In Fall 2018, the CMSA will host a Program on Mathematical Biology, which aims to describe recent mathematical advances in using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems.
The plethora of natural shapes that surround us at every scale is both bewildering and astounding – from the electron micrograph of a polyhedral virus, to the branching pattern of a gnarled tree to the convolutions in the brain. Even at the human scale, the shapes seen in a garden at the scale of a pollen grain, a seed, a sapling, a root, a flower or leaf are so numerous that “it is enough to drive the sanest man mad,” wrote Darwin. Can we classify these shapes and understand their origins quantitatively?
In biology, there is growing interest in and ability to quantify growth and form in the context of the size and shape of bacteria and other protists, to understand how polymeric assemblies grow and shrink (in the cytoskeleton), and how cells divide, change size and shape, and move to organize tissues, change their topology and geometry, and link multiple scales and connect biochemical to mechanical aspects of these problems, all in a selfregulated setting.
To understand these questions, we need to describe shape (biomathematics), predict shape (biophysics), and design shape (bioengineering).
For example, in mathematics there are some beautiful links to Nash’s embedding theorem, connections to quasiconformal geometry, Ricci flows and geometric PDE, to Gromov’s h principle, to geometrical singularities and singular geometries, discrete and computational differential geometry, to stochastic geometry and shape characterization (a la Grenander, Mumford etc.). A nice question here is to use the large datasets (in 4D) and analyze them using ideas from statistical geometry (a la Taylor, Adler) to look for similarities and differences across species during development, and across evolution.
In physics, there are questions of generalizing classical theories to include activity, break the usual Galilean invariance, as well as isotropy, frame indifference, homogeneity, and create both agent (cell)based and continuum theories for ordered, active machines, linking statistical to continuum mechanics, and understanding the instabilities and patterns that arise. Active generalizations of liquid crystals, polar materials, polymers etc. are only just beginning to be explored and there are some nice physical analogs of biological growth/form that are yet to be studied.
The CMSA will be hosting a Workshop on Morphometrics, Morphogenesis and Mathematics from October 2224 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
The workshop is organized by L. Mahadevan (Harvard), O. Pourquie (Harvard), A. Srivastava (Florida).
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Time  Speaker  Title/Abstract 
8:30 – 9:00am  Breakfast  
9:00 – 9:15am  Welcome and Introduction  
Tutorial on Morphometrics  
9:15 – 10:30  Anuj Srivastava  Geometric methods for functional data analysis 
10:30 – 10:45  Coffee Break  
10:45 – 12:15  Sayan Mukherjee  Title: Integral Geometry for Modeling Shapes Abstract: We study transformations of shapes into representations that allow for analysis using standard statistical tools. The transformations are based on Euler integration and are of interest for their mathematical properties as well as their applications to science and engineering, because they provide a way of summarizing shapes in a topological, yet quantitative, way. By using an inversion theorem, we show that both transforms are injective on the space of shapes—each shape has a unique transform. By making use of a stratified space structure on the sphere, induced by hyperplane divisions, we prove additional uniqueness results in terms of distributions on the space of Euler curves. The main theoretical result provides the first (to our knowledge) finite bound required to specify any shape in certain uncountable families of shapes, bounded below by curvature. This result is perhaps best appreciated in terms of shattering number or the perspective that any point in these particular moduli spaces of shapes is indexed using a tree of finite depth. We also show how these transformations can be used in practice for medical imaging applications as well as for evolutionary morphology questions. 
12:15 – 1:30 pm  Lunch  
1:30 – 2:15pm  Jukka Jernvall  Title: Bridging the gap between patterning and function
Abstract: Mammalian dentition is an organ system in which the basic patterning mechanisms are starting to be understood. Patterning, however, is not enough to produce a fully mature and functional organ. Growth, and especially in the case of organs such as the teeth, extracellular matrix secretion and mineralization are required before a tooth can function. Compared to patterning these final steps of organ formation are much less understood. In this talk I will discuss how the developmental shape, the result of patterning, maps to the functional shape or the mineralized enamel surface of the tooth. I will show how mathematical models on matrix secretion provide new tools and concepts to the study of development and evolution, and link work done on experimental species such as the mouse to evolutionary studies on the human lineage. 
2:15 – 3:00pm  Arkhat Abzhanov  Title: Phylogenetic principles and morphogenetic mechanisms for evolvability and biological shape change
Abstract: Understanding the origins of morphological variation is one of the chief challenges to the modern biological sciences. Cranial diversity in vertebrates is a particularly inviting research topic as animal heads and faces show many dramatic and unique adaptive features which reflect their natural history. We aim to reveal molecular mechanisms underlying evolutionary processes that generate such morphological variation. To this purpose, we employ a synergistic combination of geometric morphometrics, comparative molecular embryology and functional experimentation methods to trace cranial evolution in reptiles, birds and mammals, some of the most charismatic animals on our planet. Our research is revealing how particular changes in developmental genetics can produce morphological alterations for natural selection to act upon, for example in generating adaptive radiations and morphological novelty. 
