Morphometric analysis of ontogeny and allometry of the Middle Ordovician trilobite Triarthrus becki

Morphometric Analysis of Ontogeny and Allometry of the Middle OrdovicianTrilobite Triarthrus becki

Keonho Kim; H. David Sheets; Robert A. Haney; Charles E. Mitchell

Paleobiology, Vol. 28, No. 3. (Summer, 2002), pp. 364-377.

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Paleobiology, 28(3), 2002, pp. 364-377

Morphometric analysis of ontogeny and allometry of the Middle Ordovician trilobite Triarthrus becki

Keonho Kim, H. David Sheets, Robert A. Haney, and Charles E. Mitchell

Abstrnct.-Traditionally, the distinction between meraspis and holaspis among trilobites has been based on the achievement of the full adult complement of thoracic segments. Using a large sample (over 700 specimens collected from a single bed) we explore the utility of employing the ontogenetic trajectory of the cranidium as an alternative means to differentiate trilobite growth stages. This method is particularly useful for species represented solely by exuviae and disarticulated individ- uals. We use geometric morphometrics to examine shape change among cranidia ranging in size from 0.9 mm to 11.6 mm in cephalic length. The 114 measured specimens exhibit a rather contill- uous gradation in size in which no distinct instars are evident.

The meraspid and holaspid specimens exhibit allometry when partial warp scores and uniform components of shape derived from thin-plate spline analysis are regressed onto log centroid size. To describe allometric shape change, deformation vectors from the smallest to the largest specimen in both ontogenetic stages are presented in three different superimposition settings by using a new software program. We have concluded that a new superimposition method (the Sliding Baseline Registration) is a useful tool for visualizing allometry in organisms that contain an axis of sym- metry. As a result, we conclude that allometry is evident in meraspides and holaspides, but the degree of allometry in holaspides is very small relative to that in meraspides. The boundary be- tween meraspis and holaspis in Triartlzrus becki appears to correspond to a large change in the rate of ontogenetic change, but neither to a change in the direction of that trajectory nor to a cessation of ontogenetic change. This boundary also corresponds to a cranidium centroid size that matches well previous determinations that holaspis begins at about 2.8 mm in cephalic length.

Keoizllo Kinr, Robert A. Haney, and Charles E. Mitchell. Department of Geology, 876 Natnml Scieizcrs Conrplex, State U~ziuers i ty of New Y o ~ k nt Buffalo, Bujfaio, New York 14260. E-mail: k~~onkokim@hoti~~nil.conz, [email protected]@broz~~iz.edtl,

H . Dazlid Sheets. Department of Physics, Cuizisirts College, 2001 Main Street, Buffalo, New York 14208. E-mail: [email protected]

Accepted: 1 March 2002

Introduction of size and shape in a series of rather complete specimens that serve as a proxy for the growth

Paleobiologists have long been interested in of individuals. However, i t is not always pos-

features of evolutionary history such as phy- sible for paleontologists to collect enough ar-

logeny and in the evolutionary processes that ticulated trilobite specimens in a confined produce this history. In particular, the study stratigraphic level to make a detailed study of of ontogeny in fossiI specimens allows us to ontogeny. Instead, molting and frequent dis- describe and understand the specific evolu- articulation ensure that we most often find in- tionary history and processes that affected complete trilobite specimens. This rarity of those organisms (Jones and Gould 1999; Mc- complete individuals has inhibited the study Kinney 1999). Ontogenetic studies of fossils of ontogeny in trilobites and, consequently, are typically based on the description of further study of their evolution. changes in size and shape extracted from sin- During the last decade the geometric mor- gle individuals or from a sequence of speci- phometric approach has contributed greatly mens thought to be of different age. to the study of shape in living and fossil spec-

Trilobites, like many arthropods, under- imens (Rohlf and Bookstein 1990; Rohlf and went a dramatic transformation in shape over Slice 1990; Bookstein 1991, 1996; Rohlf and the course of their ontogeny. This fact, com- Marcus 1993; Zelditch and Fink 1995; Dryden bined with their molting habit, means that in - and Mardia 1998; Sheets 2001a,b; Webster et trilobites, most studies of its ontogenetic de- al. 2001). Although still evolving, these meth- velopment have been based on the description ods offer several powerful tools for the study

0 2002 The Paleontological Society. All rights reserved.

365 TRILOBITE ONTOGENY AND ALLOMETRY

of evolution. We present here a further elabo- ration of the geometric morphometric tech- niques and use them to solve the problems posed by the study of ontogenetic shape change in incomplete fossil materials, partic- ularly in disarticulated trilobite sclerites.

