle moment (effort) and the moment arm of the biting moment
(load), and is otherwise known as the force advantage (Westneat, 1994). MA quantifies
the degree in which effort can be translated into load and is an indicator of mechanical
efficiency. A high MA relates to high efficiency in force translation from effort to
load. Conversely, a low MA means low efficiency in force translation, but high
rotational speed at the point of load (high speed advantage or displacement advantage
(Westneat, 1994)).
Based on anatomical observations of modern avian and crocodilian jaw
adductor musculature (pers. obs. but also Holliday and Witmer, 2007; Holliday, 2009),
corresponding muscle attachment sites were digitally landmarked on lateral view
images of 50 skulls (from published sources, photographs, or novel composite
reconstructions), including 41 theropod taxa and the outgroup Plateosaurus.
Landmarks were plotted on published figures (lateral-view photographs or interpretive
diagrams of well-preserved skull specimens and skull reconstructions).
Lateral-view
photographs, taken by the author, of theropod skull replicas from various museum
collections were also used.
In some taxa, composite skull reconstructions were used
that were created for this study from images of disarticulated cranial materials.
Jaw
muscles were simplified into three major groups, the MAME group (M. adductor
mandibulae externus including pars profundus + M. pseudotemporalis superficialis), the
MAMP group (M. adductor mandibulae posterior/M. pseudotemporalis profundus), and
MPT group (M. pterygoideus dorsalis + M. pterygoideus ventralis) following Rayfield
et al. (2001) (Fig. S1).
Attachment sites for each muscle group were marked for the
origins and insertions on the cranium and mandible (Fig. S2). Each attachment site
was represented by at least two landmarks, at the anterior-most and posterior-most
extents of the attachment site (Fig. S2). Biting positions taken at the tip of each tooth
along the tooth row and the fulcrum (jaw joint) were also marked (Fig. S2).
X,Y
coordinates of muscle attachments, biting points and jaw joint were recorded using
ImageJ (Abramoff et al., 2004). The distance between the jaw joint and the line of
action for each muscle was taken as the moment arm of the muscle (mM) and calculated
using the following formula:
mM = ( |aXJ + bYJ + c| )/ (a2 + b2)
(S1)
where XJ and YJ are the X,Y coordinates of the jaw joint, and a, b, and c are the
parameters for the straight line connecting the muscle landmarks (in the form ax + by +
c = 0; Fig. S3). This is a direct application of a fundamental relationship between a
point (X0, Y0) and a straight line, ax + by + c =0, where the distance between the point
and straight line l is:
l = (|aX0 + bY0 + c|) / (a2 + b2).
(S2)
If the straight line were to be the major axis through the body of the muscle and the
point were to be the jaw joint (XJ, YJ) then the moment arm of that muscle (mM) would
be the perpendicular distance from the straight line and the jaw joint, or the distance
from a point to a straight line l of Equation S2, hence Equation S1.
The coefficients a, b and c can be derived from the X,Y coordinates of the
landmarks. First the straight line ax + by + c =0 can be converted to the form:
y = (-a/b)x + (-c/b)
(S3)
which is the familiar form of a straight line, y = slope x + intercept.
and intercept is -c/b.
Thus slope is -a/b
From the equation of a straight line through two points,
y = {[(Y2 – Y1) / (X2 – X1)] × (x – X1)} + Y1
(S4)
the slope and intercept can be calculated from the coordinates of the insertion (XI,YI)
and origin (XO,YO) points as:
slope = (YI – YO) / (XI – XO)
and
intercept = -XO [(YI – YO) / (XI – XO)] + YO.
A common factor b in the slope (-a/b) and intercept (-c/b) will determine the
coefficients a, b and c; i.e. a = slope × factor, b = -factor, c = intercept × factor.
Moment arms were calculated for each origin/insertion pair for the anteriorand posterior extents of the muscle (Fig. S2) the mean value of which taken as the
moment arm for each muscle group. Moment arms for each muscle group were then
divided by the load arm (direct distances between the jaw joint and bite points) to
compute mechanical advantages (MA).
along the tooth row.
MA were computed at each biting point
Potentially valuable three-dimensional information is lost by using simple
two-dimensional models, but the benefits of using simple models outweigh the
drawbacks.
Firstly, simple models are explicit in the information represented. This
enables explicit hypothesis testing (the ability to test a single hypothesis relating to a
single biomechanical performance).
Sophisticated models on the other hand have
inherent uncertainties (many variables with uncertain properties) so the information
represented is not explicit. This makes hypothesis testing tricky; due to the complexity
of the model, there is no certainty that the analysis conducted actually is testing the
hypothesis of interest (or in other words, there are too many variables).
