TREE vol. 15, no. 6 June 2000 0169-5347/00/$ – see front matter © 2000 Elsevier Science Ltd. All rights reserved. PII: S0169-5347(99)01775-9 2 1 7
T
he shape of a structure, organ orwhole organism is the outcome of processes that operate at different scales and levels of organizational complexity. At the molecular and cellular scales, genes have been identified to code for extracellular proteins that establish sig-nalling between epithelial and mesenchy-mal cells, and also to code for nuclear proteins that act to control gene expres-sion in response to the molecular sig-nals1. Interactions between such gene
products and cell movement in the extra-cellular matrix are important mech-anisms in the dynamic process of mor-phogenesis in which development takes place2. Nevertheless, the realized
pheno-types revealed in the final shape of struc-tures, organs and organisms arise from the interfacing of, and complex mappings between, morphogenetic rules, ecologi-cal phenomena, and deterministic and stochastic evolutionary forces3,4. The
ob-served pattern of complex morphologies in the final shape of organisms is, as is the case with any complex system, the result of mechanisms that operate at dif-ferent scales from those on which the patterns are observed3,4. Because there
is no single natural scale at which mor-phogenetic phenomena should be studied, a perceptual bias is imposed per forceby the observer4. Ecologists and
evolution-ary biologists, as ‘observers’ of whole organisms, have long been fascinated by complex morphologies (e.g. mollusc shells and vertebrate skulls generated by the process of morphogenesis) and its interactions with ecological and evolu-tionary forces. Consequently, the de-scription and the measurement of shape has remained an enduring endeavour5–7.
The formalism to describe and to measure shape (Box 1), and to infer its causes – the discipline of morphometrics – has recently undergone a major shift in paradigm, which was reviewed in 1993 by Rohlf and Marcus8in TREE. As outlined
by these authors, morphometrics essen-tially deals with methodology for the stat-istical study of shape variation and the covariation of shape with intrinsic (mor-phogenetic) and extrinsic (ecological and evolutionary) causes. For example, fundamental questions asked include: the role of morphogenetic and historical determinants of the shape of single morphological structures, such as the mammalian scapula9; and the evolution
and diversification of whole body shape in stickleback fishes as a response to predation and environmental effects10.
Approaches to the geometric study of shape variation
The major development reported by Rohlf and Marcus8in 1993 was the
estab-lishment of a geometric framework for the description of shape as envisioned by Thompson11and formalized in the
path-breaking work of Kendall12, Bookstein13
and Goodall14. In the new formalism of
geometric morphometrics, the measure-ment of shape (Box 1) is derived from morphological landmarks (Box 1), which are points of correspondence on each object that match between and within populations15. During the development of
this area, two rather different geometric approaches to the statistical analysis of shape using landmark data have appeared. In the first approach, the distribution of superimposed landmark coordinates (Box 1) are used to study statistically the variation and covariation of shape with causal factors. This approach is based on the Procrustes distance, which is a shape distance (Box 1) defined as the square root of summed squared distances be-tween least-square superimposed land-marks. The Procrustes metric is the basis for Kendall’s shape space, which is a mani-fold16(Box 1). The analysis of shape
vari-ation within Kendall’s curved shape space requires special statistical methods that account for the space curvature15. A linear
approximation to Kendall’s shape space can be created by projecting each object into a tangent space (Box 1). The tangent space approximation is an important step for geometric morphometrics, because it allows the use of standard multivariate techniques, which are important for ana-lysing the structure of shape covariation in the data set16.
NEWS & COMMENT
Shape distances, shape spaces and the
comparison of morphometric methods
Box 1. Glossary
Interlandmark distance:euclidean distance measured between two landmarks within the same object. Landmark:Bookstein13defines a landmark as a point with a biological label and a geometric lo-cation. Dryden and Mardia15define it as a point of correspondence on each object that matches between and within populations. Although Bookstein’s definition is sufficient for biological stud-ies, Dryden and Mardia’s definition is more general and is even useful for the statistical analysis of shapes of nonbiological objects.
Landmark coordinates:cartesian rectangular coordinates that locate the position of a landmark relative to an axis system. The landmark coordinates can be defined also as single complex numbers. Manifold:a generalized curved surface with more than three dimensions, which looks like a linear euclidean space in localized regions. We can trace a parallel with the earth’s surface for compari-son. If we are measuring distances in a small neighbourhood we can use simple euclidean dis-tances, because we perceive the earth as a two-dimensional surface in our vicinity. However, if we are to measure distances over wider regions, we always have to take its geodesic curvature into consideration. The region occupied by a data set on a shape manifold is a function of the amount of variation in the landmark coordinates. For most biological applications, including variation in the shape of biological structures within taxonomical categories as high as a class, the shape variation is so small that the region within the shape manifold occupied by the object shapes is small enough for its curvature to be ignored.
