Tooth counts through growth in diapsid reptiles: implications for interpreting individual and size‐related variation in the fossil record

Tooth counts are commonly recorded in fossil diapsid reptiles and have been used for taxonomic and phylogenetic purposes under the assumption that differences in the number of teeth are largely explained by interspecific variation. Although phylogeny is almost certainly one of the greatest factors influencing tooth count, the relative role of intraspecific variation is difficult, and often impossible, to test in the fossil record given the sample sizes available to palaeontologists and, as such, is best investigated using extant models. Intraspecific variation (largely manifested as size‐related or ontogenetic variation) in tooth counts has been examined in extant squamates (lizards and snakes) but is poorly understood in archosaurs (crocodylians and dinosaurs). Here, we document tooth count variation in two species of extant crocodylians (Alligator mississippiensis and Crocodylus porosus) as well as a large varanid lizard (Varanus komodoensis). We test the hypothesis that variation in tooth count is driven primarily by growth and thus predict significant correlations between tooth count and size, as well as differences in the frequency of deviation from the modal tooth count in the premaxilla, maxilla, and dentary. In addition to tooth counts, we also document tooth allometry in each species and compare these results with tooth count change through growth. Results reveal no correlation of tooth count with size in any element of any species examined here, with the exception of the premaxilla of C. porosus, which shows the loss of one tooth position. Based on the taxa examined here, we reject the hypothesis, as it is evident that variation in tooth count is not always significantly correlated with growth. However, growth trajectories of smaller reptilian taxa show increases in tooth counts and, although current samples are small, suggest potential correlates between tooth count trajectories and adult size. Nevertheless, interspecific variation in growth patterns underscores the importance of considering and understanding growth when constructing taxonomic and phylogenetic characters, in particular for fossil taxa where ontogenetic patterns are difficult to reconstruct.

