Topological Metric Detects Hidden Order in Disordered Media

Recent advances in microscopy techniques make it possible to study the growth, dynamics, and response of complex biophysical systems at single-cell resolution, from bacterial communities to tissues and organoids. In contrast to ordered crystals, it is less obvious how one can reliably distinguish two amorphous yet structurally different cellular materials. Here, we introduce a topological earth mover’s (TEM) distance between disordered structures that compares local graph neighborhoods of the microscopic cell-centroid networks. Leveraging structural information contained in the neighborhood motif distributions, the TEM metric allows an interpretable reconstruction of equilibrium and nonequilibrium phase spaces and embedded pathways from static system snapshots alone. Applied to cell-resolution imaging data, the framework recovers time ordering without prior knowledge about the underlying dynamics, revealing that fly wing development solves a topological optimal transport problem. Extending our topological analysis to bacterial swarms, we find a universal neighborhood size distribution consistent with a Tracy-Widom law.

Figure 1

Key conceptual steps for calculating the TEM distance illustrated for an epithelial cell layer. (a) Experimental image of epithelial cell layer from Drosophila embryo (adapted with permission from Refs. [8, 37]) with Delaunay triangulation overlayed (blue, red). The local neighborhood network of radius $r=2$ (red) is shown for a selected vertex (white circle). (b) A topological $T1$ transition, corresponding to an exchange of neighbors, is reflected in a change of the local radius-2 neighborhood. Voronoi cells (red, blue) with Delaunay triangulation overlayed (black, gray) are shown. (c) Nodes of the flip graph correspond to networks and are connected by an edge if the networks are one $T1$ transition (or flip) away from each other. (d) The map ${\gamma }_{ij}$ transports the neighborhood motif distribution of material $A$ to the motif distribution of material $B$.

Figure 2

Two-dimensional poly- and monodisperse packings are distinguished by the TEM distance. (a) Two alternative paths from aspect ratio $1:1$ to aspect ratio $1:3$ ellipsoid packings. Top is monodisperse with varying aspect ratios; bottom is polydisperse with a mixture of $1:1$ and $1:3$ aspect ratios. (b) Distance matrix where each pixel represents the pairwise distance between simulated 2D planar ellipsoid packings, each containing 10 000 ellipsoids. (c) Simulations are embedded in 2D using MDS, recovering two distinct paths, one for monodisperse simulations (hexagons) and one for polydisperse simulations (circles). (d) The residual variance plateaus after embedding dimension 2, correctly identifying the true embedding dimension of the phase space (arrow).

Figure 3

Developing Drosophila embryos solve topological optimal transport problem, with MDS embedding recovering temporal order. (a)–(e) Enlarged images showing epithelial cells at unknown times. Three experiments and 40 time frames per experiment were used. (f) Matrix of TEM distances between unsorted experimental images has no apparent structure. (g) TEM distance matrix sorted by hierarchical clustering shows approximately three phases. (h) 2D MDS embedding recovers the temporal order as the principal component. Included in the embedding are intermediate stages of a dissipation-minimizing [23] geodesic between average start and end states, which the data fall on. (i) The correct temporal ordering of the image time series is recovered. The white boxes show the source of images (a)–(e). Data adapted with permission from Refs. [8, 37].

Figure 4

Motif sizes in heterogenous bacterial swarms exhibit universal nonequilibrium statistics. (a) An azimuthally symmetric bacterial swarm expands as motile cells grow and divide [27]. We analyzed configurational snapshots at different space-time positions in the swarming monolayer phase (blue dots; select example images at the top); the central multilayered biofilm phase (red) was excluded. (b) Aided by a machine learning algorithm, cell centroids (yellow circles) and their Delaunay tessellation (blue lines) were determined. Scale bar: $10\mu \mathrm{m}$. (c)–(e) The distribution of motif sizes, defined as the number of vertices per $r=2$ neighborhood, is heterogenous in space and time. The mean motif size increases, on average, with radial distance and time (c),(d). The variance decreases with area filling fraction (e), reflecting more ordered cell packings in dense swarming regions. (f) After normalizing to zero mean and unit variance, the combined histogram over all space-time snapshots (circles) follows a universal Tracy-Widom distribution (line). Horizontal error bars represent the standard deviation within a bin. Inset: log scale plot.