Modelling and analysing biological oscillations in quorum sensing networks

Abstract

Recent experiments have shown that the biological oscillation of quorum sensing (QS) system play a vital role not only in the process of bacterial synthesis but also in the treatment of cancer by releasing drugs. As known, these five substances TetR, CI, LacI, AiiA and AI are the core components of the QS system. However, the effects of AiiA and protein synthesis time delay on QS system are often ignored in the theoretical model, which is taken as a priority in the proposed research. Therefore, the authors developed a new mathematical model to explore the effects of AiiA and time delay on the dynamical behaviour of QS system theoretically and numerically. The results show that time delay can induce oscillation of QS system. Concretely, there exists a time delay threshold ${\tau }_0$. When time delay is less than ${\tau }_0$, the system is stable. With the increasing of time delay and once it passes ${\tau }_0$, oscillation behaviour occurs. Moreover, the length of time delay determines the amplitude and period of the QS oscillation. In addition, the value of ${\tau }_0$ is sensitive to AiiA. These results may enhance the understanding of QS oscillations and provide new insights for bacterial release drugs to treat cancer.

1 Introduction

Several bacterial taxa can modify their behaviour and regulate gene expression by secretion of chemical signalling molecules, which are called autoinducers. This phenomenon is the consequence of cell–cell communication and is referred as quorum sensing (QS) in biology [1–4]. The QS actively participates in the regulation of various living bacterial lifestyles and physiological processes such as bioluminescence, plasmid binding, biofilm, sporulation, cell differentiation, exercise and antibiotic production etc. [5, 6]. Specifically, the bacteria produces and releases autoinducers and perceive the change in the number of bacteria via the concentration of autoinducers, when this reaches a critical value, the autoinducer can initiate the expression of the relevant gene and coordinate the behaviour of bacteria population [7]. The autoinducer is a special type of water‐soluble small molecule compound, also called a chemical signalling molecule in the QS system, whose concentration will rise with the increasing of bacterial density in the surrounding environment [8]. However, the generation and existence of this signal molecule is completely different at low cell density and high cell density. When the bacterial population density is low, the autoinducer does not accumulate in the internal environment, but rapidly spreads to the external environment. When the bacterial population density is high, the autoinducer accumulates in the internal environment due to the extracellular autoinducer concentration gradient. Once the intracellular autoinducer concentration reaches a certain threshold, it can initiate the gene express and carry out certain behaviours in a collective manner. Therefore, it is almost impossible for a single bacterium to perform certain physiological functions and their mechanics of adjustment, but it can be achieved by a bacterial population [9].

Both Gram‐positive and negative bacteria communicate with the surrounding environment through QS. However, in this study, we only focus on the biological oscillatory behaviour of QS system in gram‐negative bacteria. The following characteristics can be found in almost all gram‐negative bacteria QS system. First, AI is the most universal signalling molecule in gram‐negative bacteria, which is synthesised by the expression of LuxI protein, which can diffuse through the cell membrane thereby mediate cell‐to‐cell coupling. Second, when this signal molecule reaches a certain concentration, it binds to the LuxR produced in the cell and the LuxR‐AI complex is a transcriptional activator of the luxI promoter. Third, AiiA becomes a major negative regulator of luxI promoter by catalysing the degradation of AI. Finally, Gram‐negative bacteria can often alter the expression of dozens or even hundreds of genes designed for multiple biological processes through QS. Just because of these characteristics, Din et al. [10] and Zhou [11] constructed a QS loop in Salmonella typhimurium using the most typical synthetic biology methods of gram‐negative bacterium enabling bacteria to serve as a drug delivery carrier to cure cancer. When the target cells are reached, these bacteria spontaneously decompose and release the drug, which is also described in Fig. 1. The flexible manner to regulate and control the release of drugs is what an ideal drug delivery system should have for cancer treatment. To this end, some researchers proposed that the oscillation pattern of QS system has connection to the way for drugs delivery [12, 13]. In addition, the time and dose of the drug released by the bacteria depends on the period and amplitude of the oscillation of the QS system, respectively. Precisely, these properties and important applications have urged a large number of scholars to propose mathematical models about QS system [12, 14–20].