Shape:the geometrical properties of an object that are invariant to effects of translation, scaling and rotation.
Shape distance:a dissimilarity measure, D, between two shapes, X and Y, that satisfies the following properties:
D(X,Y ) 5 D(Y,X )
D(X,Y ) . 0 if XÞY
D(X,Y ) 5 0 if X5Y
D(X,Y ) < D(X,Z ) 1D(Z,Y )
where Z is any other intermediate point in shape space. The last property relates to the triangle inequality, which states that for any set of three shapes, X, Y, and Z (XÞYÞZ ), it must be possible
to construct a triangle with sides D(X,Y ), D(X,Z ), and D(Z,Y ).
Shape space:the set of all possible shapes for given landmark configurations with the same num-ber of landmarks and dimensions in figure space. The shape space is highly dependent on the distances used to measure the amount of shape difference between objects.
2 1 8 TREE vol. 15, no. 6 June 2000
By contrast to the morphometric methods that are based on landmark coordinates, the second approach encom-passes methods based on interlandmark distances (Box 1), such as the eucli-dean distance matrix analysis17(EDMA-I),
size-corrected interlandmark distances18
(EDMA-II), the shape variables proposed by Rao and Suryawanshi19, and angles
from triangulations of landmarks20. These
methods use various distances to charac-terize shape differences, such as the
difference between the largest and the smallest ratio of corresponding interland-mark distances (EDMA-I procedure), and the simple euclidean distance using linear combinations of shape variables (Rao and Suryawanshi’s19 procedure). The
distribution of shapes in the spaces im-plied by such techniques have not been studied21, although most of these
meth-ods are based on the assumption that their shape variables are approximately multivariate normal.
The two different approaches (i.e. those based on landmark coordinates and those on interlandmark distances) have been applied to landmark data to address questions of evolutionary and ecological significance9,10,22. Nevertheless,
morpho-metricians have disagreed about which type of shape variable (i.e. coordinates, interlandmark distance ratios or angles) is most suitable for the statistical analysis of shape. So far, the statistical consequences of using either of the two approaches have not been evaluated thoroughly and little has been done to compare methods effectively on a common ground23.
The comparison of morphometric methods by shape spaces
Now, in two new papers, Rohlf21,24 has
devised a most ingenious and elegant simulation approach to compare mor-phometric methods by visualizing the spaces implied by each technique in each of the two approaches. Rohlf21 uses
tri-angles of landmarks, because for configu-rations with more than three points, the shape spaces have more than three dimensions and cannot be drawn. The comparative procedure conceived by Rohlf involves three important steps.
First, the space topologies are visual-ized by simulating a uniform sample from the distribution of all possible shapes. The shape space for the approach that uses Procrustes distances can be depicted using analytical techniques, although the interlandmark distance-based approach requires numerical procedures.
NEWS & COMMENT
Fig. 1.Shape spaces, implied by different metrics that are used by the different morphometric approaches to characterize shape differences, are displayed. (a) Kendall’s tangent space with tri-angles plotted at the positions corresponding to their shapes. The point of tangency with Kendall’s shape space (centre of plot) is set at the equilateral triangle for convenience when com-paring with the shape spaces based on interlandmark distances. (b) Shape space based on Rao and Suryawanshi19variables. The equilateral triangle is at the centre and other triangular shapes are located in the arms, depending on which edge is relatively shorter.
(a) (b)
Trends in Ecology & Evolution
Fig. 2.Simulation of Gaussian error around each landmark, for a triangular shape showing the effect of the restriction and boundaries of shape spaces in the scatter of random shapes. (a) A triangle is used as a mean to simulate digitizing error in shape space. (b) Kendall’s shape space showing the position of the mean shape (red circle) and the random triangles generated by simulation (green dots in the scatter) is shown. In this plot, the equilateral triangle would be located at the North Pole. (c) Shape space implied by euclidean distance matrix analysis (EDMA-I). The random noise around the mean triangle (red circle) is no longer circular, but elliptical. A principal components analysis performed in this space would show a major axis of variation, suggesting the presence of structure in the data, although there is only random error in the sample.
