MS09 - MFBM-08

Mathematical methods for biological shape data analysis (Part 2)

Friday, July 18 at 3:50pm

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Note: this minisymposia has multiple sessions. The other session is: and Part-1.

Wenjun Zhao (UBC/Wake Forest University), Khanh Dao Duc (UBC)

Description:

Advances in imaging have for the past few years revolutionized our understanding of biological processes, illustrated by the recent mapping of trillions of human cells, or the explosive rate at which thousands of new protein 3D structures are now determined every year. This unprecedented surge of new biological data yields various mathematical problems and challenges at multiple spatial and time scales for interpreting and analyzing morphological shapes and shape dynamics, which finds close connections to machine learning, statistics, and physics. In this context, our proposed mini-symposium focuses on the mathematical, physical, and statistical aspects of shapes that describe the morphologies of biological structures—from molecules and intracellular organelles to cells, tissues, and organs—and how new modeling tools in this field can help elucidate fundamental biological questions. Intended for a diverse audience of mathematicians, physicists, and computational and experimental biologists, the symposium will feature complementary talks covering multiple aspects of imaging data analysis, ranging from the interplay between cell shapes and fate to new discoveries in protein dynamics through Riemannian geometry. To foster collaboration, share insights, and educate junior scientists, we have invited a diverse panel of participants and speakers from various institutes and career stages, including graduate students, postdocs, and faculty members.

Laurent Younes

JHU

"Aligning measures using large deformation diffeomorphic mapping for spatial transcriptomics"

Signed measures provide a powerful and flexible object representation for registration problems as they cover both continuous and singular objects and are naturally transformed by diffeomorphisms. They can, in particular, be used as tools describing functions taking values on arbitrary feature space, which makes them well adapted to the representation of spatial transcriptomic images. In this presentation, we will summarize the theoretical foundations of measure registration using large deformation diffeomorphic measure mapping, and provide applications to spatial transcriptomics, within and across modalities.

Luis F Pereira

UCSB

"Statistical shape analysis with Geomstats"

Geomstats is an open-source Python package for computations and statistics on Riemannian manifolds. It provides object-oriented and extensively unit-tested implementations. Manifolds can be equipped with Riemannian metrics with associated exponential and logarithmic maps, geodesics, and parallel transport. Building on this general framework, the shape module implements widely used shape spaces, such as the Kendall shape space and elastic spaces of discrete curves and surfaces, by leveraging the abstract mathematical structures of group actions, fiber bundles, and quotient spaces. The Riemannian geometry tools enable users to compare, average, and interpolate between shapes belonging to a given shape space. These essential operations can then be used to perform statistics on shape data. In this talk, we will present the object-oriented implementation of the shape module along with illustrative examples and demonstrate its use in performing statistics on shape spaces.

Qiyu Wang

UBC

"Studying SARS-CoV2 spike protein heterogeneity from large Cryo-EM dataset with linear subspace method and path analysis"

Recent advances in single particle cryogenic electron microscopy (cryo-EM) have allowed to capture biomolecules in various conformations through large image datasets. However, interpreting and quantifying such conformational heterogeneity remain computationally challenging, leading to a variety of recent methods. In the context of SARS-CoV-2, we developed and implemented a pipeline to process large datasets (~ millions) of 2D images of spike proteins, and apply REgularized COVARiance estimator (RECOVAR), to project the images into a latent linear subspace. Our pipeline also includes new methods for trajectory inference and transport-based segmentation that facilitate data analysis, revealing specific transitions between multiple conformations of the receptor binding domains (RBDs) in SARS-Cov2 spike protein. Our study notably led us to discover a state with three RBDs up, as well as finding a cooperativity mechanism from states with one RBD up, that goes towards the closed state before transiting to the state with two RBD’s up, offering valuable insights into the conformational landscape of SARS-CoV-2.

Willem Diepeveen

UCLA

"Curvature corrected tangent space-based approximation of manifold-valued data and applications in protein dynamics analysis"

When generalizing schemes for real-valued data approximation or decomposition to data living in Riemannian manifolds (widely used for modelling biological shapes), tangent space-based schemes are very attractive for the simple reason that these spaces are linear. An open challenge is to do this in such a way that the generalized scheme is applicable to general Riemannian manifolds, is global-geometry aware and is computationally feasible. Existing schemes have been unable to account for all three of these key factors at the same time. In this work, we take a systematic approach to developing a framework that is able to account for all three factors. In addition, we consider applications of our theory to analysis of protein dynamics data.

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Annual Meeting for the Society for Mathematical Biology, 2025.