Smooth real-valued functions are ubiquitous in the modeling and simulation of physical phenomena, and therefore also in structural biology. Examples are the electron-density of a protein defined on three-dimensional Euclidean space and the electrostatic potential of a protein restricted to its 2-manifold boundary. The structural properties of such functions are studied in Morse theory, which rests on the insight that every smooth function is near a generic smooth function whose critical points are isolated. We take another step and extend the differential notions from classical to combinatorial Morse theory, which studied piecewise linear real-valued functions on manifolds. This step is necessary because we are limited to finite measurements of smooth functions. A common scenario consists of measurements at finitely many points and a triangulation of the manifold in which the points are the vertices and the function is extended beyond the vertices by linear interpolation.

We have developed algorithms for recognizing critical points of piecewise linear functions. For 2- and 3-manifolds, we have also developed algorithms that decompose the manifold into regions of similar flow behavior. We call this the Morse-Smale complex. For 2-manifolds, it consists entirely of quadrangles in which the points flow from a minimum in one corner to the maximum in the opposite corner. For 3-manifolds, the complex consists of crystals, which are generalizations of the cube in which the points flow again from a minimum at one vertex to the maximum at the opposite vertex. We combine these complexes with the concept of topological persistence to obtain a hierarchy of progressively simpler versions. The medial axis of a surface in three-dimensional Euclidean space can be modeled as a substructure of the Morse-Smale complex for the minimum distance function. We use estimates of the medial axis to aid extraction of ligand structure from x-ray crystallography data.