Geometric estimates of heritability in biological shape

The recently developed geometric morphometrics methods represent an important contribution of statistics and geometry to the study of biological shapes. We propose simple protocols using shape distances that incorporate geometric techniques into linear quantitative genetic models that should provide insights into the contribution of genetics to shape variation in organisms. The geometric approaches use Procrustes distances in a curved shape space and distances in tangent spaces within and among families to estimate shape heritability. We illustrate the protocols with an example of wing shape variation in the honeybee, Apis mellifera. The heritability of overall shape variation was small, but some localized components depicting shape changes on distal wing regions showed medium to large heritabilities. The genetic variance-covariance matrix of the geometric shape variables was significantly correlated with the phenotypic shape variance-covariance matrix. A comparison of the results of geometric methods with the traditional multivariate analysis of interlandmark distances indicated that even with a larger dimensionality, the interlandmark distances were not as rich in shape information as the landmark coordinates. Quantitative genetics studies of shape should greatly benefit from the application of geometric methods.     

Vesicle-like biomechanics governs important aspects of nuclear geometry in fission yeast

It has long been known that during the closed mitosis of many unicellular eukaryotes, including the fission yeast (Schizosaccharomyces pombe), the nuclear envelope remains intact while the nucleus undergoes a remarkable sequence of shape transformations driven by elongation of an intranuclear mitotic spindle whose ends are capped by spindle pole bodies embedded in the nuclear envelope. However, the mechanical basis of these normal cell cycle transformations, and abnormal nuclear shapes caused by intranuclear elongation of microtubules lacking spindle pole bodies, remain unknown. Although there are models describing the shapes of lipid vesicles deformed by elongation of microtubule bundles, there are no models describing normal or abnormal shape changes in the nucleus. We describe here a novel biophysical model of interphase nuclear geometry in fission yeast that accounts for critical aspects of the mechanics of the fission yeast nucleus, including the biophysical properties of lipid bilayers, forces exerted on the nuclear envelope by elongating microtubules, and access to a lipid reservoir, essential for the large increase in nuclear surface area during the cell cycle. We present experimental confirmation of the novel and non-trivial geometries predicted by our model, which has no free parameters. We also use the model to provide insight into the mechanical basis of previously described defects in nuclear division, including abnormal nuclear shapes and loss of nuclear envelope integrity. The model predicts that (i) despite differences in structure and composition, fission yeast nuclei and vesicles with fluid lipid bilayers have common mechanical properties; (ii) the S. pombe nucleus is not lined with any structure with shear resistance, comparable to the nuclear lamina of higher eukaryotes. We validate the model and its predictions by analyzing wild type cells in which ned1 gene overexpression causes elongation of an intranuclear microtubule bundle that deforms the nucleus of interphase cells.     