Fig. 1.

Diagram of cyclic lysis of bacteria and drug release

(a) Binding of AI to the protein LuxR induces activation of the promoter $P_{\text{luxI}}$. This promoter promotes the expression of the LuxI gene, which in turn catalyses the synthesis of AI. In this way, a positive feedback loop is formed. Moreover, the promoter $P_{\text{luxI}}$ can also drive drugs that kill cancer cells and cell lysate E. In addition, AI can diffuse freely between the inside and outside of the cell through the cell membrane, which is described as red arrows, (b) Bacterial lysis and drug release. The bacteria that survived in the previous step re‐entered the next bacterial lysis and drug release

However, on the one hand, in previous research work, the synthesis of most default proteins was usually transient, and the inherent time delays in the transcription and translation steps involved in the emergence of gene fragments to mature proteins were of little concern. However, the dynamic principle for numerous biological functions is strongly associated with the time delay required in gene expression, such as the biological oscillations corresponding to biological rhythms and the chaos corresponding to the frequent discharge of neurons [21–28]. Previously, it was proposed that ignoring the time delay in gene expression is questionable unless their sum is much smaller than other important timescales that characterise the genetic system [29]. However, this constraint is usually unsatisfied in most proteins.

On the other hand, from a biological point of view, a large number of literature works [30–33] suggest that AiiA protein plays a crucial role in the QS system. The main function of AiiA in the QS system is to suppress signal molecule AI. It is worth noting that the important influence of AiiA protein on QS system was also ignored in most previous studies.

As above, it is meaningful and significant topic to discuss how the time delay and AiiA coordinates the oscillatory behaviour of the QS system. Therefore, in this study, we develop a new mathematical model that introduces the variable AiiA and time delay based on the previous model [14, 16–20] to be more comprehensive and deeply capture the underlying oscillation mechanisms of QS system. First, the effect of AiiA and AI on the QS system is studied. Next, how the protein synthesis time delay affects the oscillatory kinetics of the QS system is analysed. Finally, the combinatorial effect including AiiA, AI and time delay is further discussed.

2 Model formulation

In this study, we are mainly concerned about QS pathway involved in multiple feedforward loops and negative feedback loops. The schematic diagram of this QS network interaction is shown schematically in Fig. 2. Specifically, these three proteins TetR, CI and LacI inhibit each other and establish a negative feedback loop. The four substances CI, LacI, AI and AiiA constitute a feedforward loop. The protein CI plays a dual role of directing the inhibition of the synthesis of protein LacI and promoting the synthesis of signal molecule AI. These two substances LacI and AiiA promote each other, and the main function of AiiA is negative regulatory signal molecule AI.

Fig. 2.

Simplified schematic representation of a QS system that contains five major biomolecules: TetR, CI, LacI, AiiA and AI. This red solid line arrows denotes the positive regulator relation (promotion/synthesis), the black arrows with the blunt ends denote the negative regulator relation (inhibition). The blue arrows represent that the autoinducer AI can diffuse freely between the inside and outside of the cell through the cell membrane, and l is the diffusion coefficient. The yellow frame (AiiA) is an important variable added to the original model, which is a major repressor of autoinducer AI

Mathematical model of the gene regulatory network including the QS pathway has been previously proposed [14, 16–20]. However, the important AiiA and time delay are not mentioned in the above QS pathway. Meanwhile, the main regulatory role of AiiA is to inhibit the signal molecule AI, which determines whether QS can be performed. In addition, it is worth mentioning that the protein synthesis process takes a certain time, so the time delay in the gene expression process is ubiquitous and inevitable. Theoretically, time delay usually leads to the system oscillations [34–36]. Specifically, the process from the transcription factor acting on the gene promoter to the appearance of mature mRNA often takes an average of about 10–20 min [37]. At the same time, the synthesis process from mRNA to a mature protein usually takes about 1–3 min [38]. It can be seen that the gene expression process from gene fragment to a mature protein takes about 11–23 min. Hence, based on the above considerations, we developed a new model that has two key differences compared with the previous model. The first difference is that the variable AiiA is added in order to describe more closely to the actual biological significance. The second difference is that the influence of protein synthesis time delay on the QS pathway is considered in our new model. Here, it is assumed that the synthetic time delay of TetR, CI, LacI and AiiA proteins are identical, which is defined as $\tau$. Inspired by the above‐mentioned vital properties, the mathematical model describing the QS network can be directly converted from Fig. 2, which is shown in the following non‐linear delay differential equations.