There have been many trilobite ontogenetic studies since Beecher (1895) coined the term "protaspis" for a common early trilobite lar- val form that is usually a convex and discoid body less than 1 mm in length and has ce- phalon and protopygidium without thorax. He also described the larval stages of several trilobite species, including Triarthvus entoni, for the first time. Raw (1925) proposed the terms meraspid and holaspid periods for later successive developmental stages of trilobites. He defined the meraspid period as the series beginning with subdivision of the protaspis into cephalon and transitory pygidium, and the holaspid period as the development after completion of the thorax. Therefore, a critical criterion for distinguishing meraspis from ho- laspis is the completion of the addition of tho- racic segments. However, it is commonly not possible to recognize "the completion of the thorax" because the species is known from few complete specimens. In such cases, it is not easy to distinguish meraspides from ho- laspides (see, for instance, Mansson 1998). Furthermore, the thoracic segment criterion is not always correlated with transition from meraspis to holaspis. Stubblefield (1926) had a different view of holaspis. He suggested that segments were generated at the anterior of the terminal postcephalic segment in some trilo- bites after completion of the thorax, and he thus defined holaspis as ontogeny after the development of all postcephalic segments. Hughes and Chapman (1995) found that Au- lacoplelira konincki shows continuous addition of thoracic segments throughout growth, even in the holaspid growth stage, and they con- cluded that neither Raw's (1925) nor Stubble- field's (1926) definition of holaspis applies to A. konincki. In the case of A. konincki, it is better to use growth trajectories rather than thoracic segment number in defining ontogenetic stag- es. As we demonstrate below, the growth tra- jectory approach appears to work well in Tuiarthrus becki as well.

Several papers have described the ontogeny of Triauthuus beclci (Walcott 1879; Whittington 1957) and 7: eatoni (Beecher 1893, 1894, 1895; Cisne 1973, 1981). All of them distinguished meraspides from holaspides on the basis of thoracic segment number and mentioned little about shape change of the cranidium during the transition from the meraspid period to ho- laspid period. We have found numerous dis- articulated cranidia of fiiartlznls becki without protaspides or complete specimens. In this pa- per, we use cranidia of T. becki and geometric morphometric analyses (Procrustes distances and the thin-plate spline) to differentiate mer- aspid specimens from holaspid specimens and to study allometry in cranidium shape. We have followed the systematics of Tviarthvus used by Ludvigsen and Tuffnell (1983, 1994).

Geologic Setting

We have collected over 700 specimens of Tuinutlzvus becki from a single bed of the lower Dolgeville Formation in the Mohawk River Valley, New York State. The outcrop is located on New Turnpike Road, Montgomery County, 8.4 km north of the town of Canajoharie and 7.5 km east of St. Johnsville (Fig. 1).

The Dolgeville Formation is a sparsely fos- siliferous, interbedded limestone-shale facies and represents a transitional unit between the western carbonate shelf facies (Trenton Group) and eastern deep basinal facies (Utica Group) of the Taconic Foreland Basin (Kay 1937,1943,1953; Cisne and Rabe 1978; Fisher 1979; Cisne et al. 1982; Hay and Cisne 1989; Mehrtens 1989; Baird et al. 1992; Mitchell et al. 1993). The Dolgeville Formation is shale-rich with alternating carbonate mud turbidite and shale beds of roughly equal thickness (Mehr- tens 1989; Mitchell et al. 1993) and has been interpreted as a laterally extensive submarine fan or sheet deposit (Mitchell et al. 1993,1994). The most common fossils are graptolites of the Outhoguaptus vuedeinanizi Zone (Goldman et al. 1994, 1999; Mitchell et al. 1994).

The outcrop is approximately 3.5 m thick. We found a Tuiartlzrus-rich carbonate layer, 1 cm thick, at 1 m above the base of the outcrop and could find no other Tuiarthrus-bearing lay- ers in the outcrop. The layer contains disartic- ulated fiiarthrus becki with abundant cranidia

366 KEONHO KIM ET AL.

FIGURE 1. Locality map of the Mohawk River Valley, New York State, showing location of the New Turnpike Road fossil site.

and few free cheeks and pygidia (Fig. 2). No plete specimens and all specimens in the Triar- other fossils, even graptolites, are preserved thrus layer are disarticulated (articulation is in the layer. It is not an obrution (Lagerstatten) the most distinctive feature of obrution de- deposit (Speyer and Brett 1985; Brett and Sei- posits, which are formed by mass mortality lacher 1991) because it does not contain com- with rapid burial). A possible cause of the

Triarthrus layer may be mass mortality with- -- ,---.- - a-.- out rapid burial. The low-diversity fauna, only one species in this case, with a wide range of specimen sizes may indicate in s i b burial or slight transportation (Brett and Baird 1993) of an ephemeral or opportunistic population of IT: becki.

Materials and Methods

For this study, we chose 114 cranidia of the , over 700 specimens that were well preserved

and not distorted. All specimens are reposited in the New York State Museum (NYSM) Pale- ontology Collection. Non-illustrated mea- sured and unmeasured specimens are stored under NYSM locality numbers 9560A and 9560B, respectively.

