Oscillatory dynamics mechanism induced by protein synthesis time delay in quorum-sensing system

Recent experimental evidence reports that the oscillatory behavior of quorum sensing plays an extremely important role in the process of bacterial synthesis and release drug to fight cancer. As we know, the six substances AiiA, LuxI, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$ are the core parts of the quorum-sensing system. Here, the effects of several important factors, including time delay, variable ${\mathrm{H}}_2{\mathrm{O}}_2$, AHL synthesis rate induced by LuxI, and AHL degradation rate induced by AiiA on the oscillatory behavior of the quorum-sensing system are studied theoretically based on a part of mathematical model describing the interaction of the above six substances proposed by Prindle et al. [Nature 508, 387 (2014)]. The results show that the time delay is a prerequisite for inducing oscillation of the quorum-sensing system. Furthermore, the length of time delay can determine the amplitude and period of oscillation. As a further matter, the change of ${\mathrm{H}}_2{\mathrm{O}}_2$ concentration can induce the oscillatory behavior of the quorum-sensing system. In addition, under the regulation of ${\mathrm{H}}_2{\mathrm{O}}_2$, the period robustness of the quorum-sensing system is increased. Similarly, the quorum-sensing system exhibits periodic oscillation when AHL synthesis rate induced by LuxI less than a certain critical value, unless it displays a steady state. Additionally, a too-high or too-low level of AHL degradation rate induced by AiiA will fail to generate oscillation of the quorum-sensing system, only the intermediate level will cause oscillation. Finally, the two and three parameter regions in which the quorum-sensing system exhibits oscillation behavior are generated.

Figure 1

Periodic lysis and drug release. (a) Binding of AHL to the receptor protein LuxR leads to activation of the promoter DNA sequence ${\mathrm{P}}_{\text{luxI}}$. This promoter drives the expression of the gene encoding LuxI, which catalyzes the synthesis of AHL, thus form a feed-forward loop. The promoter also drives the expression of drugs that kill cancer cells, as well as cell lytic protein E. AHL can freely diffuse out of cells (red arrow). When bacterial population density is low, AHL mainly diffuses out the extracellular and the circuit is inactive. When the bacterial density increases, the intracellular AHL accumulation reaches a certain threshold that activates the circuit in most cells. (b) Bacterial lysis and releases the drug. A small number of bacteria that survived enter the next cycle.

Figure 2

Schematic diagram of the QS system. Our model consists of six main components: LuxI, AiiA, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$. The main functions of LuxI and AiiA are synthesis and hydrolysis AHL, respectively. When AHL reaching a certain threshold concentration in the cell, which can diffuse out of the cell membrane and regulate cell-to-cell coupling. AHL substrate is able to promote the generation of LuxI by consuming itself. ClpXP is a major negative regulator of LuxI and AiiA, which can inhibition the activity of LuxI and AiiA proteins. ${\mathrm{H}}_2{\mathrm{O}}_2$ can promote the production of AiiA and LuxI proteins, but at the same time also leading to increased degradation rate of AiiA and LuxI by reducing the load of ClpXP. Note that there are two time delays in the process resulting from the synthesis of AiiA and LuxI, denoted as ${\tau }_1$ and ${\tau }_2$, respectively. In this paper, transcription, translation, and maturation of functional AiiA protein and LuxI protein are integrated into a single time-delay parameter $\tau$.

Figure 3

Time evolution of the levels of AiiA, LuxI, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$ without time delay. This displays that the QS system lingers on a stable level state when $\tau =0$.

Figure 4

Time evolution of AiiA, LuxI, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$ under the influence of time delay. This shows that the QS system is still in a in a stable state when $\tau =6.5<{\tau }_0=7.049539$.

Figure 5

Time evolution of the levels of AiiA, LuxI, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$ under the influence of time delay. This indicates that the QS system can generate sustained oscillations when $\tau =7.5>{\tau }_0=7.049539$.

Figure 6

The impact of the length of the time delay on amplitude and period of oscillations. The blue, green, and cyan lines, respectively, represent the time history with $\tau =8$, $\tau =9,$ and $\tau =10$. It is shown that both the amplitudes and periods of the QS oscillations increase with the increasing of time delay $\tau$.

Figure 7

The impact of the length of the time delay on the oscillatory amplitude and period of the QS system in the absence of ${\mathrm{H}}_2{\mathrm{O}}_2$ regulation. The blue, green, and cyan lines, respectively, represent the time history with $\tau =15$, $\tau =16,$ and $\tau =17$.

Figure 8

The impact of the ${\mathrm{H}}_2{\mathrm{O}}_2$ production due to QS fluorophores on oscillatory behavior of the QS system. Time course of the levels of AiiA, LuxI, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$ with $k_{14}=0.6$ and $\tau =8$ min.

Figure 9

The impact of the ${\mathrm{H}}_2{\mathrm{O}}_2$ production due to QS fluorophores on oscillatory behavior of the QS system. Time course of the levels of AiiA, LuxI, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$ with $k_{14}=0.8$ and $\tau =8$ min.

Figure 10

The impact of the ${\mathrm{H}}_2{\mathrm{O}}_2$ production due to QS fluorophores on amplitude and period of oscillations. The blue, green, and cyan lines, respectively, represent the time history with $k_{14}=1$, $k_{14}=1.2,$ and $k_{14}=1.5$. It is shown that the amplitudes of the QS system oscillations increase with the increasing of $k_{14}$, but the periods decrease with the increasing of $k_{14}$.

Figure 11

The impact of the parameter $k_{10}$ (AHL synthesis rate by LuxI) on the QS system oscillations. Time course of the levels of AiiA, LuxI, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$ with $k_{10}=0.95$ and $\tau =7.2$ min.

Figure 12

The impact of the parameter $k_{10}$ (AHL synthesis rate by LuxI) on the QS system oscillations. Time course of the levels of AiiA, LuxI, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$ with $k_{10}=1.1$ and $\tau =7.2$ min.

Figure 13

The impact of the parameter $k_{12}$ (AHL degradation rate by AiiA) on the QS system oscillations. Time course of the levels of AiiA, LuxI, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$ with $k_{12}=0.9$ and $\tau =7.2$ min.

Figure 14

The impact of the parameter $k_{12}$ (AHL degradation rate by AiiA) on the QS system oscillations. Time course of the levels of AiiA, LuxI, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$ with $k_{12}=1.05$ and $\tau =7.2$ min.

Figure 15

The impact of the parameter $k_{12}$ (AHL degradation rate by AiiA) on the QS system oscillations. Time course of the levels of AiiA, LuxI, internal AHL, external AHL, AHL substrate, and ${\mathrm{H}}_2{\mathrm{O}}_2$ with $k_{12}=1.9$ and $\tau =7.2$ min.

Figure 16

(a) Oscillation region generated by simultaneous changes of $k_{14}$ and $\tau$. (b) Oscillation region generated by simultaneous changes of $k_{10}$ and $\tau$. (c) Oscillation region generated by simultaneous changes of $k_{12}$ and $\tau$. (d) Oscillation region generated by simultaneous changes of $k_{10}$, $k_{12}$, and $\tau$.