Syst. Biol. 50(5):640–656, 2001
Comparing Ontogenetic Trajectories Using Growth Process Data
PAUL M. M AGWENE
Department of Ecology and Evolutionary Biology, Yale University, P.O. Box 208106, New Haven, Connecticut 06520,
USA; E-mail: [email protected]
Abstract.—Ontogenetic trajectories are commonly quantiŽed by characterizing changes in the sizes
and shapes of organisms over the course of development. This formulation of ontogenetic transformations can be misleading in that it ignores critical aspects of the biological processes responsible
for constructing morphology. Hypothetical examples are used to illustrate some of the shortcomings
of methods that rely exclusively on size and shape data for ontogenetic analyses. By characterizin g
growth as a vector Želd, and representing growth vectors as complex numbers, one can simultaneously analyze size, shape, and growth processes. The utility of such an approach is demonstrated in
a study of shape and growth process variation in turtle shells. [Growth; morphometrics; ontogeny;
size; shape; Testudines.]
The idea that morphological diversity
among taxa arises from diversity in the underlying developmental processes responsible for “building” morphology is well accepted (Woodger, 1945; Gould, 1977; Alberch
et al., 1979; Atchley and Hall, 1991; Raff,
1996), if not well understood. A desire to
understand the relationship between developmental process and morphological evolution has, in large part, been the driving
impetus behind the ever-growing body of literature on topics such as heterochrony, allometry, and developmental integration.
A particularly suggestive framework for
analyzing variation in developmental parameters and its effect on morphology is the
concept of ontogenetic trajectories (Alberch
et al., 1979; Atchley, 1987). This approach is
based on the notion of quantifying changes
in form over the course of development. The
path taken by a particular organism, or the
mean path of a population of organisms,
through some multivariate space (“ontogenetic space”) that describes changes in form
is called an ontogenetic trajectory. If parameterized reasonably, such trajectories may
be compared among individuals or between
taxonomic groups. A common application
has been to try to interpret differences in ontogenetic trajectories among related taxa as
indicative of various heterochronic processes
(e.g., Alberch et al., 1979; McKinney and
McNamara, 1991). How particular differences in trajectories correspond to various
categories of heterochronic phenomena is
controversial (Reilly et al., 1997; Rice, 1997).
The study of ontogenetic trajectories is
usually framed in the context of changes in
the size and shape of organisms. This formulation is useful but, by focusing attention exclusively on patterns of size and shape
change, has resulted in a tendency for students of the Želd to ignore variation in the
processes and patterns of growth underlying
those changes.
The following pages represent an attempt
to reevaluate and expand the methodological
bases by which ontogenetic transformations
are characterized, quantiŽed, and compared.
I begin with a brief review of the dominant
analytical framework for studying ontogeny
that focuses on characterizing changes in the
size and shape of organisms. The types of
problems one may encounter in relying exclusively on parameters of shape and size
are illustrated with simple hypothetical examples. I then demonstrate that by using
vector-valued variables to quantify “growth
Želds,” one can incorporate process information into an analytical framework that
facilitates the concurrent analysis of size,
shape, and growth process variation. This
type of analysis provides a richer set of
metrics for characterizing ontogenetic differences among taxa. The efŽcacy of the approach is demonstrated by applying it to an
analysis of plastral scute growth in various
turtle species.
T HE S TANDARD APPROACH: CHANGES
IN F ORM
Organismal form is usually characterized
as consisting of two components: size and
shape. This is often represented by a simple
formula Žrst advanced by Needham (1950):
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MAGWENE—COMPARING GROWTH PROCESS ES
Form D Size C Shape. Needham’s formula
for organismal form has been implicit (e.g.,
Gould, 1977; Alberch et al., 1979) or explicit
(e.g., Atchley, 1987; Atchley and Hall, 1991) in
most quantitative explorations of morphology during the last three decades. Alternative deŽnitions of form have been advanced
that incorporate other aspects of the phenotype such as texture and color (Lestrel,
1997), but such formulations have not been
widely applied in practice. Working from
Needham’s formula, transformations of morphology are usually characterized as change
in the constituent parts, that is, 1Form D
1Shape C 1Size (Needham, 1950). This formulation is both convenient and useful and
may be applied equally to ontogenetic as well
as evolutionary transformations of morphology. Depending on the context, changes in
form are construed to apply to either individuals (i.e., Growth D 1Form over ontogeny)
or higher units (i.e., Morphological Evolution
= 1Form over the evolutionary history of a
population or other taxonomic group). The
analyses detailed herein concentrate primarily on ontogenetic transformations.
Despite the appeal of Needham’s formulation for describing changes in organismal
form, a simple reliance on shape and size
can obscure the relationship between morphological pattern and the growth processes
responsible for its production. In many respects, the growth processes responsible for
the production of form, rather than form itself, are of primary interest in understanding
patterns of morphological evolution (Rice,
1998). If one endeavors to quantify and characterize growth for the purpose of understanding how developmental variation leads
to morphological diversity, then simply describing how size and shape changes over the
course of ontogeny is not sufŽcient. One must
dig deeper and consider explicitly the mechanisms and processes that produce those
patterns.
641
sively on size and shape may often misspecify changes in development.
Hypothetical Examples: Growing Boxes
Figure 1 illustrates some of the problems
that can arise when one relies on simple
shape and size parameters for analyzing ontogenetic and evolutionary transformations.
This Žgure represents morphological structures of three hypothetical taxa, each structure consisting of two subunits (S1 and S2).
All three taxa begin ontogeny with identical
morphologies. The nested boxes represent
the shape and size of the subunits at various ontogenetic stages; the innermost boxes
represent the youngest stages, which are initially adjacent to each other. Such patterns
of growth are commonly recorded in the tissues of organismal structures that grow by
accretion or addition (e.g., epidermal scutes
of turtles, calcitic plates of echinoderms;
Raup, 1960). Figure 1a depicts the ancestral
morphology. In the ancestor, both morphological subunits are identical in shape and
size, and growth is isometric. At every stage
of ontogeny, the shape of the subunits remains constant and overall shape remains
unchanged; consequently, size is the only
aspect of form that changes over ontogeny.