Secondly,
simple models enable direct comparisons across a wide range of taxa (simple
biomechanical metrics are directly comparable) while it is difficult in sophisticated
models. Thirdly, simple models readily allow one to amount a large enough sample
size for rigorous statistical treatment. This is especially important when dealing with
fossil taxa as three-dimensionally preserved specimens are rare and such data is limited.
For the purposes of this study (biomechanical disparity and phylogenetic comparative
analyses) it is imperative that a large standard dataset be compiled. Simple
biomechanical metrics from simple biomechanical models are most appropriate for this.
To enable direct comparisons amongst taxa with differing tooth counts, biting
positions were adjusted as percentages along the tooth row with 0% being the
posterior-most position and 100% being the anterior-most position of the tooth row (Fig.
S2).
Note, that this standardisation of tooth row position is preceded by MA
computation; MA is computed from the absolute distances, not standardised positions.
For each taxon, MA was plotted against tooth row positions and polynomial functions
of either the second order (y = β0 + β1x1 + β2x22), third order (y = β0 + β1x1 + β2x22 +
β3x33) or fourth order (y = β0 + β1x1 + β2x22 + β3x33 + β4x44) were fitted using R (R Core
Development Team, 2009) (Fig. S4).
The best order of polynomial for each profile
was determined using a weighted AIC-based test in the paleoTS library (Hunt, 2006) in
R. Although MA profiles look simple, and intuitively, a second order polynomial may
best represent them, a vast majority of theropods had MA profiles that could not be
described sufficiently by second order polynomials (second order polynomial curves
had such poor fit to the profiles that they were even visually obvious; this was
substantiated by AIC-based comparisons). Thus in most theropod profiles, either third
or fourth order polynomials were fitted.
This introduces a discrepancy in the number
of coefficients amongst taxa but the three common coefficients (i.e. β0, β1 and β2) were
sufficient in explaining the multivariate variability in the profiles (multivariate
ordination using three and five coefficients respectively were nearly identical), and they
were taken as independent variables and subjected to a correlation-based principal
components analysis (PCA) to visualise “function space” (Anderson, 2009) occupation
of different theropod and non-theropod clades.
A correlation matrix was employed
instead of a variance-covariance matrix because the absolute magnitudes differed on
orders of magnitudes across the variables (Jolliffe, personal communications).
In this
way, much of the variance could be explained by PC1 and PC2 (89.9% and 9.9%
respectively).
Loadings for PC1 show that each polynomial coefficient is represented
in nearly equal proportion.
An alternative to using the polynomial coefficients to describe the
biomechanical profiles would be to fit MA values along the polynomial curves (or to
interpolate between the observed MA values along the different positions of the tooth
row). For instance, MA values along the curves from 0% to 100% of the tooth row
can be predicted by values fitted onto the polynomial functions; values fitted at an
increment of 1 would yield 101 fitted MA values per taxon. Thus each taxon would be
represented by 101 “variables”. A correlation-based PCA on these “variables” would
result in PC1 and PC2 explaining 93.1% and 6.9% respectively of the total variance.
A plot of PC1 against PC2 shows a different rotation of axes as compared to that on
polynomial coefficients but the relative distribution in “function space” is nearly
identical.
One problem with this approach of course is that the “variables” are not
independent of each other but form a continuum.
Moreover, interpretation of the PC
axes with respect to the original variables is difficult in the 101 fitted MA values but
simple in polynomial coefficients (i.e. intercept, slope, parabolic curvature).
Therefore,
polynomial coefficients were preferred over 101 fitted MA values for use in PCA but
also subsequent phylogenetic comparative analyses.
Fig. S1.
Identification of muscle attachments and reconstruction of jaw adductor
muscles in theropods. (a) Based on anatomical observations of jaw myology in
modern archosaurs, corresponding muscle attachments were identified in fossil
materials.
MAME, red; MAMP, green; MPT, purple. (b) - (d) Muscles were
reconstructed by major groups: (b) MAME group; (c) MPT group; and (d) MAMP
group.
Fig. S2. Computation of moment arms using digital landmarks. (a) Bite points at the
tip of each tooth along the entirety of the tooth row and muscle attachment points were
landmarked. Each muscle was recognised by at least four landmarks, a pair of origin
and insertion points for the anterior and posterior extents of the muscles. (b)
Landmarks and moment arms for the MAME.
Moment arms for the line of action of
the muscle at the anterior and posterior extents of the MAME were computed as the
perpendicular distance from the jaw joint to the line of action of the muscle. Total
moment arm for that muscle (mE) was taken as the mean of the anterior and posterior
moment arms (mant and mpost respectively). (c) and (d) Moment arms for the MPT and
MAMP respectively.