$$\left\{\begin{array}{rl}& \dot{x}=a\left[-x(t)+\frac{e}{1+{(z(t-\tau ))}^n}\right],\\ & \dot{y}=b\left[-y(t)+\frac{e}{1+{(x(t-\tau ))}^n}\right],\\ & \dot{z}=c\left[-z(t)+\frac{e}{1+{(y(t-\tau ))}^n}+\frac{o{(v(t-\tau ))}^n}{j^n+{(v(t-\tau ))}^n}+\frac{kp(t)}{1+p(t)}\right],\\ & \dot{v}=m\left[-v(t)+\frac{u{(z(t-\tau ))}^n}{q^n+{(z(t-\tau ))}^n}\right],\\ & \dot{p}=-fp(t)+gy(t-\tau )-\frac{rv(t-\tau )p(t)}{h+p(t)}-l(p(t)-U).\end{array}\right.$$ (1)

Here, the meaning of all the variables is given in Table 1. The corresponding meaning of all the parameters and their values are described in Table 2, which will be used in the next numerical calculations unless otherwise stated.

Table 1.

State variable description of the system (1)

Name	Description	Reference
x	the amount of TetR	[16]
y	the amount of CI	[16]
z	the amount of LacI	[16]
v	the amount of AiiA	[15]
p	the amount of AI	[16]

Table 2.

Parameters values in the system (1)

Parameter	Interpretation	Value	Reference
a	the ratio between TetR protein and mRNA decay rate	0.1	[16]
b	the ratio between CI protein and mRNA decay rate	0.1	[16]
c	the ratio between LacI protein and mRNA decay rate	0.1	[16]
m	the ratio between AiiA protein and mRNA decay rate	0.1	[16]
e	the maximum transcription rate in the absence of an inhibitor	8	[17]
o	production rate of LacI by AiiA	1	estimated
j	Michaelis constant of LacI production by AiiA	0.5	estimated
k	the maximal contribution to LacI transcription	20	[18]
u	production rate of AiiA by LacI	1	estimated
q	Michaelis constant of AiiA production by LacI	0.3	estimated
n	hill coefficient	2	[16]
f	the ratio of AI decay rate to mRNA decay rate	1	[16]
g	the production rate of autoinducer AI	0.025	[16]
r	AI degradation rate induced by AiiA	1	[15]
h	AI binding affinity to AiiA	0.1	[15]
l	the AI diffusion coefficient	2	[16]
U	the extracellular concentration of AI	0	[17]
$\tau$	the time delay of synthesis protein	11–23	[37, 38]

3 Results

In this section, we will use a combinational way of theoretical prediction and numerical simulation to comprehensively study the dynamic behaviour of the QS system. Particular attentions are paid to the oscillatory behaviour generated by Hopf bifurcation. Hopf bifurcation, including supercritical and subcritical Hopf bifurcation, is a key principle in the design of biochemical oscillators [39]. In the field of biology, oscillatory behaviours are generated by mutual coupling among multiple genes within a cell. They are used to control various aspects of cellular physiology, from signalling, movement and development to growth, division and death [40, 41]. Furthermore, as described in the Section 1, the oscillation of the QS system is of great significance in the treatment of cancer by bacterial release of drugs. Therefore, it is a tempting topic to explore the Hopf bifurcation resulting from time delay and parameters related to AiiA and signal molecule AI. In addition, how AiiA, AI and time delay cooperate to execute their biological function also is analysed in this section.