M ORE THAN S HAPE AND S IZE
By concentrating solely on aspects of size
and shape change, many studies attempting
to quantify ontogeny fail to consider a key aspect of developmental programs—variation
in the processes that produce morphologies.
In the following paragraphs, hypothetical examples are used to illustrate the fact that
a methodological approach focusing exclu-
FIGURE 1. Hypothetical examples illustratin g problems that can arise when growth is characterized with
reference only to changes in shape and size. See text for
further explanation.
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S YSTEMATIC BIOLOGY
Figure 1b shows a derived morphology, Descendant X, in which the shape of the subunits remains constant over ontogeny, but the
pattern of nested boxes suggests an asymmetry (within subunits) has arisen in the underlying growth process, resulting in a greater
rate of accretion on the rightmost and bottommost surfaces of each subunit. Because
the growth process has been altered in a similar fashion for both subunits, there is no overall change in the shape of the structure and
the Žnal outcome of growth is again a simple
change in size. A second derived morphology, Descendant Y, is depicted in Figure 1c.
Here as well, the shape of each of the subunits
remains constant over ontogeny, but in this
case the nested patterns suggest an asymmetry of growth process both within and between subunits. Subunit S1 has a greater rate
of accretion on the rightmost and bottommost surfaces; subunit S2 has greater rates of
accretion on the rightmost and topmost surfaces. As a result of these asymmetries, the
overall shape of the structure changes even
though the shapes of the individual subunits
remain constant.
An analysis relying exclusively on considerations of size and shape would conclude
that no signiŽcant change has occurred (i.e.,
shape and size are identical) in the evolution
of Descendant X from the ancestral state, despite the fact that interesting changes have
occurred in the growth processes that produce the morphology. With respect to the
comparison between Descendant Y and the
ancestor, an analysis focusing purely on the
shape of the structure as a whole would
conclude that there had been a change in
shape but would be unable to specify the
source of this change; alternatively, focusing
on the subunits, an analysis based on shape
and size would conclude there is no difference between the ancestral and descendant
subunits. What these Žgures reveal is that although the form (shape C size) of an organism at time t1 is the sum of its form at time t0
plus the growth processes over the interval
1t, knowing the form at t0 and t1 is not sufŽcient to infer the growth process. There is no
unique solution for inferring growth based
on size and shape alone.
Structures and Subunits
The hypothetical examples presented
above are based on simple geometric struc-
VOL. 50
FIGURE 2. A hypothetical example demonstrating
that even in the case of simple structures, similar
forms may be attained by way of very different growth
processes. Dotted curves represent initial form, solid
curves represent Žnal form. Solid arrows represent accretionary processes, arrow heads represent resorption.
(Left) Structure attains Žnal form through a process of
simple accretion. (Right) Final form results from a combination of accretion and resorption.
tures. The analogy to particular organismal
structures (e.g., Žsh scales, turtle scutes) is
obvious and direct, but the connection with
other types of morphologies may not be
so clear. These types of problems can arise
when considering any organismal structure
composed of subunits whose growth processes are autonomous or semiautonomous.
They may also apply to single structures that
have multiple centers of growth. Phrased
in this way, such problems can be seen to
apply to most morphological structures “in
which Žnal form results from the integration
of a number of separate component parts”
(Atchley and Hall, 1991).
The problem of understanding growth
from a time-ordered series of “snapshots”
may be particularly acute in the case of complex morphologies, but in even the simplest
of systems, similar forms may be generated by radically different growth processes
(Fig. 2). Unfortunately, studying growth processes often requires much more effort than
studying the resulting patterns. Because of
its time-dependent nature, process is difŽcult
to assess from preserved specimens or single slices of ontogenetic time. In the discussion section I recommend several approaches
and techniques for facilitating the study of
growth processes. For the moment, let us assume that growth processes are observable;
below, I present an analytic framework for
their quantiŽcation and comparison.
G ROWTH AS A V ECTOR FIELD
The previous section illustrated some of
the problems of interpretation that may arise
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MAGWENE—COMPARING GROWTH PROCESS ES
when analysis of ontogenetic change is formulated in terms of the shape and size
of structures without regard to underlying
developmental process. A useful method
for characterizing and comparing growth
processes is to treat growth as a vector Želd
acting on the morphological structure or
structures of interest. I will demonstrate below that this characterization is useful from
both analytical and biological perspectives.
Growth Fields
Many types of growth can be characterized by using growth Želds—vector Želds
in which the vectors indicate movements
of particular cells, pieces of tissue, or other
structures over ontogenetic intervals. The
treatment of growth processes as vector Želds
may be construed to imply that changes
in morphological form result from morphogenetic factors (biochemical, mechanical, or
other) that cause structures to move from an
initial position x0 to a subsequent position x1
over a time interval 1t D t1 ¡ t0 . This “movement” may represent the actual movement
of individual cells or alternatively may represent the relative changes in the positions of
topologically homologous points. Consider,
for example, the growth of the epidermal
scutes of turtles (Fig. 3). A piece of tissue at
the corner of a scute at time t0 is not the same
643
as that at the equivalent corner of the same
scute at time t1 (assuming 1t is long enough
for meaningful growth to have occurred);
nonetheless, one presumes a structural correspondence. By characterizing changes in the
position of multiple points of biological correspondence, one can represent the growth
of a structure over the time interval 1t as a
vector Želd (Fig. 3). For particular types of
tissues, growth vectors might represent processes of accumulation (accretion, volumetric increase) and tissue migration as well as
such processes as resorption.
The term vector Želd is used rather loosely
here. The typical mathematical deŽnition of
a vector Želd is “a rule assigning a unique
value of a [vector] to each point” in space
(Borisenko and Tarapov, 1979). Whereas it is
certainly possible to estimate a function that
assigns a vector to every point on a growing structure (see discussion of related approaches below), the emphasis of the method
described herein rests purely on the observed
data. The term growth Želd will be used to
refer to a Žnite set of observed vectors in the
plane that serve as an estimate of the growth
process.