3:00 – 3:30pm  Coffee Break  
3:30 – 4:15pm  Karen Sears  
Time  Speaker  Title/Abstract 
8:30 – 9:00am  Breakfast  
Tutorial on Morphometrics (methods and applications)  
9:00 – 10:30 am  David Gu  Geometric flows for shape matching 
10:30 – 11:00am  Coffee Break  
11:00 – 11:45pm  Anjali Goswami  Title: From Development to Deep Time: Phenomic Analysis of Tetrapod Cranial Integration and Evolution
Abstract: Phenotypic integration is a pervasive characteristic of organisms. Interactions among morphological traits, termed phenotypic integration, can be readily identified through quantitative analysis of geometric morphometric data from living and extinct organisms. These interactions have been hypothesized to reflect genetic, developmental, and functional relationships and to be a fundamental influence on morphological evolution on small to large time scales. Simulations using covariance matrices derived from landmark data for diverse vertebrate taxa confirm that trait integration can influence the trajectory and magnitude of response to selection. At a macroevolutionary scale, high phenotypic integration produces both more and less disparate organisms, and most often the latter, than would be expected under unconstrained evolution, thereby increasing morphological range, but also homoplasy and convergence. However, this effect may not translate simply to evolutionary rates. While most largescale studies of phenotypic integration and morphological evolution utilise relatively limited descriptors of morphology, such as lengths or a small set of homologous landmarks, surface sliding semilandmark analysis allows for detailed quantification of complex 3D shapes, even across highly disparate taxa. We conducted the largest analysis to date of morphological evolution across diverse tetrapod clades using a dense dataset of landmarks and sliding semilandmarks spanning the entire cranium and nearly 300 million years of evolution. Crania are highly modular, but this pattern varies across tetrapods. Modules also have disparate magnitudes of trait integration, which reflect developmental complexity in some, but not all clades. Tempo and mode are similarly highly variable, with some modules, such as the basicranium of birds, showing early bursts of shape evolution, while other regions, such as the rostrum, show sustained change throughout clade evolution. Leveraging this high density morphometric data, we further demonstrate that variation is unequally distributed across the cranium and that distinct patterns of variation characterize different tetrapod clades. 
11:45 – 12:30  Siobhan Braybrook  Title: The evolution and development of epidermal cell shape in plants Abstract: The outer surfaces of leaves have been examined microscopically since microscopes were invented. The cell shapes presented by leaf epidermides can be diverse: from complex multicellular ‘hairs’ to flat tabular ‘pavement’ cells. We have recently begun to explore the diversity of pavement cell shape across the plant kingdom – with an eye towards understanding both how cell shapes are generated and what, if any, the function of shape might be. Using traditional morphometrics, in the first instance, we have found little phyllogenetic signal for cell shape in plant leaf epidermides. One of the interesting shape features in these cells is degree of margin undulation; solidity has been the best traditional morphometric to describe undulation thus far in our work. However, we are expanding our methods to include analysis of Total Absolute Curvature. Data examining periodic undulations in the maize leaf epidermis will be presented and explored. Further questions about cell shape in 3D will be presented as well. 
12:30 – 2:00pm  Lunch  
2:00 – 2:45pm  Eric Klassen  Title: Comparing Shapes of Curves, Surfaces, and Higher Dimensional Immersions in Euclidean Space
Abstract: Comparing shapes and treating them as data for statistical analyses has many applications in biology and elsewhere. Certain elastic metrics on spaces of immersions have proved very effective for comparing curves and surfaces. The elastic metrics which have proved most useful for computation have been first order metrics, i.e., they compare tangent vectors to the shapes rather than points on the shapes. In this talk I will present a unifying view of these metrics, shedding new light on old methods and, I hope, suggesting new methods for analyzing surfaces and higher dimensional shapes. 