Fig. S3. Computation of moment arms from X-Y coordinates. Moment arms are
computed as the distance between a point (jaw joint, J) and the straight line (ax + by + c
= 0) connecting the origin and insertion points (Mo and Mi respectively).
for both anterior and posterior extents of the muscles.
This is done
Fig. S4.
Biomechanical profile plot.
The mean MA at each biting position is
standardised along the tooth row as a percentage scale, with the posterior extent as 0%
and anterior extent as 100%.
MA are plotted against respective percent tooth row
positions and a polynomial function is fitted using least squares regression.
Phylogeny
The phylogeny of the Lloyd et al. (2008) dinosaur supertree formed the basis of the
phylogenetic comparative analyses.
The position of Monolophosaurus was shifted to a
more basal position to reflect more recent understanding (Smith et al., 2007; Brusatte et
al., in press). For branch length estimation, a tree with 168 taxa (primarily theropods
but with some sauropodomorphs and basal ornithischians) was used (Fig. S5).
In
estimating branch lengths, a larger tree is preferable over a smaller tree (i.e., n = 42)
because additional stratigraphic information from taxa interspersed amongst those of the
smaller tree help to more accurately estimate the ages of internal and more basal nodes.
The tree was dated using the first occurrence dates of each terminal taxon (compiled
from various sources but primarily the Dinosauria II and converted to millions of years
ago using the International Stratigraphic Chart, 2008).
After this initial dating some
zero duration branches existed (an inevitable consequence in the standard dating method
used by palaeontologists) and at this point the method of Brusatte et al. (2008) was
applied such that zero duration branches were lengthened by sharing time equally with a
directly ancestral branch of positive duration.
This is a modification of the approach
of Ruta et al. (2006) where sharing is proportional to character changes, which was not
possible or desirable in the present context (R code for implementing both methods is
available from http://www.graemetlloyd.com/).
Terminal branches were extended to
fit the last occurrence dates for each terminal taxon.
Taxa not represented in the
biomechanical analysis were pruned out after branch lengths were estimated, and the
reduced tree (42 taxa) was used for subsequent phylogenetic comparative analyses (Fig.
S6).
The recent phylogenetic hypothesis with Proceratosaurus as a basal
tyrannosauroid (Rauhut et al., 2010) is not tested here as this alternate position is
difficult to reflect in the 168-taxon supertree topology (there is limited overlap in taxa
between this tree and that of Rauhut et al., 2010).
Thus conforming to an objective
consensus phylogenetic position, i.e. the Lloyd et al. (2008) supertree, was preferred.
Fig. S5. The phylogeny of Theropoda taken from the supertree of Lloyd et al. (2007)
with additional basal sauropodomorph and ornithischian taxa (n = 168).
The inclusion
of Late Triassic to Early Jurassic basal sauropodomorphs and ornithischians supply
additional information on the possible ages of the basal divergence dates, i.e. for
Saurischia.
Fig. S6. Phylogenetic relations of the 42 extinct saurischian taxa. The tree of Fig. S5
was pruned after nodes were dated and branch lengths estimated to include the taxa
under study for biomechanical profiling.
Testing for phylogenetic signals
One method of detecting phylogenetic signal in the biomechanical variable is to employ
the phylogenetic eigenvector regression (PVR; Diniz-Filho et al., 1998). PVR is a test
based on multiple linear regression in which a phenotypic variable is the response
variable with phylogeny as the predictor variable represented as principal coordinates
(PCo) axes extracted from a phylogenetic distance matrix (pair wise Euclidean distances
computed in R).
From the total of 41 PCo axes, the first 34 were retained for multiple
regression analysis because they explain 95% of the total variance in phylogeny.
Since the response in this case is a multivariate dataset (i.e, the three polynomial
coefficients, 0, 1 and 2), PVR was modified slightly by using multivariate multiple
regression (MMR) instead of a standard multiple linear regression. MMR is an
extension of multiple regression, where the response variable is a matrix of multiple
variables instead of a vector of a single variable.
Another method for detecting phylogenetic signal in a phenotypic dataset is to
employ the method of Blomberg et al. (2003).
This method uses phylogenetically
independent contrasts (Felsenstein, 1985) and compares the variances of the contrasts
computed from a given variable on a particular tree topology with those computed from
permutations of that variable across the same tree (i.e. randomly reshuffling the values
amongst the OTUs while keeping the tree topology constant).
If the variances in the
contrasts for the data in the real phylogenetic positions are lower than those from the
permutations, then there is a significant phylogenetic signal in that data (Blomberg et al.,
2003). This test was conducted in R using the picante library (Kembel et al., 2009)
which also computes Blomberg et al.’s (2003) K statistic.