Principles and methods of geometric morphometrics

The basic concepts, notions and methods of geometric morphometrics (GM) are considered. This approach implies multivariate analysis of landmark coordinates located following certain rules on the surface of a morphological object. The aim of GM is to reveal differences between morphological objects by their shapes as such, the "size factor" being excluded. The GM is based on the concept of Kendall's space (KS) defined as a hypersphere with points distributed on its surface. These points are the shapes defined as aligned landmark configurations. KS is a non-Euclidian space, its metrics called Procrustes is defined by landmark configuration of a reference shape relative to which other shapes are aligned and compared. The differences among shapes are measured as Procrustes distances between respective points. For the linear methods of multivariate statistics to be applied to comparison of shapes, the respective points are projected onto the tangent plane (tangent space), the tangent point being defined by the reference. There are two principal methods of shape comparisons in GM: the Procrustes superimposition (a version of the least squares analysis) and thin-plate spline analysis. In the first case, Procrustes residuals are the outcome shape variables which remain after isometric alignment of the shapes being compared. Their summation over all landmarks yields Procrustes distances among these shapes. The Procrustes distances can be used in multivariate analyses just as the Euclidian distances. In the second case, the shapes are fitted to the references by stretching/compressing and shearing until complete identity of their landmark configurations. Eigenvectors of resulting bending energy matrix are defined as new shape variables, principal warps which yield another shape space with the origin defined by the reference. Projections of the shapes being compared onto principal warps yield partial warps, and their covariance matrix decomposition into eigenvectors yields relative warps which are similar to principal components (in particular, they are mutually orthogonal). Both partial and relative warps can be used in many multivariate statistic analyses as quantitative shape variables. Results of thin-plate spline analysis can be represented graphically by transformation grid which displays type, amount and localization of the shape differences. Basis rules of sample composition and landmark positioning to be used in GM are considered. At present, rigid (with minimal degrees of freedom) 2D morphological objects are most suitable for GM applications. It is important to recognize three type of real landmarks, and additionally semi-landmarks and "virtual" landmarks. Some procedures of thin-plate spline analysis are considered exemplified by some study cases, as well as applications of some standard multivariate methods to GM results. They make it possible to evaluate correlation between different shapes, as well as between a shape and some non-shape variables (linear measurements etc); to evaluate the differences among organisms by shape of a morphological structure; to identify landmarks which most accounted for both correlation and differences between the shapes. An annotated list of most popular softwares for GM is provided.     

Shape disparity of bovid (Mammalia,Artiodactyla) horn sheaths and horn cores allows discrimination by species in 3D geometric morphometric analyses

The bony cranial structures of even‐toed hoofed mammals are important for understanding ecology and behavior of ruminants. Horns, the cranial appendages of the family Bovidae, are covered in a layer of keratin that is often not preserved in the fossil record; however, this keratin sheath is intimately involved in the processes that influence horn shape evolution. To understand the relationship between these two components of horns, we quantified both core and sheath shape for four extant species using three‐dimensional geometric morphometric analyses in separate, core‐ and sheath‐specific morphospaces as well as a combined morphospace. We assessed correlations between the horn and sheath morphospaces using two‐block partial least squares regression, a Mantel test of pairwise distances between species, and Procrustes ANOVA. We measured disparity in the combined morphospace as Procrustes distances between mean shapes of cores and sheaths within and between species and as Procrustes variance. We also tested whether core and sheath shapes could be discriminated by taxon with a canonical variate analysis. Results show that horn core and sheath morphospaces are strongly correlated. The differences in shape between a species' core and sheath were statistically significant, but not as great as those between the cores and sheaths of different species when close relatives were not considered, and core and sheath Procrustes variances are not significantly different within species. Cores and sheath shapes were highly identifiable and were assigned to the correct clade 93% of the time in the canonical variate analysis. Based on these tests, horn cores are distinguishable in geometric morphometric analyses, extending the possibility of using geometric morphometrics to study the ecology and evolution of bovid horns to the fossil record.     

Cross-talk between Rho and Rac GTPases drives deterministic exploration of cellular shape space and morphological heterogeneity

One goal of cell biology is to understand how cells adopt different shapes in response to varying environmental and cellular conditions. Achieving a comprehensive understanding of the relationship between cell shape and environment requires a systems-level understanding of the signalling networks that respond to external cues and regulate the cytoskeleton. Classical biochemical and genetic approaches have identified thousands of individual components that contribute to cell shape, but it remains difficult to predict how cell shape is generated by the activity of these components using bottom-up approaches because of the complex nature of their interactions in space and time. Here, we describe the regulation of cellular shape by signalling systems using a top-down approach. We first exploit the shape diversity generated by systematic RNAi screening and comprehensively define the shape space a migratory cell explores. We suggest a simple Boolean model involving the activation of Rac and Rho GTPases in two compartments to explain the basis for all cell shapes in the dataset. Critically, we also generate a probabilistic graphical model to show how cells explore this space in a deterministic, rather than a stochastic, fashion. We validate the predictions made by our model using live-cell imaging. Our work explains how cross-talk between Rho and Rac can generate different cell shapes, and thus morphological heterogeneity, in genetically identical populations.     