In contrast to squamates, much less is known regarding tooth count variation in the other major radiation of extant diapsids, the archosaurs (including Pterosauria, Crocodylia and Dinosauria). Studies on the initial development of teeth in embryos and hatchlings of Alligator mississippiensis indicate that tooth count is variable and highly correlated with jaw growth (Westergaard & Ferguson, 1986, 1987. Iordan-sky (1973) reported that the second premaxillary tooth is lost during the juvenile stages of post-hatching ontogeny in Crocodylus cataphractus, Crocodylus porosus, Crocodylus siamensis, and Tomistoma schlegelii. Yet, it is unclear when exactly in ontogeny this tooth loss occurs, and how strongly its loss is correlated with growth.  reported individual variation of up to two tooth positions in the maxilla and dentary, and one tooth position in the premaxilla in a sample of 20 species of extant crocodylians, and also noted that no ontogenetic trends were present in an analysis of 41 A. mississipiensis skulls.
Tooth replacement patterns in crocodylians have been studied extensively and suggest that teeth are replaced in a uniform regular pattern when animals are young, but the pattern becomes less uniform through life, and that the mesial-most teeth are replaced more quickly than the distal-most teeth (Edmund, 1960(Edmund, , 1962. Younger individuals shed their teeth quickly and replace them with larger teeth through growth, and the growth rate of tooth-bearing elements decreases with age (McIlhenny, 1935;Edmund, 1962). However, it is unknown how tooth counts vary with age and size in most species of crocodylians. The large change in body size that crocodylians undergo throughout ontogeny makes them good models for understanding the relationship between tooth count variation and growth in extinct dinosaurian archosaurs.
Here we investigate tooth count variation in crocodylians by documenting tooth counts in the maxilla, dentary, and premaxilla of A. mississippiensis and C. porosus across their recorded intraspecific size range. We also add to the growing body of literature on squamate tooth count variation by providing data for the largest extant squamate, Varanus komodoensis. We test the hypothesis that variation in tooth counts is driven by growth, as was observed in previous studies (e.g. Westergaard & Ferguson, 1986;Greer, 1991), and predict significant positive relationships between size (a proxy for relative age) and number of tooth positions. Secondly, we test for complementary changes in relative tooth size through growth. In amniotes with multiple tooth generations, we define two distinct models for how a growing tooth row can be filled with teeth: (i) increasing tooth size or (ii) increasing tooth number (tooth count). In the former, tooth size (anteroposterior length of teeth) will increase, either isometrically or positively allometrically with jaw length; tooth count can remain constant (or even decrease). In the second model, the number of tooth positions increases positively with jaw length, and tooth size can remain constant (i.e. no relationship to jaw length) or be negatively allometric. It is important to note that, biologically, these models are not entirely mutually exclusive; different combinations of absolute increase in tooth size and count are possible. However, given the limited space within a jaw, a concomitant increase in relative tooth size and tooth number seems doubtful. Therefore, these models should, in theory, oppose each other when filling the avail-able space in a tooth row, resulting in an upper limit to their combined effect. A similar lower limit may not exist, as these would merely create diastema in the tooth row; teeth could decrease in size, either relatively or absolutely, and also decrease in number. These models are generally applicable to relatively homodont dentitions and do not make predictions regarding taxa with a heterodont dentition. Given our initial prediction of positive correlations between tooth counts and jaw size, we predict that tooth size should remain relatively constant throughout growth (i.e. tooth size increases isometrically relative to tooth row length). We also survey the literature and present current data on tooth count change within many species of extant squamates and extinct dinosaurs to illustrate the patterns exhibited by these clades. Testing the predictions outlined above will provide insights into the nature of variation of tooth counts in diapsids (intra-vs. interspecific variation) with implications for interpreting taxonomy and diversity in the fossil record.  Tables S1-S3) were taken for both the left and right side of the premaxilla, maxilla, and dentary. Measurements of tooth row length were taken for each maxilla and dentary, as well as the anteroposterior diameter of the fourth (A. mississippiensis), fifth (C. porosus) or seventh (V. komodoensis) maxillary tooth and fourth (A. mississippiensis), fifth (C. porosus) or seventh (V. komodoensis) dentary tooth (Tables S1-S3). These tooth positions were chosen as they represent the largest tooth within the dental series, and as a result were easy to identify consistently if the number of teeth varied between individuals. Basal skull lengths, measured from the anterior-most tip of the premaxilla to the posterior-most tip of the occipital condyle, were also taken for each skull. These skulls range in size from 35 to 689 mm (A. mississippiensis), 107 to 578 mm (C. porosus), and 130 to 221 mm (V. komodoensis). All measurements were taken to the nearest millimeter. All length measurements were log-transformed (base 10) prior to analysis, and count data were left as integers. Left and right tooth rows were analyzed independently, and the results were not pooled for counts, regressions or correlations.

Materials and methods
Within each taxon, the modal tooth count was determined for the premaxilla, maxilla, and dentary. Variation in tooth counts was calculated by the number and frequency of deviations from the modal value for each element (independent) and for bilateral element pairs. Deviations from modal tooth count were categorized into either asymmetric deviations (a situation where the two sides showed differing tooth counts) or symmetric deviations (both left and right showing equal deviation from the mode). Tooth counts in each element were regressed against basal skull length using an ordinary least squares (OLS) regression. Correlations between basal skull length, a proxy for size and relative growth stage, and tooth count were tested using the Kendall tau rank correlation coefficient.
For both the maxilla and dentary, tooth counts and the anteroposterior length of the tooth (fourth for A. mississippiensis, fifth for C. porosus, and seventh for V. komodoensis) for both the left and right side (right only for V. komodoensis) were regressed (OLS) against tooth row length. To account for possible outliers in the regression analysis we also performed a robust regression on the same datasets, but our discussion will concentrate on the basic OLS regression. All references to 'growth' within the present study reflect increases in size along a continuous intraspecific size-series, and do not necessarily reflect relative age.
In addition to the new data presented here, we surveyed the existing literature and collected data on tooth count change through growth for a variety of living and extinct diapsid taxa, including dinosaurs (Supporting Information Table S4). For historical papers, where only the plots were illustrated and the raw data were not presented in table form, the data were extracted digitally using the program PLOTDIGITIZEr (V. 2.6.3) (Huwaldt, 2014). Although this method will introduce some error in the measure of tooth row length, counts are less sensitive to such errors (due to rounding to the nearest whole number), and the overall pattern of tooth count change through increasing size should be preserved. In addition to published extant tooth count data, we also collected tooth count data for a selection of extinct non-avian dinosaur species, for which relatively complete size/growth series are known (Table S4). Data for non-avian dinosaurs were collected using the same methods as those for the extant samples, but due to small sample sizes, the pattern for left and right sides were averaged. Data were utilized to place our detailed case studies in a broader evolutionary context, and to present the overall patterns of tooth count change through growth in diapsids, as well as the possible effect of overall size on this pattern (all data obtained using PLOTDIGITIZER are provided in Table S4).
All analyses were conducted using standard packages in the R Statistical Language (V 3.0.2; R Core Development Team, 2013). The complete R code used for these analyses can be found in Supporting Information Data S1.

