## 4 Materials And Methods
### Computational Model
#### 4.1.3 Cell Growth and Cell Cycle Length  
We follow the standard model of eukaryotic cell division and consider the cell cycle to have two distinct growth phases: \(G1\) and \(G2\). At the time of separation, mother and new daughter cells are unequal in size. Thus, new daughter cells undergo an extended \(G1\) phase in order to grow to a mature adult size before producing their own bud [81]. (Figure 14.) In our model, the average cell cycle length for mother cells is \(\approx\) 90 min (\(\sim\)15 min in \(G1\) and \(\sim\)75 min in \(G2\)) and the average cell cycle length for new daughters cells is \(\approx\) 120 min (\(\sim\)75 min for the \"Budding\" phase while attached to their mother and \(\sim\)45 min growing on their own). To depict more realistic cell cycle dynamics, we introduce an element of stochasticity to the cell division times.  
The variable **Cell Progress**\((CP\in[0,1])\) is used to track the progress of individual cells through the \(G1\) and \(G2\) phases. In our model, the progress of cell \(i\) at time \(t\) is given by:  
\[CP_{i}(t)=CP_{i}(t-\Delta t)+CI_{i}\times\Delta t \tag{7}\]  
where \(CI_{i}=(G1_{i}+G2_{i})^{-1}\). The length of \(G2_{i}\) is computed once for all cells:  
\[G2_{i}=(1+\textbf{U}[-0.1,0.1])\ G2_{\text{avg}}.\]  
(In the previous expression \(\textbf{U}[-0.1,0.1]\) is a uniformly distributed random variable on the interval \([-0.1,0.1]\).) To represent the longer \(G1\) phase of daughter cells, the length of \(G1_{i}\) is assigned once upon creation of a new bud  
\[G1_{\text{new daughter}}=(1+\textbf{U}[-0.1,0.1])\ G1_{\text{avg daughter}}\]  
Figure 13: Selecting a Bud Site. The choice of the next bud location for a cell depends on whether it is a mother cell (left) or a new daughter cell (right). (Left) For mother cells, the next bud location will be chosen either adjacent to the previous bud scar with probability 0.5 (**A**) or opposite to the previous bud scar with probability 0.5 (**B**). In the case when the location of the new bud site overlaps with a previous bud site (**A**), we adjust the location of the new bud site by increments of \(10^{\circ}\) in either the clockwise (probability 0.5) or counterclockwise (probability 0.5) direction until we arrive at a location with no previous bud scar. (Right) For new daughter cells the next bud location (red) is chosen opposite to the previous birth scar (black) with probability 1 (**C**). New daughter cells that have successfully completed a full cell cycle are considered mother cells for the remainder of the simulation. (See Section 4.1.3.)  
and then updated once the new daughter cell completes its first \(G1\) phase and forms a bud of its own  
\[G1_{\text{mother}}=(1+\mathbf{U}[-0.1,0.1])\;G1_{\text{avg}_{\text{mother}}}.\]  
In the \(G1\) phase, mother cells have already reached their adult size, so the \(G1\) phase is simply a waiting time until entering the \(G2\) phase where they form a bud (or new daughter). Every new bud is initiated with a radius of size \(0\mu\)m and grows for \(\approx 75\) minutes while attached to its mother. (This \(75\) minutes of attachment accounts for the entire \(G2\) phase of the mother and make up the first part of the \(G1\) phase for the daughter.) After this phase, the mother and bud are physically separated resulting in a new daughter cell. At separation, the mother cell enters the \(G1\) phase, and the new daughter cell stays in its \(G1\) phase and continues to grow for \(\approx 45\) minutes until it reaches its adult radius size (Figure 14). At this time, the daughter cell transitions into a mother cell and begins to produce its first bud.  
The adult size, corresponding to a maximum radius \(R_{\text{max}}\), is assigned to each cell upon creation and set to  
\[R_{i,\text{max}}=(1+\mathbf{U}[-0.1,0.1])\;R_{\text{avg}}.\]  
The radius of cell \(i\) at time \(t\) is given by:  
\[R_{i}(t)=\begin{cases}R_{i}(t-\Delta t)+\frac{R_{i,\text{max}}}{G1_{i}}\times \Delta t&R_{i}(t)\leq R_{i,\text{max}}\\ R_{i,\text{max}}&\text{else}.\end{cases} \tag{8}\]