Shape analysis of symmetric structures: quantifying variation among individuals and asymmetry

Abstract.— Morphometric studies often consider parts with internal left-right symmetry, for instance, the vertebrate skull. This type of symmetry is called object symmetry and is distinguished from matching symmetry, in which two separate structures exist as mirror images of each other, one on each body side. We explain a method for partitioning the total shape variation of landmark configurations with object symmetry into components of symmetric variation among individuals and asymmetry. This method is based on the Procrustes superimposition of the original and a reflected copy of each landmark configuration and is compatible with the two-factor ANOVA model customary in studies of fluctuating asymmetry. We show a fully multivariate framework for testing the effects in the two-factor model with MANOVA statistics, which also applies to shapes with matching symmetry. We apply the new methods in a small case study of pharyngeal jaws of the Neotropical cichlid fish Amphilophus citrinellus . The analysis revealed that the symmetric component of variation in the pharyngeal jaws is dominated by the contrast between two alternative trophic morphs in this species and that there is subtle but statistically significant directional asymmetry. Finally, we provide some general recommendations for morphometric studies of symmetric shapes.     

Beyond bilateral symmetry: geometric morphometric methods for any type of symmetry

Background

Studies of symmetric structures have made important contributions to evolutionary biology, for example, by using fluctuating asymmetry as a measure of developmental instability or for investigating the mechanisms of morphological integration. Most analyses of symmetry and asymmetry have focused on organisms or parts with bilateral symmetry. This is not the only type of symmetry in biological shapes, however, because a multitude of other types of symmetry exists in plants and animals. For instance, some organisms have two axes of reflection symmetry (biradial symmetry; e.g. many algae, corals and flowers) or rotational symmetry (e.g. sea urchins and many flowers). So far, there is no general method for the shape analysis of these types of symmetry.

Results

We generalize the morphometric methods currently used for the shape analysis of bilaterally symmetric objects so that they can be used for analyzing any type of symmetry. Our framework uses a mathematical definition of symmetry based on the theory of symmetry groups. This approach can be used to divide shape variation into a component of symmetric variation among individuals and one or more components of asymmetry. We illustrate this approach with data from a colonial coral that has ambiguous symmetry and thus can be analyzed in multiple ways. Our results demonstrate that asymmetric variation predominates in this dataset and that its amount depends on the type of symmetry considered in the analysis.

Conclusions

The framework for analyzing symmetry and asymmetry is suitable for studying structures with any type of symmetry in two or three dimensions. Studies of complex symmetries are promising for many contexts in evolutionary biology, such as fluctuating asymmetry, because these structures can potentially provide more information than structures with bilateral symmetry.     

Are geometric morphometric analyses replicable? Evaluating landmark measurement error and its impact on extant and fossil Microtus classification

Geometric morphometric analyses are frequently employed to quantify biological shape and shape variation. Despite the popularity of this technique, quantification of measurement error in geometric morphometric datasets and its impact on statistical results is seldom assessed in the literature. Here, we evaluate error on 2D landmark coordinate configurations of the lower first molar of five North American Microtus (vole) species. We acquired data from the same specimens several times to quantify error from four data acquisition sources: specimen presentation, imaging devices, interobserver variation, and intraobserver variation. We then evaluated the impact of those errors on linear discriminant analysis‐based classifications of the five species using recent specimens of known species affinity and fossil specimens of unknown species affinity. Results indicate that data acquisition error can be substantial, sometimes explaining >30% of the total variation among datasets. Comparisons of datasets digitized by different individuals exhibit the greatest discrepancies in landmark precision, and comparison of datasets photographed from different presentation angles yields the greatest discrepancies in species classification results. All error sources impact statistical classification to some extent. For example, no two landmark dataset replicates exhibit the same predicted group memberships of recent or fossil specimens. Our findings emphasize the need to mitigate error as much as possible during geometric morphometric data collection. Though the impact of measurement error on statistical fidelity is likely analysis‐specific, we recommend that all geometric morphometric studies standardize specimen imaging equipment, specimen presentations (if analyses are 2D), and landmark digitizers to reduce error and subsequent analytical misinterpretations.     