The 114 cranidia chosen for this ontogenetic study exhibit a rather continuous size change - (Fig. 3) from the smallest (0.9 mm in cephalic length) to the largest (11.6 mm in cephalic - - - ' length). A series of photos of T becki from

FIGURE 2. Upper surface of rock slab containing nu- small to large specimens is shown in 12igure 4. merous cranidia of Triarthrus becki from sample studied herein. Scale units are 1 cm. Note predominantly con- landmarks On One side (left Or right) vex-up orientation and wide range of cranidium size. on the cranidium of T becki including four

TRILOBITE ONTOGENY A N D ALLOMETRY

Cephalic Length (mm)

FIGURE3. Plot of bivariate relationships between ce-phalic length (length between landmark 1 and 4) and glabellar width at I s glabellar furrow (width between landmark 8 and sagittal axis) of Trinrthrzrs becki.

landmarks on the sagittal axis (landmarks 1 through 4) were selected (Table 1, Fig. 5) be-cause trilobites show bilateral symmetry. This procedure increased the number of specimens useful for this study by allowing the inclusion of some specimens well preserved only on one side or the other. We digitized the landmarks in each specimen using the BioScan OPTIMAS software program, and the x-y coordinates of twelve landmarks were recorded for each specimen. We used the twelve left-side land- marks of all 114 specimens for statistical anal- ysis and graphical presentations after reflect- ing eight landmarks of right side to the left side of the sagittal axis.

Centroid size (the square root of the sum of squared distances of all landmarks from their centroid) was calculated for all 114 specimens. In the following analyses, we used log-trans- formed centroid size (LCS) as our size vari- able. It is natural to use logarithms because growth is usually better thought of in relative or proportional terms.

To examine the overall ontogenetic shape trends, we used two measures of shape dif- ference: (1)Procrustes distance from a refer- ence form and (2) the partial warp scores plus uniform components from a thin-plate spline decomposition. Procrustes distance is the square root of the sum of squared differences between the positions of the landmarks in two optimally superimposed (by least-squares)

configurations. A consensus specimen of the five smallest specimens was used as a refer- ence specimen in calculating the Procrustes distances of all 114 specimens from the refer- ence, and the Procrustes distances from the reference were plotted against LCS.

We used the thin-plate spline (TPS) to mod- el shape change as a deformation between landmarks (see Bookstein 1991 for technical details), relative to a grand mean consensus form. The net shape change can be decom- posed into uniform and nonuniform compo- nents. The uniform component describes ho- mogeneous, affine transformation (stretching or shearing) with no bending energy; i.e., par- allel lines remain parallel after transforma- tion. The nonuniform component describes transformations that are different in different parts of the form. The thin-plate spline meth- od uses minimum bending energy to describe the nonuniform component of changes in landmark position of a specimen relative to a reference form. The way to think simply about this is to imagine that the landmarks of one specimen are placed on a flexible thin metal plate, which is then manipulated by bending until the vertical deformation of the metal plate at each of the landmarks on the plate cor- responds to the extent of the deformation of the first specimen relative to the second spec- imen. The degree of change in height of the plate at each landmark as it is deformed to match the reference form encodes the amount of difference in shape present at that land- mark. During this bending process, low bend- ing energy is required to bend the thin metal plate between widely spaced landmarks. In contrast, it takes relatively more bending en- ergy to bend the metal plate between closely spaced landmarks. This means that different amounts of bending energy are required in different parts of the form. Principal warps are eigenvectors of this bending-energy ma- trix, and partial warps are projections (shad- ows) of the principal warps on the tangent space defined by the nonuniform components of the thin-plate spline. Any deformation is the sum of its uniform and nonuniform com- ponents (see Zelditch et al. 1992, 1993; Zeld- itch and Fink 1995 for more detailed expla- nation about uniform and nonuniform com-

368 KEONHO KIM ET AL.

FIGURE 4. Small (A) to large (L) specimens of Tviavthrus becki. A, Meraspis, NYSM 16781, Cephalic length (CL) = 0.97 mm. B, Meraspis, NYSM 16782, CL = 1.13 mm. C, Meraspis, NYSM 16783, CL = 1.31 mm. D, Meraspis, NYSM 16784, CL = 1.41 mm. E, Meraspis, NYSM 16785, CL = 1.49 mm. F, Meraspis, NYSM 17058, CL = 1.72 mm. G, Meraspis, NYSM 17059, CL = 1.91 mm. H, Meraspis, NYSM 17060, CL = 2.13 mm. I, Holaspis, NYSM 17061, CL = 3.10 mm. J, Holaspis, NYSM 17062, CL = 4.25 mm. K, Holaspis, NYSM 17063, CL = 5.39 mm. L, Holaspis, NYSM 17064, CL = 6.42 mm. These 12 specimens are reposited in the New York State Museum as hypotypes.