Tooth count change through growth
In A. mississippiensis, the modal premaxillary tooth count is five, maxillary tooth count is 15, and dentary tooth count is 20 across the entire size range (Fig. 1, Tables 1 and 2), with no change in tooth count correlated with size, all P-values > 0.05 (Supporting Information Table S5). Several skulls show deviations from this pattern, with asymmetrical (single side) and symmetrical (both side and same direction) deviations from this modal count exhibited (Tables 1 and  2), but these deviations do not correlate with skull size. The proportion of specimens with tooth counts that differ from the modal count is different between the tooth-bearing elements (Table 1). Deviations from the modal tooth count for each element are largely (68%) explained by asymmetry alone, rather than a deviation in tooth count from the norm on both left and right sides of the jaw (Fig. 1A, Table 1). Deviation from tooth count is also restricted in magnitude, with all but three deviant skulls (88%) showing deviations of only one tooth position. The proportion of specimens with tooth counts differing from the modal count is different between the tooth-bearing elements ( Table 1).
The data for C. porosus show a similar pattern to that of A. mississippiensis, with the modal tooth count of 14 and 15 for the maxilla and dentary, respectively (Fig. 2, Tables 3 and 4), and with neither element showing an increase or decrease though growth (Table S5). The premaxilla, however, shows a significant (P < 0.05) decrease in tooth Symmetric frequency 0 (0%) 6 (10%) 8 (15%) 9 (18%) 9 Proportion asymmetric ( count from five to four through the size series (Fig. 2B, Table S5). As with A. mississippiensis, the rate of deviation from modal tooth count is relatively low (43%) and largely explained by asymmetry (50-100%, depending on the element) ( Fig. 2A, Table 3). Again, similar to A. mississippiensis, the magnitude of tooth count deviations is quite low, with all deviations being only one tooth position. Finally, tooth count data for V. komodoensis also show similar patterns to that of the two crocodylians, with tooth number in all three major tooth-bearing elements (premaxilla, maxilla, and dentary) showing no correlation with skull length (Fig. 3, Tables 5, 6 and S5). The modal tooth count is four for the premaxilla, 13 for the maxilla, and 13 for the dentary (Tables 5 and 6). All deviations from the modal tooth count in the premaxilla and maxilla, and 60% of those in the dentary are due to asymmetry in tooth counts between left and right sides (Fig. 3A, Table 5).