(c) (b)
(a)
TREE vol. 15, no. 6 June 2000 2 1 9
Second, a circular scatter (simulating Gaussian error) is generated around each landmark of a mean shape to assess the constraints on the statistical distribution of shapes imposed by the topology of the shape space itself under the simplest null model.
Third, the statistical power of differ-ent morphometric methods is compared for the simple case of testing for shape differences between two groups.
Uniform distributions in shape spaces
In the first step (the visualization of the shape spaces simulating uniform distri-bution of triangles), Rohlf21demonstrates
that Kendall’s shape space corresponds to the surface of a sphere, as already shown analytically by Kendall12. In the
case of triangles, the linear tangent space is a plane that approximates the sphere of Kendall’s space at the position of the mean shape (Fig. 1a). The shape spaces implied by EDMA-I, EDMA-II, and the shape variables defined by Rao and Suryawanshi19,20are Y-shaped, with arms
or a triangular shape (Fig. 1b). The shape space topology (Box 1) has strong influ-ences on the results of analyses per-formed within them, as revealed in the next two comparison steps.
Normal distributions in shape spaces
The second step21, which involves the
comparison of error dispersion, is accom-plished by simulating circular digitizing error in triangles around each landmark using Goodall’s14perturbation model. This
model corresponds to circular bivariate normally distributed scatter around each landmark in the mean shape (Fig. 2a).
When these simulated shapes are plot-ted in Kendall’s shape space and its tan-gent spaces, a circular scatter around the point corresponding to the mean shape is obtained. This is the expected result for these simulations (Fig. 2b). When these same shapes are plotted with respect to the shape spaces implied by the methods based on interlandmark distances, they showed elliptical scatter around the mean shape (Fig. 2c). This is a serious problem for any multivariate analysis that might use results from techniques using the interlandmark distances, because the shape space they span is constrained if the mean triangle is not equilateral (Fig. 1b).
For biological applications, there will usually be more than three landmarks in a plane or more than four landmarks in a space. In this case, it is virtually impos-sible for all the interlandmark distances to be equal, and the morphometric meth-ods based on interlandmark distances or angles will show a distorted distribu-tion of random shapes. An unfortunate consequence is that variance maximizing
analyses, such as principal components analysis, will show major trends of vari-ation where there is none – only inde-pendent random error around mean landmark positions.
Power comparisons
Continuing the sets of comparisons among morphometric methods, in the third step, Rohlf24performed power
analy-ses contrasting geometric methods based on Kendall’s shape space and its tangent approximations, and the interlandmark distance-based and angle-based meth-ods24. He found that statistically the most
powerful methods (i.e. those with greater capacity to detect true shape differences) are those based on Kendall’s shape space. The power difference between these methods and those based on interland-mark distance-based methods is often strikingly high. The EDMA methods pre-sented low power even for comparisons of different shapes and the type I error rates for EDMA methods were often not correct. The programs developed by Rohlf to visualize shape spaces and to calculate the power for morphometric methods, are available to download from the Internet at the morphometrics page at Stony Brook (http://life.bio.sunysb.edu/morph/).
Prospects
The simulation approach introduced by Rohlf21,24 to compare geometric
morpho-metric methods represents an important advance in the evaluation of the theoretical foundations of the different morphometric approaches. Rohlf’s insightful contribu-tions16,21,24demonstrate the superiority of
geometric morphometric methods based on the Procrustes metric over the approaches that use interlandmark dis-tances and angles of triangulations.
The benefits of the mathematical rigor of the work by Kendall and Bookstein are clear. It should be an important concern for anyone proposing new morphometric approaches to study the shape space that they use to investigate shape variation. Now, the challenge for ecologists and evo-lutionary biologists seems to lie in the incorporation of these rigorous methods of statistical description of shape and of shape change into models designed to search for the causes of diversity in shape, the constraints imposed upon such diversity, and the apportionment of the causes of diversity among exogenous and endogenous determinants.
Acknowledgements
The authors would like to thank F. James Rohlf for comments on this article. Work by L.R.M. and S.F.R. is supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (grants 99/0302-8 and 99/06845-3).