Related Concepts
The notion of using vectors to describe
growth is well established in the biological
FIGURE 3. Photograph of the posterior portion of the plastron of the Asian leaf turtle, Cyclemys dentata, illustrating
the notion of growth Želds. White arrows represent growth vectors, and illustrate the “movement” of homologous
points of tissue during ontogeny. The dark, radiating lines interspersed between the growth vectors are actual
patterns of pigmentation. This pigmentation pattern lends support to the assessment of the growth Želd for this
specimen. Note that by using only two ontogenetic stages to characterize growth Želds, nonlinear trajectories may
be misrepresented.
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S YSTEMATIC BIOLOGY
VOL. 50
literature. Several authors concerned with
both plant (e.g., Richards and Kavanagh,
1943, 1945; Erickson, 1966, 1976; Silk and
Erickson, 1978) and animal (e.g., Cox and
Peacock, 1978, 1979; Skalak, 1981; Skalak
et al., 1982, 1997) growth have utilized vector
Želds, or related concepts (e.g., streamlines,
ux), to describe how cells or other units of
tissue move or are produced over the course
of development. Many of these approaches
utilize concepts from continuum mechanics
(Skalak, 1981; Silk, 1984). Several interesting models for describing particular types of
growth have emerged from such approaches.
For example, using a model that incorporates information about “generating cells”
and their orientation, Skalak et al. (1997) were
able to provide a reasonable explanation for
how various antler morphologies might be
generated. Despite the usefulness of such
models for describing particular organismal
structures, accurately estimating the parameters of the models for large numbers of
individual organisms is often an overly burdensome task. As a result, statistical analyses
based on such models are rare, and the emphasis of such approaches is on describing a
few representative forms, rather than populations of individuals.
thin-plate spline family of methods, as currently formulated, is meant solely for analyzing patterns of shape variation. As Bookstein
notes (1996:145),
A Note on Transformation Grids
CHARACTERIZING G ROWTH FIELDS
Other approaches, using a variety of
mathematical frameworks, have been applied to the characterization of growth.
D’Arcy Thompson’s (1917) transformation
grids immediately come to mind. Characterizing changes between ontogenetic stages as
deformations of a Cartesian grid provides a
suggestive picture of growth (Richards and
Kavanagh, 1943; Zelditch and Fink, 1995). If
the deformations are accurately calculated,
then such transformation grids might be construed as simply a different visualization
of the same information present in a diagram of vector displacements. For nearly 70
years, the study of transformation grids suffered a fate similar to that of the continuum
mechanics–based approach: an inability to
quantify and compare transformations in a
statistical sense. Bookstein (1989, 1991) overcame this problem with the introduction of
morphometric methods based on decompositions of a thin-plate spline. However, Bookstein’s solution, though elegant, does not directly apply to the problem at hand. The
If growth over a time interval 1t(D t1 ¡ t0 )
is characterized as a growth Želd, one can
compare and analyze both changes in shape
as well as changes in the processes that
produce that shape. The advantage of using growth Želds to characterize ontogenetic
processes is that the vectors of such Želds
can be quantiŽed by using complex numbers
(that is, numbers with a “real” and “imaginary” component). These complex-valued
variables may be combined with complex
variables representing morphometric landmarks. Complex covariance matrices may
be constructed that can be used in multivariate techniques such as principal components analysis or canonical variates analysis. Below, I briey review the methodology underlying this approach (Magwene and
Chernoff—unpublished manuscript).
The splines themselves are only a suggestive visual
metaphor; don’t talk about them as if they speciŽed real changes of little bits of organism. They reexpress the landmark data by lines drawn in-between
the landmarks, where there really isn’t any additional
information.
As I argued above, an understanding of
how the “little bits of organism” move is precisely the type of information that is critical if
one wishes to analyze growth processes and
understand how their end product, biological shape, is produced over the course of
ontogeny.
Transformation grids may still serve the
role of useful visual metaphor in growth
studies, but as noted above, they have the
unfortunate property of implying pattern
where there are no data to support the suggestion. In the rare case when growth data are
sampled very densely, this approach seems
justiŽed. In general, it seems best to use a
technique that focuses one’s attention on precisely those patterns for which information is
available.
Vectors as Complex Variables
Vectors in the Euclidean plane, R2 , are
characterized by two parameters: magnitude
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645
MAGWENE—COMPARING GROWTH PROCESS ES
and direction. Additionally, one may associate a third parameter with a vector: location, the point from which a vector emanates.
Vectors with location are referred to as bound
vectors (Borisenko and Tarapov, 1979). Location, however, is not critical to the characterization of a vector. Two-dimensional vectors
may be represented by complex numbers
because there is an isomorphism (a one-toone mapping) between the Euclidean plane
and the Želd of complex numbers (Andersen
et al., 1995; Pedoe, 1988). Ignoring location,
a vector in the Euclidean plane with magnitude m and direction µ (relative to some
coordinate axis) may be characterized as a
complex number, z D m(cos µ C i sin µ), a
representation usually referred to as the polar form of a complex number (see Fig. 4).
The modulus (also called magnitude) of a
complex number characterizes the length
of the vector and is written, jzj D m. The
argument of a complex number, arg(z) D
µ, is a measure of the angular displacement of the vector relative to the real axis
in the complex plane. The conjugate of
z, z D m(cos µ ¡ i sin µ ), is equivalent to a
reection of z across the real axis. DeŽnitions of other common operations on complex numbers are given in Magwene and
Chernoff (unpubl. manuscript) or in any
standard text on complex analysis (e.g.,
Stewart and Tall, 1983).
In addition to representing vectors, complex numbers can be used to represent twodimensional points. Any point (x, y) in the
Euclidean plane may be represented by the
complex number x C i y. Complex numbers
are suitable for representing both points and
vectors (Pedoe, 1988), a fact I will exploit in
the subsequent analyses.
