## 2 Results
#### 2.1.3 Budding Division Maintains Closeness between Mothers and Daughters after Physical Separation  
To better quantify the differences in local spatial arrangement and organization between budding and non-budding colonies, we analyzed the spatial graph (\(G_{\text{S}}\)) and lineage graph (\(G_{\text{L}}\)) for each colony. The spatial graph has an edge between cells that are adjacent according to the Delaunay triangulation of their centers. (Figure 16A). The lineage graph has an edge between mother and daughter cell pairs regardless of their spatial position (Figure 2 (top row) and Figure 16B). We construct both graphs from our simulation output following the procedures defined in Section 4.2.2. Together, these graphs provide a framework to quantify how spatial relationships between mother-daughter cell pairs dynamically evolve during colony growth. Colony connectivity is a measure of how many mother-daughter cell pairs are adjacent in the colony. We determine this by looking at the fraction of mother-daughter edges in \(G_{\text{S}}\). (See Section 4.2.2 for more details.) First, we compared colony connectivity between budding and non-budding colonies after 12, 18 and 24 h of growth (Figure 5A). While colony connectivity decreases over time for both budding and non-budding colonies, it is significantly higher in budding colonies starting after 18 h of growth. We hypothesized that since cells in budding colonies remain physically attached to their mother until separation, high connectivity of the cells still attached to their mother could explain the observed difference. To investigate this hypothesis, we compared the connectivity between mother-daughter cell pairs in two distinct phases: when they are attached during budding cell division (\"budded cells\", Figure 5B) and after separation (\"un-budded cells\", Figure 5C). We observe that the connectivity of budding colonies is significantly higher in each phase (Figure 5B,C). Namely, in the case of budded cells, connectivity stays close to 1 during the entire 24 h time period for budding colonies whereas it decreases between 12-18 h and then again between 18-24 h for non-budding colonies (Figure 5B). Surprisingly, in the case of un-budded cells, the difference in connectivity is significantly different between budding and non-budding colonies starting at 18 h (Figure 5C). In fact, the absolute difference in mean connectivity between budding and non-budding colonies remains unchanged from the overall colony connectivity computed for all cells. To further address the mechanism driving our observed differences in colony connectivity between budding and non-budding colonies, we hypothesized that mother-daughter cell pairs remain physically close for a longer period of time after separation in budding colonies. To investigate this hypothesis, we generated Kaplan-Meier survival curves for the lifetime of mother-daughter edges in \(G_{\text{S}}\) (lifetime being the length of time after separation). We observe that the probability that a given mother-daughter edge will remain in \(G_{\text{S}}\) for longer than \(t\) hours after separation is higher for budding colonies (Figure 5D). In addition, the restricted mean survival time for a given mother-daughter edge in \(G_{\text{S}}\) is 80 min compared to 66 min for non-budding colonies. Furthermore, we observe distinctly different behavior between the two survival curves from 0-50 min after separation, whereas after 50 min, the two curves display more similar behavior. This change in behavior is marked by a \"sharp\" decrease in survival probability for both types of colonies. The timing of this sharp decrease is consistent with the appearance of a second bud from the mother cell and we conjecture that the new bud \"pushes away\" the previous daughter. Figure 5: Colony Connectivity in Nutrient-Rich Colonies.