ponents). The nonuniform (partial warp consensus as a reference specimen in calculat- scores) and uniform components were de- ing the partial warp scores. rived from TPS analysis by using the tpsRelw In the analysis of shape, it is generally held program (Rohlf 1998). The tpsRelw software that shape is the information in the landmark program calculates a consensus specimen coordinates that is not altered by arbitrary from all input data by using a Generalized translations, rotations, and rescaling of all of Least Squares (GLS) procedure and uses this the data points (Kendall 1977; Dryden and

369 TRILOBITE ONTOGENY AND ALLOMETRY

TABLE1. Locations of the 12 landmarks in left side on the cephalon of T r i n r t k r u s beck1 in Figure 5. Landmarks 1to 4 lie on the plane of bilateral symmetry.

Landmark Location

Anteriormost point of cephalon on sagittal axis Anteriormost point of glabellar anterior lobe on sagittal axis Anterior point of occipital ring on sagittal axis Posteriormost point of cephalon on sagittal axis Anterior endpoint on eye Posterior endpoint on eye Adjoining point between axial furrow and 2s glabellar furrow Adjoining point between axial furrow and I s glabellar furrow Adjoining point between axial furrow and anterior edge of occipital ring Adjoining point between axial furrow and posterior edge of occipital ring Adaxial end point of 2s glabellar furrow Adaxial end point of I s glabellar furrow

Mardia 1998). The GLS superimposition method fits all specimens in a data set to an estimated mean (reference) specimen by min- imizing the sum-of-squared distances be-tween landmarks of each specimen and the corresponding landmarks of a reference spec- imen (Rohlf and Slice 1990; Slice 1993). To ob- tain shape coordinates using Bookstein's (1991) two-point registration method (re-ferred to as BTR, hereafter), two landmarks are chosen to form the endpoints of a baseline and their coordinates are set to (0,O) and (0,l). In this study, we chose landmarks 1 and 4 as the endpoints of the baseline because the tri- lobite shows bilateral symmetry and it is rea- sonable to take two endpoints on the sym- metry axis to fix the baseline. Then landmark locations of all specimens can be transformed to the same baseline orientation and length (Bookstein 1991; Dryden and Mardia 1998).

When there is an axis of symmetry in the organism being studied (or more generally, a

FIGERE5. Twe11.e landmark points on the cranidiuln exoskeleton of T r i a i ~ t l i ~ ~ l t shecki .

longitudinal body axis, such as a dorsoventral plane in an elongate body), the operations that may be carried out when comparing two or more shapes become more restricted than in the general case of shape comparison. When we desire to make a biologically meaningful comparison of the shape of two organisms with an axis, the shapes are not comparable under arbitrary rotations and translations, but are comparable only when the symmetry axes remain parallel and coincident. In other words, when the superimposition is carried out, the landmarks can only be translated par- allel to the axis, and rotation of the sample must align the symmetry axes. Given these re- strictions on the allowed operations used in the superimposition, we may use BTR, as ex- plained above, or a new method, the Sliding Baseline Registration (SBR). Let us assume that the symmetry axis is parallel to the y-axis. After the size differences have been removed by setting specimens to unit centroid size, the SBR may be done by sliding all landmarks parallel to the symmetry axis until the y-co- ordinates of the centroids are same. This alignment minimizes the summed squared distance between landmarks, given the con- dition that the baselines are coincident. Fur- ther study of the SBR method is ongoing, and we are preparing a paper about the mathe- matical and statistical aspects of the method with applications to both fossil and modern organisms. We prefer the SBR method for axis- restricted alignments because of the variance- mapping issue: BTR produces spurious pat- terns of covariance among landmarks (Dry-

370 KEONHO KIM ET AL

den and Mardia 1998) that are avoided by SBR.

In the study of allometry it is appropriate to use nonuniform (partial warp scores) and uni- form components of shape in a regression analysis of the shape change as a function of log centroid size (Bookstein 1991, 1996; Rohlf and Marcus 1993; Zelditch and Fink 1995). For this multivariate regression analysis, we used the partial warp scores and uniform compo- nents calculated above.

For the description and graphical presen- tation of allometric shape change during mer- aspid and holaspid periods, we used the pro- gram Regress6 (Sheets 2001a). After comput- ing partial warp scores and uniform compo- nents on the basis of GLS superimposition, the program allows regression of those scores on LCS. We used a mean specimen of all 114 specimens as a reference in calculating partial warp scores and uniform components. Defor- mation vectors derived from the regression re- sults were displayed from the smallest (lowest LCS) to the largest (highest LCS) specimen in both meraspid and holaspid periods. Re-gress6 can display deformation vectors in three different alternative superimposition settings (GLS, SBR, and BTR) for comparison purposes. By comparing the deformation pat- terns under different superimposition meth- ods, it is possible to develop an understanding of which features of the illustrated change are produced by the superimposition, and which are features of the ontogenetic trajectory. The Regress6 morphometric program is part of the Integrated Morphometrics Package (IMP) de-veloped by David Sheets and is available to download from a web site (http://www. canisius.edu/ -sheets/morphsoft.html).

