BACKGROUND
mammalian dentition is complex and variable, adapting to the dietary, environmental, and social demands of a species’ niche.
despite the considerable range of variability in dental forms, certain morphological and evolutionary patterns emerge repeatedly in phylogenetically-distant taxa.
for example, the hypocone  evolved independently more than  <dig> times  <cit> , hypsodonty  evolved in multiple species of ungulates  <cit> , lagomorphs  <cit> , and south american rodents  <cit> , and tribosphenic molars may have evolved twice in mammals  <cit>  .
during evolution, teeth are more frequently lost than gained  and the last developing teeth within a particular dental field are almost always lost first  <cit> .
while many of these patterns likely evolved in response to similar selective pressures  <cit> , there is evidence that developmental constraints limit molar shape  <cit> .
thus it is reasonable to assume that some of the dental variability of mammalian teeth is constrained by developmental processes.
in the past decade, several studies have explored developmental mechanisms potentially responsible for dental morphological variation .
kavanagh and colleagues  <cit>  developed the dental inhibitory cascade  model to explain the evolution and development of molar proportions.
through experimental alteration of developing mouse molars, the authors suggested that much of mammalian molar proportion diversity could be attributed to a simple, highly conserved pattern of differences in the timing and concentration of molecules that activate  or inhibit  molar initiation and proliferation.
when the enamel knot of the earliest developing mandibular molar  is initiated, it expresses inhibitory signals  that inhibit, or greatly delay, the subsequent formation of m <dig> and m <dig>  a change in the expression or timing of these inhibitory signals was proposed to determine the eventual size of m <dig> and m <dig>  by altering the relative timing of molar initiation.
according to kavanagh et al.
<cit> , therefore, the patterning of molar sizes works as an inhibitory cascade, so that the timing of m <dig> development cumulatively affects the timing of m <dig> and m <dig> development, as the dic mechanism operates along the developing tooth row.
decreased m <dig> inhibition allows m <dig> and m <dig> to form earlier and ultimately grow larger than m <dig>  while increased inhibition restricts the size of subsequently developing molars and may eventually lead to their loss.
the dic model can be characterized by the following equation:
<dig> y=1+/ix− <dig>
where y is molar area, as determined by position along the tooth row , and by the relative strength of activators  and inhibitors , defined by the a/i ratio.
molar areas are derived from eq.
<dig> as m1= <dig> m2=a/i,m3=2− <dig>  thus, balance between activation and inhibition leads to equal sized molars, while decreases in inhibition result in larger distal molar areas with a cumulative effect from m <dig> to m <dig>  since inhibition is hypothesized to cascade distally along the molar row, it is possible to predict the relative size of each molar using the following relationship:
<dig> m3/m1=2− <dig>
as a test of the dic model’s predictions in a macroevolutionary context, kavanagh et al.
<cit>  demonstrated that most of the molar patterns seen in murid rodents can be explained by alteration of the ratio of activation and inhibition molecules expressed during early tooth development.
one extension of the model is that in morphospace regions of high inhibition , we would predict minimal temporal overlap between m <dig> and m <dig> calcification.
as the level of inhibition decreases, however, the amount of overlap should increase, so that in morphospace regions with the lowest levels of inhibition  temporal overlap would be substantial.
the dic model highlights the importance of developmental timing in determining dental metrics, by establishing the pattern of hierarchical dependence within a particular developmental system and by asserting broad applicability across mammals .
while the model does not predict variation in cusp height, enamel thickness, or eruption schedule, one implication of the model is that molar size proportions are independent from these and other traits determined later in development.
for example, our interpretation is that under this model, fruit bats, golden moles, and tree shrews not only exhibit similar molar size proportions, but proportions that were determined by fundamentally the same developmental interactions.
the dic model is innovative in suggesting that development may also drive molar size variation in addition to other variables, such as dietary adaptation, phylogenetic inertia, and allometry .
researchers have begun to test the predictions of the dic model in taxa, such as rodents and carnivores, which have shown rapid expansion of dental morphospace over their recent evolutionary history.
a study of mesozoic and cenozoic mammals suggested that the dic model could be plesiomorphic for this clade  <cit> .
yet, while some studies support the original findings of the dic model  <cit> , others have presented evidence that calls into question its broad phylogenetic applicability .
for example, voles show an expansion of m <dig> size that drives molar proportions into morphospace  previously thought unoccupiable  <cit> , while canids exhibit a reduced major axis regression slope  between m3/m <dig> and m2/m <dig> that is much smaller than predicted by the dic model  <cit> .
though the authors of both the above studies explain deviations from the dic model as the result of changes in differential evolvability of the m <dig>  the inability of the dic model to explain all relationships between molar size proportions across mammalia is worth further study.
in particular, the model may not generalize to species with longer life histories, as the decay rates of activation and inhibition molecules may be sufficient enough to lessen the effect of the model  <cit> .
bernal et al.
<cit>  assessed molar size proportion variation among platyrrhines, an extant radiation of south american primates, and found limited support for the dic model.
while the results of a phylogeny-dependent regression analysis corresponded with the predictions of the dic model, an examination of platyrrhine molar proportions using ordinary least squares and reduced major axis regression, as well as an assessment of intraspecific variability in molar proportions, deviated from the dic model’s predictions.
the authors attribute these findings to the highly derived dentition of some platyrrhine clades, including the loss of the third molar in marmosets and tamarins, and reduction of the third molar in cebinae  <cit> .
we suggest that a broader consideration of the anthropoid dentition reveals multiple other morphological observations that are incongruent with the predictions of the dic model.
for example, great apes exhibit a m1<m2>m <dig> molar pattern, in contrast to the m1<m2<m <dig> pattern of cercopithecoids .
the m1<m2>m <dig> pattern is difficult to explain under the dic model.
moreover, m <dig> agenesis in humans, which unlike in dwarfed marmosets and tamarins is polymorphic, has been reported to occur without large changes in relative m <dig> size .
our aim in this paper is to test three predictions, and two extensions, of the dic model using a broad sample of extant anthropoid primates.
if anthropoid molar proportions follow the dic model, this represents further evidence of the model’s robustness across mammalian taxa.
if, however, anthropoids deviate from the dic model’s predictions, it is possible that other factors play an equally important role in determining their relative molar sizes.
previous studies that have sought to test the dic model’s developmental predictions have relied solely on measurements of molar crown size .
anthropoid primates offer a unique opportunity to test the developmental predictions of the dic model, as data are available on the timing of calcification in successive molars, allowing ontogenetic patterns to be evaluated.
to test the model’s predictions we employ an integrative approach that considers dental metrics, development, diet, and function.
in the following section we report the results of testing three macroevolutionary predictions , and two extensions , of the dic model:
molar area relationships: the relationship among molar areas is: m1= <dig> m2=a/i,m3=2− <dig>  balance between a/i leads to equal area molars, while increasing inhibition enlarges distal molar areas with a cumulative effect from m <dig> to m <dig>  the relationship between m3/m <dig> and m2/m <dig> molar proportions can best be described by the linear equation: m3/m1=2− <dig>  three regions of molar proportion morphospace are consistent with the dic model: a) a high a/i region where m1<m2<m <dig>  b) a low a/i region where m1>m2>m <dig>  and c) a region where m3/m1=2− <dig>  for mammals with three molars, m <dig> accounts for one-third of total lower molar area.
molar area variability:m <dig> is predicted to have the greatest size variability of any lower molar.
m <dig> agenesis: m <dig> agenesis occurs when m <dig> is less than half the size of m <dig>
developmental timing: in regions of molar proportion morphospace subject to high inhibition  there is minimal temporal overlap between m <dig> and m <dig> calcification, while in morphospace regions of low inhibition  temporal overlap is substantial.
primary dietary category and molar area proportions: molar proportions can be a measure of diet  <cit> .