COVARIANCE M ATRICES FOR
G ROWTH FIELDS
Given a set of k growth vectors, which
adequately characterize a growth Želd, one
can represent those vectors as a set of k
complex numbers as described above. A
comparable set of vectors may be characterized for a sample, size n, of individuals,
and an n £ k matrix of complex numbers
thus represents the sample population to be
analyzed. This representation summarizes
information about the growth vectors but
ignores information about their location. If
the initial position of the vectors from specimen to specimen is invariant, or not of interest, one can proceed to characterize the
covariance matrix of the growth vectors
alone. This corresponds to a “pure” analysis
of growth process. If, on the other hand, one
is also interested in patterns of variation of
vector location (essentially the shape of the
set of points at time t0 ), then this information can be included by simply appending
an additional k complex variables that represent the point locations of the vectors. If
these additional points are appended, then
the covariance matrix will include information about shapes at time t0 , the growth processes over 1t D t1 ¡ t0 , and shape at time
t1 . Such an analysis represents a simultaneous consideration of growth process and
shape information and possibly size as well,
if that has not been removed from the point
data.
After reexpressing the observations on
each variable as deviations from their respective means, D, the complex covariance matrix
is deŽned as:
VD
FIGURE 4. Illustration of the correspondence between points/vectors in the Euclidean plane and complex numbers. The vector with magnitude m and direction µ can be represented by the complex number
z1 D m(cos µ C i sin µ ). The point (x,y) is represented by
the complex number z2 D x C i y.
1
DD¤
n
where D¤ represents the conjugate transpose
of a complex matrix (Anderson et al., 1995).
This complex covariance matrix is
Hermitian (i.e., V D V¤ ), which means that
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S YSTEMATIC BIOLOGY
the elements below the diagonal are the
conjugates of the corresponding elements
above the diagonal. The fact that this complex covariance matrix is Hermitian means
that it has real values on the diagonal (variances) and complex values for off-diagonal
elements (covariances). The familiar real
symmetric matrices are just a special case
of Hermitian matrices (Barnett, 1990). Both
Hermitian and real symmetric matrices
share a number of features that make them
convenient for statistical analysis. For the
purposes of this discussion, the most important feature is that both have real-valued
sets of eigenvalues that are greater than or
equal to zero.
This complex covariance matrix may be
subjected to a variety of multivariate statistical analyses (Andersen et al., 1995; Magwene
and Chernoff, unpubl. manuscript). One
such analysis is demonstrated below: complex principal component analysis (xPCA) of
growth patterns in turtles.
Why Use Complex Variates?
Either a pair of x-and y-coordinates or a
single complex number can be used to represent two-dimensional points and vectors in
the plane. Manipulating real-valued variates
is more familiar to a biological audience; why
bother to use complex variates at all?
From a mathematical perspective, using
complex variables to characterize twodimensional points is natural; complex numbers are often deŽned as ordered pairs
(Stewart and Tall, 1983), and the geometric
representation of complex numbers in the
complex plane is easy to intuit. Furthermore,
complex numbers have the algebraic structure of a Želd. That means that both addition and multiplication are commutative and
associative; multiplicative and additive identities exist as do inverses; and complex numbers satisfy the distributive law. As such,
complex numbers can be treated in much
the same way as real numbers are. Each of
these operations also has an interpretation
in terms of Euclidean geometry. For example, multiplication by a complex number can
be geometrically depicted as a combination
of rotation and scaling. Many morphometric
maneuvers, such as Procrustes analysis, are
most easily formulated in terms of complex
variates (Bookstein, 1991; Kent, 1994; Dryden
and Mardia, 1998).
From an analytical perspective, using complex numbers provides several advantages
that relate to interpreting patterns of association. For example, complex covariances are invariant to how one chooses
to orient the sample of specimens (following Procrustes superimposition); covariances
based on treating x,y-coordinates as separate
variables are not. It is important that interpretation of patterns of association among
biological landmarks or growth vectors not
be affected by how one chooses to depict the
specimens under analysis.
Another shortcoming of x,y-coordinates
treated separately relates to the inability of
the method to deal with symmetrical conŽgurations of landmarks or growth vectors. If the landmarks or growth vectors
under consideration include two or more
landmarks that differ primarily by reection, then a covariance matrix based on
real variates will be singular (or nearly so).
This means that any multivariate procedures involving matrix inversion will have
to be modiŽed to take symmetry into account (Bookstein, 1996). Complex covariance matrices are not susceptible to this
problem.
ANALYSIS OF G ROWTH PROCESS ES :
T URTLE S HELL G ROWTH
In the following pages I describe an analysis of growth processes, utilizing a subset
of data from a larger study (Magwene, unpubl. data) on turtle shell morphology. This
analysis focuses on the growth of the keratinous scutes that cover the bony elements of
the turtle shell.
Turtle Scutes
Turtle scutes are keratinized, epidermal
structures arranged in an array that covers
the underlying bony plates of the shell. The
formula and arrangement of scutes is variable among taxa, and scutes are absent in
some groups (Zangerl, 1969). Scute development precedes bony shell development. At
the time of hatching, the largely unossiŽed
bony shell is covered with epidermal scutes
that are typically granular in appearance. As
posthatching growth proceeds, germinal epithelium underlying the granular scute lays
down a new layer of keratinized epidermal
tissue (Ernst et al., 1994). Growth in natural
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647
MAGWENE—COMPARING GROWTH PROCESS ES
populations is typically periodical, and shell
growth may cease during periods of hibernation or estivation. When growth resumes,
a new layer of tissue is generated by the germinal epithelium, which bulges out from underneath the periphery of the layer from the
previous growing season. These bulges of tissue form distinct rings (Ernst et al., 1994).
Older layers of scutes may be shed in some
taxa. Rings on the scutes of nonshedding
species may be worn smooth. Growth rings
on plastral scutes, which come in regular contact with the substrate, are more likely to be
obscured than are rings from the carapace
(Magwene, pers. observation). When scutes
are not shed, or remain relatively unworn,
the rings represent a record of growing seasons, which may or may not be annular in na-
ture (Cagle, 1946). Growth rates vary among
taxa and among populations growing under different environmental conditions; consequently, aging specimens by their growth
rings remains somewhat controversial (Germano, 1998; Litzgus and Brooks, 1998). Regardless, any specimen that retains the granular infantile scute and successive growth
rings can be used to provide an estimate
of growth processes by analyzing relative
rates of growth within and between different
scutes.