Two-sample tests for comparing intra-individual genetic sequence diversity between populations

Consider a study of two groups of individuals infected with a population of a genetically related heterogeneous mixture of viruses, and multiple viral sequences are sampled from each person. Based on estimates of genetic distances between pairs of aligned viral sequences within individuals, we develop four new tests to compare intra-individual genetic sequence diversity between the two groups. This problem is complicated by two levels of dependency in the data structure: (i) Within an individual, any pairwise distances that share a common sequence are positively correlated; and (ii) for any two pairings of individuals which share a person, the two differences in intra-individual distances between the paired individuals are positively correlated. The first proposed test is based on the difference in mean intra-individual pairwise distances pooled over all individuals in each group, standardized by a variance estimate that corrects for the correlation structure using U-statistic theory. The second procedure is a nonparametric rank-based analog of the first test, and the third test contrasts the set of subject-specific average intra-individual pairwise distances between the groups. These tests are very easy to use and solve correlation problem (i). The fourth procedure is based on a linear combination of all possible U-statistics calculated on independent, identically distributed sequence subdatasets, over the two levels (i) and (ii) of dependencies in the data, and is more complicated than the other tests but can be more powerful. Although the proposed methods are empirical and do not fully utilize knowledge from population genetics, the tests reflect biology through the evolutionary models used to derive the pairwise sequence distances. The new tests are evaluated theoretically and in a simulation study, and are applied to a dataset of 200 HIV sequences sampled from 21 children.     

The Impact of Tooth Wear on Occlusal Shape and the Identification of Fossils of New World Porcupines (Rodentia: Erethizontidae)

Although isolated dental remains are a central source of data in mammalian paleontology, the teeth of many taxa remain unidentifiable at low taxonomic levels because of intraspecific variation. Extant New World porcupines, Erethizontinae, illustrate this problem. Since being declared unusually variable over 50 years ago, molar occlusal morphology has been neglected as a potential source of diagnostic traits. Here, we use geometric morphometrics to re-evaluate the occlusal morphology of the third lower molar of erethizontines as the basis for identification of genera while accounting for variation, particularly changes in shape caused by progressive tooth wear. We model the impact and predictability of this source of variation by using CT-based digital sectioning. We use modelled occlusal shapes to analytically search for traits that can be used to improve identification of isolated dental remains, as well as training machine-learning models to assign occlusal surfaces to genus. Nearly half of all shape variation can be explained by variables such as wear and size. Resulting models of shape predict six univariate traits that may diagnose the two extant genera, especially when specimens are compared at the same wear stage. However, high levels of variation previously noted qualitatively also influence quantitative analyses, rendering machine learning models unreliable. Overall, although the identity of some specimens that are highly worn or contain certain combinations of traits will remain ambiguous, large segments of the gnathic erethizontine fossil record have renewed utility when analyzed using these methods.

     

Developmental plasticity, morphological variation and evolvability: a multilevel analysis of morphometric integration in the shape of compound leaves

The structure of compound leaves provides flexibility for morphological change by variation in the shapes, sizes and arrangement of leaflets. Here, we conduct a multilevel analysis of shape variation in compound leaves to explore the developmental plasticity and evolutionary potential that are the basis of diversification in leaf shape. We use the methods of geometric morphometrics to study the shapes of individual leaflets and whole leaves in 20 taxa of Potentilla (sensu lato). A newly developed test based on the bootstrap approach suggests that uncertainty in the molecular phylogeny precludes firm conclusions whether there is a phylogenetic signal in the data on leaf shape. For variation among taxa, variation within taxa, as well as fluctuating asymmetry, there is evidence of strong morphological integration. The patterns of variation are similar across all three levels, suggesting that integration within taxa may act as a constraint on evolutionary change.     

