1. Morphological Evolution of the Turtle Shell and Its Mechanical Implications
Part 2: Theoretical
Connor McLaughlin, Taylor Wise, and C. Tristan Stayton Department of Biology, Bucknell University, Lewisburg, PA 17837
Abstract
Evolutionary biologists have considered understanding the relationship between organismal morphology and functional
performance to be fundamental for understanding phenotypic diversification. Numerous studies have investigated
performance and morphological evolution within lineages. However, far less attention has been paid to the relationship
between performance and morphological diversification among lineages. This study develops the turtle shell as a model
system for studying morphological and performance diversification within a comparative context. Original data consisted of
3D landmark coordinates digitized on 1962 turtle shells representing 254 separate species. Data were aligned using a
Generalized Procrustes fit and ordinated with principal component (PC) analysis. High scores on PC1 and 2 indicate taller,
more domed shells. To explore the functional implications of this variation, theoretical shell shapes corresponding to 117
evenly-spaced points in morphospace were extracted. Finite element models were built for all theoretical shapes to assess
mechanical performance. Heat transfer ability was quantified using surface area to volume ratios (SA/V), and three shape
indices were used to assess righting ability. Turtle shells with high PC1 and 2 scores were stronger, possessed greater righting
ability, and had lower SA/V. Shells with low PC1 and 2 scores were more streamlined, and had higher SA/V. Terrestrial and
aquatic turtles did not differ in shell shape (see part I), but terrestrial turtles showed a tendency to evolve towards higher
PC1 and 2 scores. Similar values for all performance measures could be found in large areas of morphospace, suggesting
many-to-one mapping of form onto function; thus, turtle shells can diversify morphologically without necessarily sacrificing
performance for all shell functions.
Part 2 of this project will address the following questions: “What shell shapes are characteristic of certain regions of
morphospace?”, “What are the possible functional implications of a given theoretical shape?”, and “Can turtle shells diversify
without sacrificing their performance for a particular function?”. These questions must be answered in order to better
understand not only what kinds of turtle shell shapes are plausible, but also what types of shell morphology optimize
particular functions.
Results
Morphospace
• The first 10 PCs together accounted for 85% of the total variation in the data set, with the first two PCs
accounting for 21% and 11% of variation, respectively.
• High scores on PC1 were associated with shells with relatively tall and domed carapaces, while low scores were
associated with much flatter shells (Figure 3).
• PC2 also separated highly-domed shells (positive scores) from relatively flat shells (negative scores).
• No actual turtle specimens seemed to have moderately high PC1 scores and high PC2 scores.
Performance surfaces
• R and S Indices
 Higher R and S index scores were associated with higher values of PC1 and PC2.
 More variation in R index scores occurred along PC1, while S index scores had approximately equal variation
on both PC axes.
 Theoretical models produced R indices from 0.4-1.1 and S indices from 0.62-0.78.
 However, most specimens only occurred in regions characterized by R index scores of 0.6-0.9 (Figure 4A) and
S index scores of 0.66-0.74 (Figure 4B).
• Average Overall Stress
 Higher stresses were associated with lower values of PC1 and higher values of PC2.
 Stress had approximately equal variation on both PC axes.
 Theoretical models produced average overall stresses from 4-13 MPa.
 However, most specimens only occurred in regions with average overall stresses from 5-8 MPa
(Figure 4C).
• SA/V Ratio
 PC1 values from -0.15-0 and corresponding PC2 values from -0.2-0 were associated with higher SA/V ratios,
while PC scores outside these respective ranges corresponded to lower SA/V ratios.
 More variation in SA/V ratios occurred along PC1.
 Theoretical models produced SA/V ratios from 160-220 mm-1.
 However, most specimens only occurred in regions with SA/V ratios from 170-200 mm-1 (Figure 4D).
Discussion
• Although there were no turtle shells occupying areas of morphospace with moderately high PC1 scores and high
PC2 scores, the theoretical shell shapes associated with that region appear to be perfectly cromulent and
reasonable for turtles to evolve.
• The theoretical models indicated that as PC1 and PC2 increased, R and S indices also increased. Additionally, as
PC1 decreased and PC2 increased, average overall stress increased.
 This suggests that theoretical models in regions of morphospace characterized by high values of PC1 and
PC2 (most similar to tortoises) are more spherical and possess greater righting ability compared to those
that are found in regions with low values of PC1 and PC2 (map turtles, for example).
 Furthermore, theoretical models in regions of morphospace characterized by low values of PC1 and PC2 are
much flatter compared to those that are found in regions with high values of PC1 and PC2.
