# Mathematical Modeling of Bacteria Communication in Continuous Cultures

^{1}

^{2}

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## Abstract

**:**

## 1. Background

## 2. Methods

#### 2.1. Compartmental Models

#### 2.1.1. Regulatory Pathway in One Cell

#### 2.1.2. Population Dynamics

#### 2.1.3. Lactonase Regulates AHL Degradation

#### 2.1.4. Full Model

#### 2.1.5. Reduced Model

#### 2.2. Experimental Data

#### 2.3. Parameter Estimation

## 3. Results

#### 3.1. Existence of Solutions

**Theorem 1.**

**Proof.**

#### 3.2. Fixed Points

#### 3.3. The case $\mathsf{\tau}=0$

#### 3.4. The Case $\mathsf{\tau}>0$

- no positive intercept with the horizontal axis, if ${a}^{2}{d}^{2}-{b}^{2}{c}^{2}>0$, i.e., if $\left|ad\right|>-bc$;
- one positive intercept (${\phi}_{+}$) with the horizontal axis, if $\left|ad\right|<-bc$.

**Theorem 2.**

#### 3.5. Numerical Simulations and Data Fitting

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Nealson, K.H.; Platt, T.; Hastings, J.W. Cellular control of the synthesis and activity of the bacterial luminescent system. J. Bacteriol.
**1970**, 104, 313–322. [Google Scholar] [PubMed] - Fuqua, W.C.; Winans, S.C.; Greenberg, E.P. Quorum sensing in bacteria: The LuxR-LuxI family of cell density-responsive transcriptional regulators. J. Bacteriol.
**1994**, 176, 269–275. [Google Scholar] [PubMed] - Williams, P.; Winzer, K.; Chan, W.C.; Camara, M. Look who’s talking: Communication and quorum sensing in the bacterial world. Philos. Trans. R. Soc. Lond. Biol.
**2007**, 362, 1119–1134. [Google Scholar] [CrossRef] [PubMed] - Yarwood, J.M.; Bartels, D.J.; Volper, E.M.; Greenberg, E.P. Quorum sensing in Staphylococcus aureus biofilms. J. Bacteriol.
**2004**, 186, 1838–1850. [Google Scholar] [CrossRef] [PubMed] - Whitehead, N.; Barnard, A.; Slater, H.; Simpson, N.; Salmond, G. Quorum-sensing in Gram-negative bacteria. FEMS Microbiol. Rev.
**2001**, 25, 365–404. [Google Scholar] [CrossRef] [PubMed] - Rumbaugha, K.; Griswold, J.; Hamood, A. The role of quorum sensing in the in vivo virulence of Pseudomonas aeruginosa. Microbes Infect.
**2000**, 2, 1721–1731. [Google Scholar] [CrossRef] - Gustafsson, E.; Nilsson, P.; Karlsson, S.; Arvidson, S. Characterizing the dynamics of the quorum sensing system in Staphylococcus aureus. J. Mol. Microbiol. Biotechnol.
**2004**, 8, 232–242. [Google Scholar] [CrossRef] [PubMed] - Cooley, M.; Chhabra, S.R.; Williams, P. N-Acylhomoserine lactone-mediated quorum sensing: A twist in the tail and a blow for host immunity. Chem. Biol.
**2008**, 15, 1141–1147. [Google Scholar] [CrossRef] [PubMed] - Steidle, A.; Sigl, K.; Schuhegger, R.; Ihring, A.; Schmid, M.; Gantner, S.; Stoffels, M.; Riedel, K.; Givskov, M.; Hartmann, A.; et al. Visualization of N-acylhomoserine lactone-mediated cell-cell communication between bacteria colonizing the tomato rhizosphere. Appl. Environ. Microbiol.
**2001**, 67, 5761–5770. [Google Scholar] [CrossRef] [PubMed] - Dockery, J.D.; Keener, J. A mathematical model for quorum sensing in Pseudomonas aeruginosa. Bull. Math. Biol.
**2001**, 63, 95–116. [Google Scholar] [CrossRef] [PubMed] - Müller, J.; Kuttler, C.; Hense, B.A.; Rothballer, M.; Hartmann, A. Cell-cell communication by quorum sensing and dimension-reduction. J. Math. Biol.
**2006**, 53, 672–702. [Google Scholar] [CrossRef] [PubMed] - Ward, J.; King, J.; Koerber, A.; Williams, P.; Croft, J.; Sockett, R. Mathematical modelling of quorum sensing in bacteria. Math Med. Biol.