Results

The centroid size of all 114 specimens rang- es from 1.088 (an LCS value of 0.00366) to 15.601 (an LCS value of 1.1931). Procrustes distance displays a strong linear dependence on LCS within both meraspis and holaspis (Fig. 6). This linear dependence exhibits a rather sharp break around an LCS value of 0.6. This marked change in the ontogenetic trajec- tory of Triarthrus becki suggests comparison with the meraspid and holaspid growth phas-

0

0 0 0 2 0 4 0 6 0 8 1 0 1 2 1 4

Log Centroid Size

FIGURE6. Bivariate plot of the Procrustes distances of all 114 specimens against their log (base 10) centroid sizes. Note that the boundary between meraspid and ho- laspid periods lies near 0.6 on the x-axis (log centroid size).

es. To determine the size of specimens at the transition boundary between meraspid and holaspid periods, we calculated cephalic length from original pictures of the previous ontogenetic studies of T becki (Walcott 1879; Whittington 1957). Walcott (1879) and Whit- tington (1957) recognized 7: becki specimens with from 1 to 16 thoracic segments and, us- ing the number (from 1 to 15) of thoracic seg- ments in the specimens, divided the meraspid period into 15 degrees. They regarded speci- mens with 16 thoracic segments as holaspides. Whittington (1957) reported that the largest meraspis with 15 thoracic segments he found was 8.8 mm in length of the exoskeleton, but the smallest holaspis with sixteen thoracic segments (Whittington 1957: Fig. 10 in Plate 116) is 23 mm in length. These specimens ex- hibit cephalic lengths of 2.1 and 6.8 mm, re- spectively, and correspond to LCS values of about 0.42 and 0.97 (Fig. 7). Thus, there is a significant size gap between the latest meras- pis and the first holaspis in Whittington's col- lections in the size interval at which we find a significant break in the allometric slope shown in Figure 6. The ontogeny of Triarthrus eatorzi is more completely known (Walcott 1879), and in that species the holaspid stage is first seen in specimens of about 9 mm in total length (Cisne 1973). If the same ratio of ce-phalic length to whole body length of the smallest holaspis in Whittington (1957) ap- plies to this 9.0-mm boundary in 7: eatoni and if the size at commencement of holaspis is the

371 TRILOBITE ONTOGENY AND ALLOMETRY

-0.2 0 0 0.2 0.4 0.6 0.8 1.0 1.2

Log-Cephalic Length

FIGURE7. Bivariate plot of the log centroid sizes (LCS) of all 114 specimens against their log cephalic lengths.

same in the these two closely related species, then the holaspid stage of 7: becki started at around 2.7 mm in cephalic length. A specimen drawn by Walcott (1879: Fig. 14 in Plate 2) sug- gests that the holaspid stage of T becki may be- gin at about 2.8 mm in cephalic length. Those two values, 2.7 and 2.8 mm in cephalic length, correspond to around 0.57 in LCS. We there- fore set the boundary between the meraspid and the holaspid stages at an LCS value of 0.6, the same place at which the break in slope shown in Figure 6 also occurs, and conclude that the two ontogenetic phases shown in Fig- ure 6 may likely correspond to meraspid and holaspid phases.

By carrying out linear regression analysis of Procrustes distances on LCS, we can test for allometry in the meraspid and holaspid spec- imens. Because the ontogenetic trajectories of the meraspid (LCS < 0.6) and holaspid (LCS 2 0.6) are markedly different, we conducted separate regression analyses for meraspid and holaspid specimens to test for a significant de- pendence of shape upon size (LCS). As a re- sult, the slope of the Procrustes distance against LCS is significantly different from zero (slope = 0.4240, p < 0.001) for the mer- aspid specimens at the 95% confidence level. This result indicates that net change in the Procrustes distances is attributable to changes in LCS during the meraspid stage. Regression analysis for the holaspid specimens shows that allometry is evident during this phase as well (slope = 0.0688, p < 0.01).

TABLE Results of multivariate regression analyses to 2. test for allometry in the meraspid and the holaspid on- togenetic stages. n indicates number of specimens in the ontogenetic stage. Adjusted R2 is used as a measure of effect of sample size to account for possible inflation of the multiple correlation in the relatively small samples used for the analysis.

