## 4 Materials And Methods
### Computational Model
#### 4.2.2 Colony Organization Metrics  
Next we introduce new metrics related to the organization of cells within the colony. We define a graph \(G(V,E)\) as a set of nodes \(V\) and edges \(E\). In each graph, all cells are represented as nodes. To analyze the colony organization, we consider two undirected graphs which evolve dynamically along with the colony (Figure 16). The first graph, \(G_{S}\), is based on the Delaunay triangulation [85; 86] and encodes spatial relationships between cells (Figure 16A). The second graph, the lineage graph which we denote as \(G_{L}\), encodes mother-daughter relationships between cells (Figure 16B) Both graphs are constructed from the same vertex set, namely, all cells in the colony at time \(t\):  
\[V=\{c_{1},c_{2},\dots c_{N(t)}\}.\]  
The edge set \(E_{S}\) is constructed based on the Delaunay triangulation. The Delaunay triangulation is the dual graph of the Voronoi diagram for cell centers which consists of all points in the plane that are equidistant to their two nearest sites [87]. In our case, the edge set \(E_{S}\) is defined as:  
\[E_{S}=\{(c_{i},c_{j})\mid\text{ cell }i\text{ and }j\text{ share an edge in the Delaunay triangulation.}\}. \tag{17}\]  
The edge set \(E_{L}\) consists of only edges between immediate mother-daughter cell pairs. Each cell, \(c_{i}\), has a unique mother, \(m(c_{i})\). Thus, the edge set \(E_{L}\) consists of the following edges:  
\[E_{L}=\{(c_{i},m(c_{i}))\}\quad\text{for }i\in\{2,\dots,N(t)\}. \tag{18}\]  
Figure 15: Colony Sparsity and Expans. We first compute the colony center of mass (represented as a green square) using Equation (13). Then we determine the radius (length of the blue line) of the smallest circle which surrounds the entire colony centered at the center of mass (shown in red) using Equation (14). The colony sparsity is then computed using Equation (16) with the result of Equation (14), the area of the circle, and the total area of the cells using Equation (15). (Left): The space within the circle is more dense, thus covering more area within the circle with radius equal to the colony expanse, resulting in a small colony sparsity. (Right): The space that cells occupy within the circle is less dense, resulting in a higher colony sparsity. (Note that \(m(c_{1})\) is not defined because \(c_{1}\) is the founder cell.) Together \(G_{S}=\{V,E_{S}\}\) and \(G_{L}=\{V,E_{L}\}\) can be used to quantify how closely the spatial organization of the colony relates to mother-daughter cell pair interactions within the colony. To do this, we first define the intersection graph, \(G_{I}\) (Figure 16 C). \(G_{I}\) is constructed from the same vertex set containing all cells in the colony, but the edge set, \(E_{I}\), only includes edges belonging to both \(E_{S}\) and \(E_{L}\), namely:  
\[E_{I}=E_{S}\cap E_{L}. \tag{19}\]  
Our first colony organization metric, **colony connectivity**, is the fraction of mother-daughter edges that are also in the intersection graph:  
\[\text{Colony Connectivity}:=\frac{|E_{I}|}{|E_{L}|}. \tag{20}\]  
Figure 16: Colony Spatial Graphs. The vertex set for all three colony graphs is the same (all cell centers). The founder cell is designated in black. The edge set differs depending on the relationships between cells; **(A)**: The edge set for the spatial graph \(G_{S}\) (blue edges) are those induced by the the Delaunay triangulation applied to the cell centers (Equation (17)): **(B)**: The edges for the lineage graph \(G_{L}\) (red edges) correspond to mother-daughter pairs (Equation  
We define \(G_{\text{sub},d_{k}}\) to be the subgraph of \(G_{S}\) whose vertex set is \(V_{\text{sub},d_{k}}\) and whose edge set is \(E_{\text{sub},d_{k}}\) (see Figure 17B,C).