GEOMETRIC MORPHOMETRICS OF DEVELOPMENTAL INSTABILITY: ANALYZING PATTERNS OF FLUCTUATING ASYMMETRY WITH PROCRUSTES METHODS

Although fluctuating asymmetry has become popular as a measure of developmental instability, few studies have examined its developmental basis. We propose an approach to investigate the role of development for morphological asymmetry by means of morphometric methods. Our approach combines geometric morphometrics with the two-way ANOVA customary for conventional analyses of fluctuating asymmetry and can discover localized features of shape variation by examining the patterns of covariance among landmarks. This approach extends the notion of form used in studies of fluctuating asymmetry from collections of distances between morphological landmarks to an explicitly geometric concept of shape characterized by the configuration of landmarks. We demonstrate this approach with a study of asymmetry in the wings of tsetse flies (Glossina palpalis gambiensis). The analysis revealed significant fluctuating and directional asymmetry for shape as well as ample shape variation among individuals and between the offspring of young and old females. The morphological landmarks differed markedly in their degree of variability but multivariate patterns of landmark covariation identified by principal component analysis were generally similar between fluctuating asymmetry (within-individual variability) and variation among individuals. Therefore there is no evidence that special developmental processes control fluctuating asymmetry. We relate some of the morphometric patterns to processes known to be involved in the development of fly wings.     

Geometric morphometrics and leaf phenotypic plasticity: assessing fluctuating asymmetry and allometry in European white oaks (Quercus)

Because plants are unable to move away from unfavourable habitats and environmental perturbations, leaf phenotypic plasticity facilitates light absorption and gas exchange. Oaks (Quercus spp.) are particularly known for their adaptability and plastic phenotypes, and leaf allometry and developmental instability may represent important mechanisms for their adaptation to environments and evolution. Because of its important role in the adaptation of plant populations to different environments, allometry can be involved in diversifying selection. Developmental instability is related to environmental perturbations and stresses by producing random deviations in structures characterized by bilateral symmetry, such as oak leaves. In addition, developmental instability can also arise from genetic bottlenecks or as a result of hybridization. The splitting of symmetric and asymmetric components of variation and their separate analysis allows the variability in leaf shape traits to be summarized, reducing the variation produced by developmental instability. The geometric morphometric approach is a useful method for the study of leaf asymmetry and allometric patterns. This method provides an important tool for the visualization of shape attributes that characterize species with highly variable leaf phenotypic patterns. In this study, leaf shape and size variability of three white oak species was investigated by means of a two‐dimensional landmark‐based method providing improved knowledge of variance partitioning, species discrimination, fluctuating asymmetry and allometric patterns of variation resulting from the different analyses. © 2015 The Linnean Society of London, Botanical Journal of the Linnean Society, 2015, 179 , 335–348.     

Getting into Shape: An Empirical Comparison of Traditional Truss-Based Morphometric Methods with a Newer Geometric Method Applied to New World Cichlids

Body shape is a difficult, but important, trait to quantify. Researchers have traditionally used multivariate analysis of several linear measures ('trusses') across the body form to quantify shape. Newer geometric morphometric methods claim to better estimate shape because they analyze the geometry among the locations of all landmarks simultaneously rather than the linear distances between pairs of landmarks. We tested this claim by comparing the results of several traditional morphometric analyses against a newer geometric analysis involving thin-plate splines (TPS), all applied to a common data set of morphologically variable new world cichlids Amphilophus citrinellus and A. zaliosus. The TPS method yielded slightly stronger evidence of morphological differences among forms, although traditional methods also distinguished the two species. Perhaps our most important result was the idiosyncratic interpretation of shape variation among the traditional truss-based methods, whereas the generation of deformation grids using the TPS approach yielded clear and visually interpretable figures. Our results indicate that geometric morphometrics can be a more effective way to analyze and interpret body form, but also that traditional methods can be relied upon to provide statistical evidence of shape differences, although not necessarily accurate information about the nature of variation in shape.     

Geometric craniometric analysis of sexual dimorphism and ontogenetic variation: A case study based on two geographically disparate species, Aethomys ineptus from southern Africa and Arvicanthis niloticus from Sudan (Rodentia: Muridae)

Non-geographic morphometric variation, particularly at the level of sexual dimorphism and ontogenetic (age-related) variation, has been documented in rodents, and useful for establishing whether to analyse sexes separately or together, and for selecting adult specimens for subsequent data recording and analysis. However, such studies have largely been based on traditional morphometric analyses of linear measurements that mainly focus on overall size, rather than shape-related morphometric variation. Unit-free, landmark/outline-based geometric morphometric analyses are considered to offer a more appropriate tool for assessing shape-related morphometric variation. In this study, we used geometric cranial morphometric analysis to assess the nature and extent of sexual dimorphism and age variation within the Tete veld rat, Aethomys ineptus (Thomas and Wroughton, 1908) from southern Africa and the African Nile rat, Arvicanthis niloticus (Desmarest, 1822) from Sudan. The results obtained were in turn compared with previously published results based on independent geometric and traditional cranial morphometric data from the same sampled populations examined in the present study. While our geometric morphometric results detected statistically significant sexual dimorphism in cranial shape within Ar. niloticus only, previously published results based on traditional morphometric data failed to detect significant sexual dimorphism within this species. However, similar to previously published traditional morphometric data, our geometric morphometric results detected statistically significant age-related variation in cranial shape and size within both Ae. ineptus and Ar. niloticus, with individuals of age classes 5 and 6 being considered to represent adult specimens. Our results highlight the importance of carefully evaluating both size- and shape-related non-geographic morphometric variation prior to the analysis of geographic variation and the delineation of species. Erroneous conclusions of non-geographic variation may have implications in the interpretation of geographic and evolutionary processes that may be responsible for morphological differences at both the inter- and intra-specific levels.     

