I will introduce how concepts of Riemannian geometry for representing 2D and 3D shape space provides a natural framework to study cell morphological heterogeneity. The lecture will introduce to the basic concepts of Riemannian geometry through the classical definition of the Kendall shape space, before focusing on quantifying cancer cell morphological variations under various drug treatment using elastic metrics on curves with the Geomstats Python package
While the two previous lectures demonstrate the benefit of non linear metrics to study biological shape variation, they also highlight the need for tools to represent biological shape or conformations of a biomolecules in a low dimensional space (i.e. learn a low-dimensional embedding). Besides the standard PCA, UMAP or t-SNE algorithms that are commonly used to visualize biological data, I will give an overview of various manifold learning techniques, with a focus on diffusion maps and variants leveraging the concept of fiber bundle to analyze shape datasets.
Recent advances in cryogenic electron microscopy (cryo-EM) and tomography (cryo-ET) have allowed to image proteins and biomolecular complexes at an unprecedented resolution, fueling initiatives for cryo-EM guided drug design, including in cancer therapeutics. In this context, I will present the basics of cryo-EM/ET and mathematical challenges associated with it. I will then present and show how variants of the Wasserstein distances arising from Optimal Transport theory can be leveraged to improve current methods in 3D alignment and atomic model building