## 2 Methods
### Formulation of the Multiple Resource Limitation
In case (2), the cell division time can be approximated by \(T=T_{N}+T_{P}\), which implies that a plausible lower bound for the cell growth rate is approximately:  
\[\mu_{m}=\frac{1}{T}=\frac{1}{T_{N}+T_{P}}=\frac{1}{\frac{1}{P_{N}}+\frac{1}{P _{P}}}=\frac{\mu_{N}\mu_{P}}{\mu_{N}+\mu_{P}}. \tag{3}\]  
We obtain the following theoretical bounds on growth rate \(\mu\) under limitations by \(N\) and \(P\):  
\[\mu_{m}<\mu<\mu_{l}. \tag{4}\]  
If \(\mu_{N}\) is much larger than \(\mu_{P}\) (or the other way around), then the difference between \(\mu_{m}\) and \(\mu_{l}\) is much smaller, which gives a tighter range for the true growth rate \(\mu\). For more specific considerations of co-limitation by \(N\) and \(P\), we refer the readers to previous studies [29; 30; 31]. The theoretical forms of the multiple resource limitation and [PERSON]'s law of the minimum resemble two extreme instances of a production chain. Think of [PERSON]'s law in case (1) as a single large factory that takes in \(N\) and \(P\) and processes them simultaneously to make parts of a product (the \(NP\)-factory). In this case, there is no order for the processing of \(N\) or \(P\). Once the relative numbers of the \(N\)-part and \(P\)-part are produced, they are then combined to make the final product. Thus, in this scenario, the rate that a complete product is produced is limited by the rate of the slower sub-production chain. On the other hand, the proposed multiple resource limitation in case (2) would be akin to two smaller factoriesin a production chain, where one exclusively handles \(N\) (the \(N\)-factory) to make product parts out of \(N\), and moves these \(N\)-parts to the \(P\)-exclusive factory (the \(P\)-factory). The \(P\)-factory, then builds on these \(N\)-parts with \(P\)-parts to complete the process. Of course, the order may be reversed, but the key difference is that there is a strict order of operation in the second case. Thus, in this scenario, the uptake and processing times of \(N\) and \(P\) would then be additive, so the growth is much slower than the simultaneous process in case (1). Figure 2 describes this analogy. We remark that it is possible to formulate the growth equations for [PERSON]'s law of the minimum and the proposed multiple resource limitation based on these analogies. A straightforward extension of this approach can yield a (lower) growth rate expression under MRL with more than two complementary limiting resources. Assume a cell grows under limitation by \(n\) resources \(N_{i}\), \(i=1,2,\ldots,n\), and the time to take up and use resource \(N_{i}\) is \(T_{i}\). Then, the growth rate with only an \(i\)-th limiting resource is \(\mu_{i}=1/T_{i}\). As in case (1) and case (2), we have  
\[\mu_{L}=\min\{\mu_{i}:i=1,2,\ldots n\}, \tag{5}\]  
and \(T=T_{1}+T_{2}+\ldots+T_{n}\), which implies that a plausible lower bound for the cell growth rate is approximately  
\[\mu_{M}=\frac{1}{T}=\frac{1}{T_{1}+T_{2}+\ldots+T_{n}}=\frac{1}{\frac{1}{\mu _{1}}+\ldots+\frac{1}{\mu_{n}}} \tag{6}\]  
Figure 2: An analogy for [PERSON]’s law of the minimum and the proposed multiple resource limitation. (**A**) An \(NP\)-factory that concurrently processes both \(N\) and \(P\) to make the final product as an analogy to [PERSON]’s law of the minimum. The production rate of the final product is constrained by the slowest rate of making sub-parts from \(N\) or \(P\). (**B**) Two resource-specific ones (\(N\)-factory and \(P\)-factory) in a strict production chain as an analogy to the proposed multiple resource limitation. The time to make a final product is then the combined time to make \(N\)- and \(P\)-sub-parts. Note that there is no order of production in A, but a strict order of production in B. We thus arrive at the following theoretical bounds on growth rate \(\mu\) under the limitations of multiple complementary resources:  
\[\mu_{M}<\mu<\mu_{L}. \tag{7}\]  
Note that Equation (7) is the generalized version of Equation (4).