## 4 Materials And Methods
### Computational Model
#### 4.1.1 Cell-Cell Interaction and Spatial Arrangement Of Cells  
We assume the resting shape of individual yeast cells is circular, and track the size and location of cell \(i\) at time \(t\) by its radius \(R_{i}(t)\) and center \(\vec{x}_{i}(t)=(x_{i}(t),y_{i}(t))\) (Figure 12). We track the total number of cells at time \(t\), \(N(t)\), and index cells by their birth order \(i\in\{1,2,\dots,N(t)\}\). Since all cells are of the same type, the mass of each cell, \(m_{i}(t)\), is proportional to the area of each cell with the same constant. Yeast cells interact through different potentials that we use to represent biologically-relevant processes seen in experiments (Figure 12). For example, yeast cells in physical contact form adhesive bonds that result in an attractive force [41; 42; 44; 72; 73]. However, due to the incompressibility of their cell wall, yeast cells also resist compression from neighboring cells with a repulsive force [41; 42; 44; 75; 47]. We represent the combination of repulsive and attractive interactions between cells using a modified Hertz-model, as has been previously done [48; 51; 76; 77], where the potential \(E_{ij}^{CC}(t)\) between two cells \(i\) and \(j\) is given by:  
\[E_{ij}^{CC}(t)=\frac{(R_{i}(t)+R_{j}(t)-d_{ij}(t))^{5/2}}{5\tilde{E}_{ij}}\sqrt {\frac{R_{i}(t)R_{j}(t)}{R_{i}(t)+R_{j}(t)}}+E_{ij}^{\text{adh}}(t). \tag{1}\]  
The first term of Equation (1) depicts the repulsive interaction and \(d_{ij}(t)=\|\vec{x}_{i}(t)-\vec{x}_{j}(t)\|\). In this equation \(\tilde{E}_{ij}(t)\) is defined by:  
\[\tilde{E}_{ij}=\frac{3}{2}\left(\frac{1-\sigma^{2}}{E}\right) \tag{2}\]  
Figure 12: Biophysical Model of Cell-Cell Interactions. We simulate budding (**A**) and non-budding (**B**) colonies using a 2D center-based modeling approach where cells interact through different potentials. (**A**) For budding colonies, the mechanical interactions of all cell pairs (\(E^{CC}\)) are governed by a combination of repulsive and attractive interactions using a modified Hertz-model described in Equation (1). We use a linear spring to model the additional adhesive force between mother cells and new daughter cells during the budding phase (\(E^{MB}\)) as in Equation (4): (**B**) For non-budding colonies, the mechanical interactions are similar to that of cells in budding colonies except that the adhesive force between mother cells and new daughter cells during the budding phase (\(E^{MB}\)) is neglected (see Section 4.1.1 for details). where \(E\) and \(\sigma\) are the Young's moduli and Poisson ratios of cells, respectively. The second term of Equation (1) models the adhesive interaction between cells and is given by:  
\[E_{ij}^{\text{adh}}(t)=\phi W_{s}A_{ij}(t) \tag{3}\]  
where \(\phi\) is the density of surface adhesion molecules in the contact area, \(W_{s}\) is the single bond bind energy and and \(A_{ij}(t)=(R_{i}(t)+R_{j}(t))\times 0.5\) is the contact area between cells \(i\) and \(j\). In simulations of non-budding colonies, the mechanical interactions of all cell pairs are given by Equation (1). However, in simulations of budding colonies, we explicitly model an additional force between mother cells and new daughter cells during the budding phase (Section 4.1.2, Figure 12A and Figure 14). To do this, we represent the adhesive interaction caused by attachment of the new daughter cell to its mother using a linear spring potential given by:  
\[E_{\text{mb}}^{\text{MB}}(t)=K_{\text{bud}}\left(d_{mb}(t)-\left(R_{m}(t)+R_{ b}(t)\right)\right)^{2} \tag{4}\]  
where \(d_{mb}(t)=\|\vec{x_{m}}(t)-\vec{x_{b}}(t)\|\), \(R_{m}(t)\) is the radius of the mother cell at time \(t\), \(R_{b}(t)\) is the radius of daughter cell at time \(t\) and \(K_{\text{bud}}\) is a spring constant chosen large enough to ensure that the new daughter cell \(b\) remains attached to its mother for the duration of the budding phase and is not pushed away due to forces from other neighboring cells.