Onto-genetic Adjusted Signifi-stage n R2 R2 cance

Meraspid 72 0.957 0.940 0.000 Holaspid 42 0.795 0.601 0.001

In contrast to the univariate approach men- tioned above, a multivariate approach to al- lometry within both ontogenetic stages may be carried out by regressing the partial warp plus uniform components scores obtained from thin-plate spline analysis on LCS. Table 2 shows the results of the multivariate regres- sion analyses in the meraspid and holaspid periods. It is evident from these analyses that specimens in both the meraspid and the ho- laspid stages show allometry. Therefore, we conclude that allometry (shape change with size) is present in both ontogenetic stages of Trinrthrus becki. However, as shown in Figure 6, the rate of change in shape in the holaspid stage is much less than that in the meraspid stage. This is reflected also by the lower per- cent of variance explained by the holaspid re- gression (Table 2), which yields an adjusted R2 of 0.6 compared with 0.94 for the meraspid re- gression.

What is the nature of the allometric shape change within the meraspid and the holaspid stages? Does the allometric shape change in the meraspid stage show the same trend as that in the holaspid stage? We will describe the shape change using deformation vectors within three different superimposition set- tings (GLS, SBR, and BTR) and then discuss the differences and effects of studying shape changes in the three settings. Deformation vectors from the smallest to the largest spec- imen within the meraspid (upper three fig- ures in Fig. 8) and the holaspid (lower three figures in Fig. 8) stages were calculated by us- ing three different superimposition settings (GLS, SBR, and BTR from left to right in each ontogenetic stage).

KEONHO KIM ET AL

Meraspid Stage

B

GLS

% SBR BTR

Holaspid Stage

GLS A SBR a BTR

FTGLRE8 . Plots of deformation vectors from the predicted smallest to the predicted largest specimen according to regression model ~vithinthe meraspid (upper three figures) and the hoiaspid (Ion-er three figures) stages; vectors calculated frorn three different superimposition settings (GLS, SBR, and BTR from left to right in each ontogenetic stage). Twel\e landmarks on the left side of Triai.tl~ui~sbecki were used for these graphical presentations.

In each of the three different superimposi- through 4 in the shape coordinates obtained tion settings used here, the choice was to ac- by GLS superimposition show some variation cept the option that best preserved bilateral perpendicular to the symmetry axis on which symmetry by minimizing dispersion of land- the four landmarks lie. Because of this distor-marks 1to 4 away from the plane of symme- tion (relative rotation and lateral offset of the try. As shown in Figure 8, landmarks 1 axis) of the four landmarks on the symmetry

TRILOBITE ONTOGENY AND ALLOMETRY

FIGLRE9. Scatter plots of landmarks of meraspid specimens in three superimposition settings (GLS, SBR, and BTR). The big black dots indicate the mean of the five smallest meraspid specimens in each landmark.

axis, the shape coordinates provided by GLS superimposition provide a poor basis for bi- ological interpretation of shape vectors. This distortion, however, is in part a product of our choice to use landmarks from only one side of the bilaterally symmetric Trinrflzrusbecki cran-idium.

Deformations of on-axis landmarks perpen- dicular to the axis of symmetry are not a prob- lem in the shape coordinates provided by the BTR and SBR methods, because the symmetry axis, as a baseline, is fixed in the two methods. There is no variation of landmarks 1and 4 in Bookstein's shape coordinates because the landmarks used are fixed endpoints on the baseline. Points located well away from the baseline, however, (e.g., landmarks 5 and 6) show increased variance in their location (Figs. 8 and 9). Variance in the location of the fixed baseline points (landmarks 1and 4) is in- stead spread among the other landmarks (Dryden and Mardia 1998). In other words, by fixing the location of landmark 1, i.e., remov- ing the variance in position of landmark 1,the variation that was present in landmark 1 is dispersed to the other landmarks, which in- creases the degree of variation of other land- marks. As a consequence, shape transforma- tion vectors show exaggerated and complex motions that express shape changes originat- ing far from these landmarks. Thus, BTR-

based shape vectors are, in this case, also dif- ficult to interpret biologically.

In contrast, landmarks 1 and 4 still show some variation in the y direction in the shape coordinates by SBR (Fig.9). The SBR produces a tight superimposition of all landmarks while allowing the baseline points to exhibit variation parallel to the symmetry axis. This appears to us to be the most biologically rea- sonable description of landmark variation in T.becki because the symmetry axes of the spec- imens are constrained to be coincident, and SBR creates smaller distortion of the variance patterns than BTR does.

When the magnitude of the vectors indicat- ing the relative changes in landmark locations are compared between the meraspid and the holaspid stages in Figure 8, one can clearly see that the magnitude of vectors in the holaspid stage is much less than that in the meraspid stage. This supports the evidence that the rate of development in the holaspid stage is much smaller than that in the meraspid stage (Fig.

