## 1 Background  
More than twenty years ago it was first discovered that even primitive single-celled organisms such as bacteria are able to communicate with each other and coordinate their behavior [1; 2]. Bacterial communication is based on the exchange of signaling molecules, or autoinducers, which are produced and released in the surrounding space. At the same time, bacteria are able to measure the autoinducer concentration in the environment, and according to this, they can coordinate and even switch their behavior, adapting to environmental changes. The term \"quorum sensing\" was coined to summarize the cell-to-cell communication mechanism thanks to which single bacteria cells are able to measure (\"sense\") the whole population density [3]. Quorum sensing was first observed for the species _Vibrio fischeri_[2], which uses such a mechanism to regulate its bioluminescence. Nowadays, it is known that many bacterial species are able to use similar regulation systems, controlling biofilm formation, swarming motility, and the production of antibiotics or virulence factors [4; 5; 6]. The basis for cell-to-cell communication is a gene regulatory network that not only controls certain target genes, but often also their own production, resulting in a positive feedback loop. Gram-positive bacteria use so-called two-component systems (see e.g., [7]), whereas Gram-negative bacteria produce autoinducers directly in the cells, release them to and take them up from the extracellular space without any further modification or transformation. In the following, we focus on the architecture of a quorum sensing system in Gram-negative bacteria, which mainly communicate via \(N\)-Acyl homoserine lactones (AHLs) [3; 8], typically produced by a synthase. AHL molecules bind to receptors, which control the transcription of target genes. The receptor-AHL complex usually induces the expression of AHL synthases in a positive feedback loop. We restrict our considerations to the bacteria species _Pseudomonas putida_, a root colonizing, plant growth-promoting organism [9]. Nevertheless, these basic principles may be easily transferred to related bacterial species. Mathematical modeling of quorum sensing systems has developed in the last decade. Basic principles for a mathematical approach can be found, for example, in [10], where quasi-steady state assumptions for mRNA and corresponding protein in _Pseudomonas aeruginosa_ were introduced, or in [11], which focuses on the basic feedback system of _Vibrio fischeri_ and the resulting bistability. Alternative approaches for Gram-negative bacteria can be found in [12] (focussing on population dynamics) and [13] (including a further feedback loop). Classical mathematical models for Gram-positive bacteria were introduced, for example, for _Staphylococcus aureus_ in [7; 14; 15]. Several model approaches have also been proposed for _Pseudomonas putida_, in closed systems (batch) as well as in continuous cultures (chemostat) [16; 17; 18]. The goal of this manuscript is to review such models, investigating mathematical properties and principles underlying the equations. The interesting component of quorum sensing models of _Pseudomonas putida_ is that beside a positive feedback for the autoinducer one also finds a negative feedback via an autoinducer-degrading enzyme, a Lactanase. This is initialized with a certain time lag, leading to a system of delay differential equations (DDEs). The paper is organized as follows. In Section 2 we provide a short overview of previous modeling approaches for quorum sensing of _Pseudomonas putida_. Starting from ordinary differential equations (ODEs) for the regulatory network in one single cell, in a second step we extend to quorum sensing in populations, including signal exchange among cells and Lactanase activity. The latter component introduces delays into the system. The delay represents the activation time of the Lactanase-dependent negative feedback. Bacteria population might be considered in batch as well as in continuous cultures.