Tooth size allometry
Maxillary and dentary tooth size in A. mississippiensis, based on the fourth alveolus and relative to tooth row length, exhibits positively allometric trends (Fig. 4A,   Table 7). The mean and 95% confidence intervals for the ordinary least squares (OLS) regression are all greater than a slope of one. Positive allometry in tooth size is also noted in C. porosus (Fig. 4B, Table 7). These patterns are consistent on both the right and left maxillae and dentaries independently for A. mississippiensis and C. porosus (Fig. 4).
The dataset for V. komodoensis is smaller than those for   A. mississippiensis and C. porosus, and although the slope of the relationship for the maxilla is greater than one and the dentary is less than one, the 95% confidence intervals of both include one, and therefore cannot be statistically differentiated from isometry (Fig. 4C, Table 7). The results for the robust regression (Supporting Information Table S6) are largely consistent with that of the OLS regression. Differences between the two analyses are present only in the datasets with smaller sample sizes and likely represent differences in statistical power between the tests.

Discussion
Tooth count change through growth Although a modest body of literature exists on changes in tooth counts through growth in some squamates, crocodylians are more poorly documented, and the relationship between tooth allometry and tooth counts is not well understood. The results of this study suggest that, for A. mississippiensis and C. porosus, as well as for V. komodoensis, the tooth counts in the maxilla and dentary do not change through the size series, and most cases of deviation in tooth count are explained by left-right asymmetry in the jaw. The results for A. mississippiensis are consistent with that of a previous study that employed a smaller sample size . These findings indicate that deviations   from the modal number likely relate to individual variation, independent of size. This same pattern is characteristic of the premaxilla of A. mississippiensis and V. komodoensis.
The premaxilla of C. porosus, however, does show a distinct trend towards a loss of a tooth position (decreased tooth count from five to four teeth) that is correlated with size (Iordansky, 1973). The loss of a tooth position is not due to 'packing' of teeth in the premaxilla, but rather the growth of the first dentary tooth. As the first dentary tooth grows, it forms a prominent foramen in the tooth row of the premaxilla, which subsequently leads to the exclusion of a premaxillary tooth position (Fig. 5). One specimen (FMNH 10866) has this foramen only on the left premaxilla. In FMNH 10866, the second premaxillary tooth was lost on the left side, but was not reduced in size on the right (Fig. 5).
Therefore, it appears that the loss of this tooth is dependent on the development of the dentary tooth associated with this foramen, rather than on an independent resorption at a specific time in the animal's life. Taken together, these results falsify our initial prediction that growth and tooth counts are significantly correlated in the taxa examined here with the exception of the decrease in premaxillary tooth counts in C. porosus.
Alligator mississippiensis and C. porosus both show significant positive allometry of anteroposterior tooth length, in the measured position, relative to the length of the tooth row, indicating that the anteroposterior space occupied by the tooth increases faster than the tooth row length. This increase in relative tooth size while maintaining constant tooth count is somewhat counter-intuitive, as one would expect these two metrics to act reciprocally. This pattern may be explained by a combination of two factors. First, the measurement of the tooth row and teeth did not take into account the spacing between adjacent alveoli. If sufficient space exists between alveoli in small individuals (particularly juveniles), a decrease in spacing through growth may accommodate the allometric increase in alveolar size while maintaining a constant number of alveoli. The alternative possibility is that the allometry of the measured tooth does not adequately represent the growth patterns exhibited by other teeth within the tooth row, and although teeth examined here grow via positive allometry, other teeth may exhibit a more isometric or even negatively allometric pattern. Although our qualitative observations suggest the former, the present dataset does not allow for a quantitative test of these two factors (or any combination therein). Nevertheless, the results indicate that allometry and tooth count changes throughout growth are not strictly dependent on each other. A similar pattern was noted in Iguana iguana (Kline & Cullum, 1984), which supports this conclusion.