Leandro Rabello Monteiro
Laboratório de Ciências Ambientais – CBB, Universidade Estadual do Norte
Fluminense, Av. Alberto Lamego 2000, Horto, 28015-620 Campos dos Goytacazes, Rio de Janeiro, Brazil
Benjamin Bordin
Departamento de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970 Campinas, São Paulo, Brazil
Sérgio Furtado dos Reis Departamento de Parasitologia, Universidade Estadual de Campinas, Caixa Postal 6109, 13083-970 Campinas, São Paulo, Brazil
References
1 Tucker, A.S. and Sharpe, P.T. (1999) Molecular
genetics of tooth morphogenesis and patterning: the right shape in the right place. J. Dent. Res. 78, 826–834
2 Blelloch, R. and Kimble, J. (1999) Control of organ shape by a secreted metalloprotease in the nematode Caenorhabditis elegans. Nature 399, 586–590
3 Murray, J.D. (1990) Mathematical Biology, Springer-Verlag
4 Levin, S.A. (1992) The problem of pattern and scale in ecology. Ecology 73, 1943–1976
5 Raff, R.A. (1996) The Shape of Life: Genes,
Development, and the Evolution of Animal Form, Chicago University Press
6 Pennisi, E. (1999) From embryos and fossils, new clues to vertebrate evolution. Science 284, 575–576
7 Janvier, P. (1999) Major events in early
vertebrate evolution. Trends Ecol. Evol. 14, 298–299
8 Rohlf, F.J. and Marcus, L.F. (1993) A revolution
in morphometrics. Trends Ecol. Evol. 8, 129–132
9 Monteiro, L.R. and Abe, A.S. (1999) Functional and historical determinants of shape in the scapula of xenarthran mammals: evolution of a complex morphological structure. J. Morphol. 241, 251–263
10 Walker, J.A. (1997) Ecological morphology of
lacustrine threespine stickleback Gasterosteus aculeatus L. (Gasterosteidae) body shape. Biol. J. Linn. Soc. 61, 3–50
11 Thompson, D′A.W. (1917) On Growth and Form, Cambridge University Press
12 Kendall, D.G. (1984) Shape-manifolds, Procrustean metrics and complex projective spaces. Bull. London Math. Soc. 16, 81–121
13 Bookstein, F.L. (1991) Morphometric Tools for
Landmark Data: Geometry and Biology, Cambridge University Press
14 Goodall, C.R. (1991) Procrustes methods in the
statistical analysis of shape. J. R. Stat. Soc. 53, 285–339
15 Dryden, I.L. and Mardia, K.V. (1998) Statistical Shape Analysis, John Wiley & Sons
16 Rohlf, F.J. (1999) Shape statistics: Procrustes superimpositions and tangent spaces. J. Class. 16, 197–223
R
otifers are microscopic pseudocoelo-mates that have attracted scientific interest for many years because they are conspicuous in aquatic environments, swim slowly and can be easily cultured. Early on, many rotifer workers were ama-teurs, who were interested in cataloging different morphologies found in benthic and planktonic samples. As academic sci-entists became interested in rotifers, they focused on more general and less descriptive issues. Peculiarities of the group, such as eutely, parthenogenetic reproduction, heterogony and dormancy have been exploited in experimental investigations. Thus, rotifers have be-come useful biological models for testing ecological and evolutionary theories.Rotifer research has been signifi-cantly enhanced by a series of inter-national symposia where rotiferologists can meet, discuss, plan collaborations, and establish scientific and personal inter-actions. The themes of the latest meeting in the series*reflected the heterogeneity
of interests. However, because there were no concurrent sessions, all partici-pants were exposed to a variety of topics and different modes of interpretation. The topics ranged from evolution and phylogeny, to genetics, ecotoxicology, trophic interactions, zoogeography and biochemistry.
Rotifer evolution
Much attention focused on ancient asex-uality of bdelloid rotifers, which repre-sents an evolutionary challenge because bdelloids are obligate parthenogens. They seem to have escaped Muller’s ratchet and still retain considerable evolutionary potential. David and Jessica Mark-Welch
(Harvard University, Cambridge, MA, USA) presented work on molecular evo-lution in bdelloids. Intron structure of bdelloids is particularly informative and fluorescent in situ hybridization showed that, although diploid, bdelloids do not possess homologous chromosomes. This reinforces the hypothesis that recombin-ation has been suppressed for a long time and that bdelloids are indeed ancient asexuals.