6). To examine how similar the directions of

shape change are between the two ontogenetic stages, we computed an angle between the two growth vectors determined by multivari- ate regression of partial warp scores and land- mark coordinates (expressed as shape coor- dinates produced by SBR). The null hypothe-

KEONHO KIM ET AL.

Species 1 A Species 2

Y

FIC;UKI.10. Example of basic approach of the multivar-iate regression analysis of landmark coordinates. The x-and y-coclrdinates of landmarks are plotted on x- and y-axes, and time or LCS on z-axis. V1 and V2 are growth vectors of species 1 and 2, respectively. is an angle be-tween species 1 and 2. a1 and u2 are angles within spe-cies 1 and 2, respectively, at YS'%, confidence range that are calculated from 1600 bootstrapped sets formed by a resampling procrdure.

sis used is that the angle between two growth vectors is zero. This means that the two tra-jectories are statistically different if the angle between two growth vectors is significantly different from zero. This hypothesis can be tested statistically by comparing the angle be-tween two growth vectors from different groups to the 95'2, confidence interval of an-gles produced by comparisons of growth vec-tors within each group. If the between-groups angle is greater than the range of within-group angles, the null hypothesis should be rejected at 95'21 confidence level. To obtain 95'2) confidence intervals of angles between trajectories within groups, we formed 1600 pairs of bootstrapped data sets and corre-sponding growth vectors, by bootstrapping the residuals of the multivariate regression of landmark coordinates (see Zelditch et al. 2000 for details) and calculating the angle between the resulting pairs of bootstrapped data sets.

If there are k landmarks, the growth vector lies in 2k dimensional shape space for Pro-crustes shape coordinates, 2k - 2 for SBR, and 2k - 4 for partial warp scores and Bookstein's shape coordinates. For the convenience of ex-

TABLE3. Angles between growth vectors of meraspid and holaspid trajectories and within each vector of the two trajectories; calculations based on partial warp scores and shape coordinates by SBR. 1600 bootstrap sets were used to calculate 95% confidence interval.

Within Within Between meraspis holaspis

Partial warp scores 41.2 17.0 60.3 SBR shape coordinates 37.9 17.7 65.0

planation, let us examine a three-dimension case. Assume that the two growth vectors (VI and V2) of species 1and 2, respectively, are in the three-dimensional shape space (Fig. 10). In this case, x- and y-coordinates of landmarks are plotted on the x- and y-axes, and time or LCS on the z-axis. P is an angle between spe-cies 1 and 2, and a1 and a 2 are angles within species 1 and 2, respectively, at 95% confi-dence range that are calculated from 1600 bootstrapped sets formed by the resampling procedure described above. So, the null hy-pothesis is that P is zero. If P exceeds a 1 and a2, the null hypothesis is rejected at 95'21con-fidence interval. Therefore, one can say that the two trajectories are different, i.e., that the shape change over time or LCS takes different paths in the shape space (Zelditch et al. 2000).

Growth vector angles of meraspis and ho-laspis were calculated (Table 3) by the pro-gram VecComparel2 (Sheets 2001b). Our re-sults indicate that meraspid and holaspid tra-jectories do not exceed the range of growth vector angles within the holaspid trajectory in calculations based on both partial warp scores and shape coordinates. As a result, the null hypothesis cannot be rejected at 95% confi-dence intervals. In other words, the direction of growth of meraspis is not significantly dif-ferent from that of holaspis.

In sum, it is clear that there is allomctric shape change in T bccki. However, the ability to interpret the shape change depends on the choice of how to superimpose landmarks. Care must be taken in choosing a suitable su-perimposition method for one's own study. It is evident from the preceding analysis that the several different methods available for align-ing specimens in order to calculate shape co-ordinates independent of specimen size have subtly different effects on the shape descrip-

TRILOBITE ONTOGENY AND ALLOMETRY

tors produced. In particular, they apportion variance among landmarks differently. GLS superimposition may be inappropriate for sets of landmarks that possess a plane of sym- metry, and that have numbers of landmarks or landmark variance that is not balanced across the symmetry plane. Such sets may be han- dled more effectively by constraining align- ment along the symmetry plane by BTR or the new SBR method. The latter will provide bet- ter results than the fixed registration of Book- stein's method when one or both of the base- line landmarks also experience localized var- iation in its position, as in the case for Tviav-fkrus becki.

Although the deformation vectors present- ed by the three different superimposition methods examined here (Fig. 8) differ slightly from each other in their directions and mag- nitudes, we can form a general description of allometric shape change in the meraspid and holaspid stages: the cranidium becomes rela- tively shorter and wider, and the eyes become slightly smaller relative to the other land- marks during the course of size increase. This deformation, especially the relative shape change in eyes, is less clearly shown by vectors based on Bookstein's two-point registration (BTR). The general character of allometric shape change in the holaspid stage is the same as in the meraspid stage (Table 3), and the de- gree of shape change is quite small relative to the distance among landmarks. Thus, holas- pid allometry, although real, is quite subtle.