Gait asymmetry: composite scores for mechanical analyses of sprint running

Gait asymmetry analyses are beneficial from clinical, coaching and technology perspectives. Quantifying overall athlete asymmetry would be useful in allowing comparisons between participants, or between asymmetry and other factors, such as sprint running performance. The aim of this study was to develop composite kinematic and kinetic asymmetry scores to quantify athlete asymmetry during maximal speed sprint running. Eight male sprint trained athletes (age 22±5 years, mass 74.0±8.7 kg and stature 1.79±0.07 m) participated in this study. Synchronised sagittal plane kinematic and kinetic data were collected via a CODA motion analysis system, synchronised to two Kistler force plates. Bilateral, lower limb data were collected during the maximal velocity phase of sprint running (velocity=9.05±0.37 ms(-1)). Kinematic and kinetic composite asymmetry scores were developed using the previously established symmetry angle for discrete variables associated with successful sprint performance and comparisons of continuous joint power data. Unlike previous studies quantifying gait asymmetry, the scores incorporated intra-limb variability by excluding variables from the composite scores that did not display significantly larger (p<0.05) asymmetry than intra-limb variability. The variables that contributed to the composite scores and the magnitude of asymmetry observed for each measure varied on an individual participant basis. The new composite scores indicated the inter-participant differences that exist in asymmetry during sprint running and may serve to allow comparisons between overall athlete asymmetry with other important factors such as performance.     

Multiple 2D approaches to human sexual dimorphism of the distal end of femur

Various studies on the difference of distal femoral condyles between genders have been reported recently in order to provide anatomic information for knee prosthesis design and surgical planning in total knee arthroplasty.They also had the objective to be used as a sex recognition character,as may be needed in forensic medicine.Except for a recent 3D approach on the distal femur,most of the studies used dimensional information or aspect ratio but not shape.Our 2D study aimed to determine the size and shape variation of femoral condyles in Thais,considering age,sex and sides.One hundred and twenty-four cadaveric femurs (male 84 legs and female 40 legs) were dissected.The specimens were photographed by digital camera and images were analyzed using three geometric techniques:(i) the landmark-based method (5 landmarks),(ii) with or without addition of 23 sliding semilandmark and (iii) the outline-based methods.From the resulting geometric coordinates,size and shape were extracted for comparisons between genders and sides.Between sides,directional asymmetry could be detected only for shape variation,and only when introducing curves in the analyses (either through the semilandmarks technique or through the outline-based one).Non-directional asymmetry,probably fluctuating asymmetry,was detected for size,as well as for shape,in both genders.Sex discrimination was performed for each geometric technique using two classification methods:the Mahalanobis distance classification and the Maximum likelihood classification.The latter provided much more satisfactory gender validated reclassification (87％) than shape (72％).     

Assessment of rampant genitalic variation in the spider genus Homalonychus (Araneae, Homalonychidae)

Abstract. Animal genitalia are often complex and thought to vary little within species but differ between closely related species making them useful as primary characters in species diagnosis. Spiders are no exception, with nearly all of the 40,462 (at the time of this writing) described species differentiated by genitalic characteristics. However, in some cases, the genitalia of putative species are not uniform, but rather vary within species. When intraspecific variation overlaps interspecific variation, it can be difficult (if not impossible) to place a name on a specimen. The quantification of shape variation in genitalia has not often been attempted, probably because until recently it was not a methodologically and computationally simple process. In the two currently recognized species of the spider genus Homalonychus , genitalic variation is rampant in both male and female structures, with some parts of the genitalia (e.g., the retrolateral tibial apophysis) differing in each specimen examined. In this study, geometric morphometric analysis employing landmark data is used to quantify both intra- and interspecific variation in this genus. The large amount of variation is condensed into two or three groups depending on the structures examined, and these groups correspond to either the two species or to previously established mitochondrial DNA clades within one of the species. The results also show that analyses of female structures do not separate the groups as readily as the analyses of the male structures. The large amount of variation present in some structures is not correlated with geography or population genetic structure.     