The molecular phylogeny of rotifers was reviewed by David Mark-Welch, who warned against the use of a single species as a representative of entire taxa because some groups possess a high GC content biasing any comparisons. He concluded that Acanthocephala, long considered as a separate taxon, is a highly derived class of Rotifera, and that there is molecular support for joining the two classes Mono-gononta and Bdelloidea into a single monophyletic clade, Eurotatoria, in agree-ment with morphological cladistics. A pu-tative relative of rotifers was suggested by Martin Sørensen (Copenhagen University, Denmark), who compared the Gnamoth-ostomulida jaws with the rotifer trophi.
Rotifers inhabit environments with high temporal heterogeneity and diverse life cycles have evolved to cope with this, thus providing opportunities to under-stand the adaptive significance and consequences of dormancy. Claudia Ricci (Milan University, Italy) compared the two forms of dormancy in rotifers, dia-pause and quiescence, focusing on condi-tions that elicit dormancy. She concluded that these are alternative, mutually exclu-sive responses to disturbance. Bdelloids face unpredictable, fine-grained environ-ments and respond deterministically by entering a fast, short-lasting dormancy (anhydrobiosis). Monogononts produce a long-lasting dormant stage, in response to a more predictable coarse-grained
environment. Substantial stochasticity exists in the timing of the initiation of sex-ual reproduction and resting egg produc-tion. This was illustrated independently by Thomas Schroeder (Berlin Free Univer-sity, Germany), who examined the plank-tonic rotifers of the Oder River floodplains, and by Eduardo Aparici et al.(Valencia University, Spain), who studied clones iso-lated from a single rotifer population. Both showed that rotifers vary the start of the sexual phase and thus dormancy, fol-lowing bet-hedging strategies to cope with temporally unpredictable environments.
Dormant forms are viewed as a bank of biodiversity – genetic variability is introduced upon hatching. Tomonari Kotani et al.(Nagasaki University, Japan) hatched resting eggs of Brachionus rotundiformis, collected from sediment cores of different depths from a Japanese lake, and performed mating assays to test for reproductive isolation among the tem-porally separated populations. The oldest resting eggs recovered were 65-years-old and were still viable. Mating tests and binding assays, using an antibody for a mate-recognition pheromone, showed that the four temporally isolated popu-lations were reproductively compatible. Africa Gómez (University of Hull, UK) applied microsatellite analysis to the study of polymorphic loci in rotifers. She found evidence of discrepancies between clonal diversity in the resting egg bank and that of the early planktonic popu-lation of Brachionus plicatilis. Her results suggest that clonal diversity decreases during parthenogenetic reproduction, that mating produces a resting egg bank in Hardy–Weinberg equilibrium and that hatching is not synchronized among the different genotypes. In contrast with the extensive genetic variation at microsatel-lite markers, strong clonal selection was suggested by the theoretical analysis of Charles King (Oregon State University, Corvallis, USA), who had previously re-ported low allozyme variation among clones at the same sampling date.
Rotifer ecology
Some contributions tested the theory of ecological interactions at individual,
Small, beautiful and sexy: what rotifers
tell us about ecology and evolution
*The IX International Rotifer Symposium,
Khon Kaen, Thailand, 16–23 January 2000.
2 2 0 0169-5347/00/$ – see front matter © 2000 Elsevier Science Ltd. All rights reserved. PII: S0169-5347(00)01867-X TREE vol. 15, no. 6 June 2000
17 Lele, S. and Richtsmeier, J.T. (1991) Euclidean distance matrix analysis: a coordinate free approach for comparing biological shapes using landmark data. Am. J. Phys. Anthropol. 86, 415–427
18 Lele, S. and Cole, T.M. (1996) A new test for shape differences when variance–covariance matrices are unequal. J. Hum. Evol. 31, 193–212
19 Rao, C.R. and Suryawanshi, S. (1996) Statistical analysis of shape of objects based on landmark
data. Proc. Natl. Acad. Sci. U. S. A. 93, 12132–12136
20 Rao, C.R. and Suryawanshi, S. (1998) Statistical analysis of shape through triangulation of landmarks: a study of sexual dimorphism in hominids. Proc. Natl. Acad. Sci. U. S. A. 95, 4121–4125
21 Rohlf, F.J. (2000) On the use of shape spaces to compare morphometric methods. Hystrix Int. J. Mamm. 11, 8–24
22 Richtsmeier, J.T. et al. (1993) The role of postnatal growth in the production of facial morphology. Syst. Biol. 42, 307–330
23 Coward, W.M. and Conathy, D.M. (1996) A Monte Carlo study of the inferential properties of three methods of shape comparison. Am. J. Phys. Anthropol. 99, 369–377