**2001**, 18, 263–292. [Google Scholar] [CrossRef] - Williams, J.; Cui, X.; Levchenko, A.; Stevens, A. Robust and sensitive control of a quorum-sensing circuit by two interlocked feedback loops. Mol. Syst. Biol.
**2008**, 4. [Google Scholar] [CrossRef] [PubMed] - Jabbari, S.; King, J.; Koerber, A.; Williams, P. Mathematical modelling of the agr operon in Staphylococcus aureus. J. Math. Biol.
**2010**, 61, 17–54. [Google Scholar] [CrossRef] [PubMed] - Koerber, A.; King, J.; Williams, P. Deterministic and stochastic modelling of endosome escape by Staphylococcus aureus: Quorum sensing by a single bacterium. J. Math. Biol.
**2005**, 50, 440–488. [Google Scholar] [CrossRef] [PubMed] - Fekete, A.; Kuttler, C.; Rothballer, M.; Hense, B.A.; Fischer, D.; Buddrus-Schiemann, K.; Lucio, M.; Müller, J.; Schmitt-Kopplin, P.; Hartmann, A. Dynamic regulation of N-acyl-homoserine lactone production and degradation in Pseudomonas putida IsoF. FEMS Microbiol. Ecol.
**2010**, 72, 22–34. [Google Scholar] [CrossRef] [PubMed] - Barbarossa, M.V.; Kuttler, C.; Fekete, A.; Rothballer, M. A delay model for quorum sensing of Pseudomonas putida. Biosystems
**2010**, 102, 148–156. [Google Scholar] [CrossRef] [PubMed] - Buddrus-Schiemann, K.; Rieger, M.; Mühlbauer, M.; Barbarossa, M.V.; Kuttler, C.; Hense, A.B.; Rothballer, M.; Uhl, J.; Fonseca, J.R.; Schmitt-Kopplin, P.; et al. Analysis of N-acylhomoserine lactone dynamics in continuous cultures of Pseudomonas putida IsoF by use of ELISA and UHPLC/qTOF-MS-derived measurements and mathematical models. Anal. Bioanal. Chem.
**2014**, 406, 6373–6383. [Google Scholar] [CrossRef] [PubMed] - Smith, H.L. An Introduction to Delay Differential Equations with Applications to the Life Sciences; Springer: New York, NY, USA, 2011. [Google Scholar]
- Goryachev, A.B.; Toh, D.J.; Lee, T. Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants. Biosystems
**2006**, 83, 178–187. [Google Scholar] [CrossRef] [PubMed] - Kuttler, C.; Hense, B.A. Interplay of two quorum sensing regulation systems of Vibrio fischeri. J. Theor. Biol.
**2008**, 251, 167–180. [Google Scholar] [CrossRef] [PubMed] - Pearson, J.P.; van Delden, C.; Iglewski, B. Active efflux and diffusion are involved in transport or Pseudomonas aeruginosa cell-to-cell signals. J. Bacteriol.
**1999**, 181, 1203–1210. [Google Scholar] [PubMed] - Smith, H.L.; Waltman, P. The Theory of the Chemostat: Dynamics of Microbial Competition; Cambridge University Press: Cambridge, UK, 1995; Volume 13. [Google Scholar]
- Kuang, Y. Delay Differential Equations: With Applications in Population Dynamics; Academic Press: Cambridge, MA, USA, 1993. [Google Scholar]
- Bellen, A.; Zennaro, M. Numerical Methods for Delay Differential Equations; Oxford University Press: Oxford, UK, 2013. [Google Scholar]
- Ellermeyer, S.F. Competition in the chemostat: Global asymptotic behavior of a model with delayed response in growth. SIAM J. Appl. Math.
**1994**, 54, 456–465. [Google Scholar] [CrossRef] - Freedman, H.I.; So, J.W.H.; Waltman, P. Coexistence in a model of competition in the chemostat incorporating discrete delays. SIAM J. Appl. Math.
**1989**, 49, 859–870. [Google Scholar] [CrossRef]

**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.

**Figure 2.**Experimental data and numerical solution of the mathematical model (4). Picture adapted from [18]. Copyright 2014, Springer-Verlag Berlin Heidelberg. The cell population reaches its equilibrium after approximatively 20 h from the beginning of the experiment.