A K less than one would
indicate that closely related OTUs have values that are less similar than expected under
Brownian motion evolution (or departure from Brownian motion, such as adaptive
evolution), while a K greater than one would indicate that closely related OTUs have
values more similar than expected (Blomberg et al., 2003).
Because the independent contrasts method (Felsenstein, 1985) assumes that
character evolution can be modelled as a random walk (i.e. a Brownian motion model of
evolution) and that characters change at a uniform rate per unit branch length in all
branches (i.e. variance accumulation is assumed to be equal per unit time), the data
and/or selected branch lengths must conform to these assumptions.
Diagnostic checks
of branch lengths available in the PDAP module (Midford et al., 2005) of Mesquite
(Maddison & Maddison, 2009) revealed that the combination of raw data and initial
branch lengths did not meet the assumptions. Therefore data were logarithmically
transformed, log100, log10(-1/1), and log102. The coefficient 1 required logarithmic
transformation of the negative reciprocal instead of just logarithmic transformation of
the raw values because the raw values were all negative (negative values cannot be log
transformed) and simply reversing the sign would also reverse the magnitude hence the
negative reciprocal.
In the case that data transformation was not enough, then branch
lengths were adjusted by assigning minimum internal branch lengths (Laurin, 2004;
Laurin et al., 2009) while keeping the ages of the terminals constant using the
Stratigraphic Tools (Josse et al., 2006) module of Mesquite until contrasts were
adequately standardised (Garland et al., 1992).
Minimal branch lengths of 3 million
years and 4.5 million years were necessary for adequate standardisation of log100 and
log10(-1/1) contrasts respectively.
Contrasts for log102 did not require any branch
length adjustment for adequate standardisation so the initial dated tree was used.
Even
after branch length adjustments, there was still a strong correlation between the
estimated nodal values against their respective ages in all three variables.
However,
this may not necessarily be a statistical artefact when using fossil phylogenies (these
tests were devised with ultrametric trees in mind) and instead may indicate the presence
of a trend (Laurin, pers. comm.).
The RMesquite library (Lapp & Maddison, 2010)
was used to enable the graphical tree manipulation devices of Mesquite within R prior
to the Blomberg et al. (2003) test.
Tracing the evolution of function space occupation
To trace the evolution of function space occupation, nodal values (ancestors) were
estimated for the three coefficients (i.e. β0, β1 and β2) using the maximum likelihood
(ML) method of ancestor character estimation (Schluter et al., 1997) available in the ape
library (Paradis et al., 2004) in R (this is the equivalent of the weighted squared-change
parsimony method of Maddison (1991)). Because the maximum likelihood method
also assumes a Brownian motion model of evolution, the transformed data and adjusted
branch lengths were used (see above).
Data transformation is not only necessary for
conformation to the assumptions of Brownian motion evolution, but also for accurate
estimation of ancestor values on small-value variables (β1 is in the order of 10-3 while β2
is in the order of 10-5) which without transformation tend to produce spurious results.
Ancestor estimates were back-transformed and along with the original coefficients
formed the basis for a multivariate ordination to visualise function space occupation
using a correlation-based PCA.
Ancestor values were estimated for the polynomial
coefficients and included in the post hoc PCA (along with the OTU values) instead of
being estimated for PC scores of the OTUs from an a priori PCA (using only the OTU
polynomial coefficients) because the latter computes ancestral function space
coordinates (PC scores) from OTU function space which could be variable depending
on the ordination method (PCoA instead of PCA, or a variance-covariance matrix basis
as opposed to a correlation matrix), whereas the former is more or less concrete (there is
no variability in the OTU values). Thus, while the former results in one set of ancestor
values, the latter results in variable ancestor values (depending on the ordination method
employed on the OTUs).
Fig. S7. ML ancestor estimates of β0. Branch lengths were adjusted so that
minimum internal branch length is 3 Ma and ML ancestor estimates were computed for
log10β0. Ancestor estimates were back-transformed to the arithmetic scale.
Fig. S8. ML ancestor estimates of β1. Branch lengths were adjusted so that
minimum internal branch length is 4.5 Ma and ML ancestor estimates were computed
for log10(-1/β0). Ancestor estimates were back-transformed to the arithmetic scale and
shown at ×103 because values are small on the level of 10-3.
Fig. S9. ML ancestor estimates of β2. ML ancestor estimates were computed for
log10β2. Ancestor estimates were back-transformed to the arithmetic scale and shown
at ×105 because values are small on the level of 10-5.
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