Studying Developmental Variation with Geometric Morphometric Image Analysis (GMIA)

The ways in which embryo development can vary across individuals of a population determine how genetic variation translates into adult phenotypic variation. The study of developmental variation has been hampered by the lack of quantitative methods for the joint analysis of embryo shape and the spatial distribution of cellular activity within the developing embryo geometry. By drawing from the strength of geometric morphometrics and pixel/voxel-based image analysis, we present a new approach for the biometric analysis of two-dimensional and three-dimensional embryonic images. Well-differentiated structures are described in terms of their shape, whereas structures with diffuse boundaries, such as emerging cell condensations or molecular gradients, are described as spatial patterns of intensities. We applied this approach to microscopic images of the tail fins of larval and juvenile rainbow trout. Inter-individual variation of shape and cell density was found highly spatially structured across the tail fin and temporally dynamic throughout the investigated period.     

Detecting shape differences in species of the Littorina saxatilis complex by morphometric analysis

We investigated variation in shell size and shape of marinesnail species of the Littorina saxatilis complex (L. saxatilis,L. compressa and L. arcana) using geometric morphometric methods.These morphologically similar periwinkle species that are commonin the European intertidal have presented many problems fordiagnosis based on morphology alone. A discriminant analysisdemonstrated that geometric morphometrics is very efficientfor diagnosing individuals to species among sympatric populations.We successfully diagnosed an average of 96% of the specimens(with 85.7–100% correct diagnosis for specific comparisons).The diagnosis capability of this method is absolute at the populationlevel. This makes the technique potentially useful for the designof manipulative field or laboratory experiments. Moreover, ageometric-morphometric analysis was also accomplished in twosnail ecotypes (H and M) of L. saxatilis from rocky shores ofNE England which are apparently adapted to different degreesof wave exposure. We found that the H (exposed) ecotype hasa relatively rounded shell shape with a bigger aperture, whereasthe M (protected) ecotype has a smaller aperture. (Received 23 March 2006; accepted 25 January 2007)     

Extent of structural asymmetry in homodimeric proteins: prevalence and relevance

Most homodimeric proteins have symmetric structure. Although symmetry is known to confer structural and functional advantage, asymmetric organization is also observed. Using a non-redundant dataset of 223 high-resolution crystal structures of biologically relevant homodimers, we address questions on the prevalence and significance of asymmetry. We used two measures to quantify global and interface asymmetry, and assess the correlation of several molecular and structural parameters with asymmetry. We have identified rare cases (11/223) of biologically relevant homodimers with pronounced global asymmetry. Asymmetry serves as a means to bring about 2:1 binding between the homodimer and another molecule; it also enables cellular signalling arising from asymmetric macromolecular ligands such as DNA. Analysis of these cases reveals two possible mechanisms by which possible infinite array formation is prevented. In case of homodimers associating via non-topologically equivalent surfaces in their tertiary structures, ligand-dependent mechanisms are used. For stable dimers binding via large surfaces, ligand-dependent structural change regulates polymerisation/depolymerisation; for unstable dimers binding via smaller surfaces that are not evolutionarily well conserved, dimerisation occurs only in the presence of the ligand. In case of homodimers associating via interaction surfaces with parts of the surfaces topologically equivalent in the tertiary structures, steric hindrance serves as the preventive mechanism of infinite array. We also find that homodimers exhibiting grossly symmetric organization rarely exhibit either perfect local symmetry or high local asymmetry. Binding of small ligands at the interface does not cause any significant variation in interface asymmetry. However, identification of biologically relevant interface asymmetry in grossly symmetric homodimers is confounded by the presence of similar small magnitude changes caused due to artefacts of crystallisation. Our study provides new insights regarding accommodation of asymmetry in homodimers.     

Visualizing patterns of craniofacial shape variation in Homo sapiens

The geometric morphometric analysis of shape variation in complex biological structures such as the human skull poses a number of specific challenges: the registration of homologous morphologies, the treatment of bilateral symmetry, the graphical representation of form variability in three dimensions and the interpretation of the results in terms of differential growth processes. To visualize complex patterns of shape change, we propose an alternative to classical Cartesian deformation grids in the style of D'Arcy W. Thompson. Reference to the surface structures of the organism under investigation permits a comprehensive visual grasp of shape change and its tentative interpretation in terms of differential growth. The application of this method to the analysis of human craniofacial shape variation reveals distinct modes of growth and development of the neurocranial and viscerocranial regions of the skull. Our data further indicate that variations in the orientation of the viscerocranium relative to the neurocranium impinge on the shapes of the face and the cranial vault.     

