Topology in Biology

Fueled by increasing computing power and ever-growing datasets, novel methods for complex systems analyses seem to emerge daily. With this whirlwind flows excitement and opportunity as new methods allow us to view our system of interest with a fresh perspective. One set of approaches that has offered particularly deep insights into complex systems is that of applied topology, also known as the field of topological data analysis (TDA). Topological data analysis specializes in representing and analyzing the entirety of the system and therefore can often remove the need for thresholds and filtering when selecting data, as is commonly done when studying biological systems. In this chapter we will briefly introduce the topological perspective that makes the above possible, and then we will dive into three methods within TDA through a biological lens. First, we cover persistent homology, which detects evolving topological cavities within data. Next, we join data to graphs in order to construct sheaves or structures that can help us understand systems with linear relations between adjacent nodes. We then describe path signatures, which allow us to detect lead-lag behavior in time series. While detailing each of these three main concepts, we note how we can view each as a perspective shift from an analysis we might already perform to a similar analysis using the topological language. As we discuss, this reframing into the language of topology can provide extensive additional insight into our system of interest, and at present, much of this added information remains ripe for interpretation and application.