## 4 Materials And Methods
### Computational Model
#### 4.1.4 Nutrient-Limited Growth  
Until now, we assumed that the environment cells were in contained an inexhaustible nutrient supply and cell maturation and division occurs at the same rate rate no matter how many cells were present. We now revisit this assumption by modeling the growth of individual yeast cells as dependent on a local nutrient supply. That is, a depletion in nutrient concentration slows down individual cell growth by prolonging the cell cycle length. Previous studies have incorporated the effect of enzyme and/or nutrient concentration on individual cell behaviors in ABM models of microbial colony growth [8; 29; 42; 48; 49; 50; 51; 52; 53]. The majority of these studies use reaction-diffusion equations that include the uptake of growth substrate by each cell to compute spatial gradients of enzyme or nutrient concentration. For simplicity, we consider each region of our simulation domain to have a maximal  
Figure 14: Cell Cycle Length. (Left): The \(G1\) phase for mother cells is approximately \(15\) min. Since mother cells have already reached their adult size, the \(G1\) phase serves as a waiting period before the mother cell enters \(G2\) and forms a bud. When the mother cell enters \(G2\), the new daughter cell forms as a bud and stays attached for \(\sim\)\(75\) min as it grows. After \(\sim\)\(75\) min, the mother and new daughter physically separate resulting in two unevenly sized cells. At this time, the mother cell enters \(G1\) and begins a new cell cycle. (Right:) The new daughter cell continues to grow until it reaches its adult size (\(\sim\)\(45\) min) and forms its own bud. Under nutrient-limited conditions, the length of the \(G1\) and \(G2\) phases are increased for both mother and daughter cells (see Section 4.1.4 for details).  
possible biomass (i.e. carrying capacity). We divide the simulation domain into smaller subdomains and adjust the cell cycle progression \(CP(t)\) for cells in each subdomain \(j\) at time \(t\) as follows.  
First, for each cell \(i\) we track the subdomain the cell is in denoted \(D_{i}(t)\) and compute the total mass of cells in each subdomain \(j\) where \(m_{i}(t)=\pi R_{i}(t)^{2}\):  
\[M_{j}(t)=\begin{cases}\frac{\sum_{i=1}^{N(t)}m_{i}(t)\mathbb{I}_{j}(D_{i}(t))} {M_{j,\max}},&\text{if }\sum_{i=1}^{N(t)}m_{i}(t)\mathbb{I}_{j}\left(D_{i}(t) \right)\leq M_{j,\max}\\ 1&\text{else}\end{cases} \tag{9}\]  
where \(\mathbb{I}_{j}\) is an indicator variable that equals 1 if \(D_{i}(t)=j\) and 0 otherwise. Note that in practice \(M_{j,\max}\) is chosen large enough that \(\frac{M_{j}(t)}{M_{j,\max}}\) is always less than 1.  
Next, we define a growth-rate adjustment factor for each subdomain that is initialized to 1 at the beginning of simulations and decreases in time according to the following equation:  
\[GR_{\text{adjust}}(t)=\begin{cases}GR_{\text{adjust}}(t-\Delta t,j)-rM_{j}(t) \Delta t,&\text{if }\frac{GR_{\text{adjust}}(t-\Delta t,j)}{\Delta t}\geq-rM_{j}(t)\\ 0&\text{else}\end{cases} \tag{10}\]  
where \(M_{j,\max}\) is the carrying capacity for the \(j\)-th subdomain and \(r\) is the per capita rate of decrease of \(GR_{\text{adjust}}\). We then re-scale the cell cycle increment:  
\[\widehat{CI_{i}}(t)=CI_{i}(t)\times GR_{\text{adjust}}(t,D_{i}(t)) \tag{11}\]  
and thus the cell progression in nutrient-limited growth becomes  
\[CP_{i}(t)=CP_{i}(t-\Delta t)+\widehat{CI_{i}}(t)\times\Delta t. \tag{12}\]  
As the colony grows, cells move, new cells are born and cells are displaced into new subdomains. To account for this we calculate the growth rate adjustment factor for each subdomain at each timestep, and use it to update \(\widehat{CI_{i}}(t)\) for each cell according to the unique subdomain it occupies at time \(t\).