RESULTS
to test macroevolutionary predictions of the dic model we used previously published lower molar occlusal area data from  <dig> extant anthropoid primates .
we analyzed these data using bayesian phylogenetic generalized linear mixed models , and employed a markov chain monte carlo  approach to sample the posterior distribution of parameter space.
phylogenetic signal for each model was quantified using pagel’s λ parameter  <cit> .
all models achieved convergence to a stationary posterior distribution, as determined by hiedelberger and welch’s convergence diagnostic  <cit>  and low levels  of autocorrelation.
our mcmc sampling strategies resulted in effective sample sizes of  <dig>  for each parameter.
detailed descriptions of the anthropoid sample, phylogenetic tree, morphometric variables, and regression models used are provided in the methods section.
to evaluate if dic model expectations were credible, given our anthropoid sample, we calculated  <dig> % highest density intervals  of posterior distribution parameter values.
the dic model specifies point predictions, rather than interval predictions, of parameter values.
for any empirical dataset, the probability is therefore extremely small that molar proportions would exactly match dic model expectations.
to provide a more useful way of calculating the probability that dic model predictions were correct, given our data, we created a region of practical equivalence  around each dic model parameter expectation.
a rope is an interval that encloses values of the parameter that are, for practical purposes, negligibly different from the point value  <cit> .
we used the rope as a decision tool for determining whether dic model predicted parameter values were credible and/or probable for the sampled anthropoid taxa.
to determine a range of slope/intercept values that might be deemed practically equivalent to the dic mathematical model’s predictions, we used, as a starting point, experimental evidence reported in kavanagh et al.
<cit> .
this experimental evidence yielded a slope of  <dig>  and intercept of - <dig> , thus deviating slightly from the dic mathematical model’s predictions of  <dig> and − <dig>  for each pglmm, we calculated posterior probabilities for ropes of several sizes.
the inclusion of ropes of any size in our analyses was a conservative measure, increasing the chance of dic model predictions being corroborated, compared with using only the point predictions of the strict mathematical model.
molar area corrected estimates
the product of maximum linear breadth and length measurements often overestimates molar occlusal area .
we employed a correction  to make these rectangular areas more comparable to areas calculated by tracing outlines around molar occlusal perimeters.
we found that the average difference between outline areas and corrected areas was much smaller  than that between outline areas and rectangular areas .
this indicated that corrected areas provided a better approximation to molar outline occlusal area .
molar area relationships
molar area proportions
the dic model predicts that the relationship between m3/m <dig> and m2/m <dig> is best characterized by a line with a slope of  <dig> and an intercept of − <dig>  in addition, two other regions of molar proportion morphospace are also consistent with the dic model: a) a high a/i region where m1<m2<m <dig>  and b) a low a/i region where m1>m2>m <dig>  we calculated the posterior probability that anthropoid lower molar proportions can best be described by a line with parameters contained within rope intervals for the interspecific slope  and intercept .
we also determined whether these ropes contained credible parameter values for our anthropoid sample by estimating  <dig> % hdis for the interspecific slope and intercept.
we found that  <dig> % of the sampled anthropoid species’ centroids fell inside regions of molar proportion morphospace consistent with the dic model .
however, cercopithecins, hominids, and hylobatids occupied the m1<m2>m <dig> region of morphospace, indicating that their dental phenotypes cannot have originated from alterations in the a/i ratio.
as in other mammalian clades  <cit> , we found that the m1>m2<m <dig> region of morphospace was unoccupied, suggesting that some dental morphologies result from developmental mechanisms that rarely occur in mammalian evolution.
fig.
<dig> anthropoid lower molar proportions .
points are species mean molar proportions.
convex hulls denote the range of species values for each higher ranked taxon.
solid line and surrounding polygon ribbon are the estimated mean and  <dig> % highest density interval for interspecific slope and intercept.
dashed line indicates dic model’s predicted relationship between molar proportion ratios: m
3/m
1=2− <dig>  white regions are locations in molar proportion morphospace consistent with the dic model: a high a/i region where m
1<m
2<m
<dig> and a low a/i region where m
1>m
2>m
3
for the entire anthropoid sample, we found that the posterior probability  of the interspecific slope, ℙ= <dig> , or intercept, ℙ= <dig> , being within the rope was low.
the rope was partly encompassed by the  <dig> % hdi , while the posterior mean was  <dig>  .
for clades within anthropoidea there was marked heterogeneity in slope and intercept estimates.
some taxa  had posterior mean slope values closely mirroring dic model predictions, while others  had much shallower slopes .
hominoids and cercopithecins both had  <dig> % hdis that excluded the rope, indicating that dic model predicted molar proportions are not credible for these taxa.
all of the clades analyzed, however, exhibited low probabilities of slope or intercept values being within the dic molar proportion rope .
fig.
<dig> lower molar proportions for clades within anthropoidea.
points are species mean molar proportions.
convex hulls denote the range of species values for each higher ranked taxon.
solid line and surrounding polygon ribbon are the estimated mean and  <dig> % highest density interval for interspecific slope and intercept.
dashed line indicates dic model’s predicted relationship between molar proportion ratios: m
3/m
1=2− <dig>  white regions are locations in molar proportion morphospace consistent with the dic model: a high a/i region where m
1<m
2<m
<dig> and a low a/i region where m
1>m
2>m
3
λ
lhd
uhd
ℙ
lhd
uhd
ℙ
lhd
uhd
dic
mean; lhd; uhd posterior mean, lower and upper bound  <dig> % highest density levels
ℙposterior probability of parameter estimate being inside the rope: slope ℙ∈, intercept ℙ∈
*non-callitrichin platyrrhines
these results indicate that anthropoids likely exhibit a shallower slope for molar proportion relationships than the dic model predicts, suggesting that on average m <dig> increases in size less than twice as rapidly as m <dig>  just over one-half of sampled anthropoid species’ centroids fall within regions of molar proportion morphospace consistent with the dic model, but several lower ranked clades  fall into the m1<m2>m <dig> region of morphospace and display molar proportion slopes that are substantially shallower than the dic predicted slope.
catarrhines as a whole deviate more from the dic model, and occupy regions of molar proportion morphospace with lower inhibition, than platyrrhines.
within the new world monkeys, only alouattines exhibited larger distal molars as a result of low inhibition.
this variability highlights considerable clade-specific differences in molar proportion relationships among anthropoid primates.
while the dic model may provide a plausible explanation for molar proportions in platyrrhines, colobines, and papionins, the probability is low that the inhibitory cascade is a substantial factor driving relative molar size in hominoids and cercopithecins.
in the above analysis, we tested macroevolutionary predictions of the dic model related to molar proportion morphospace.
however, this analysis is potentially sensitive to the inclusion of ratios as variables in the regression model  <cit> .
therefore, as a robustness check, we tested two mathematically equivalent predictions of the dic model that relate to the morphospace of raw molar areas, using regression models that included only raw molar area variables.
m <dig> relative area
for mammals with three molars, the dic model predicts that m <dig> accounts for one-third of total lower molar area .
we defined a rope of  around this one-third point value.
for anthropoids, we found that the probability of relative m <dig> area being one-third of mt was extremely low: ℙ< <dig> .
the  <dig> % hdi  did not overlap the rope.
this indicates that, for anthropoids, the proportion of lower molar total area attributable to m <dig> is likely larger than  <dig>  .
several lower ranked clades also followed this pattern of having relatively large m2s.
both hominoids  and cercopithecins  exhibited relatively larger m2s than anthropoids as a whole.
in contrast, colobines  and papionins  deviated from this pattern, with  <dig> % hdis that encompassed most of the rope .
overall, these results echo those of the molar proportions analysis.
hominoids and cercopithecins deviate most from dic model predictions, while papionins, colobines, and platyrrhines are most consistent with the inhibitory cascade.
lucas et al.
