Paweł Dąbrowski-Tumański: From Lasso Proteins to Medical Imaging: Spatial Graphs in Biology and Medicine

Topological structures such as knots and links are now well established in proteins, DNA, and other biomolecules. In many cases, however, biological systems are more naturally described not as curves in space, but as spatial graphs—graphs embedded in physical, inferred, or abstract spaces. Examples include proteins with disulfide-closed loops pierced by one or more chains, theta-curves and handcuff graphs realized in protein structures, chromatin organization inferred from Hi-C data, neuronal and vascular networks reconstructed from imaging, and molecular graphs defined by atomic connectivity. In this seminar, I will present a unified perspective on spatial graphs across scales in biology and medicine, emphasizing how the notion of “space” can be progressively generalized—from three-dimensional Euclidean embeddings to high-dimensional representation spaces, and finally to similarity spaces that need not satisfy metric axioms. I will show that this generality can be naturally studied using persistent homology, which therefore can be applied to study problems in chemistry, biology, and medicine making persistent homology as a versatile, biologically meaningful tool for analyzing complex spatial and similarity-based graphs.