Hyperbolic Geometry in Biological Systems

Abstract

Across different scales of biological organization, biological networks often exhibit hierarchical tree‐like organization. For networks with such structure, hyperbolic geometry provides a nnatural metric because of its exponentially expanding resolution. In this talk, I will describe how the use of hyperbolic geometry can be helpful for visualizing and analyzing high‐dimensional datasets. Our examples will include data from plant and animal volatiles that serve as input for the sence of smell (Zhou et al Science Advances 2018) and patterns of gene expression in diverse mammalian cell types (Zhou and Sharpe, iScience 2021). We find that local noise causes data to exhibit Euclidean geometry on small scales, but that at broader scales hyperbolic geometry becomes visible and pronounced. The hyperbolic maps are typically larger for datasets of more diverse and differentiated cells, e.g. with a range of ages. We find that adding a constraint on large distances according to hyperbolic geometry improves the performance of t‐SNE algorithm to a large degree causing it to outperform other leading methods, such as UMAP and standard t‐SNE. I will conclude with a discussion of new opportunities for identifying age‐ and disease‐related factors afforded by hyperbolic maps.