<cit>  have previously reported that m <dig> accounts for one-third of mt area in anthropoids.
in contrast, our results indicate that m <dig> is relatively larger in anthropoidea, accounting for slightly more than one-third of mt area, though we note the strong phylogenetic component driving variability in relative m <dig> size.m
<dig> area as proportion of total lower molar area , pglmm regression results for anthropoidea and subclades
λ
lhd
uhd
ℙ
lhd
uhd
ℙ
lhd
uhd
dic
platyrrhini ∗
mean; lhd; uhd posterior mean, lower and upper bound  <dig> % highest density levels
ℙposterior probability of parameter estimate being inside the rope: slope ℙ∈, intercept ℙ∈
*non-callitrichin platyrrhines
molar areas
the dic model predicts that the relationship among molar areas is: m1= <dig> m2=a/i,m3=2− <dig>  equal area molars result when the a/i ratio is balanced, while decreasing inhibition results in larger distal molars with a cumulative effect from m <dig> to m <dig>  since the effect of relative a/i levels cascades distally along the molar field, one implication of this model is that the effect of m <dig> to m <dig> is repeated from m <dig> to m <dig>
we modeled the tripartite relationship of lower molar areas using path analysis to estimate how much the relationship between m <dig> and m <dig> area was mediated by m <dig> area.
the ‘direct’ path  was the simple relationship between m <dig> and m <dig>  while the ‘indirect’ path  was the product of a proximal path  between m <dig> and m <dig> and a distal path  between m <dig> and m <dig>  <cit> .
our effect size measure was the proportion of total lower molar size variability accounted for by m <dig> acting as mediator: prm=ab/ab+|c′|.
under a strict interpretation of the dic model, m <dig> size influences m <dig> size only through the size of m <dig> .
we defined a rope of  for the dicprm expectation.
fig.
<dig> path diagram of lower molar area relationships, illustrating the theoretical tripartite relationship between m
<dig> m
<dig> and m
<dig> molar areas
for anthropoids, the probability was extremely low that prm was within the rope: ℙ< <dig> .
the prm posterior mean  was just over two-thirds, with a  <dig> % hdi  that did not encompass the rope .
there was some heterogeneity in prm among clades within anthropoidea.
posterior means for all taxa fell well below dic model predictions.
the  <dig> % hdis for platyrrhines and catarrhines excluded the rope, though some upper bound highest density limits approached  <dig> for catarrhine subclades.
for cercopithecoids the prm was much higher than in other anthropoids  and partially overlapped the rope.
overall, the path analysis suggests, therefore, that old world monkeys are more likely than other anthropoids to have molar sizes determined by an inhibitory cascade mechanism.
however, even for cercopithecoids, the posterior probabilities of parameter estimates being inside the rope were moderate: ℙ<= <dig>  .
<dig> , pglmm regression results for anthropoidea and subclades
lhd
uhd
ℙ
dic
mean; lhd; uhd posterior mean, lower and upper bound  <dig> % highest density levels
ℙposterior probability of parameter estimate being inside the rope: ℙ∈
*non-callitrichin platyrrhines
only about two-thirds of the total effect of m <dig> size on m <dig> size was mediated through m <dig> size in anthropoids; much less than expected under the dic model.
while some of the inter-taxon heterogeneity in prm may be driven by body size , both the largest and smallest taxa in our study seem to deviate most from the dic model expectation of prm.
surprisingly, those taxa for whom m <dig> is the largest lower molar  display the greatest mediation through m <dig> .
overall, the path model either implies that there is a mechanism by which m <dig> size directly affects m <dig> size, or in some taxa a secondary developmental mechanism exists that solely affects m <dig> size.
renvoise et al.
<cit>  conducted mediation analysis on arvicoline rodents, finding that m <dig> size predicted m <dig> size  much more reliably than m <dig> size predicted m <dig> size .
this is consistent with the dic model, but renvoise et al.
did not calculate the overall effect of the indirect path through m <dig> , which is a more appropriate summary of the overall mediation effect and a better comparator to the direct path   <cit> .
it is therefore difficult to compare renvoise et al.’s results with our own.
however, both analyses suggest that while m <dig> size plays a large role in predicting m <dig> size, factors external to the dic model are also involved in determining distal molar size.
body mass and fit/deviation from the dic model
while not a prediction of the dic model, the relationship between body size  and fit/deviation from the dic molar proportion line is of interest because long life histories  may provide a mechanism through which developmental constraints of the a/i pathway can be relaxed in a broad mammalian context.
we tested the association between species’ absolute perpendicular distance from the dic molar proportion line and log body mass.
for our entire sample of anthropoids, the  <dig> % hdi  for the specific-level slope was negative, indicating that increased deviation from the dic model line is associated with a reduction in body size .
since callitrichins have extremely small body mass and deviate markedly from the dic model line, we also fitted this model including only non-callitrichin anthropoids, to see if marmosets and tamarins were driving this negative relationship.
we found that removing callitrichins reduced the likelihood of a negative association between body size and dic model deviation, as the resulting  <dig> % hdi  overlapped zero .
in addition, among anthropoids, cercopithecins exhibit the greatest deviation from dic model expectations and yet are relatively small-bodied.
we therefore fitted a third model, including only non-cercopithecin and non-callitrichin anthropoids, to see how the small-bodied and extremely deviant cercopithecins were influencing the relationship between body mass and dic deviation.
we found that removing both of these clades resulted in a positive association between body size and dic model deviation, as the posterior mean of the interspecific slope was positive , though the  <dig> % hdi  overlapped zero .
molar area variability
the dic model predicts that m <dig> will have the greatest size variability of any lower molar.
we modeled the difference in average molar area variance  between each lower molar type across the anthropoid sample.
we found that m <dig> size was indeed more variable than either m <dig> size  or m <dig> size .
this is consistent with both the dic model and previously theorized patterns of dental development in mammals .
m <dig> agenesis
the dic model makes predictions about expected m2/m <dig> proportions when the third molar is absent.
specifically, m <dig> agenesis is predicted to occur when m <dig> is less than half the size of m <dig>  in anthropoids, we instead found that m <dig> agenesis occurred when the m2/m <dig> ratio was much larger  and that the expected m2/m <dig> ratio under the dic model had very low probability: ℙ= <dig> .
within a sample of modern humans  polymorphic for m <dig> agenesis , we also found that relative m <dig> size was much larger than predicted under the dic model in individuals with congenitally missing m3s  and that the dic expected m2/m <dig> ratio was improbable: ℙ= <dig> .
in addition, the mean m2/m <dig> proportion for individuals not exhibiting m <dig> agenesis  was similar to those with congenitally missing m3s .
fig.
<dig> lower molar proportions for modern humans polymorphic for m
<dig> agenesis .
points are individual specimen molar proportions.
dashed line indicates dic model’s predicted relationship between molar proportion ratios: m
3/m
1=2− <dig>  white regions are locations in molar proportion morphospace consistent with the dic model: a high a/i region where m
1<m
2<m
<dig> and a low a/i region where m
1>m
2>m
3
third molar variations are seen with some frequency among primates.
modern humans, for example, are polymorphic in third molar number  <cit> .
evans et al.
<cit>  have recently proposed that the dic is a mechanism that explains both the relative and absolute sizes of permanent molars in modern humans, including the relative size of m <dig> when the third molar is absent.
our results demonstrate, in contrast, that modern humans have second molars subequal in size to first molars, whether or not the m <dig> is congenitally absent  <cit> .
in marmosets and tamarins, m <dig> agenesis is likely to have occurred independently three times  <cit>  and there is some reduction of m <dig> relative to other platyrrhines.
however, our results agree with those of bernal et al.
