Zelditch M.L. (и др.) Geometric Morphometrics for Biologists: a primer

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338 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

1.0

2.0

3.0

4.0

5.0

log(Y )

4.5

3.5

2.5

1.5

2.5 3.5 4.5 5.5

log(X )

0.5

1.0

1.5

2.0

2.5

3.0

3.5

2.5 3.5 4.5 5.5

Figure 13.7 Channeling when growth and development are decoupled, depicted by a bivariate

plot of log(Y)onlog(X). Two plots are required to depict such cases, such as that of neoteny

(a decrease in developmental rate) because positively and negatively allometric coefficients are dif-

ferently modified. The figured case is an example of neoteny; compared to the ancestor (circles),

the descendant (squares) has allometric coefficients that are closer to isometry – the descen-

dant’s adult proportions more closely resemble those of the shared juvenile morphology. Positively

allometric coefficients decrease (towards 1.0) whereas negatively allometric coefficients increase

(towards 1.0).

developmental stage, i.e. {Y

, Y

,…Y

} at that stage. That comparison is made by

predicting {Y

, Y

,…Y

} at the appropriate values of X for each species, using species-

specific regression equations. The residuals from each regression are then added to the

species-specific mean, and the two samples are compared. That is precisely what we did

when size-standardizing geometric data in Chapter 10.

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THE RELATIONSHIP BETWEEN ONTOGENY AND PHYLOGENY 339

Shape

Size

Age

Figure 13.8 Gould’s (1977) clock model. The descendant’s size, shape and age at one developmental

stage are compared to the ancestor’s ontogeny of shape, size and age. The hands of the clock show

the change from ancestral to descendant values, pointing from the descendant’s age-specific shape

and size to the corresponding ancestral values (no hand is shown if there is no change). The shape

hand points to the left, so the descendant adult has the morphology of a younger stage in the ancestral

ontogeny.

When channeling is found and age data are available (or collected), it is possible to go

further and identify the changes in developmental rate and/or timing. Both Gould (1977)

and Alberch et al. (1979) provide analytic formalisms for that purpose, and it is worth

understanding them because they are so widely used in the literature.

Formalisms for heterochrony

There are two formalisms for the study of heterochrony: Gould’s (1977) “clock-model”

and Alberch et al’s. (1979) scheme. Although the clock model is rarely used in the modern

literature, understanding it is important because it supplied the context for Alberch et al.’s

scheme. Alberch et al. retained the meaning of the concepts and terms defined by Gould

(excepting those they explicitly redefined).

The face of the clock contains two arcs and one bar (Figure 13.8). One arc is a shape

axis. The values of the ancestral shape are plotted along the arc, with the values for the

youngest age on the left. The second arc is the size axis; values of the ancestral size are

plotted on this axis so that the size and shape for each age match up. Age is represented by

the bar at the bottom. Although the entire ontogeny of the ancestral shape is represented

on the clock, the descendant is analyzed at a single (static) stage. Not surprisingly, the need

to single out one stage for comparison prompted much discussion about the appropriate

standard for comparison. That standard could be a chronological age, a developmental

age, or even a size. Whatever standard is used, the objective is to find the matching ancestral

size and shape at that point. When found, the hands of the clock are arranged to point

to it; if the matching shape occurs at an earlier stage in the ancestor, the “shape hand”

of the clock will point to the left. Similarly, if the matching size occurs at an earlier stage

in the ancestor, the “size hand” also points to the left. Differences between ancestor and

descendant in chronological age at the developmentally comparable stages are indicated

by cross-hatching on the age bar.

The clock provides diagnostic tests for types of heterochrony, which Gould defines

both verbally and in terms of the patterns revealed by the clock model. For example,

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340 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

Shape (σ)

Size

(

)

⫺δk

Age (A)

Figure 13.9 Alberch et al. formalism. The clock is redrawn by representing the ancestral shape,

size and age as three mutually orthogonal axes. Species are compared with respect to the age at

onset of development (α), rate of development (k

), rate of growth (k

), and age at termination of

development (β). See Table 13.4 for the names of the heterochronic perturbations defined by changes

in these three parameters.

neoteny is retardation in the development of shape, and it is evident from the shape hand

pointing to the left. However, Gould’s classification was soon replaced by the one devised

by Alberch et al., and the two schemes are not completely consistent. Thus, we will detail

the types of heterochrony based on the Alberch et al. scheme.

Alberch et al. (1979) redesigned Gould’s formalism, using a more conventional rep-

resentation of a three-dimensional space: three mutually orthogonal axes (Figure 13.9).