Using heterogeneous data to identify signatures of dengue outbreaks at fine spatio-temporal scales across Brazil

Dengue virus remains a significant public health challenge in Brazil, and seasonal preparation efforts are hindered by variable intra- and interseasonal dynamics. Here, we present a framework for characterizing weekly dengue activity at the Brazilian mesoregion level from 2010–2016 as time series properties that are relevant to forecasting efforts, focusing on outbreak shape, seasonal timing, and pairwise correlations in magnitude and onset. In addition, we use a combination of 18 satellite remote sensing imagery, weather, clinical, mobility, and census data streams and regression methods to identify a parsimonious set of covariates that explain each time series property. The models explained 54% of the variation in outbreak shape, 38% of seasonal onset, 34% of pairwise correlation in outbreak timing, and 11% of pairwise correlation in outbreak magnitude. Regions that have experienced longer periods of drought sensitivity, as captured by the “normalized burn ratio,” experienced less intense outbreaks, while regions with regular fluctuations in relative humidity had less regular seasonal outbreaks. Both the pairwise correlations in outbreak timing and outbreak trend between mesoresgions were best predicted by distance. Our analysis also revealed the presence of distinct geographic clusters where dengue properties tend to be spatially correlated. Forecasting models aimed at predicting the dynamics of dengue activity need to identify the most salient variables capable of contributing to accurate predictions. Our findings show that successful models may need to leverage distinct variables in different locations and be catered to a specific task, such as predicting outbreak magnitude or timing characteristics, to be useful. This advocates in favor of “adaptive models” rather than “one-size-fits-all” models. The results of this study can be applied to improving spatial hierarchical or target-focused forecasting models of dengue activity across Brazil.     

Signals in Stochastically Generated Neurons

To incorporate variation of neuron shape in neural models, we developed a method of generating a population of realistically shaped neurons. Parameters that characterize a neuron include soma diameters, distances to branch points, fiber diameters, and overall dendritic tree shape and size. Experimentally measured distributions provide a means of treating these morphological parameters as stochastic variables in an algorithm for production of neurons. Stochastically generated neurons shapes were used in a model of hippocampal dentate gyrus granule cells. A large part of the variation of whole neuron input resistance RN is due to variation in shape. Membrane resistivity Rm computed from RN varies accordingly. Statistics of responses to synaptic activation were computed for different dendritic shapes. Magnitude of response variation depended on synapse location, measurement site, and attribute of response.     

Computational studies of the extraction of visual spatial information from binocular and motion cues

This paper reviews some of the contributions that work in computational vision has made to the study of biological vision systems. We concentrate on two areas where there has been strong interaction between computational and experimental studies: the use of binocular stereo to recover the distances to surfaces in space, and the recovery of the three-dimensional shape of objects from relative motion in the image. With regard to stereo, we consider models proposed for solving the stereo correspondence problem, focussing on the way in which physical properties of the world constrain possible methods of solution. We also show how critical observations regarding human stereo vision have helped to shape these models. With regard to the recovery of structure from motion, we focus on how the constraint of object rigidity has been used in computational models of this process.     

ESPRIT-Tree: hierarchical clustering analysis of millions of 16S rRNA pyrosequences in quasilinear computational time

Taxonomy-independent analysis plays an essential role in microbial community analysis. Hierarchical clustering is one of the most widely employed approaches to finding operational taxonomic units, the basis for many downstream analyses. Most existing algorithms have quadratic space and computational complexities, and thus can be used only for small or medium-scale problems. We propose a new online learning-based algorithm that simultaneously addresses the space and computational issues of prior work. The basic idea is to partition a sequence space into a set of subspaces using a partition tree constructed using a pseudometric, then recursively refine a clustering structure in these subspaces. The technique relies on new methods for fast closest-pair searching and efficient dynamic insertion and deletion of tree nodes. To avoid exhaustive computation of pairwise distances between clusters, we represent each cluster of sequences as a probabilistic sequence, and define a set of operations to align these probabilistic sequences and compute genetic distances between them. We present analyses of space and computational complexity, and demonstrate the effectiveness of our new algorithm using a human gut microbiota data set with over one million sequences. The new algorithm exhibits a quasilinear time and space complexity comparable to greedy heuristic clustering algorithms, while achieving a similar accuracy to the standard hierarchical clustering algorithm.     