<cit>  that m <dig> is still larger than predicted by the dic model.
one other platyrrhine, the extinct genus xenothrix, has third molar agenesis, though it too has an m2/m <dig> proportion  much larger than the dic model expectation  <cit> .
the departure of callitrichins and modern humans from the expectations of the dic developmental model suggests that mechanisms other than alterations in the a/i ratio may regulate variation in molar number and relative size.
with the exception of certain syndromes, the developmental mechanisms leading to third molar agenesis are unknown, even in humans  <cit> .
there is some developmental evidence that supports a lack of association between relative m <dig> size and m <dig> agenesis, as cai et al.
<cit>  found that tooth size and tooth number were largely independent in rodents.
overall, anthropoids with third molar agenesis have m2/m <dig> proportions that are much larger than predicted by the dic model, which contrasts with carnivora, whose m2/m <dig> proportions range from larger  to much smaller  than expected under the model, with many forms being congruent with the model  <cit> .
developmental timing
one extension of the dic model concerns the relative timing of molar development, specifically that temporal overlap in molar calcification initiation will increase with higher a/i.
an assessment of nine primate species for which molar calcification data were available suggests the opposite trend.
we predicted that temporal overlap would be highest in species with small m1s.
instead we found that genera with relatively large m1s, such as varecia, had the greatest amount of temporal overlap between the development of m <dig> and m <dig>  while genera like macaca, with the smallest m1s, had the least amount of temporal overlap .
though other factors, such as growth rates of subsequent molars, may also contribute to relative molar proportions, the evidence suggests that species with large posterior molars initiate them long after their smaller anterior molar crowns are complete.
fig.
<dig> developmental timing of lower molar calcification in primates .
points are species mean molar proportions.
colors denote degree of temporal overlap in m
2– m
<dig> calcification start times.
dashed line indicates dic model’s predicted relationship between molar proportion ratios: m
3/m
1=2− <dig>  white regions are locations in molar proportion morphospace consistent with the dic model: a high a/i region where m
1<m
2<m
<dig> and a low a/i region where m
1>m
2>m
3
a large m <dig> was associated with greater temporal separation between m <dig> and m <dig> initiation, even when controlling for m <dig> size.
though this pattern could also be consistent with a weak correlation between early stage  and late stage  developmental events, there is no histological evidence to support marked deviation in the relative schedule of these events.
this suggests that, for anthropoids, the dic model cannot fully explain variation in molar proportions.
primary dietary category and molar area proportions
kavanagh et al.
<cit>  suggested that molar proportions could be associated with dietary preference.
we tested this hypothesis by estimating whether anthropoid species that primarily eat fruit are more likely to occupy the m1<m2>m <dig> region of molar proportion morphospace than anthropoids that primarily eat other food items.
we found that frugivorous anthropoids had much higher odds of occupying the m1<m2>m <dig> region of molar proportion morphospace .
this equates to a  <dig> %  increase in the odds of an anthropoid species being frugivorous if m <dig> is its largest lower molar.
fig.
<dig> anthropoid primary dietary category and lower molar proportions .
points are species mean molar proportions.
colors denote species’ primary dietary category.
dashed line indicates dic model’s predicted relationship between molar proportion ratios: m
3/m
1=2− <dig>  white regions are locations in molar proportion morphospace consistent with the dic model: a high a/i region where m
1<m
2<m
<dig> and a low a/i region where m
1>m
2>m
3
four taxonomic groups  that predominantly occupy the m1<m2>m <dig> molar proportion morphospace have diets that are primarily frugivorous, with low-crowned bunodont molar cusps.
though some researchers have argued that molar proportions represent adaptations to processing food with different material properties  <cit> , there is minimal evidence to support the m1<m2>m <dig> pattern as an adaptation for frugivory.
there is, however, evidence to suggest that different cusp shapes and relief profiles have large effects on processing food with different mechanical properties  <cit> .
broadly, folivores tend to have high-crowned or lophed teeth that aid the shearing of leafy plant material, while frugivores have lower, more bulbous, cusps that aid in mastication of soft pulpy fruit and the emission of juices  <cit> .
if the strong selective pressure on reducing cusp height acts on genes that are involved in both the establishment of cusp relief and the regulation of molar sizes, then selection for lower cusp height would constrain the developmental pathways through which molars could evolve.
CONCLUSIONS
our results provide only limited support for the hypothesis that anthropoid molar proportions are partially governed by the dental inhibitory cascade model.
we found a large amount of variability in relative and absolute molar sizes across anthropoid groups.
some clades  showed patterns of relative molar size consistent with changes in a/i concentrations, while others  diverged markedly from the expectations of the inhibitory cascade.
while the dic model may have been the plesiomorphic model for mammalian molar proportions, in anthropoid primates and other taxa it is likely that secondary developmental pathways have influenced relative molar sizes  <cit> .
we argue that the molar proportions of anthropoids result from the influence of two constraints in different directions: 1) co-option of the same signaling molecules  for different developmental processes throughout dental evolution constrains the relationship between cusp height and relative molar proportion; and, 2) relaxation of constraints between activation and inhibition molecules caused by lengthening of absolute developmental time.
co-option of genes occurs throughout dental evolution, including the development of toothed jaws  <cit> , regionalization in the dentition  <cit>  and dental complexity  <cit> .
dental features are interdependent, both within a tooth  <cit>  and along a tooth row  <cit> , and it is likely that a similar relationship exists throughout developmental time between molar proportions and cusp relief.
for example, shh is known to regulate sostdc <dig>  and alter the size of the enamel knot, influencing the ultimate size of cusps  <cit> .
similarly, altering concentrations of pdgf can change both molar size by  <dig> % and cusp height by  <dig> %  <cit> .
thus it is probable that selection for change in cusp relief has a related effect on molar proportions that drives species into regions of morphospace that are unexpected under the dic model.
we propose that a relaxation of the constraints of the a/i pathway brought on by long life histories may allow the evolution of molar proportions that are inconsistent with the dic model.
activation molecules are expressed only for a short period of time and these up-regulate inhibitory molecules.
while larger-bodied individuals will typically have longer and more intense expression of both types of genes, there is an absolute limit on how long they can be expressed because of their molecular decay rate  <cit> .
smaller-bodied mammals that have slowed rates of embryonic development, such as callitrichins  <cit> , may similarly be more prone to molar loss.
thus, while small-bodied and quick developing species such as mice may be greatly constrained by a/i genes, larger-bodied taxa may evolve in ways that are not consistent with the dic model.
this possibility reiterates long-known concerns about using a small-bodied species with fast life history to generalize aspects of evo-devo across mammalia  <cit> .
heterochronic change in the eruption timing of mammalian molars may also relax these developmental constraints, though this has not been tested in the developmental literature.
this is potentially the case for many of the subfossil and living lemurids, who have a fast dental development schedule with early eruption of the molars  <cit> , in addition to slow-developing callitrichins  <cit> .
in this study, our aim was to investigate macroevolutionary predictions of the dic model in anthropoid primates, specifically testing whether the developmental mechanisms proposed by the model matched observed morphological variation in relative molar areas.
we found some congruence between the results of our analysis and the dic model.
over one-half of the sampled species’ centroids fell within dic model expected regions of molar proportion morphospace.
in particular, platyrrhines, colobines, and papionins showed patterns of relative molar size consistent with changes in a/i concentrations.
in anthropoids, however, it seems relatively easy to diverge from the strict predictions of the inhibitory cascade, which was particularly the case for hominoids and cercopithecins who occupied the m1<m2>m <dig> region of molar proportion morphospace.
diverse groups of non-primate taxa, for example canids  <cit> , notoungulata and astrapotheria  <cit> , and arvicoline rodents  <cit> , also show divergence from dic model expectations.
though we found only weak evidence of a positive association between deviation from the dic model and body size, this could potentially be a result of the majority of the sampled anthropoid taxa having medium body size.
a more robust test of this hypothesis will require a broader mammalian sample.
while the dic model explains some of the variation in mammalian molar proportions, there are likely other important developmental pathways being co-opted to produce molar rows with larger m2s than expected or those missing m3s.
further studies investigating the ontogenetic and molecular processes underlying different species will shed more light on the specific developmental pathways that determine molar proportions across mammals.