Many fossils have suffered some degree of deformation. Arthropods with mineralized exoskeletons, like this trilobite, often have fractures or cracks or both that were produced during compaction. Such compaction may cause shape deformation by lateral or vertical expansion or contraction. In the study of mor- phology of fossils, the possible effects of de- formation should be considered (e.g., Hughes 1999; Webster and Hughes 1999). There are two major styles of compactional deforma- tion: flattening and shear (Hughes 1999). None of the specimens used here show any sign of flattening because they all still have high con- vexity, although a very few large specimens have small cracks in the frontal area. Relative- ly large specimens typically experience more

prominent shape change by deformation than small specimens (Webster and Hughes 1999). If large (holaspid) specimens of 7: becki have suffered from greater deformation than small (meraspid) specimens, the direction of defor- mation vectors in the large specimens would be different from those in the small speci- mens. However, there is difference only in the magnitude of deformation vectors, not in their direction (Fig. 8). Thus, we conclude that the minor cracking exhibited by some of our larg- er specimens has not significantly affected their shape.

Our material also exhibits no obvious sign of shear effects (Fig. 4). Hughes (1999) noted that shearing deformation changes the rela- tionship between length and width, and that plots of length versus width measurements consequently exhibit higher degrees of scatter than plots of length versus length measure- ments. However, the plot of bivariate relation- ship between cephalic length and glabellar width of T becki in Figure 3 shows a low de- gree of scatter. Indeed, it exhibits even lower degree of scatter than that illustrated by Hughes (Hughes 1999: Fig. 4.13A) for what he considers to be undeformed material. We con- clude therefore that this morphometric study of 7: becki has not been affected seriously by compaction.

Summary

1. The 114 specimens of cranidia of Triar-tlzvus becki collected from a single bed of the lower Dolgeville Formation in the Mohawk River Valley, New York State, were measured for morphometric analyses.

2. The specimens can be grouped into mer- aspid and holaspid specimens on the basis of (a)comparison with cephalic lengths reported in previous ontogenetic studies of 7: becki and (b) a distinct change in the regression slope of shape on size. The boundary is located around 0.6 in log centroid size (LCS).

3. Regression of Procrustes distances on LCS reveals that allometry is present in the meraspid and the holaspid stages. The rate of shape change in the meraspid stage is much greater than that in the holaspid stage.

4. Allometry is also evident when partial warp scores and uniform components are re-

KEONHO KIM ET AL.

gressed on LCS in both the meraspid and the holaspid stages.

5. Comparison of growth vectors deter-mined by regressing partial warp scores and uniform components, or shape variables pro- duced by SBR, on LCS indicates that the di- rection of shape change is not statistically dif- ferent in the meraspid and holaspid phases.

6. The ontogenetic patterns described here for 7: becki suggest that the cranidium be- comes shorter and wider as the individuals grew, and eyes become slightly smaller. The new shape registration method, the Sliding Baseline Registration (SBR), is preferable for the visualization of allometry of organisms that contain an axis of symmetry.

Acknowledgments

We would like to thank R. Chapman, N. Hughes, G. Hunt, and M. Zelditch for their helpful suggestions and critical reviews that substantially improved the manuscript. Part of this project was funded by a research grant from the New York State Museum and the Graduate Group in Evolutionary Biology and Ecology at SUNY Buffalo for Kim.

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Morphometric Analysis of Ontogeny and Allometry of the Middle Ordovician TrilobiteTriarthrus beckiKeonho Kim; H. David Sheets; Robert A. Haney; Charles E. MitchellPaleobiology, Vol. 28, No. 3. (Summer, 2002), pp. 364-377.Stable URL:

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K-Bentonites and Graptolite Biostratigraphy in the Middle Ordovician of New York State andQuebec: A New Chronostratigraphic ModelDaniel Goldman; Charles E. Mitchell; Stig M. Bergström; John W. Delano; Steven TicePALAIOS, Vol. 9, No. 2. (Apr., 1994), pp. 124-143.Stable URL:

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Direct Measurement of Age in Fossil Gryphaea: The Solution to a Classic Problem inHeterochronyDouglas S. Jones; Stephen Jay GouldPaleobiology, Vol. 25, No. 2. (Spring, 1999), pp. 158-187.Stable URL:

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Compaction-Related Deformation in Cambrian Olenelloid Trilobites and Its Implications forFossil MorphometryMark Webster; Nigel C. HughesJournal of Paleontology, Vol. 73, No. 2, Papers from the Second International Trilobite Conference,August 1997. (Mar., 1999), pp. 355-371.Stable URL:

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