The study detailed here is concerned with
growth processes of plastral scutes. The plastron is an appropriate structure to analyze
using two-dimensional representations because it is relatively at. The taxa considered in this study have six plastral scutes;
from anterior to posterior, these are gular,
humeral, pectoral, abdominal, femoral, and
anal (Fig. 5; Zangerl, 1969).
M ETHODS
Patterns of epidermal plastral scute
growth were characterized for 20 species
of cryptodiran turtles in three putative
clades: Emydidae (New World pond turtles), Bataguridae (Old World pond turtles), and Testudinidae (tortoises) (Table 1).
Collectively, these taxa are referred to as
the Testudinoidea (Gaffney and Meylan,
1988). There is some debate about the phylogenetic relationships among these taxa.
TABLE 1. Turtle species included in analyses of plastron growth.
FIGURE 5. Landmarks and vectors used to characterize turtle plastron growth are shown on the left half of
the diagram and illustrate the growth process and shape
variation. Forty-four landmarks were digitized as taken
from the right half of the plastron of each specimen.
Landmarks 1 to 22 (solid circles) represent the shape
of the hatching plastron (stage t0 ); landmarks 23 to 44
(ends of vectors) represent the same points at a later ontogenetic stage, t1 . Vectors have been drawn between homologous landmarks at stages t0 and t1 , representing the
growth process at a particular position over the course
of ontogeny. Typical arrangement and nomenclature of
plastral scutes for testudinoid turtles are shown on the
right half of the diagram. Anterior is towards the top of
the Žgure.
Label
Species
Family
n
E1
E2
B1
B2
B3
B4
B5
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
T12
T13
Clemmys marmorata
Emydoidea blandingi
Chinemys reevesi
Cuora amboiensis
Cuora avomarginata
Cyclemys dentata
Malayemys subtrijuga
Chersina angulata
Geochelone carbonaria
Geochelone pardalis
Geochelone radiata
Gopherus berlandieri
Gopherus polyphemus
Homopus signatus
Kinixys belliana
K. erosa
Psammobates oculifer
Ps. tentorius
Testudo graeca
T. kleinmanni
Emydidae
Emydidae
Bataguridae
Bataguridae
Bataguridae
Bataguridae
Bataguridae
Testudinidae
Testudinidae
Testudinidae
Testudinidae
Testudinidae
Testudinidae
Testudinidae
Testudinidae
Testudinidae
Testudinidae
Testudinidae
Testudinidae
Testudinidae
2
2
1
2
1
2
1
1
1
2
1
4
1
1
1
1
2
1
1
2
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S YSTEMATIC BIOLOGY
Haiduck and Bickham (1982) support a sister
group relationship between the Emydidae
and Bataguridae, with the Testudinidae as
the next most closely related group. Other
recent analyses (Gaffney and Meylan, 1988;
Lamb, 1994; Shaffer et al., 1997) support a
closer relationship between the Bataguridae
and the Testudinidae, and even suggest that
the Bataguridae may be paraphyletic with
respect to the Testudinidae (Gaffney and
Meylan, 1988; Shaffer et al., 1997). A hypothesis of phylogenetic relationships among
these taxa, based on Gaffney and Meylan
(1988), is shown in Figure 6.
The majority of the samples analyzed here
consists of species within the Testudinidae. In
large part this reects a sampling bias against
those taxa that shed their scutes. Many emydid species shed scutes, making it difŽcult
or impossible to characterize scute shapes
at earlier ontogenetic stages. A second bias
in the sample is towards juvenile animals,
which are more likely to show good scute
preservation. I used the criterion of a minimum of four nested scutes when selecting
specimens appropriate for analysis. In some
cases several specimens per species met the
criteria of age and scute preservation. In
these cases, the mean species values for shape
and growth data were used. In all, 30 individual specimens were pooled by species to
arrive at a Žnal sample representing 20 taxa
FIGURE 6. Phylogenetic hypothesis for the turtle genera included in this analysis, based on Gaffney and
Meylan (1988).
VOL. 50
(Table 1). All specimens analyzed in this report were obtained from the collections of the
Field Museum of Natural History, Chicago.
Specimens were photographed using a
coupled-charge display camera, and 44 landmarks were digitized on the right half of the
plastron of each specimen. The landmarks
represent the junctions of pairs or triplets of
scutes. Landmarks 1–22 represent scute morphology at the earliest discernible growth increment (t0 , typically the granular hatching
scute); landmarks 23–44 represent the same
topological scute positions at the time of the
animal’s death (t1 ). Growth vectors were estimated as vectors from a point xi at stage
t0 (xi 0 ), to the homologous point at stage t1
(xi 1 ; Fig. 5).
The landmark conŽguration of each specimen was superimposed on a baseline
deŽned by the anteriormost and posteriormost points along the midline at ontogenetic
stage t1 . Growth Želds were deŽned by using
the unbound vectors, gi D xi 1 ¡ xi 0 . Analyses of shape variation were conducted by using the landmark conŽgurations deŽned by
xi 1 . I chose to characterize shapes from the
landmark conŽgurations at stage t1 rather
than t0 because accretionary growth gradually “pushed” the scutes of earlier stages
away from each other. Although one can
back-calculate to reconstruct the initial conŽguration of subunits and their landmarks
(Magwene, unpubl. data), using the t1 stage
is less error prone. In combined analyses of
shape and growth variation, the information about shape at stage t0 is still implicitly
included.
Three analyses were conducted: (1) analysis of shape variation using the variables
xi 1 , for i D 1, : : : , k; (2) an analysis of growth
Želds deŽned by gi , for i D 1, : : : , k; and (3) a
simultaneous analysis of shape and growth
variation. Each of these data sets is of high
dimension (20, 22, and 42 dimensions, respectively). xPCA (Magwene and Chernoff,
unpubl. manuscript) was used to produce a
subspace projection of lower dimensionality
that summarizes essential patterns of variation. xPCA is essentially the same as PCA of
real-valued variables. Because the complex
covariance matrix (deŽned above) is Hermitian, it has real eigenvalues (the sum of which
equals the total variance in the sample) and
complex-valued eigenvectors (xPCs). One
can evaluate xPC loadings and scores resulting from xPCA similar to the way one would
2001
649
MAGWENE—COMPARING GROWTH PROCESS ES
interpret such quantities in a standard PCA.