 mammalian dentition is complex and variable, adapting to the dietary, environmental, and social demands of a species’ niche. despite the considerable range of variability in dental forms, certain morphological and evolutionary patterns emerge repeatedly in phylogenetically-distant taxa. for example, the hypocone evolved independently more than times, hypsodonty evolved in multiple species of ungulates, lagomorphs, and south american rodents, and tribosphenic molars may have evolved twice in mammals. during evolution, teeth are more frequently lost than gained and the last developing teeth within a particular dental field are almost always lost first. while many of these patterns likely evolved in response to similar selective pressures, there is evidence that developmental constraints limit molar shape. thus it is reasonable to assume that some of the dental variability of mammalian teeth is constrained by developmental processes. in the past decade, several studies have explored developmental mechanisms potentially responsible for dental morphological variation. kavanagh and colleagues developed the dental inhibitory cascade model to explain the evolution and development of molar proportions. through experimental alteration of developing mouse molars, the authors suggested that much of mammalian molar proportion diversity could be attributed to a simple, highly conserved pattern of differences in the timing and concentration of molecules that activate or inhibit molar initiation and proliferation. when the enamel knot of the earliest developing mandibular molar is initiated, it expresses inhibitory signals that inhibit, or greatly delay, the subsequent formation of m and m a change in the expression or timing of these inhibitory signals was proposed to determine the eventual size of m and m by altering the relative timing of molar initiation. according to kavanagh et al., therefore, the patterning of molar sizes works as an inhibitory cascade, so that the timing of m development cumulatively affects the timing of m and m development, as the dic mechanism operates along the developing tooth row. decreased m inhibition allows m and m to form earlier and ultimately grow larger than m while increased inhibition restricts the size of subsequently developing molars and may eventually lead to their loss. the dic model can be characterized by the following equation: y=1+/ix− where y is molar area, as determined by position along the tooth row, and by the relative strength of activators and inhibitors, defined by the a/i ratio. molar areas are derived from eq. as m1= m2=a/i,m3=2− thus, balance between activation and inhibition leads to equal sized molars, while decreases in inhibition result in larger distal molar areas with a cumulative effect from m to m since inhibition is hypothesized to cascade distally along the molar row, it is possible to predict the relative size of each molar using the following relationship: m3/m1=2− as a test of the dic model’s predictions in a macroevolutionary context, kavanagh et al. demonstrated that most of the molar patterns seen in murid rodents can be explained by alteration of the ratio of activation and inhibition molecules expressed during early tooth development. one extension of the model is that in morphospace regions of high inhibition, we would predict minimal temporal overlap between m and m calcification. as the level of inhibition decreases, however, the amount of overlap should increase, so that in morphospace regions with the lowest levels of inhibition temporal overlap would be substantial. the dic model highlights the importance of developmental timing in determining dental metrics, by establishing the pattern of hierarchical dependence within a particular developmental system and by asserting broad applicability across mammals. while the model does not predict variation in cusp height, enamel thickness, or eruption schedule, one implication of the model is that molar size proportions are independent from these and other traits determined later in development. for example, our interpretation is that under this model, fruit bats, golden moles, and tree shrews not only exhibit similar molar size proportions, but proportions that were determined by fundamentally the same developmental interactions. the dic model is innovative in suggesting that development may also drive molar size variation in addition to other variables, such as dietary adaptation, phylogenetic inertia, and allometry. researchers have begun to test the predictions of the dic model in taxa, such as rodents and carnivores, which have shown rapid expansion of dental morphospace over their recent evolutionary history. a study of mesozoic and cenozoic mammals suggested that the dic model could be plesiomorphic for this clade. yet, while some studies support the original findings of the dic model, others have presented evidence that calls into question its broad phylogenetic applicability. for example, voles show an expansion of m size that drives molar proportions into morphospace previously thought unoccupiable, while canids exhibit a reduced major axis regression slope between m3/m and m2/m that is much smaller than predicted by the dic model. though the authors of both the above studies explain deviations from the dic model as the result of changes in differential evolvability of the m the inability of the dic model to explain all relationships between molar size proportions across mammalia is worth further study. in particular, the model may not generalize to species with longer life histories, as the decay rates of activation and inhibition molecules may be sufficient enough to lessen the effect of the model. bernal et al. assessed molar size proportion variation among platyrrhines, an extant radiation of south american primates, and found limited support for the dic model. while the results of a phylogeny-dependent regression analysis corresponded with the predictions of the dic model, an examination of platyrrhine molar proportions using ordinary least squares and reduced major axis regression, as well as an assessment of intraspecific variability in molar proportions, deviated from the dic model’s predictions. the authors attribute these findings to the highly derived dentition of some platyrrhine clades, including the loss of the third molar in marmosets and tamarins, and reduction of the third molar in cebinae. we suggest that a broader consideration of the anthropoid dentition reveals multiple other morphological observations that are incongruent with the predictions of the dic model. for example, great apes exhibit a m1<m2>m molar pattern, in contrast to the m1<m2<m pattern of cercopithecoids. the m1<m2>m pattern is difficult to explain under the dic model. moreover, m agenesis in humans, which unlike in dwarfed marmosets and tamarins is polymorphic, has been reported to occur without large changes in relative m size. our aim in this paper is to test three predictions, and two extensions, of the dic model using a broad sample of extant anthropoid primates. if anthropoid molar proportions follow the dic model, this represents further evidence of the model’s robustness across mammalian taxa. if, however, anthropoids deviate from the dic model’s predictions, it is possible that other factors play an equally important role in determining their relative molar sizes. previous studies that have sought to test the dic model’s developmental predictions have relied solely on measurements of molar crown size. anthropoid primates offer a unique opportunity to test the developmental predictions of the dic model, as data are available on the timing of calcification in successive molars, allowing ontogenetic patterns to be evaluated. to test the model’s predictions we employ an integrative approach that considers dental metrics, development, diet, and function. in the following section we report the results of testing three macroevolutionary predictions, and two extensions, of the dic model: molar area relationships: the relationship among molar areas is: m1= m2=a/i,m3=2− balance between a/i leads to equal area molars, while increasing inhibition enlarges distal molar areas with a cumulative effect from m to m the relationship between m3/m and m2/m molar proportions can best be described by the linear equation: m3/m1=2− three regions of molar proportion morphospace are consistent with the dic model: a) a high a/i region where m1<m2<m b) a low a/i region where m1>m2>m and c) a region where m3/m1=2− for mammals with three molars, m accounts for one-third of total lower molar area. molar area variability:m is predicted to have the greatest size variability of any lower molar. m agenesis: m agenesis occurs when m is less than half the size of m developmental timing: in regions of molar proportion morphospace subject to high inhibition there is minimal temporal overlap between m and m calcification, while in morphospace regions of low inhibition temporal overlap is substantial. primary dietary category and molar area proportions: molar proportions can be a measure of diet.  to test macroevolutionary predictions of the dic model we used previously published lower molar occlusal area data from extant anthropoid primates. we analyzed these data using bayesian phylogenetic generalized linear mixed models, and employed a markov chain monte carlo approach to sample the posterior distribution of parameter space. phylogenetic signal for each model was quantified using pagel’s λ parameter. all models achieved convergence to a stationary posterior distribution, as determined by hiedelberger and welch’s convergence diagnostic and low levels of autocorrelation. our mcmc sampling strategies resulted in effective sample sizes of for each parameter. detailed descriptions of the anthropoid sample, phylogenetic tree, morphometric variables, and regression models used are provided in the methods section. to evaluate if dic model expectations were credible, given our anthropoid sample, we calculated % highest density intervals of