Factors affecting the errors in the estimation of evolutionary distances between sequences

Phylogenetic methods that use matrices of pairwise distances between sequences (e.g., neighbor joining) will only give accurate results when the initial estimates of the pairwise distances are accurate. For many different models of sequence evolution, analytical formulae are known that give estimates of the distance between two sequences as a function of the observed numbers of substitutions of various classes. These are often of a form that we call "log transform formulae". Errors in these distance estimates become larger as the time t since divergence of the two sequences increases. For long times, the log transform formulae can sometimes give divergent distance estimates when applied to finite sequences. We show that these errors become significant when t approximately 1/2 |lambda(max)|(-1) logN, where lambda(max) is the eigenvalue of the substitution rate matrix with the largest absolute value and N is the sequence length. Various likelihood-based methods have been proposed to estimate the values of parameters in rate matrices. If rate matrix parameters are known with reasonable accuracy, it is possible to use the maximum likelihood method to estimate evolutionary distances while keeping the rate parameters fixed. We show that errors in distances estimated in this way only become significant when t approximately 1/2 |lambda(1)|(-1) logN, where lambda(1) is the eigenvalue of the substitution rate matrix with the smallest nonzero absolute value. The accuracy of likelihood-based distance estimates is therefore much higher than those based on log transform formulae, particularly in cases where there is a large range of timescales involved in the rate matrix (e.g., when the ratio of transition to transversion rates is large). We discuss several practical ways of estimating the rate matrix parameters before distance calculation and hence of increasing the accuracy of distance estimates.     

Modeling and Inferring Cleavage Patterns in Proliferating Epithelia

The regulation of cleavage plane orientation is one of the key mechanisms driving epithelial morphogenesis. Still, many aspects of the relationship between local cleavage patterns and tissue-level properties remain poorly understood. Here we develop a topological model that simulates the dynamics of a 2D proliferating epithelium from generation to generation, enabling the exploration of a wide variety of biologically plausible cleavage patterns. We investigate a spectrum of models that incorporate the spatial impact of neighboring cells and the temporal influence of parent cells on the choice of cleavage plane. Our findings show that cleavage patterns generate “signature” equilibrium distributions of polygonal cell shapes. These signatures enable the inference of local cleavage parameters such as neighbor impact, maternal influence, and division symmetry from global observations of the distribution of cell shape. Applying these insights to the proliferating epithelia of five diverse organisms, we find that strong division symmetry and moderate neighbor/maternal influence are required to reproduce the predominance of hexagonal cells and low variability in cell shape seen empirically. Furthermore, we present two distinct cleavage pattern models, one stochastic and one deterministic, that can reproduce the empirical distribution of cell shapes. Although the proliferating epithelia of the five diverse organisms show a highly conserved cell shape distribution, there are multiple plausible cleavage patterns that can generate this distribution, and experimental evidence suggests that indeed plants and fruitflies use distinct division mechanisms.     

Simple Rho GTPase Dynamics Generate a Complex Regulatory Landscape Associated with Cell Shape