Furthermore, xPC loadings can be projected
back into the two-dimensional space of the
original observations (the space of landmark
conŽgurations and growth Želds) to provide a visual summary of shape and growth
variation (Magwene and Chernoff, unpubl.
manuscript).
Implementing Complex Statistical Analyses
Conducting analyses based on complex
variance–covariance matrices is relatively
straightforward. A wide variety of statistical
and mathematical software packages include
facilities for dealing with eigenanalyses of
complex matrices. A sampling of commercial software packages with these facilities
includes S-PLUS, Matlab, Mathematica, and
Mathcad.
All analyses described in this report were
performed with algorithms written in the
computer programming language Python
(http://www.python.org), using a numerical
extension module based on LAPACK routines. Python is freely available and runs
on a variety of computing platforms. I have
made source code for these algorithms available via the World Wide Web at http://
pantheon.yale.edu/ »pmm34/xpca.html.
R ESULTS
Shape Variation
As noted above, shape variation was assessed on the basis of the conŽguration of
plastral scute landmarks at time t1 . The data
set in this analysis includes what appears, on
visual inspection, to be a fairly diverse population of shapes. Inspection of a regression
of Procrustes distances from the reference
conŽguration (GLS mean) against distance in
the tangent space of Procrustes residuals (a
Euclidean distance) shows high correlation
(r D 0:99); hence, the tangent approximation
for shape space, implicit because of the way
specimens were aligned, was judged to be
acceptable.
Plastron shapes were subjected to
xPCA (Magwene and Chernoff, unpubl.
manuscript), utilizing residuals around the
mean Bookstein coordinates resulting from
baseline Žtting (Dryden and Mardia, 1998).
The total shape variance in the sample
is 0.04661 units. The percentage of shape
variation summarized by each of the Žrst 11
TABLE 2. Comparison of variance summarized by
each of the Žrst 11 xPCs: shape, growth, and shape plus
growth analyses.
xPC
1
2
3
4
5
6
7
8
9
10
Shape
Growth
63.2
17.3
5.7
4.7
3.9
1.9
1.2
0.8
0.6
0.2
34.8
21.5
12.3
8.9
5.6
4.6
3.5
2.4
1.6
1.3
Shape C growth
50.9
13.8
11.1
6.0
3.5
2.9
1.9
1.3
1.0
0.8
xPCs is listed in the Žrst column of Table 2.
The three largest xPCs summarize »86%
of the shape variation within the sample.
Figure 7 is an ordination plot of the sample
based on shape xPC1 scores. Superimposed
on this plot is the minimum spanning tree for
this ordination of specimens. The minimum
spanning tree is useful for highlighting close
neighbors in an ordination (Krzanowski,
1988) and will be discussed below.
Growth Variation
Growth variation was analyzed by xPCA
of the growth vector Želds as described
above. The total growth vector variance in
the sample is 0.02252 units, roughly half that
of the shape data. The percentage of variation
summarized by the 11 largest growth xPCs
is detailed in the second column of Table 2.
FIGURE 7. Ordination of testudinoid specimens on
xPC1, analysis of plastron shape. Labels correspond to
those in Table 1. Edges drawn between taxa represent
the minimum spanning tree for the ordination.
650
VOL. 50
S YSTEMATIC BIOLOGY
FIGURE 10. Ordination of turtle genera based on
xPC1 of combined shape and growth data, with the
phylogeny of Gaffney and Meylan (1988; see Fig. 6)
superimposed.
FIGURE 8. Ordination of testudinoid specimens on
xPC1, growth analysis. Labels correspond to those in
Table 1. Edges drawn between taxa represent the minimum spanning tree for the ordination.
Growth vectors covary less strongly than do
the shape variables: Five xPCs are required to
summarize roughly 83% of the growth vector
variance. An ordination of specimens based
on scores on the Žrst growth xPC is presented
in Figure 8.
Variation of Shape plus Growth
A combined analysis of variation in shape
and growth is the most informative of
the analyses, allowing simultaneous explo-
ration of patterns of variation in shape
and growth processes. The last column
of Table 2 details the shape plus growth
variation summarized by xPCs. Approximately 51% of the shape plus growth
variation is captured on the Žrst xPC.
Substantial portions of variance are also
summarized by xPCs 2 through 5. The total variance in the shape plus growth analysis (0.06914 units) is simply the sum of
the variances in the separate shape and
growth analyses. A plot of xPC1 scores for
the combined shape plus growth analysis
is most similar to the shape subspace projection (Fig. 9). The shape variates have a
stronger signal because the shape data are
about twice as variable as the growth data.
Subsequent xPCs indicate relatively equal
contributions of both shape and growth
parameters.
Figure 10 shows an identical shape plus
growth ordination over which the phylogeny
of Gaffney and Meylan (1988) has been superimposed (branches of the tree in Fig. 6
have been rotated around their respective
nodes so as to minimize crossing over). As
in the previous Žgure, shape variables, which
have a strong taxonomic signal, dominate the
ordination.
I NTERPRETATION OF S HAPE AND
G ROWTH V ARIATION
Shape vs. Growth
FIGURE 9. Ordination of testudinoid specimens
based on xPC1 of combined shape and growth process
data. Labels correspond to those in Table 1. Edges drawn
between taxa represent the minimum spanning tree for
the ordination.
The ordination of turtle specimens in the
subspace of the largest shape xPC (Fig. 7)
suggests that shape variation carries a strong
taxonomic signal. Arrayed from left to right
across Figure 7 the specimens indicate a transition from testudinid to batagurid and emydid taxa. There is some minor overlap between the groups, but for the most part
2001
MAGWENE—COMPARING GROWTH PROCESS ES
the three clades can be distinguished on the
shapes of their plastra alone.