**Figure 3.**Comparison between the numerical solution of the dynamical systems and experimental data. Red curve: solution of the reduced system (8); Blue curve: solution of the full model (4) in [18]. Initial data for the reduced system are $x\left(t\right)={x}_{0}\left(t\right),\phantom{\rule{0.166667em}{0ex}}y\left(t\right)={y}_{0}\left(t\right)$, $t\in [18,20]$, where ${y}_{0}\left(t\right)\equiv 5.2\times {10}^{-13}$ was fitted and ${x}_{0}\left(t\right)\equiv 5.4044\times {10}^{-7}$ is the mean value of ELISA and UHPLC measurements at 19 h from the beginning of the experiment. When the cell population has reached its stationary level, the reduced model provides a good approximation of the dynamics. Parameter values used for the reduced model are given in Table 1.

**Table 1.**Variables and parameters in model (8), with values used for data fit in Figure 3.

Symbol | Description | Value (Unit) | Comments/Source |
---|---|---|---|

${N}^{*}$ | Cell density at equilibrium | $4.5929\times {10}^{11}$ (cells/lit) | [18] |

α | Basic AHL production rate | $1.0564\times {10}^{-7}$ (mol/(lit${}^{2}\xb7$ h)) | $={\alpha}_{A}*{N}_{equi}$, [18] |

γ | AHL decay rate (includes washout) | 0.105 (1/h) | $={\gamma}_{A}+D$, [18] |

δ | Lactonase-dependent degradation rate | $1.5000\times {10}^{-4}$ (lit/(mol · h)) | [18] |

β | Feedback-regulated AHL production rate | $1.0564\times {10}^{-6}$ (mol/(lit${}^{2}\xb7$ h)) | $={\beta}_{A}*{N}_{equi}$, [18] |

n | Hill coefficient for x | 2.3 (dimensionless) | [18] |

${x}_{th}$ | Critical threshold for positive-feedback in x | $3.597\times {10}^{-13}$ (mol/lit) | estimated |

ω | Lactonase decay rate (includes washout) | 0.105 (1/h) | $={\gamma}_{e}+D$, [18] |

ρ | Lactonase production rate | $5.0521\times {10}^{3}$ (mol/(lit${}^{2}\xb7$ h)) | $={\alpha}_{e}*{N}_{equi}$, [18] |

τ | Delay in the release of y | 2 (h) | [18] |

m | Hill coefficient for x | 2.5 (dimensionless) | [18] |

${y}_{th}$ | Critical threshold for positive-feedback in y | $3.597\times {10}^{-13}$ (mol/lit) | estimated |

${x}_{0}\left(t\right)$ | AHL concentration (initial data) $t\in [18,20]$ | $5.4044\times {10}^{-7}$ (mol/lit) | mean of exp. data |

${y}_{0}\left(t\right)$ | Lactonase (initial data) $t\in [18,20]$ | $5.2\times {10}^{3}$ (mol/lit) | estimated |

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Barbarossa, M.V.; Kuttler, C. Mathematical Modeling of Bacteria Communication in Continuous Cultures. *Appl. Sci.* **2016**, *6*, 149.
https://doi.org/10.3390/app6050149

**AMA Style**

Barbarossa MV, Kuttler C. Mathematical Modeling of Bacteria Communication in Continuous Cultures. *Applied Sciences*. 2016; 6(5):149.
https://doi.org/10.3390/app6050149

**Chicago/Turabian Style**

Barbarossa, Maria Vittoria, and Christina Kuttler. 2016. "Mathematical Modeling of Bacteria Communication in Continuous Cultures" *Applied Sciences* 6, no. 5: 149.
https://doi.org/10.3390/app6050149