# A Computational Study of Amensalistic Control of Listeria monocytogenes by Lactococcus lactis under Nutrient Rich Conditions in a Chemostat Setting

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

- ${N}_{1}$: population size of L. monocytogenes,
- ${N}_{2}$: population size of L. lactis,
- C: concentration of lactic acids,
- P: concentration of hydrogen ions,
- M: malate concentration.

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

## 3. Analysis of the Single Species Pathogen Sub-Model

^{7}CFU/mL. The simulations were conducted for various flow rates q, ranging from $q=0.00014$ to $q=0.15$. For the highest of these values $q>{q}_{2}$, whence (12) predicts washout, for the smallest value $q<{q}_{1}$, whence (12) predicts extinction due to self-inhibition.

## 4. Numerical Experiments for the Complete Dual Species Biocontrol Model

#### 4.1. Batch Cultures

#### 4.2. Longterm Behavior of the Full Chemostat Model

#### 4.3. Continuously Adding Control Agents to Eradicate the Pathogens

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Piecewise linear, continuous net growth rate ${g}_{i}(C,P)$: The population grows, $g>0$, for small values of C and P and decays, $g<0$, if either C or P becomes large; the region in between marks the neutral, stationary phase.

**Figure 2.**Simulation of the single species model (8)–(10) for various flow rates q.

**Top panel**: population size ${N}_{1}(t)$;

**Bottom panel:**lactic acid concentration $C(t)$ and hydrogen ion concentration $P(t)$ in the C-P-plane. The vertical and horizontal lines at $C={k}_{7}$ and $C={k}_{8}$ and $P={k}_{9}$ and $P={k}_{10}$ mark the transition from growth to neutral to decay regimes.

**Figure 3.**Simulation of (1)–(5) with $q=0$ and ${N}_{1}(0)={N}_{2}(0)={10}^{7}$ CFU/mL, $C(0)={C}_{0}=0.1$ mM, $P(0)={P}_{0}=0.0001$ mM and $M(0)=4$ mM.

**Figure 4.**Simulation of model (1)–(5) with $q=0$, for initial data ${N}_{1}(0)={10}^{7}$ CFU/mL, ${N}_{2}(0)=k\ast {10}^{7}$ CFU/mL, $C(0)=0.1$ mM, $P(0)=0.0001$ mM, $M(0)=4$ mM. The initial amount of control agent is varied by picking different values for k. The left plot shows the population size of L. monocytogenes for different initial population sizes of L. lactis ($k=0,1,2,4)$. In the right figure the decay time ${t}_{d}$ for L. monocytogenes is plotted for different initial population sizes of L. lactis ($k=1,2,4,8,16,32$).

**Figure 5.**Simulation of model (1)–(5), with ${C}_{0}=0.1mM,\text{}{P}_{0}=0.00001mM$ and $q=0.13$. Note that $q<{q}_{\infty}$ but $q>{\mu}_{1,2}{g}_{1,2}({C}^{\ast},{P}^{\ast})$.

**Figure 7.**Simulation of model (1)–(5), with ${C}_{0}=0.1mM,\text{}{P}_{0}=0.00001mM$ and $q=0.04<{q}^{\ast}$.

**Figure 8.**Population sizes ${N}_{1}^{\ast}$ and ${N}_{2}^{\ast}$ at steady state for model (1)–(5) for varying flow rate q and ${C}_{0}=0.1mM,\text{}{P}_{0}=0.00001mM$.

**Figure 9.**Exploration of the three dimensional parameter space q (x-axis), ${C}_{0}$ (y-axis), ${P}_{0}$ (z-axis). Shown are the sizes of the microbial populations ${N}_{1}$ and ${N}_{2}$ at steady state (4 different views of the same simulation data set).

**Figure 10.**Population dynamics for the control model (16)–(20) with continuously added control agents. For (

**a**) small values of $q<{q}^{\ast}$; and values $q>{q}^{\ast}$ for (

**b**) small and (

**c**) larger amount of control agents added; In (

**d**) eradication time is plotted as a function of the dosage value of the control agent ${N}_{2}^{0}$ for various $q>{q}^{\ast}$.

**Table 1.**Reaction parameters used in this study, from [2].

Parameter | Symbol | Unit | L. lactis | L. monocytogenes |
---|---|---|---|---|

Specific growth rate | ${\mu}_{2},{\mu}_{1}$ | h^{−1} | 0.1049 | 0.1471 |

Protonated acid production rate | $\delta ,\gamma $ | Milimoles CFU^{−1}·h^{−1} | 1.7 * 10^{−10} | 2.95 * 10^{−10} |

MIC acid (growth) | ${k}_{3},{k}_{7}$ | Milimolar | 5.2 | 4.058 |

MIC acid (metabolism) | ${k}_{4},{k}_{8}$ | Milimolar | 8.907 | 8.908 |

Maximum acid concentration | ${k}_{1},{k}_{2}$ | Milimolar | 11.5 | 11.65 |

MIC proton ion (growth) | ${k}_{5},{k}_{9}$ | Milimolar | 10^{−1.405} | 10^{−1.892} |

MIC proton ion (metabolism) | ${k}_{6},{k}_{10}$ | Milimolar | 10^{−1.147} | 10^{−1.151} |

Maximum proton ion concentration | ${k}_{11}$ | Milimolar | 10^{−1.12} | 10^{−1.132} |

Malate decay rate | θ | Milimole CFU^{−1}·h^{−1} | 1.69 * 10^{−10} | 0 |

Malate utilization rate | κ | Milimole^{−1} | 10^{−5.33} | 10^{−5.33} |

proton concentration change rate | ρ | Moles CFU^{−1}·h^{−1} | 10^{−5.472} | 10^{−5.472} |

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**MDPI and ACS Style**

Khassehkhan, H.; Eberl, H.J.
A Computational Study of Amensalistic Control of *Listeria monocytogenes* by *Lactococcus lactis* under Nutrient Rich Conditions in a Chemostat Setting. *Foods* **2016**, *5*, 61.
https://doi.org/10.3390/foods5030061

**AMA Style**

Khassehkhan H, Eberl HJ.
A Computational Study of Amensalistic Control of *Listeria monocytogenes* by *Lactococcus lactis* under Nutrient Rich Conditions in a Chemostat Setting. *Foods*. 2016; 5(3):61.
https://doi.org/10.3390/foods5030061

**Chicago/Turabian Style**

Khassehkhan, Hassan, and Hermann J. Eberl.
2016. "A Computational Study of Amensalistic Control of *Listeria monocytogenes* by *Lactococcus lactis* under Nutrient Rich Conditions in a Chemostat Setting" *Foods* 5, no. 3: 61.
https://doi.org/10.3390/foods5030061