They also replaced Gould’s static comparative framework by a dynamic one; the descen-

dant ontogeny (not just one point along it) is analyzed in conjunction with the ancestral

ontogeny. Each ontogeny is represented as a vector in the three-dimensional space defined

by the ancestral values of size and shape. The comparisons are made with respect to four

parameters: (1) α, the age at the onset of development; (2) β, the age at offset of devel-

opment, (3) k

, the rate of development (i.e. the rate of change in shape); and (4) k

the

rate of growth (i.e. the rate of change in size). Each parameter can differ in two directions,

yielding the eight pure heterochronic perturbations (Table 13.4). Two of them, propor-

tional giantism and dwarfism, are not usually considered to be heterochronic perturbations

because they do not yield either paedomorphic or peramorphic descendants, but they are

usually included for the sake of completeness. Of course, combinations of these pure cases

are also possible. However, if we found a combination of +k

and −k

, for example, we

would not construct a compound name from the labels for each one – that would result

in “proportional giantism plus neoteny” when, by definition, a proportional giant is a

giant replica of the ancestral morphology, and neoteny necessarily signifies a difference

in shape.

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THE RELATIONSHIP BETWEEN ONTOGENY AND PHYLOGENY 341

Table 13.4 Definitions of the eight pure heterochronic perturbations and

their morphological expression, as defined by Alberch et al. (1979)

Control Incremental Process Morphological

parameter change expression

α −δα Predisplacement Peramorphosis

+δα Postdisplacement Paedomorphosis

β −δβ Progenesis Paedomorphosis

+δβ Hypermorphosis Peramorphosis

−δk

Neoteny Paedomorphosis

+δk

Acceleration Peramorphosis

−δk

Proportional giantism

+δk

Proportional dwarfism

Changes in ontogenetic trajectories

Changes in ontogenetic trajectories produce novel morphologies, not just descendants

that resemble ancestors at an older or younger developmental stage. Although all involve

changes in the spatiotemporal organization of development, they produce different rela-

tionships between ontogeny and phylogeny as well as different ontogenetic patterns in

disparity.

Changes confined to early morphogenesis

When species diverge very early in development they will differ in proportions at the outset

of allometric growth, but they subsequently follow the same ontogeny. Consequently, the

descendant allometric vector parallels the ancestral one but starts at a different shape

(Figure 13.10A). This pattern is often termed “transpositional allometry” because, as is

evident in Figure 13.10A, the descendant ontogeny is merely translated up or down the

Y-axis (when reading the picture, it is important to recall that it is a log–log plot). The

evolutionary direction of change is the same whether we look at juveniles or adults (Figure

13.10B). For example, if the descendant ends larval development with a head twice as

long for its body as the ancestor’s, it will be twice as long for its body (compared to an

ancestor at the same size) throughout the whole of ontogeny. Neither the magnitude nor

the structure of disparity changes over ontogeny.

Transpositional allometry means that early development is more labile than later. It

also means that divergence occurs very early – the differences are evident at X =1 (after

which the species follow the same ontogeny).

Empirical criteria for documenting changes confined to early morphogenesis

Detecting this pattern is relatively straightforward: the expectation is that species do not dif-

fer in {k

, k

,…k

} but do in {b

, b

,…b

}, which is easily tested by MANCOVA.

Because the differences are in {b

, b

,…b

}, which is the Y-intercept, the divergence

in shape is manifest when X =1, which is when log(X) =0. That all change must occur

prior to that point is why transpositional allometry indicates a divergence in, and solely

in, very early development.

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342 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

log(X )

log(Y )log(Y )

(A)

(B)

Figure 13.10 Changes confined to early morphogenesis, depicted by a bivariate plot of log(Y)on

log(X). Species differ in proportions early in development (shown by differences in the Y-intercept)

but subsequently follow the same allometric vector. A plot of a single measurement suffices to show

the general case, even though some might not differ in the Y-intercept. No measurements differ in

slope. (A) Ontogenetic allometries of ancestor (circles) and descendant (squares); (B) comparison of

directions of ontogenetic change (O), and evolutionary change (E), to divergence of shape during

larval (i.e. early) morphogenesis (L).

Changes in the spatiotemporal organization of development from the outset

to the end of allometric growth

If species differ in their ontogenetic allometries from the outset of larval growth, they

undergo different ontogenetic transformations in shape. Whether they diverge progres-

sively as they develop, or instead converge, is not specified by the hypothesis; it simply

claims that the difference results from a particular kind of change in development. To

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THE RELATIONSHIP BETWEEN ONTOGENY AND PHYLOGENY 343

log (head length)

log (head depth)

log (body length)

Figure 13.11 Changes in the spatiotemporal organization of development from the outset to end of

allometric growth, depicted by two bivariate plots of log(Y) on log(X). Both the ancestor (circles) and

the descendant (squares) begin development with the same shape, but undergo different changes in

proportion. To show this pattern we need at least two plots because it involves a change in the ratios

among allometric coefficients. Directions of ontogenetic change and evolutionary change cannot be

compared in either two-dimensional space; such a comparison requires a multidimensional space.

consider a simple example, we can use head length: head depth proportions in two species

(Figure 13. 11). The ancestral head is positively allometric in both length and depth, and

the two allometric coefficients are nearly equal, so head shape is isometric (but the head

enlarges relative to the body). The descendant head, like the ancestral head, is positively

allometric in length, but depth is more nearly isometric. Therefore head depth is nega-

tively allometric relative to head length, even if it is isometric relative to body length. Over

ontogeny, head length increases relative to body length, and also relative to head depth.