posterior distribution parameter values. the dic model specifies point predictions, rather than interval predictions, of parameter values. for any empirical dataset, the probability is therefore extremely small that molar proportions would exactly match dic model expectations. to provide a more useful way of calculating the probability that dic model predictions were correct, given our data, we created a region of practical equivalence around each dic model parameter expectation. a rope is an interval that encloses values of the parameter that are, for practical purposes, negligibly different from the point value. we used the rope as a decision tool for determining whether dic model predicted parameter values were credible and/or probable for the sampled anthropoid taxa. to determine a range of slope/intercept values that might be deemed practically equivalent to the dic mathematical model’s predictions, we used, as a starting point, experimental evidence reported in kavanagh et al.. this experimental evidence yielded a slope of and intercept of -, thus deviating slightly from the dic mathematical model’s predictions of and − for each pglmm, we calculated posterior probabilities for ropes of several sizes. the inclusion of ropes of any size in our analyses was a conservative measure, increasing the chance of dic model predictions being corroborated, compared with using only the point predictions of the strict mathematical model. molar area corrected estimates the product of maximum linear breadth and length measurements often overestimates molar occlusal area. we employed a correction to make these rectangular areas more comparable to areas calculated by tracing outlines around molar occlusal perimeters. we found that the average difference between outline areas and corrected areas was much smaller than that between outline areas and rectangular areas. this indicated that corrected areas provided a better approximation to molar outline occlusal area. molar area relationships molar area proportions the dic model predicts that the relationship between m3/m and m2/m is best characterized by a line with a slope of and an intercept of − in addition, two other regions of molar proportion morphospace are also consistent with the dic model: a) a high a/i region where m1<m2<m and b) a low a/i region where m1>m2>m we calculated the posterior probability that anthropoid lower molar proportions can best be described by a line with parameters contained within rope intervals for the interspecific slope and intercept. we also determined whether these ropes contained credible parameter values for our anthropoid sample by estimating % hdis for the interspecific slope and intercept. we found that % of the sampled anthropoid species’ centroids fell inside regions of molar proportion morphospace consistent with the dic model. however, cercopithecins, hominids, and hylobatids occupied the m1<m2>m region of morphospace, indicating that their dental phenotypes cannot have originated from alterations in the a/i ratio. as in other mammalian clades, we found that the m1>m2<m region of morphospace was unoccupied, suggesting that some dental morphologies result from developmental mechanisms that rarely occur in mammalian evolution. fig. anthropoid lower molar proportions. points are species mean molar proportions. convex hulls denote the range of species values for each higher ranked taxon. solid line and surrounding polygon ribbon are the estimated mean and % highest density interval for interspecific slope and intercept. dashed line indicates dic model’s predicted relationship between molar proportion ratios: m 3/m 1=2− white regions are locations in molar proportion morphospace consistent with the dic model: a high a/i region where m 1<m 2<m and a low a/i region where m 1>m 2>m 3 for the entire anthropoid sample, we found that the posterior probability of the interspecific slope, ℙ=, or intercept, ℙ=, being within the rope was low. the rope was partly encompassed by the % hdi, while the posterior mean was. for clades within anthropoidea there was marked heterogeneity in slope and intercept estimates. some taxa had posterior mean slope values closely mirroring dic model predictions, while others had much shallower slopes. hominoids and cercopithecins both had % hdis that excluded the rope, indicating that dic model predicted molar proportions are not credible for these taxa. all of the clades analyzed, however, exhibited low probabilities of slope or intercept values being within the dic molar proportion rope. fig. lower molar proportions for clades within anthropoidea. points are species mean molar proportions. convex hulls denote the range of species values for each higher ranked taxon. solid line and surrounding polygon ribbon are the estimated mean and % highest density interval for interspecific slope and intercept. dashed line indicates dic model’s predicted relationship between molar proportion ratios: m 3/m 1=2− white regions are locations in molar proportion morphospace consistent with the dic model: a high a/i region where m 1<m 2<m and a low a/i region where m 1>m 2>m 3 λ lhd uhd ℙ lhd uhd ℙ lhd uhd dic mean; lhd; uhd posterior mean, lower and upper bound % highest density levels ℙposterior probability of parameter estimate being inside the rope: slope ℙ∈, intercept ℙ∈ *non-callitrichin platyrrhines these results indicate that anthropoids likely exhibit a shallower slope for molar proportion relationships than the dic model predicts, suggesting that on average m increases in size less than twice as rapidly as m just over one-half of sampled anthropoid species’ centroids fall within regions of molar proportion morphospace consistent with the dic model, but several lower ranked clades fall into the m1<m2>m region of morphospace and display molar proportion slopes that are substantially shallower than the dic predicted slope. catarrhines as a whole deviate more from the dic model, and occupy regions of molar proportion morphospace with lower inhibition, than platyrrhines. within the new world monkeys, only alouattines exhibited larger distal molars as a result of low inhibition. this variability highlights considerable clade-specific differences in molar proportion relationships among anthropoid primates. while the dic model may provide a plausible explanation for molar proportions in platyrrhines, colobines, and papionins, the probability is low that the inhibitory cascade is a substantial factor driving relative molar size in hominoids and cercopithecins. in the above analysis, we tested macroevolutionary predictions of the dic model related to molar proportion morphospace. however, this analysis is potentially sensitive to the inclusion of ratios as variables in the regression model. therefore, as a robustness check, we tested two mathematically equivalent predictions of the dic model that relate to the morphospace of raw molar areas, using regression models that included only raw molar area variables. m relative area for mammals with three molars, the dic model predicts that m accounts for one-third of total lower molar area. we defined a rope of around this one-third point value. for anthropoids, we found that the probability of relative m area being one-third of mt was extremely low: ℙ<. the % hdi did not overlap the rope. this indicates that, for anthropoids, the proportion of lower molar total area attributable to m is likely larger than. several lower ranked clades also followed this pattern of having relatively large m2s. both hominoids and cercopithecins exhibited relatively larger m2s than anthropoids as a whole. in contrast, colobines and papionins deviated from this pattern, with % hdis that encompassed most of the rope. overall, these results echo those of the molar proportions analysis. hominoids and cercopithecins deviate most from dic model predictions, while papionins, colobines, and platyrrhines are most consistent with the inhibitory cascade. lucas et al. have previously reported that m accounts for one-third of mt area in anthropoids. in contrast, our results indicate that m is relatively larger in anthropoidea, accounting for slightly more than one-third of mt area, though we note the strong phylogenetic component driving variability in relative m size.m area as proportion of total lower molar area, pglmm regression results for anthropoidea and subclades λ lhd uhd ℙ lhd uhd ℙ lhd uhd dic platyrrhini ∗ mean; lhd; uhd posterior mean, lower and upper bound % highest density levels ℙposterior probability of parameter estimate being inside the rope: slope ℙ∈, intercept ℙ∈ *non-callitrichin platyrrhines molar areas the dic model predicts that the relationship among molar areas is: m1= m2=a/i,m3=2− equal area molars result when the a/i ratio is balanced, while decreasing inhibition results in larger distal molars with a cumulative effect from m to m since the effect of relative a/i levels cascades distally along the molar field, one implication of this model is that the effect of m to m is repeated from m to m we modeled the tripartite relationship of lower molar areas using path analysis to estimate how much the relationship between m and m area was mediated by m area. the ‘direct’ path was the simple relationship between m and m while the ‘indirect’ path was the product of a proximal path between m and m and a distal path between m and m. our effect size measure was the proportion of total lower molar size variability accounted for by m acting as mediator: prm=ab/ab+|c′|. under a strict interpretation of the dic model, m size influences m size only through the size of m. we defined a rope of for the dicprm expectation. fig. path diagram of lower molar area relationships, illustrating the theoretical tripartite relationship between m m and m molar areas for anthropoids, the probability was extremely low that prm was within the rope: ℙ<. the prm posterior mean was just over two-thirds, with a % hdi that did not encompass the rope. there was some heterogeneity in prm among clades within anthropoidea. posterior means