## 2 Methods  
### Formulation of the Multiple Resource Limitation  
We first postulate that growth rates can depend on multiple resources. To present our thoughts on growth rate as a function of multiple limiting resources, we recall that [PERSON]'s growth [26; 27] rate equation takes the form of  
\[\mu=\mu_{max}\bigg{(}1-\frac{q}{Q}\bigg{)}. \tag{1}\]  
Here, \(Q\) is the internal cell quota of the most growth-limiting resource (nmol cell\({}^{-1}\)); \(q\) is the minimum or subsistence quota of the limiting resource at which the growth rate  
Figure 1: Two examples of colimitation with LLM as a special case. \(S_{1}\) and \(S_{2}\) are two resources that co-limit the growth of the organism. Light colors indicate higher growth and darker colors indicate lower growth. Instances of LLM are denoted by the red arrows. (**A**) No adverse effect and growth become infinite at infinite resources. (**B**) The inhibitory effect is incorporated at low and high resources. The maximal growth exists in case B, but not in A. These functional forms and additional ones have been studied previously [5; 7; 8]. is zero; and \(\mu_{max}\) is the theoretical maximum per capita population growth rate when all resources are abundant. Previous studies have also looked at possible mechanistic derivations of the Droop growth [28]. In the following, we shall formulate the co-limitation of multiple resources using nitrogen (\(N\)) and phosphorus (\(P\)) as an example due to their relevance in cell growth; however, the framework is also applicable to other resources, for example, the supply of light. Since cells are made of proteins that are rich in \(N\) and protein production requires ribosomes that are rich in \(P\), cell growth rates are functions of cellular \(N\) and \(P\) quotas. In the following, we assume \(q_{N}\) and \(q_{P}\) are the subsistence quota for \(N\) or \(P\), respectively, while \(Q_{N}\) and \(Q_{P}\) are cellular \(N\) and \(P\) quotas, respectively. In reality, both \(N\) and \(P\) can be simultaneously limiting. According to LLM, cell growth rates take the form of  
\[\mu_{l}=\min\{\mu_{N},\mu_{P}\}=\mu_{max}\min\biggl{\{}\biggl{(}1-\frac{q_{N}} {Q_{N}}\biggr{)},\biggl{(}1-\frac{q_{P}}{Q_{P}}\biggr{)}\biggr{\}}, \tag{2}\]  
where \(\mu_{N}\) and \(\mu_{P}\) are the Droop growth function under the \(N\) and \(P\) limitation, respectively. Intuitively, since the production of ribosomes requires both \(N\) and \(P\) at the same time (e.g., ribosomes comprise both ribosomal proteins and RNA), while the production of non-ribosomal proteins does not involve \(P\), one can argue that the protein production speed will be slower when both are in short supply, since it is highly dependent on the allocation of \(N\) in the translation phase. Hence, the true cell growth rate will be less, or at most, equal to \(\mu_{l}\). There are two possible extreme scenarios: (1) The cell takes up and uses \(N\) and \(P\) simultaneously; or (2) the cell takes up and uses \(N\) and \(P\) independently and sequentially. A real cell is likely to fall somewhere in between these two extreme scenarios, where there may be multiple factories that contribute to growth, but they all take up and process \(N\) and \(P\) at different rates. In case (1), the cell resource uptake process probably takes the least amount of time, and the cell growth rate may indeed take the form of \(\mu_{L}\) assuming other resources may be sufficient for the growth process. In case (2), the resource uptake process probably takes the longest time, which may yield the slowest growth rate. Assume that the time to take up and use \(N\) is \(T_{N}\), and the time to take up and use \(P\) is \(T_{P}\). If \(P\) is abundant, then the cell's \(N\)-limited growth rate is \(\mu_{N}=1/T_{N}\). Likewise, the cell's \(P\)-limited growth rate is \(\mu_{P}=1/T_{P}\) when \(N\) is abundant.