Use of a geometric morphometric method to determine the developmental stability of <Emphasis Type="Italic">Betula pendula</Emphasis> Roth

A geometric morphometric method has been proposed to test the fluctuating asymmetry and the developmental stability of Betula pendula Roth populations. The main factors affecting the developmental stability were the industrial emission level, the relief altitude, and the interaction of both these factors. A strong correlation between the fluctuating asymmetry indices obtained by the normalizing difference method and geometric morphometric method has been revealed. The fluctuating asymmetry determined by the geometric morphometric method is sensitive to the presence of directional asymmetry, which makes it possible to use this method for precise bioindication mapping of the developmental stability.     

A new cryptic species of Aphaenops (Coleoptera: Carabidae: Trechinae) from a French Pyrenean cave: Congruence between morphometrical and geographical data confirm species isolation

A new species of cave-dwelling beetle from the pyrenean massif, Aphaenops (Cerbaphaenops) jauzioni n. sp. (Coleoptera: Carabidae), is described. Geometric morphometric analyses based on five morphological structures and comparisons with two populations of the closely related species Aphaenops (Cerbaphaenops) cerberus Dieck 1869 support the uniqueness of the new species. The specific status of the new taxon is also confirmed by the male genitalia structures. A putative scenario of population isolation is proposed.     

A complete survey of normal pores on a smooth shell ostracod (Crustacea): Landmark‐based versus outline geometric morphometrics

Pores and sensilla on ostracod shell have often been used in studies of ontogeny, taxonomy, and phylogeny of the group. However, an analysis of sexual dimorphism and variation between valves in the number and distribution of pores is lacking. Also, such studies have never been done on a widely distributed, morphologically variable, and weakly ornamented freshwater ostracod. Here, we survey pores in one such species, Physocypria kraepelini . We choose 27 homologous pores as landmarks for 2D‐geometric morphometric analysis, with the aim to assess intersexual and between valves variation in size and shape relative to the Fourier outline analysis. This species has only simple (Type A) pores with and without a lip, and each pore carries an undivided sensory seta. Our results show that the total number of pores varies (from 270 to 296), but this is not associated with a specific valve. Males carry fewer pores than females, however no sex specific pores are found. Small intrapopulation divergence of the Cyt b molecular marker (1%) indicates that morphological variability is not species related. We found that P. kraepelini exhibits directional asymmetry of size and shape, sexual size dimorphism (SSD) but lacks sexual shape dimorphism (SShD). Two geometric morphometrics methods were congruent in the estimation of SSD, SShD, and directional asymmetry of shape but differ in the statistical evaluation of directional asymmetry of size. Contrary to other animal groups, our study suggests that ostracods have more pronounced directional asymmetry of shape compared to directional asymmetry of size.     

Fused traditional and geometric morphometrics demonstrate pinniped whisker diversity

Vibrissae (whiskers) are important components of the mammalian tactile sensory system, and primarily function as detectors of vibrotactile information from the environment. Pinnipeds possess the largest vibrissae among mammals and their vibrissal hair shafts demonstrate a diversity of shapes. The vibrissae of most phocid seals exhibit a beaded morphology with repeating sequences of crests and troughs along their length. However, there are few detailed analyses of pinniped vibrissal morphology, and these are limited to a few species. Therefore, we comparatively characterized differences in vibrissal hair shaft morphologies among phocid species with a beaded profile, phocid species with a smooth profile, and otariids with a smooth profile using traditional and geometric morphometric methods. Traditional morphometric measurements (peak-to-peak distance, crest width, trough width and total length) were collected using digital photographs. Elliptic Fourier analysis (geometric morphometrics) was used to quantify the outlines of whole vibrissae. The traditional and geometric morphometric datasets were subsequently combined by mathematically scaling each to true rank, followed by a single eigendecomposition. Quadratic discriminant function analysis demonstrated that 79.3, 97.8 and 100% of individuals could be correctly classified to their species based on vibrissal shape variables in the traditional, geometric and combined morphometric analyses, respectively. Phocids with beaded vibrissae, phocids with smooth vibrissae, and otariids each occupied distinct morphospace in the geometric morphometric and combined data analyses. Otariids split into two groups in the geometric morphometric analysis and gray seals appeared intermediate between beaded- and smooth-whiskered species in the traditional and combined analyses. Vibrissal hair shafts modulate the transduction of environmental stimuli to the mechanoreceptors in the follicle-sinus complex (F-SC), which results in vibrotactile reception, but it is currently unclear how the diversity of shapes affects environmental signal modulation.