for all taxa fell well below dic model predictions. the % hdis for platyrrhines and catarrhines excluded the rope, though some upper bound highest density limits approached for catarrhine subclades. for cercopithecoids the prm was much higher than in other anthropoids and partially overlapped the rope. overall, the path analysis suggests, therefore, that old world monkeys are more likely than other anthropoids to have molar sizes determined by an inhibitory cascade mechanism. however, even for cercopithecoids, the posterior probabilities of parameter estimates being inside the rope were moderate: ℙ<=., pglmm regression results for anthropoidea and subclades lhd uhd ℙ dic mean; lhd; uhd posterior mean, lower and upper bound % highest density levels ℙposterior probability of parameter estimate being inside the rope: ℙ∈ *non-callitrichin platyrrhines only about two-thirds of the total effect of m size on m size was mediated through m size in anthropoids; much less than expected under the dic model. while some of the inter-taxon heterogeneity in prm may be driven by body size, both the largest and smallest taxa in our study seem to deviate most from the dic model expectation of prm. surprisingly, those taxa for whom m is the largest lower molar display the greatest mediation through m. overall, the path model either implies that there is a mechanism by which m size directly affects m size, or in some taxa a secondary developmental mechanism exists that solely affects m size. renvoise et al. conducted mediation analysis on arvicoline rodents, finding that m size predicted m size much more reliably than m size predicted m size. this is consistent with the dic model, but renvoise et al. did not calculate the overall effect of the indirect path through m, which is a more appropriate summary of the overall mediation effect and a better comparator to the direct path. it is therefore difficult to compare renvoise et al.’s results with our own. however, both analyses suggest that while m size plays a large role in predicting m size, factors external to the dic model are also involved in determining distal molar size. body mass and fit/deviation from the dic model while not a prediction of the dic model, the relationship between body size and fit/deviation from the dic molar proportion line is of interest because long life histories may provide a mechanism through which developmental constraints of the a/i pathway can be relaxed in a broad mammalian context. we tested the association between species’ absolute perpendicular distance from the dic molar proportion line and log body mass. for our entire sample of anthropoids, the % hdi for the specific-level slope was negative, indicating that increased deviation from the dic model line is associated with a reduction in body size. since callitrichins have extremely small body mass and deviate markedly from the dic model line, we also fitted this model including only non-callitrichin anthropoids, to see if marmosets and tamarins were driving this negative relationship. we found that removing callitrichins reduced the likelihood of a negative association between body size and dic model deviation, as the resulting % hdi overlapped zero. in addition, among anthropoids, cercopithecins exhibit the greatest deviation from dic model expectations and yet are relatively small-bodied. we therefore fitted a third model, including only non-cercopithecin and non-callitrichin anthropoids, to see how the small-bodied and extremely deviant cercopithecins were influencing the relationship between body mass and dic deviation. we found that removing both of these clades resulted in a positive association between body size and dic model deviation, as the posterior mean of the interspecific slope was positive, though the % hdi overlapped zero. molar area variability the dic model predicts that m will have the greatest size variability of any lower molar. we modeled the difference in average molar area variance between each lower molar type across the anthropoid sample. we found that m size was indeed more variable than either m size or m size. this is consistent with both the dic model and previously theorized patterns of dental development in mammals. m agenesis the dic model makes predictions about expected m2/m proportions when the third molar is absent. specifically, m agenesis is predicted to occur when m is less than half the size of m in anthropoids, we instead found that m agenesis occurred when the m2/m ratio was much larger and that the expected m2/m ratio under the dic model had very low probability: ℙ=. within a sample of modern humans polymorphic for m agenesis, we also found that relative m size was much larger than predicted under the dic model in individuals with congenitally missing m3s and that the dic expected m2/m ratio was improbable: ℙ=. in addition, the mean m2/m proportion for individuals not exhibiting m agenesis was similar to those with congenitally missing m3s. fig. lower molar proportions for modern humans polymorphic for m agenesis. points are individual specimen molar proportions. dashed line indicates dic model’s predicted relationship between molar proportion ratios: m 3/m 1=2− white regions are locations in molar proportion morphospace consistent with the dic model: a high a/i region where m 1<m 2<m and a low a/i region where m 1>m 2>m 3 third molar variations are seen with some frequency among primates. modern humans, for example, are polymorphic in third molar number. evans et al. have recently proposed that the dic is a mechanism that explains both the relative and absolute sizes of permanent molars in modern humans, including the relative size of m when the third molar is absent. our results demonstrate, in contrast, that modern humans have second molars subequal in size to first molars, whether or not the m is congenitally absent. in marmosets and tamarins, m agenesis is likely to have occurred independently three times and there is some reduction of m relative to other platyrrhines. however, our results agree with those of bernal et al. that m is still larger than predicted by the dic model. one other platyrrhine, the extinct genus xenothrix, has third molar agenesis, though it too has an m2/m proportion much larger than the dic model expectation. the departure of callitrichins and modern humans from the expectations of the dic developmental model suggests that mechanisms other than alterations in the a/i ratio may regulate variation in molar number and relative size. with the exception of certain syndromes, the developmental mechanisms leading to third molar agenesis are unknown, even in humans. there is some developmental evidence that supports a lack of association between relative m size and m agenesis, as cai et al. found that tooth size and tooth number were largely independent in rodents. overall, anthropoids with third molar agenesis have m2/m proportions that are much larger than predicted by the dic model, which contrasts with carnivora, whose m2/m proportions range from larger to much smaller than expected under the model, with many forms being congruent with the model. developmental timing one extension of the dic model concerns the relative timing of molar development, specifically that temporal overlap in molar calcification initiation will increase with higher a/i. an assessment of nine primate species for which molar calcification data were available suggests the opposite trend. we predicted that temporal overlap would be highest in species with small m1s. instead we found that genera with relatively large m1s, such as varecia, had the greatest amount of temporal overlap between the development of m and m while genera like macaca, with the smallest m1s, had the least amount of temporal overlap. though other factors, such as growth rates of subsequent molars, may also contribute to relative molar proportions, the evidence suggests that species with large posterior molars initiate them long after their smaller anterior molar crowns are complete. fig. developmental timing of lower molar calcification in primates. points are species mean molar proportions. colors denote degree of temporal overlap in m 2– m calcification start times. dashed line indicates dic model’s predicted relationship between molar proportion ratios: m 3/m 1=2− white regions are locations in molar proportion morphospace consistent with the dic model: a high a/i region where m 1<m 2<m and a low a/i region where m 1>m 2>m 3 a large m was associated with greater temporal separation between m and m initiation, even when controlling for m size. though this pattern could also be consistent with a weak correlation between early stage and late stage developmental events, there is no histological evidence to support marked deviation in the relative schedule of these events. this suggests that, for anthropoids, the dic model cannot fully explain variation in molar proportions. primary dietary category and molar area proportions kavanagh et al. suggested that molar proportions could be associated with dietary preference. we tested this hypothesis by estimating whether anthropoid species that primarily eat fruit are more likely to occupy the m1<m2>m region of molar proportion morphospace than anthropoids that primarily eat other food items. we found that frugivorous anthropoids had much higher odds of occupying the m1<m2>m region of molar proportion morphospace. this equates to a % increase in the odds of an anthropoid species being frugivorous if m is its largest lower molar. fig. anthropoid primary dietary category and lower molar proportions. points are species mean molar proportions. colors denote species’ primary dietary category. dashed line indicates dic model’s predicted relationship between molar proportion ratios: m 3/m 1=2− white regions are locations in molar proportion morphospace consistent with the dic model: a high a/i region where m 1<m 2<m and a low a/i region where m 1>m 2>m 3 four taxonomic groups that predominantly occupy the m1<m2>m molar proportion morphospace have diets that are primarily frugivorous, with low-crowned bunodont molar cusps. though some researchers have argued that molar proportions represent adaptations to processing food with different material properties, there is minimal evidence to support the m1<m2>m pattern as an adaptation for frugivory. there is, however, evidence to suggest that different cusp shapes and relief profiles have large effects on processing food with different mechanical properties. broadly, folivores tend to have high-crowned or lophed teeth that aid the shearing of leafy plant material, while frugivores have lower, more bulbous, cusps that aid in mastication of soft pulpy fruit and the emission of juices. if the strong selective pressure on reducing cusp height acts on genes that are involved in both the establishment of cusp relief and the regulation of molar sizes, then selection for lower cusp height would constrain the developmental pathways through which molars could evolve.  