This hypothesis cannot be depicted by a single bivariate relationship because it concerns

a change in the ratios of two or more k’s, so we need at least two bivariate plots to represent

it graphically. Accordingly, we cannot draw the directions of ontogenetic and evolutionary

change on these plots. There are two different directions of ontogenetic change, and the

direction of evolutionary change will vary over ontogeny. Moreover, the hypothesis does

not explicitly state whether species resemble each other at the outset of the measured phase,

or at the outset of allometric growth (i.e. at log(X) =0) or at a later developmental stage

(meaning that the regression lines would intersect on the plot). To draw the plots, we have

incorporated an assumption not required by the hypothesis – that species resemble each

other when we first observe them.

Empirical criteria for documenting changes in spatiotemporal organization

of development from the outset to the end of allometric growth

The hypothesis predicts that species will differ in {k

, k

,…k

} and, like the hypothesis

of transpositional allometry, this is easily tested by MANCOVA. However, channeling can

also produce changes in k, so we need to document a change in the direction of {k

, k

,…k

}, not just a change in overall rate of development. To test that hypothesis, we can

show that the interspecific angle between vectors of allometric coefficients is no larger than

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344 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

(A)

(B)

log (head length)

L/P log (body length)

log (body length)

log (head length)

L/P

Figure 13.12 Changes in the spatiotemporal organization of development, confined to late stages of

development. The hypothesis predicts (A): ancestor (indicated by circles) and descendant (indicated

by squares) follow the same ontogeny up to the transition from one stage to the next (indicated

as a transition from larval (L) to postlarval (P) phases), after which they diverge. The similarity

in proportions at the transition point could, however, have a different explanation, shown by (B):

Ancestor (indicated by circles) and descendant (indicated by squares) follow divergent ontogenies

that intersect at the transitional point. The diagrams represent the two cases for a single measurement,

but the same explanation would have to hold for all measurements.

anticipated by chance (it is important to test them and not just measure them, because very

small angles are expected even when ontogenies differ considerably). We can also show

that species do not differ in developmental rate/timing, which means documenting that

neither extension nor truncation occurs. Both alter the length of the ontogenetic vector,

so we can compare those lengths (which are estimated by the square root of the summed

squared allometric coefficients).

Changes in spatiotemporal organization of development, confined to

late development

This hypothesis differs from the one described above because the modifications are

specific to late stages of ontogeny. Accordingly, species initially resemble each other and

diverge progressively as they grow. Figure 13.12 depicts such a case. This hypothesis tac-

itly assumes that the allometric model of constant relative growth rates does not hold for

the whole of ontogeny, because it claims that species share a common larval allometric

pattern but diverge later. This hypothesis thus predicts that species coincide in their regres-

sion lines during early development and later diverge, when the regression line for at least

one of the species changes its slope (Figure 13.12A). The alternative explanation, which

is not consistent with the biological hypothesis, is that both regression lines are straight

and extend all the way back to the Y-intercept, and happen to intersect at the youngest

observed age (Figure 13.12B).

This hypothesis of conserved early development and divergent later development is

consistent with von Baer’s second law, so we might anticipate that the pattern is common.

Although progressive divergence could occur by other kinds of modifications of ontogeny

as well (see below), it appears that progressive divergence is not common – or at least it is

rarely reported.

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THE RELATIONSHIP BETWEEN ONTOGENY AND PHYLOGENY 345

Empirical criteria for documenting changes in the spatiotemporal

organization of development, confined to late development

The hypothesis predicts that species will differ in {k

, k

, ...k

} but will not differ

in shape at the outset of the measured developmental phase. That is not equivalent to

predicting that they do not differ in {b

, b

,…b

}, because b is the value for Y when

X =1, which is much smaller than a larval length. Instead, the hypothesis predicts that

species do not differ in proportions at the earliest stage observed (or at the youngest stage

relevant to the hypothesis); their divergence begins after that point. The first step in testing

the hypothesis is the same as discussed above: documenting interspecific differences in

, k

,…k

}, which can be done by MANCOVA and by showing that the interspecific

angles between the vectors are larger than anticipated by chance. The second step is to show

that species are very similar in shape early in development (and diverge as they grow). This

can be done by estimating the proportions expected for each species at that stage, which is

done by predicting {Y

, Y

, ...Y

} for each. Given that we find differences in {k

, k

,…k

}, we need to base these predictions on the regression models fitted separately to

each species. We can then assess whether the expected values differ significantly, which is

done by adding the residuals from the regression model to the expected value (as calculated

for each species) and then comparing the expected shapes between species statistically. We

can also compare the lengths of the ontogenetic vectors (as described above), which are

predicted not to differ.