Body size and tooth count variation in Diapsida
The results of combining the existing literature-derived data on squamate and dinosaur tooth count changes (Kluge, 1962;Ray, 1965;Montanucci, 1968;Cooper et al. 1970;Dodson, 1976;Madsen, 1976;Arnold, 1980;Colbert, 1990;Delgado et al. 2003;Torres-Carvajal, 2007) with our new data for V. komodoensis, A. mississippiensis, C. porosus, and extinct non-avian dinosaurs are reported in Fig. 6. The majority of non-varanid squamates show distinct increases  in tooth counts through their growth. In contrast, data for V. komodoensis and the crocodylians presented here show unchanging tooth count across their respective size series. The data for ornithischians show a pattern of increasing tooth counts as body size increases, best illustrated by the hadrosaurid taxon Corythosaurus, and to a lesser degree in the ceratopsian Protoceratops. The preliminary data for non-avian theropods show a range of possible changes, with Coelophysis bauri showing an increase, Allosaurus fragilis and Gorgosaurus libratus showing a relatively stable count, and Tyrannosaurus rex (with the potentially conspecific Nanotyrannus lancensis included) showing a decrease in tooth count across the size series. Tooth count growth patterns across the diapsid size range reveal a potential trend in which the slope (tooth counts in relation to log tooth row length) is correlated, at least loosely, with overall body size. Smaller taxa have positive trajectories (increase in tooth counts throughout growth), crocodylians, Varanus, and the large-bodied theropods Allosaurus and Gorgosaurus show little or no changes in tooth counts through their respective size series, and T. rex shows a potential negative trend. When these slopes are regressed against published snout-vent lengths (SVL) for extant taxa only, the slopes of size-related tooth count changes for the dentary and maxilla are both significantly and negatively correlated with SVL (dentary: m = À0.266, r 2 = 0.34, P < 0.05; maxilla: m = À0.236, r 2 = 0.56, P < 0.01, respectively). Interestingly, despite our small sample size, the interspecific negative trajectories observed here for the dentary and maxilla conform to the theoretical expectation that size-specific rates (i.e. size-related variation in morphological or physiological properties) scale to the power À0.25 (Peters, 1983). When A. mississippiensis and C. porosus are taken out of the dataset, the significant correlation between tooth count slope and SVL is lost for the dentary (m = À0.257, r 2 = 0.20, P > 0.05) but not the maxilla  (m = À0.222, r 2 = 0.40, P < 0.05). Extrapolation of such interspecific trajectories would predict that as diapsids attain larger sizes (e.g. T. rex) tooth counts may be expected to decrease throughout growth, but this contradicts the pattern observed through growth in large ornithischian dinosaurs (Fig. 6). More data are needed to determine whether the observed patterns are truly related to body size, rather than disparate phylogenetic histories.

Conclusions
Understanding tooth count variation and the mechanisms that govern that variation is an important step towards understanding the evolution of jaw and tooth development in diapsids. This study demonstrates that no universal pattern exists for tooth count change through growth in diapsids (Fig. 6). Consequently, without additional data for a particular taxonomic group, tooth counts alone are not reliable indicators of species demarcation in the fossil record. Indeed, even taxa with the same allometric tooth count pattern may achieve that pattern through multiple underlying mechanisms. These mechanisms include changes in tooth row length, anteroposterior tooth diameter, and possibly the space between alveoli. The results of this study indicate that these variables and the allometric relationships between them should be measured and considered in future studies examining tooth growth in diapsids to determine which of these variables are determining tooth counts in particular taxa. Finally, clade-specific allometric effects of tooth counts and tooth size must be incorporated if tooth count characters are used to diagnose fossil taxa or construct phylogenetic characters, as these characters are too variable to be determined by the extant phylogenetic bracket.

Supporting Information
Additional Supporting Information may be found in the online version of this article: Table S1. Tooth counts and measurements of Alligator mississippiensis. Table S2. Tooth counts and measurements of Crocodylus porosus. Table S3. Tooth counts and measurements of Varanus komodoensis. Table S4. Tooth counts and tooth row lengths for a number of extant and extinct diapsids. Table S5. Results of correlation (Kendall) of tooth count and basal skull length for the premaxilla, maxilla and dentary of Alligator mississippiensis, Crocodylus porosus and Varanus komodoensis. Table S6. Results of the robust regression for tooth allometry plotting tooth length as a function of tooth row length in Alligator mississippiensis, Crocodylus porosus, and Varanus komodoensis. Data S1. R code used in analyses.