## 2 Methods
### Compartmental Models
#### 2.1.1 Regulatory Pathway in One Cell
}-\underbrace{\pi_{1}^{+}x_{int}R}_{ \begin{subarray}{c} \text{complex}\\ \text{formation}\end{subarray} }+\underbrace{\pi_{1}^{-}C}_{ \begin{subarray}{c} \text{complex}\\ \text{degradation}\end{subarray} }+\underbrace{d(x_{ext}-x_{int})}_{ \begin{subarray}{c} \text{exchange}\\ \text{with medium}\end{subarray} }\] \[\dot{x}_{ext} =\underbrace{d(x_{int}-x_{ext})}_{ \begin{subarray}{c}\text{ exchange}\\ \text{with medium}\end{subarray} }-\underbrace{\gamma_{A}x_{ext}}_{ \begin{subarray}{c} \text{natural}\\ \text{decay}\end{subarray} }\] \[\dot{R} =\underbrace{\alpha_{R}}_{ \begin{subarray}{c}\text{basic}\\ \text{production}\end{subarray} }-\underbrace{\pi_{1}^{+}x_{int}R}_{ \begin{subarray}{c} \text{complex}\\ \text{formation}\end{subarray} }+\underbrace{\pi_{1}^{-}C}_{ \begin{subarray}{c} \text{complex}\\ \text{degradation}\end{subarray} }-\underbrace{\gamma_{R}R}_{ \begin{subarray}{c} \text{natural}\\ \text{decay}\end{subarray} }\] \[\dot{C} =\underbrace{\pi_{1}^{+}x_{int}R}_{ \begin{subarray}{c}\text{ complex}\\ \text{formation}\end{subarray} }-\underbrace{\pi_{1}^{-}C}_{ \begin{subarray}{c} \text{complex}\\ \text{degradation}\end{subarray} }+\underbrace{\pi_{2}^{-}C_{2}}_{ \begin{subarray}{c} \text{dimer}\\ \text{degradation}\end{subarray} }-\underbrace{2\pi_{2}^{+}C^{2}}_{ \begin{subarray}{c} \text{dimer}\\ \text{formation}\end{subarray} }+\underbrace{\sum_{j=3}^{n}\pi_{j}^{-}C_{j}}_{ \begin{subarray}{c}\text{j-mer}\\ \text{degradation}\end{subarray} }-\underbrace{\sum_{j=3}^{n}\pi_{j}^{+}CC_{j-1}}_{ \begin{subarray}{c}\text{j-mer}\\ \text{degradation}\end{subarray} }\] \[\dot{C}_{i} =\underbrace{\pi_{i}^{+}CC_{i-1}}_{ \begin{subarray}{c}\text{i-mer }\\ \text{formation}\end{subarray} }-\underbrace{\pi_{i}^{-}C_{i}}_{ \begin{subarray}{c} \text{i-mer}\\ \text{degradation}\end{subarray} }+\underbrace{\pi_{i+1}^{-}C_{i+1}}_{ \begin{subarray}{c}\text{(i+1)-mer}\\ \text{degradation}\end{subarray} }-\underbrace{\pi_{i+1}^{+}CC_{i}}_{ \begin{subarray}{c}\text{(i+1)-mer}\\ \text{formation}\end{subarray} }\qquad\text{for $2\leq i\leq n-1$}\] \[\dot{C}_{n} =\underbrace{\pi_{n}^{+}CC_{n-1}}_{ \begin{subarray}{c}\text{n-mer }\\ \text{formation}\end{subarray} }-\underbrace{\pi_{n}^{-}C_{n}}_{ \begin{subarray}{c} \text{n-mer}\\ \text{degradation}\end{subarray} }\.\]  
Although this regulatory pathway seems to be well understood, experimental settings cannot provide information on the dynamics of all components described in system (1). Typically only data for the time course of AHL (and for the population dynamics of the bacteria, which will be introduced in the next step) are available. For this reason, one is interested in a model reduction, decreasing the number of variables and parameters in the system of equations. In a first step, we assume the formation of complexes and its polymers to take place on a fast time scale. Quasi steady state assumptions (\(\varepsilon\to 0\)) yield for the \(n\)-mer,  
\[\varepsilon\dot{C}_{n}=\pi_{n}^{+}CC_{n-1}-\pi_{n}^{-}C_{n}\qquad\underbrace{ \gamma_{\varepsilon\to 0}}_{\varepsilon\to 0}\qquad C_{n}=\left(\frac{\pi_{n}^{+}}{\pi_{n} ^{-}}\right)CC_{n-1}\]Consider now the \((n-1)\)-mer for \(\varepsilon\to 0\) and substitute the last expression. We find  
\[0=\pi_{n-1}^{+}CC_{n-2}-\pi_{n-1}^{-}C_{n-1}+\underbrace{\pi_{n}^{-}C_{n}-\pi_{n }^{+}CC_{n}}_{=0}\]  
It follows that  
\[C_{n}=\left(\frac{\pi_{n}^{+}\pi_{n-1}^{+}}{\pi_{n}^{-}\pi_{n-1}^{-}}\right)C^ {2}C_{n-2}\]  
and, recursively,  
\[C_{n}=\left(\prod_{j=2}^{n}\frac{\pi_{j}^{+}}{\pi_{j}^{-}}\right)C^{n}\]  
We denote \(p_{I}:=\prod_{j=2}^{n}\frac{\pi_{j}^{+}}{\pi_{j}^{-}}\) and substitute the result of the quasi-steady state assumption into the \(I\)-equation in (1), obtaining  
\[\dot{I}=\alpha_{I}+\beta_{I}\frac{p_{I}C^{n}}{I_{th}+p_{I}C^{n}}-\gamma_{I}I.\]  
Observe that the Hill coefficient \(n\) covers the fact that polymers (\(n\)-mers) of the complex PpuR-AHL are relevant for the positive feedback loop (see also [17]).