our results provide only limited support for the hypothesis that anthropoid molar proportions are partially governed by the dental inhibitory cascade model. we found a large amount of variability in relative and absolute molar sizes across anthropoid groups. some clades showed patterns of relative molar size consistent with changes in a/i concentrations, while others diverged markedly from the expectations of the inhibitory cascade. while the dic model may have been the plesiomorphic model for mammalian molar proportions, in anthropoid primates and other taxa it is likely that secondary developmental pathways have influenced relative molar sizes. we argue that the molar proportions of anthropoids result from the influence of two constraints in different directions: 1) co-option of the same signaling molecules for different developmental processes throughout dental evolution constrains the relationship between cusp height and relative molar proportion; and, 2) relaxation of constraints between activation and inhibition molecules caused by lengthening of absolute developmental time. co-option of genes occurs throughout dental evolution, including the development of toothed jaws, regionalization in the dentition and dental complexity. dental features are interdependent, both within a tooth and along a tooth row, and it is likely that a similar relationship exists throughout developmental time between molar proportions and cusp relief. for example, shh is known to regulate sostdc and alter the size of the enamel knot, influencing the ultimate size of cusps. similarly, altering concentrations of pdgf can change both molar size by % and cusp height by %. thus it is probable that selection for change in cusp relief has a related effect on molar proportions that drives species into regions of morphospace that are unexpected under the dic model. we propose that a relaxation of the constraints of the a/i pathway brought on by long life histories may allow the evolution of molar proportions that are inconsistent with the dic model. activation molecules are expressed only for a short period of time and these up-regulate inhibitory molecules. while larger-bodied individuals will typically have longer and more intense expression of both types of genes, there is an absolute limit on how long they can be expressed because of their molecular decay rate. smaller-bodied mammals that have slowed rates of embryonic development, such as callitrichins, may similarly be more prone to molar loss. thus, while small-bodied and quick developing species such as mice may be greatly constrained by a/i genes, larger-bodied taxa may evolve in ways that are not consistent with the dic model. this possibility reiterates long-known concerns about using a small-bodied species with fast life history to generalize aspects of evo-devo across mammalia. heterochronic change in the eruption timing of mammalian molars may also relax these developmental constraints, though this has not been tested in the developmental literature. this is potentially the case for many of the subfossil and living lemurids, who have a fast dental development schedule with early eruption of the molars, in addition to slow-developing callitrichins. in this study, our aim was to investigate macroevolutionary predictions of the dic model in anthropoid primates, specifically testing whether the developmental mechanisms proposed by the model matched observed morphological variation in relative molar areas. we found some congruence between the results of our analysis and the dic model. over one-half of the sampled species’ centroids fell within dic model expected regions of molar proportion morphospace. in particular, platyrrhines, colobines, and papionins showed patterns of relative molar size consistent with changes in a/i concentrations. in anthropoids, however, it seems relatively easy to diverge from the strict predictions of the inhibitory cascade, which was particularly the case for hominoids and cercopithecins who occupied the m1<m2>m region of molar proportion morphospace. diverse groups of non-primate taxa, for example canids, notoungulata and astrapotheria, and arvicoline rodents, also show divergence from dic model expectations. though we found only weak evidence of a positive association between deviation from the dic model and body size, this could potentially be a result of the majority of the sampled anthropoid taxa having medium body size. a more robust test of this hypothesis will require a broader mammalian sample. while the dic model explains some of the variation in mammalian molar proportions, there are likely other important developmental pathways being co-opted to produce molar rows with larger m2s than expected or those missing m3s. further studies investigating the ontogenetic and molecular processes underlying different species will shed more light on the specific developmental pathways that determine molar proportions across mammals. 
we tested the macroevolutionary implications of this model using anthropoid primate species, focusing on overall morphological patterns, as well as predictions made about molar size variability, direct developmental control, and diet. overall, we found considerable clade-specific differences in relative molar sizes among anthropoid primates, with hominoids and cercopithecins strongly divergent from dic model predictions, and platyrrhines, colobines, and papionins more consistent with the inhibitory cascade. molar proportionsdental inhibitory cascadeanthropoid primatesevo-devobayesian phylogenetic generalized linear mixed modelshttp://dx.doi.org/ /100000001national science foundationdge-1144152carter katherine e. issue-copyright-statement© the author 2016 recently, the dental inhibitory cascade model has been suggested to explain much of the variation in mammalian molar size proportions. furthermore, we found that some variability in third molar size could not be explained by the influence of the inhibitory cascade. of the species sampled, % had centroids that fell within regions of molar proportion morphospace consistent with the dic model. some dic model predictions were not supported, however, such as the expected proportion of m2/m when the third molar is absent. finally, we investigated reasons why some clades deviated from dic model expectations. the dic model explains some of the variation in anthropoid primate molar proportions. developmental processes that underpin morphological variation have become a focus of interest when attempting to interpret macroevolutionary patterns. however, there are interesting deviations away from this broad mammalian pattern, particularly in hominoids and cercopithecins, which suggest the model is only one of multiple mechanisms determining morphological variability in mammalian teeth.
 developmental processes that underpin morphological variation have become a focus of interest when attempting to interpret macroevolutionary patterns. recently, the dental inhibitory cascade model has been suggested to explain much of the variation in mammalian molar size proportions. we tested the macroevolutionary implications of this model using anthropoid primate species, focusing on overall morphological patterns, as well as predictions made about molar size variability, direct developmental control, and diet.  of the species sampled, % had centroids that fell within regions of molar proportion morphospace consistent with the dic model. we also found that the third molar had greater variation in size than either the first or second molars, as expected by the model. some dic model predictions were not supported, however, such as the expected proportion of m2/m when the third molar is absent. furthermore, we found that some variability in third molar size could not be explained by the influence of the inhibitory cascade. overall, we found considerable clade-specific differences in relative molar sizes among anthropoid primates, with hominoids and cercopithecins strongly divergent from dic model predictions, and platyrrhines, colobines, and papionins more consistent with the inhibitory cascade. finally, we investigated reasons why some clades deviated from dic model expectations. adaptations for frugivory appeared to be one driver of relatively larger second molars and have evolved independently in multiple lineages of anthropoids.  the dic model explains some of the variation in anthropoid primate molar proportions. however, there are interesting deviations away from this broad mammalian pattern, particularly in hominoids and cercopithecins, which suggest the model is only one of multiple mechanisms determining morphological variability in mammalian teeth. electronic supplementary material the online version of this article contains supplementary material, which is available to authorized users. keywords molar proportionsdental inhibitory cascadeanthropoid primatesevo-devobayesian phylogenetic generalized linear mixed modelshttp://dx.doi.org/ /100000001national science foundationdge-1144152carter katherine e. issue-copyright-statement© the author 2016 