Fig 1.
Geometry and topology of cell aggregates.
A Cell aggregate is modeled by a space-filling packing of polyhedral cells. Cell shapes are parametrized by vertex positions ri, which move according to mechanical forces Fi. Topology of the cell network is specified by lists ej, pk, and cl [Eqs (2)–(4)], which store head and tail vertices of edges, oriented edges within polygons, and oriented polygons within cells, respectively. B EV transition merges vertices of an edge into a single vertex, whereas VT transition resolves a vertex into a triangle. VE and TV transitions are inverse transitions of EV and VT transitions, respectively. C Polygons involved in topological transitions from panel B. D Decomposed schematic highlighting from 3 different perspectives of polygons, edges, and vertices, in topological transitions from panel B.
Fig 2.
A The hierarchical structure of GVM vertex→edge→polygon→cell is defined by a metagraph. Relationships between these nodes are labeled IS_PART_OF and carry a sign property denoted by s, σ, and Σ for vertex→edge, edge→polygon, and polygon→cell relationships, respectively. B A subgraph representing a particular local cell state is obtained by pattern matching. This subgraph is then transformed by a graph transformation. C Metagraph of graph-transformation graph connects Vertex, Edge, Polygon, and Cell nodes with green and red relationships, indicating creation and deletion of IS_PART_OF relationships in the GVM’s knowledge graph, respectively.
Fig 3.
Graph transformation for a T1 transition.
A Graphs in the left and right columns correspond to the initial and final cell configurations, respectively. Gray arrows represent relationships labeled IS_PART_OF. The graph in the middle column shows the graph transformation, which includes green and red relationships, indicating relationship creations and deletions, respectively. Additionally, the graph transformation specifies property values of the newly created relationships. B Schematic of determining the value of contextual property for a new vertex→edge relationship generated between vertex v1 and edge e3 (). After the transformation, vertex v1 assumes the same role in the context of edge e3 as the role of v2 in the context of e3 before the transformation (s2,3). This occurs because the edge e3 merely replaces v2 (blue) with v1 (red). C Schematic of determining the value of contextual property for a new edge→polygon relationship generated between edge e5 and polygon p1 (
). The calculation of
(red) relies on one of the vertices of e5, i.e., v2 (green), and the edge linked to both v2 and p1, i.e., e2 (also depicted in green). The assignment of
is determined based on the contextual properties of: (i) edge e2 in the context of polygon p1, σ2,1, (blue) and (ii) vertex v2 in contexts of edges e2,
, and e5,
, (green). In short, the contextual property
aligns or opposes that of σ2,1 depending on the similarity or dissimilarity of
and
.
Fig 4.
Graph transformations in a 3D vertex model of polyhedral packings.
A Graph transformation of an ET transition. B Graph transformation of an TE transition. In both panels, graphs in the left and the right column correspond to the initial and the final cell configuration, respectively. Gray arrows represent relationships labeled IS_PART_OF. The graphs in the middle column show graph transformations, which include green and red relationships, indicating relationship creations and deletions, respectively. Additionally, the graph transformation specifies property values of the newly created relationships. In each graph, the node indices increase from left to right in unit steps.
Fig 5.
3D graph transformations reduce to a T1 transition when applied to a 2D vertex model.
A ET transformation when applied to a four-polygon neighborhood described by a 2D vertex model. Parts of graphs shown in transparent are not matched because the corresponding elements do not exist in 2D. After relabeling the nodes (panel B) and repositioning them (panel C), we observe that the matched graph transformation exactly corresponds to , i.e., the graph transformation that performs a T1 transition (Fig 3).
Fig 6.
Order-disorder transition in 3D cell aggregates.
A Shape parameter q for “thermalized” aggregates versus active noise σ. B Distribution of edge lengths in thermalized aggregates for σ = 0.025, 0.2, and 0.5 (red, green, and blue curves, respectively). C Order parameter 1 − f14 and normalized number of cell-rearrangement events n = (NET + NTE)/Nmax versus active noise σ. D-H Thermalized aggreages at σ = 0, 0.2, 0.3, 0.4, and 0.5. I Order parameter 1 − f14 versus time for σ = 0.01, 0.17 and 0.4 (red, green, and blue curves, respectively) during aggregate ordering. J The initial (t = 0.001; orange curve) and final (t = 1000; magenta curve) distributions of edge lengths at σ = 0.17. K Shape parameter q versus time for σ = 0.01, 0.17 and 0.4 (red, green, and blue curves, respectively) during aggregate ordering.