The ordination of specimens in the subspace provided by PCA of the growth Želds
(Fig. 8) is quite different. Clusters that correspond to the major taxonomic groups can
still be delineated, but the clusters exhibit
much more overlap. Also striking is the
difference between the minimum spanning
trees in each ordination. In the shape subspace, nearest neighbors are, for the most
part, other taxa of the same clade. The growth
subspace, however, contains indications of
greater similarity between distantly related
species. For example, Psammobates and Testudo have growth Želds that are more similar to the batagurids and the emydids than to
other testudinids (points T10 , T11 and T12 , T13 ,
respectively).
If growth processes and the morphologies
they produce evolve in concert, one would
expect a strong correspondence between ordinations in their respective subspaces, and
similar patterns of connectivity for minimum
651
spanning trees in the two ordinations. A
lack of correspondence in this data set suggests such is not the case for the posthatching ontogenies analyzed herein. Unresolved
is the issue of whether this discordance between shape and growth data is reected
in other ontogenetic stages. Analyses considering wider ontogenetic intervals will be
needed to resolve this issue.
Shape plus Growth
Figure 11 summarizes the combined patterns of shape plus growth process variation
captured by the largest subspace of the combined shape and growth xPCA. Gray polygons in the Žgure represent scute shapes at
time t0 . The vectors represent the growth process over the interval 1t D t1 ¡ t0 . Figure 11b
represents the mean shape and growth Želd.
Figure 11a represents the shape/growth
combination implied by negative loadings
on xPC1 (left side of Fig. 9); Figure 11c represents the combination of shape and growth
FIGURE 11. Testudinoid plastron scute shapes and growth Želds implied by variation on xPC1, combined shape
and growth analysis. (a) Plastron shape and growth Želd, negative loadings on xPC1; (b) mean plastron shape and
growth Želd; (c) plastron shape and growth Želd, positive loadings on xPC1. The actual range of shape and growth
differences has been multiplied four fold to aid visualization . Anterior is toward the top of the Žgure.
652
VOL. 50
S YSTEMATIC BIOLOGY
implied by positive loadings on xPC1 (right
side of Fig. 9). These Žgures capture information not only about shape variation but also
about the differences in growth processes responsible for those shapes.
To visualize the shape plus growth variation summarized by this analysis, it useful
to contrast specimens from opposite ends
of the ordination. Specimens with negative
loadings on xPC1 (testudinids; Fig. 11a) have
relatively narrow pectoral scutes, relatively
small gular and anal scutes, and abdominal
scutes with substantial anterior-, posterior-,
and midline-directed growth vectors. Taxa
with positive loadings on xPC1 (emydids
and batugurines; Fig. 11c) have relatively
larger anal and gular scutes, and their abdominal scute growth pattern primarily involves new material being deposited anteriorly and towards the midline. More subtle
differences include variations in the growth
pattern of the anal scutes (accretion on anterior and midline surfaces of anal scute
relatively equal in testudinids vs. anterior
surface greater in emydids/batagurids) and
pectoral scutes.
The causal and mechanistic bases of the
differences in growth processes between
emydid, batagurid, and testudinid taxa are
unknown. Differences in scute growth patterns reect differential activity in the germinal epithelium of the growing scutes, but
the factors that control this aspect of development remain unexplored. The strong asymmetry of scute growth patterns for emydids
and batagurids relative to that for testudinids may be related to systemic differences in growth and shell shape (testudinid
shells are highly domed), habitat (testudinids
are terrestrial, whereas most of the emydids and batagurids considered in this analysis are semiaquatic), or phylogenetic history
(a shared developmental character state).
Ontogenetic Trajectories
The analysis presented above incorporates
information about the shapes of organisms
at two distinct ontogenetic stages (t0 and t1 )
and includes data describing the growth processes operating over the ontogenetic transition. Because at least two developmental
stages are considered, the results may be
interpreted as ontogenetic trajectories, even
though any particular trajectory is represented by a single point in a k-dimensional
complex multivariate space, the ontogenetic
space. A more sensitive analysis that considered changes in shape and growth process
over several such transitions (e.g., to ! t1 !
t2 ! t3 ), could be used to produce the more
familiar representation of ontogenetic trajectories as lines or curves through ontogenetic
space.
D IS CUS SION
A thorough understanding of the interplay
between developmental process and morphological diversity requires more than a
consideration of pattern. A time-ordered sequence of forms is not a sufŽcient representation of ontogenetic transformation. We
must address the processes that produce the
patterns, not just the patterns themselves.
The study of ontogenetic trajectories requires
more than a geometry of form: Ultimately,
we must arrive at some understanding of the
biogeometry underlying the production of
morphologies (Rice, 1998).
In some cases, empirical evidence, theory,
and intuition may suggest appropriate models of growth by which the morphologies in
question can be produced (e.g., Rice, 1998;
Skalak et al., 1997). It is important to distinguish this approach, which I will refer to as
generative modeling, from the geometrical
modeling approach (Raup, 1966; Okamoto,
1988; Stone, 1995). Both generative and geometrical models can be used to produce illustrative Žgures that look something like the
morphologies of real organisms. The distinction is that the parameters of a generative
model are statements of biological hypothesis about underlying biological process; the
mathematical descriptors of the geometric
models are simply useful abstractions for describing shape (Rice, 1998).
It is no coincidence, however, that the majority of generative models have been formulated for geometrically simple structures,
ones that typically can be considered to be
composed of single parts. The growth of
structures such as mollusc shells (Ackerly,
1989; Rice, 1998) or antlers (Skalak et al.,
1997) seems straightforward enough that
one may posit reasonable models that describe their growth. When considering more
complex morphological structures, however,
one’s hopes of deriving biologically reasonable models on a priori grounds are
quickly shattered. Furthermore, publications
2001
MAGWENE—COMPARING GROWTH PROCESS ES
advocating the use of generative models typically fall short of actually measuring or estimating the pertinent model parameters from
actual specimens except to produce a few
demonstrative Žgures; when instructions for
estimating or measuring parameters are provided (e.g., Rice, 1998), one Žnds that the
process is not trivial. To my knowledge, in
no published reports have generative model
parameters been estimated on a sufŽcient
number of specimens to be able to explore
patterns of growth process variation within
populations.
