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

Abstract

We study a previously introduced mathematical model of amensalistic control of the foodborne pathogen Listeria monocytogenes by the generally regarded as safe lactic acid bacteria Lactococcus lactis in a chemostat setting under nutrient rich growth conditions. The control agent produces lactic acids and thus affects pH in the environment such that it becomes detrimental to the pathogen while it is much more tolerant to these self-inflicted environmental changes itself. The mathematical model consists of five nonlinear ordinary differential equations for both bacterial species, the concentration of lactic acids, the pH and malate. The model is algebraically too involved to allow a comprehensive, rigorous qualitative analysis. Therefore, we conduct a computational study. Our results imply that depending on the growth characteristics of the medium in which the bacteria are cultured, the pathogen can survive in an intermediate flow regime but will be eradicated for slower flow rates and washed out for higher flow rates.

Keywords: Lactococus lactis; Listeria monocytogenes; biocontrol; computer simulation; mathematical model.

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 $C_0=0.1mM,P_0=0.00001mM$ and $q=0.13$. Note that $q<q_{\infty }$ but $q>{\mu }_{1,2}{g}_{1,2}(C^{\ast },P^{\ast })$.

Figure 6

Simulation of model (1)–(5), with $C_0=0.1mM,P_0=0.00001mM$ and $q=0.08$.

Figure 7

Simulation of model (1)–(5), with $C_0=0.1mM,P_0=0.00001mM$ 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 $C_0=0.1mM,P_0=0.00001mM$.

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 $q>q^{\ast }$.