Complex changes in multiple parameters and stages

Having considered several simple cases, we can begin to explore the more interesting

combinations of two or more modifications. We will consider four possibilities:

1. That both early and late morphogenesis are modified but developmental rate/timing

is not (Figure 13.13)

2. That one phase is modified in both morphogenesis and developmental rate, whereas

the other is not modified in either (Figure 13.14)

3. That early morphogenesis is modified but later development is modified solely in

developmental rate/timing (Figure 13.15)

4. That both stages are modified in morphogenesis and developmental rate/timing is also

altered (Figure 13.16).

Although all these cases are similar in that multiple developmental parameters differ,

and also all predict a complex relationship between ontogeny and phylogeny; they differ

considerably in their biological implications. The first implies that morphogenesis is more

labile than developmental rate/timing – rate/timing is conserved although morphogenesis

evolves. The next two imply that one developmental stage is more labile than the other

and that it is labile in both morphogenesis and developmental rate/timing. The fourth

implies that development is highly labile in general – everything that can change does. The

most interesting consequences of these complex modifications, aside from what they tell us

about the lability of development, are their potential impacts on disparity. The interactions

among the multiple novelties may result in greater disparity than expected from the impact

of each one, taken separately, or in less disparity than expected from a single modification.

Interactions among multiple novelties might either amplify or counterbalance each other.

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346 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

log (head length)

log (body length)

log (head depth)

log (body length)

Figure 13.13 A complex change involving a modification of early morphogenesis combined with a

change in the spatiotemporal organization of later development. The ancestral ontogeny is indicated

by circles, descendent ontogeny by squares. Like the case shown in Figure 13.11, this one cannot be

depicted by a single bivariate plot because it involves changes in the ratios of allometric coefficients.

log (head length)

log (body length)

log (head depth)

log (body length)

Figure 13.14 A complex change involving both a modification in the spatiotemporal organization

of development late in ontogeny and a modification of developmental rate/timing. The ancestral

ontogeny is indicated by circles, the descendent ontogeny by squares. The change in late devel-

opment is evident in the translation of the descendant’s ontogeny along the Y-axis; the change in

developmental rate/timing is due to an increase in adult body size. To show a change in develop-

mental rate without concomitant change in body size, two bivariate plots would be needed to depict

the contrasting changes in positively and negatively allometric coefficients. Like all cases involving

a modification of late morphogenesis, and hence a change in direction, at least two bivariate plots

are needed to depict the pattern.

Empirical criteria for documenting changes in multiple parameters and stages

To identify the parameters that differ, we need to compare: (1) shapes at the youngest

comparable stage; (2) vectors of allometric coefficients; and (3) lengths of the ontogenetic

vectors. To determine the impact of these changes on disparity we need to measure dispar-

ity at two or more stages, and also, ideally, to measure the impact of each modification

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THE RELATIONSHIP BETWEEN ONTOGENY AND PHYLOGENY 347

log(X )

log(Y ) log(Y )

(B)

(A)

Figure 13.15 A complex change involving a modification of early morphogenesis and a change in

developmental rate/timing. The ancestral ontogeny is indicated by circles, the descendent ontogeny

by squares. As in Figure 13.13, the change in developmental rate/timing is due to an increase in

adult body size. To show a change in developmental rate without concomitant change in body

size, additional bivariate plots would be needed to depict the contrasting changes in positively and

negatively allometric coefficients. Like all cases involving a modification of spatiotemporal patterns

of development, at least two bivariate plots are needed to depict the pattern.

separately. In addition we can determine if ontogenies diverge over time, remain at a con-

stant distance apart, or converge towards a similar endpoint by combining the ontogenetic

series of multiple species and analyzing them by PCA.

Applying these criteria to an empirical case: comparing ontogenies of

S. gouldingi and S. manueli

To conclude our discussion of comparing ontogenetic allometries based on traditional

morphometric data, we will compare the ontogenetic allometries of two sister species –

S. gouldingi and S. manueli. The allometric coefficients b and k are given in Table 13.5, and

the ks are plotted on the measurements to give a better appreciation of where the species