2:45 – 3:30pm  Laurent Younes  Title: Estimation of Laminar Coordinate Systems between Two Surfaces. Abstract: Given two disjoint open surfaces, we discuss the problem of estimating a parametrization (x,y,z) > F(x,y,z) of an open subset of the threedimensional space such that the set [z=0] corresponds to one of the surfaces, [z=1] to the other one, and for each t, (x,y) > F(x,y,t) is an embedding onto the surface [z=t], with the additional constraint that the derivative of F with respect to t is always normal to this surface.The resulting construction is a natural way to build a foliation between the two surfaces for which the length of the transverse lines provides a good definition of thickness of the considered volume. It is designed with the objective of analyzing cortical volumes, for which a robust definition of thickness is essential for the characterization of degenerative diseases. Our estimation algorithm uses a version of the large deformation diffeomorphic metric mapping between surfaces in which normality constraints of the velocity field are enforced.The estimated coordinate system can also be used to estimate “cortical layers’’ as laminar segmentations of the intersurface region that satisfy a local equivolumetric condition called Bok’s hypothesis. Experimental results on human and feline brains will be presented as illustrations.This is joint work with Tilak Ratnanather, Sylvain Arguillere and Kwame Kutten. 
3:30 – 4:00pm  Coffee Break  
4:00 – 4:45pm  Cassandra Extavour  Title: Insect egg evolution defies predicted allometric relationships across eight orders of magnitude in size
Abstract: Multicellular organisms commonly reproduce via a singlecelled propagule, an egg. Egg size and shape relates to manyimportant developmental and life history processes, including maternal investment in reproduction, embryogenesis, progeny size, and fitness. Multiple mechanisms have been hypothesized to drive propagule size regulation and evolution. However, testing those hypotheses requires phenotypic shape data that varies over several orders of magnitude in size, from enough species to provide adequate statistical power, and with clear phylogenetic relationships allowing distinction between convergent and homologous sizes and shapes. To address these issues, we assembled a database of >10,000 morphological descriptions of eggs from over 6,000 unique species of the most speciose animal clade: the insects. This dataset represents, to our knowledge, the most diverse dataset of animal propagule sizes to date. We found that, contrary to published hypotheses, within insect lineages, egg size and shape have an isometric relationship. Furthermore, for any given adult size, egg volumes can range across four orders of magnitude. Larger eggs are generally more ellipsoidal, but allometric relationships vary significantly among lineages, and in Coleoptera (beetles) isometry is observed over seven orders of magnitude in egg size variation. In contrast to predicted allometric relationships between shape and size, egg shape for major lineages appears to have been determined early during insect evolution, and is conserved within lineages despite radical changes in size. We suggest that developmental parameters, including the relationship between egg size and developmental rate, will be critical in future models and analyses of propagule size evolution. Samuel H. Church*, Seth Donoughe*, Bruno de Medeiros and Cassandra G. Extavour * equal contribution

4:45 – 5:30pm  Nipam Patel  Title: The Cellular and Genetic Basis for Structural Color in Butterflies
Abstract: Research into the optical properties of structurally colored butterfly scales has been remarkably successful and revealed a great deal about the physical basis of color production in a myriad of species. Most of the studies are focused on adult scales, and often driven by the desire to eventually engineer these colorproducing nanostructures using nonbiological approaches. Our emphasis has been to understand how biological systems, in this case individual scale cells, are able to produce specific structures, particularly how these cells are able to pattern intracellular membranes and other cellular component to create the templates for the nanostructures. We have focused primarily on two systems. 
Time  Speaker  Title/Abstract 
8:30 – 9:00am  Breakfast  
9:00 – 9:45am  Peter Olver  Title: Reassembly of broken objects
Abstract: The problem of reassembling broken objects appears in a broad range of applications, including jigsaw puzzle assembly, archaeology (broken pots and statues), surgery (broken bones and reassembly of histological sections), paleontology (broken fossils and egg shells), and anthropology (more broken bones). I will discuss recent progress on such problems, based on advances in the mathematical apparatus of equivalence, invariants, and symmetry, equivariant moving frames, differential and integral invariant signatures, and invariant numerical approximations. 
9:45 – 10:30am  Stephanie Pierce  
10:30 – 11:00am  Coffee Break  
11:00 – 11:45am  Alain Trouve  Title: Modular approaches and intrinsic metrics in diffeomorphometry Abstract: The regular diffeomorphometric framework put the emphasis on induced Riemannian metrics on homogeneous spaces that are coming from right invariant metric on group of transformations. However, this extrinsic point of view on shape deformation costs (i.e. induced by extrinsic space deformation costs) can be challenged when prior information on objects properties is available. The very idea of deformation modules is to incorporate this prior knowledge into explicit shape dependent deformation constraints leading to a subriemannian point of view. However, since most of the prior knowledge is dealing with local properties of the infinitesimal strain tensor, this implies some interesting adaptation of the usual explicit framework leading to new intrinsic modules. Joint work with Barbara Gris 
11:45 – 12:45 pm  Olivier Pourquie & L. Mahadevan  Summary and Discussion 
12:45 – 2:00pm  Lunch 