## 2 Methods
### Compartmental Models
#### 2.1.1 Regulatory Pathway in One Cell  
Let us start to consider the gene regulatory system for a single _Pseudomonas putida_ cell. We follow a standard approach for modeling the quorum sensing system in _Pseudomonas putida_ (_ppu_), analogous to the _lux_ system in _Vibrio fischeri_[11], where polymers of the receptor-AHL complex initiate a positive feedback loop. The autoinducer concentration in _Pseudomonas putida_ is regulated by a (self-induced) positive feedback as well as by a negative feedback via the AHL-degrading enzyme Lactonase. Transcriptional activators PpuR bind to AHLs, forming a PpuR-AHL complex which polymerizes. PpuR-AHL _n_-mers bind to the AHL-dependent quorum sensing locus (ppu-box) and synthesize PpuI. This protein is finally responsible for AHL synthesis. We neglect possible feedbacks (_cf._[13; 20]) on the transcription of PpuR, as these seem to be of minor influence [16]. Thus, just a constant basic production of the receptor PpuR is considered, as in [10; 11; 17]. Further, we assume as in [16; 17; 18] that PpuR-AHL _n_-mers induce synthesis of Lactonase molecules. A schematic representation of this regulatory pathway is given in Figure 1.  
Figure 1: Model structure for the quorum sensing system in one _Pseudomonas putida_ cell. _N_-Acyl homoserine lactone (AHL) concentration is regulated by a (self-induced) positive feedback (\(+\)) as well as by a negative feedback (\(-\)) via the AHL-degrading enzyme Lactonase. The transcriptional activator PpuR binds to AHL forming a PpuR-AHL complex, which polymerizes. PpuR–AHL _n_-mers bind to the AHL-dependent quorum sensing locus (ppu-box) and synthesize PpuI. This protein is finally responsible for AHL synthesis. Similarly, PpuR-AHL _n_-mers induce synthesis of Lactonase molecules. Feedbacks on the transcription of PpuR are neglected. Solid arrows represent activations and inhibitions. Dashed arrows indicate reactions and processes which are partially assumed to be in quasi-steady state. Dotted arrows represent the possible exchange of substances between intracellular and extracellular space. The dashed green ellipse refers to the special case in model version (4), where it is assumed that the total amount of PpuR in one cell (consisting of PpuR and the PpuR-AHL complex) is constant whereas in the other models, PpuR and the PpuR-AHL complex follow their own dynamics. Everywhere in this work, mRNA equations are assumed to be in quasi-steady state. This assumption is justified by the evidence that many proteins are more stable than their own mRNA code (_cf._[10] and references thereof). Let us denote the intracellular concentrations of PpuI and PpuR at time \(t\) by \(I(t)\) and \(R(t)\), respectively. The variables \(C\) and \(C_{i}\) indicate the concentration of the PpuR-AHL complex and of its \(i\)-mer in one bacterial cell, respectively. It is assumed that the formation of complex \(i\)-mers takes place via the combination of an \((i-1)\)-mer with a single PpuR-AHL complex, _cf._[11]. AHL concentration will be denoted by \(x\); in some cases, it might be convenient to distinguish between intracellular (\(x_{int}\)) and extracellular concentration (\(x_{ext}\)). The variable \(y\) denotes Lactonase concentration. To begin with, we consider only the positive feedback which regulates AHL. The Lactonase-degrading activity shall be included in a separate step. The positive feedback loop of the regulatory pathway on the protein level described in Figure 1 can be written in the form of an ODE system (_cf._[21]):  
\[\dot{I} =\underbrace{\alpha_{I}}_{ \begin{subarray}{c}\text{basic}\\ \text{production}\end{subarray} }+\underbrace{\beta_{I}\frac{C_{n}}{I_{th}+C_{n }}}_{ \begin{subarray}{c}\text{feedback-regulated}\\ \text{production}\end{subarray} }-\underbrace{\gamma_{I}I}_{ \begin{subarray}{c} \text{natural}\\ \text{decay}\end{subarray} }\] \[\dot{x}_{int} =\underbrace{\dot{\alpha}I}_{ \begin{subarray}{c}\text{production} \end{subarray} }-\underbrace{\gamma_{I}\alpha_{Int}}_{ \begin{subarray}{c} \text{natural}\\ \text{decay}\end{subarray}