Migratory cells exhibit a variety of morphologically distinct responses to their environments that manifest in their cell shape. Some protrude uniformly to increase substrate contacts, others are broadly contractile, some polarize to facilitate migration, and yet others exhibit mixtures of these responses. Prior studies have identified a discrete collection of shapes that the majority of cells display and demonstrated that activity levels of the cytoskeletal regulators Rac1 and RhoA GTPase regulate those shapes. Here, we use computational modeling to assess whether known GTPase dynamics can give rise to a sufficient diversity of spatial signaling states to explain the observed shapes. Results show that the combination of autoactivation and mutually antagonistic cross talk between GTPases, along with the conservative membrane binding, generates a wide array of distinct homogeneous and polarized regulatory phenotypes that arise for fixed model parameters. From a theoretical perspective, these results demonstrate that simple GTPase dynamics can generate complex multistability in which six distinct stable steady states (three homogeneous and three polarized) coexist for a fixed set of parameters, each of which naturally maps to an observed morphology. From a biological perspective, although we do not explicitly model the cytoskeleton or resulting cell morphologies, these results, along with prior literature linking GTPase activity to cell morphology, support the hypothesis that GTPase signaling dynamics can generate the broad morphological characteristics observed in many migratory cell populations. Further, the observed diversity may be the result of cells populating a complex morphological landscape generated by GTPase regulation rather than being the result of intrinsic cell-cell variation. These results demonstrate that Rho GTPases may have a central role in regulating the broad characteristics of cell shape (e.g., expansive, contractile, polarized, etc.) and that shape heterogeneity may be (at least partly) a reflection of the rich signaling dynamics regulating the cytoskeleton rather than intrinsic cell heterogeneity.     

The taxonomic implications of cranial shape variation in Homo erectus

The taxonomic status of Homo erectus sensu lato has been a source of debate since the early 1980s, when a series of publications suggested that the early African fossils may represent a separate species, H. ergaster. To gain further resolution regarding this debate, 3D geometric morphometric data were used to quantify overall shape variation in the cranial vault within H. erectus using a new metric, the sum of squared pairwise Procrustes distances (SSD). Bootstrapping methods were used to compare the H. erectus SSD to a broad range of human and nonhuman primate samples in order to ascertain whether variation in H. erectus most clearly resembles that seen in one or more species. The reference taxa included relevant phylogenetic, ecological, and temporal analogs including humans, apes, and both extant and extinct papionin monkeys. The mean cranial shapes of different temporogeographic subsets of H. erectus fossils were then tested for significance using exact randomization tests and compared to the distances between regional groups of modern humans and subspecies/species of the ape and papionin monkey taxa. To gauge the influence of sexual dimorphism on levels of variation, comparisons were also made between the mean cranial shapes of single-sex samples for the reference taxa. Results indicate that variation in H. erectus is most comparable to single species of papionin monkeys and the genus Pan, which included two species. However, H. erectus encompasses a limited range of variation given its extensive geographic and temporal range, leading to the conclusion that only one species should be recognized. In addition, there are significant differences between the African/Georgian and Asian H. erectus samples, but not between H. ergaster (Georgia+Africa, excluding OH 9 and Daka) and H. erectus sensu stricto. This finding is in line with expectations for intraspecific variation in a long-lived species with a wide, but probably discontinuous, geographic distribution.     

The adaptive dynamics of the evolution of host resistance to indirectly transmitted microparasites

We use adaptive dynamics and pairwise invadability plots to examine the evolutionary dynamics of host resistance to microparasitic infection transmitted indirectly via free stages. We investigate trade-offs between pathogen transmission rate and intrinsic growth rate. Adaptive dynamics distinguishes various evolutionary outcomes associated with repellors, attractors or branching points. We find criteria corresponding to these and demonstrate that a major factor deciding the evolutionary outcome is whether trade-offs are acceleratingly or deceleratingly costly. We compare and contrast two models and show how the differences between them lead to different evolutionary outcomes.     

Shape variation in protein binding pockets and their ligands

A common assumption about the shape of protein binding pockets is that they are related to the shape of the small ligand molecules that can bind there. But to what extent is that assumption true? Here we use a recently developed shape matching method to compare the shapes of protein binding pockets to the shapes of their ligands. We find that pockets binding the same ligand show greater variation in their shapes than can be accounted for by the conformational variability of the ligand. This suggests that geometrical complementarity in general is not sufficient to drive molecular recognition. Nevertheless, we show when considering only shape and size that a significant proportion of the recognition power of a binding pocket for its ligand resides in its shape. Additionally, we observe a "buffer zone" or a region of free space between the ligand and protein, which results in binding pockets being on average three times larger than the ligand that they bind.