Geometry of Morphogenesis
Nadya Morozova, Robert Penner

TL;DR
This paper presents a geometric formalism for modeling eukaryotic cell and organism shapes using star-convex cells, introducing a Lagrangian framework based on Gromov-Hausdorff distance to describe morphogenetic dynamics.
Contribution
It introduces a novel geometric and mathematical formalism for modeling organism shape and development, integrating epigenetic data and morphogenetic dynamics.
Findings
A formalism for representing organism shape as a metric subspace of Euclidean space.
A bundle over cell configurations encoding epigenetic information.
A Lagrangian approach to morphogenetic dynamics using Gromov-Hausdorff distance.
Abstract
We introduce a formalism for the geometry of eukaryotic cells and organisms.Cells are taken to be star-convex with good biological reason. This allows for a convenient description of their extent in space as well as all manner of cell surface gradients. We assume that a spectrum of such cell surface markers determines an epigenetic code for organism shape. The union of cells in space at a moment in time is by definition the organism taken as a metric subspace of Euclidean space, which can be further equipped with an arbitrary measure. Each cell determines a point in space thus assigning a finite configuration of distinct points in space to an organism, and a bundle over this configuration space is introduced with fiber a Hilbert space recording specific epigenetic data. On this bundle, a Lagrangian formulation of morphogenetic dynamics is proposed based on Gromov-Hausdorff distance…
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Point processes and geometric inequalities · Dermatological and Skeletal Disorders