The approach I advocate in this report represents something of a compromise. In characterizing growth Želds, one is attempting to
measure or quantify a process that may not
be fully understood. Although one might,
for example, know that growth occurs via
accretion on previous surfaces, or that cell
migrations are controlled by concentrations
of a particular protein, such knowledge falls
short of allowing one to make explicit predictions about the long-term behavior of complex systems made up of multiple subunits.
By concentrating on the aspects of growth
processes that can be measured, one hopes
the analysis will suggest something about the
rules by which complex morphological structures are produced.
A second distinction of the approach I
advocate is that it is explicitly statistical in
emphasis. By deŽning growth processes in
such a way that observations may be summarized via a covariance or correlation matrix, new avenues for exploration and analysis are made available. Additionally, because
the approach is methodologically and philosophically consistent with landmark morphometric techniques for analyzing form,
growth processes may be simultaneously
studied together with parameters describing shape and size. In doing so, we can
come to consider a richer formulation of ontogenetic trajectories (Alberch et al., 1979),
one that includes an axis describing growth
processes along with the standard considerations of shape, size, and age. Analysis
by way of xPCA, as described above, is
but one of a myriad of approaches that
might be utilized to study patterns of growth
variation. One potentially fruitful avenue
for further exploration would be to use
growth process data to test hypotheses about
the modularity of development (Zelditch
et al., 1992; Wagner and Altenberg, 1996;
653
Magwene, in press), by analyzing covariances among growth vectors with techniques
such as latent-variable or graphical modeling
(Whittaker, 1990; Loehlin, 1992).
The growth processes characterized above
are measured on individual specimens
(although specimen values were pooled to
create species means for the purpose of statistical analysis). In this respect the approach
I advocate is distinct from previous landmark morphometric analyses of ontogenetic
transformations such as those of Zelditch
et al. (1992, 2000) and Walker (1993). Figures in those papers depict ontogenetic transformations as sets of vector displacements.
However, the analysis in each of these cases
is primarily one of shape variation rather
than growth process variation, and these vectors should not be interpreted as growth
vectors. In contrast, the present analysis includes information about growth process
variation in addition to information about
shape variation.
Because the growth vectors are properties
of individuals, they may serve as “traits” in
other types of biological analyses. One might
estimate heritabilities of growth vectors or
use correlations among a large sample of
growth vectors to estimate genetic variance–
covariance matrices for developmental parameters (Atchley, 1987; Atchley and Hall,
1991). Such analyses would signify a signiŽcant step in attempts to integrate information
about developmental processes into a general evolutionary framework (Atchley, 1987).
Capturing Information About
Growth Processes
The hypothetical examples and the analysis of turtle plastron growth presented above
are based on structures in which information about size and shape at successive
ontogenetic stages is recorded in the tissues
themselves and hence some abstraction of
the growth process is directly observable.
Although interesting and illustrative, examples such as these may represent a small fraction of biological structures of interest. In organisms and structures for which a record
of growth process is not so conveniently
recorded, several experimental alternatives
are available to obtain information about
growth. “Vital” dyes such as alizarin or uoresceins (e.g., tetracycline, calcein) may be
administered at various ontogenetic stages
654
S YSTEMATIC BIOLOGY
to study growth of hard tissue in vertebrates
(Frazier, 1985; Klevezal, 1996) and other
metazoans (Märkel, 1981). These substances
are deposited in the mineral phase of many
biological hard tissues, and use of periodic
doses may aid the study of both long- and
short-term patterns of growth. Other possible approaches include labeling individual
pieces of tissue (Avery, 1933; Erickson, 1966)
or individual cells (Schoenwolf and Sheard,
1989; Siegert et al., 1994) and recording the
positions of those structures at successive ontogenetic stages. Depending on the number
of points tracked and the difŽculty in identifying points of correspondence from stage
to stage, such undertakings may be quite
labor intensive. However, advances in automated imaging and image analysis (e.g.,
Siegert et al., 1994) make such approaches
increasingly useful.
Other Aspects of Growth Processes
The characterization of tissue or cell movements by way of growth Želds is but one
aspect of the processes that can be considered in an analysis of ontogeny. Another
aspect one might wish to consider is the timing of initiation and cessation of developmental events (Gould, 1977; Alberch et al.,
1979; Atchley, 1987). One would expect that
the turning on and off of growth Želds at
distinct times in development is as important a mechanism for morphological differentiation as the growth Želds are themselves.
Such patterns were not considered in this report, but recognizing their potential importance further emphasizes the inadequacy of
a simple characterization of size and shape
changes.
Conclusions
Here I have pointed out some of the problems with popular approaches to the study
of ontogeny and suggest an approach that
addresses some of these concerns. The characterization of growth processes by growth
Želds, and their representation by complex
numbers, facilitates analysis by providing a
common framework for studying shape and
growth variation.
For the turtle taxa represented in this
study, patterns of shape variation appear
to have a relatively strong taxonomic signal. Growth vector variation, on the other
hand, less clearly reects standard taxo-
VOL.
50
nomic groupings. Whether this discordance
between shape variability and growth variability is general, or simply a function of the
specimens considered in this study, remains
to be seen.
Quantitative studies of ontogeny must encompass more than size and shape data if
they are intended to address questions about
the underlying process rather than simply
cataloging time-ordered series of organismal
form. Studying growth processes is more difŽcult and more time consuming than characterizing form but ultimately provides richer
interpretations and a better understanding of
ontogenetic transformations.
ACKNOWLEDGMENTS
I thank M. LaBarbera, B. Chernoff, and S. Rice for
their useful discussions and critiques of the methods
discussed in this paper. H. Voris and A. Resetar, Field
Museum of Natural History, facilitate d access to specimens and equipment. P. Wagner and J. Walker provided
extensive commentary and feedback as reviewers. This
research was funded in part by a National Science Foundation Doctoral Dissertation Improvement Grant (M.
LaBarbera, Principal Investigator), and by a Ford Foundation Dissertation Fellowship for Minorities.
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Received 3 August 2000; accepted 7 February 2001
Associate Editor: M. Westneat