 Overall, theoretical models associated with more spherical shapes and greater righting ability also tend to
be stronger.
• Unlike the shape indices and average overall stress, the relationship between PC1, PC2, and the SA/V ratio of the
theoretical models was not as monotonic.
 SA/V ratios were low for both very high and very low values of PC1 and PC2, while they were high only when
values on both axes were moderately low.
 It is unknown whether high or low SA/V ratios are favorable for turtles, but these ratios are often associated
with heat transfer ability, which is important to consider when studying ectotherms.
• Specimens appear to follow certain contour lines on some performance surfaces.
 For the SA/V ratio performance surface in particular, aquatic specimens lie along lines associated with values
between 185 and 200mm-1.
• The fact that similar values for all indices, average overall stress, and SA/V exist in many different areas of
morphospace demonstrates that turtle shells can have a diverse array of shapes without sacrificing performance
for a given function.
References
1R Development Core Team. 2010. R: a language and environment for statistical computing. Vienna: R
Foundation for Statistical Computing. http://www.R-project.org.
2Stayton, C.T. 2009. Application of thin-plate spline transformation to finite element models, or, how to turn a bog
turtle into a spotted turtle to analyze both. Evolution 63:1348-1355.
3Dumont, E.R., I.R. Grosse, and G.L. Slater. 2009. Requirements for comparing the performance of finite element
models of biological structures. Journal of Theoretical Biology 256:96-103.
4Strand7. 2007. Strand7 Finite Element Analysis System. Sydney: Strand7 Software.
5Domokos, G. and P.L. Várkonyi. 2008. Geometry and self-righting of turtle shells. Proc. R. Soc. B 275:11-17.
6MathWorks. 1996. MATLAB. Natick, Mass.: Mathworks.
Materials and Methods
Original data
• Primary data consisted of 3D landmark coordinates digitized on 1962 adult turtle specimens.
• 254 species (86% of all hard-shelled turtle species) in 12 families were used.
• 53 landmarks were shared among all specimens (Figure 1). Data were collected from one side of the shell only.
(Figure 1) Base shell model (Glyptemys muhlenbergii) with landmarks in ventral (left), lateral (center), and dorsal (right)
views. Anterior is to the right.
Geometric morphometric analyses
• Data were aligned using a Generalized Procrustes fit.
• Principal components (PC) analysis was conducted on fitted coordinates. 10 PC axes were retained for analysis1.
Theoretical shapes
• Shapes corresponding to 117 evenly-spaced points in morphospace were extracted.
• To determine the mechanical properties of shapes occurring throughout morphospace (including regions not occupied by
actual turtles), finite element (FE) models were built for each of the theoretical shapes.
Finite element analysis
• The method of Stayton (2009) was used to build finite element (FE) models for all theoretical shapes by warping a model
of a bog turtle (Glyptemys muhlenbergii)2. All models were scaled to the same size3.
• All models were assigned the same material properties, and the same 12 load cases and 4 restraints (Figure 2)3.
• Models were analyzed using a linear elastic model. von Mises stresses for all elements were extracted for all load cases4.
• Average stresses for each model were used as input for functional analyses.
(Figure 2) Sample FE model (Chrysemys picta) with locations of loads (arrows), restraints (Xs), and resulting VM stresses
(cooler colors indicate lower stresses, hotter colors indicate higher stresses). Forces are normal to the shell surface. Anterior
is to the right.
Shape and stability analyses
• Shell height (SH), maximum carapace width (MaxCW), and straight carapace length (SCL) were calculated for each
theoretical shape.
• R and S indices were calculated using the formulas R = (SH/MaxCW) and S = (SH*MaxCW)/SCL2) 1/3.5
• These data were used to construct performance surfaces for each functional parameter in Matlab6.
SA/V analysis
• SA/V ratios were calculated for each of the theoretical models.
• To approximate the surface area of a turtle’s carapace, the surface area of half an ellipsoid was calculated for each
specimen using the expression ({(4[apbp+apcp+bpcp]/3)1/p}/2), where a is the straight carapace length, b is the maximum
carapace width, c is carapace height, and p = 1.6075. Volumes were calculated using the expression (4/3[abc]).
• SA/V ratios were then determined and used to construct a SA/V performance surface.
(Figure 3) Lateral view of all theoretical FE models evenly-distributed in morphospace.
(Figure 4) Performance surfaces with specimens superimposed for: A) R Index
B) S Index C) Average Overall Stress and D) SA/V Ratio . Note the different scale
for PC Axis 1 in C.
MPa
mm-1