3.1 Dependence of dynamics on AiiA and AI

From a biological perspective, AiiA has a very important role in the QS system, which is mainly used to inhibit the concentration of the signal molecule AI. However, AiiA was neglected in the mathematical models proposed in [14, 16–20]. Therefore, we developed a new mathematical model by introducing AiiA. In this part, how the dynamics of the QS system without time delay depend on AiiA and AI are discussed. Specially, AiiA and related regulation reflected by three parameters including production rate of AiiA by LacI $(u)$, production rate of LacI by AiiA $(o)$, the degradation rate of AI enhanced by AiiA $(r)$. Moreover, due to the importance of the concentration of signal molecule AI which can determine whether bacteria can perform QS and launch the expression of a series of genes, we also consider other two parameters, the maximum transcription rate in the absence of an inhibitor $(e)$ and the ratio of AI decay rate to mRNA decay rate $(f)$, which indirectly affect AI. To sum up, the impact of the five parameters, including $(u)$, $(o)$, $(e)$, $(r)$ and $(f)$, on the system dynamics are mainly concerned here.

To broaden the understanding of the influence of AiiA protein on the kinetic of QS system, the dynamic properties of the QS system in response to u, o and r are studied. To show this, first, the bifurcation diagram of QS system versus Production rate of AiiA by LacI, u, is given in Fig. 3 a. It is shown that QS level decreases monotonously with increasing the value of u. Notably, there is a subcritical Hopf bifurcation point at u = 0.786, and beyond this bifurcation point QS remains in a stable steady, otherwise undergoes period oscillation. In addition, the amplitude of these oscillations decreases with the increasing of u in a wide range. Overall, from the perspective of the biomolecules concentration, as the production rate of AiiA promoted by LacI $(u)$ increases, the reverse promotion of LacI by AiiA also increases. Subsequently, the increase of LacI concentration sufficiently inhibited the concentration of TetR, showing a downward trend. Importantly, as mentioned earlier, the dose of drug released by the bacteria depends on the amplitude of the QS system oscillation. Therefore, increasing the production rate of AiiA by LacI $(u)$ can be used to reduce the dose of drug released by bacteria. Second, the dynamic properties of the QS system with regard to production rate of LacI by AiiA, o, are studied. Fig. 3 b displays the bifurcation diagram of QS when o is varied, which has similar dynamic properties to u. Particularly, there is a supercritical Hopf bifurcation point at o = 0.7929, the QS level remains in a low level stable steady state when $o>0.7929$, otherwise, QS will lose stability and a supercritical Hopf bifurcation will occur, resulting in a series of continuous oscillations. Furthermore, with the increasing of o in a wide range, the amplitude of these oscillations exhibits decrease trend. With the continuous increase of the production rate of LacI by AiiA $(o)$, the concentration of LacI also increased continuously to more fully inhibit TetR, which showed a monotonic decreasing trend. In terms of practical application, as previously stated, the dose of drug release is related to the amplitude of the QS system. Therefore, we can draw a biological conclusion from Fig. 3 b that increasing the production rate of LacI by AiiA $(o)$ helps reduce the dose of drug release. Third, it is worth to determine the role of AI degradation rate enhanced by AiiA, r, in the regulatory network of QS. To do so, Fig. 3 c shows that the bifurcation diagram of QS versus r. It can be seen that at low concentrations of r, the corresponding steady state of QS is stable, which suggested that too little r is against the drug release. With the increasing concentrations of r and more than the supercritical Hopf bifurcation point 1.17, QS system loses its stable and undergoes period oscillation. Furthermore, in this case, it is worth noting that the continuous oscillations show almost the same amplitude of r. As AI degradation rate induced by AiiA $(r)$ increases the promotion effect of AI on LacI is weakened and the inhibitory effect of LacI on TetR is also weakened, so that the concentration of TetR is increased. In practical terms, this conclusion means that changing the AI degradation rate induced by AiiA $(r)$ has little effect on changing the dose of drug release.

Fig. 3.

Effects of important model parameters on the QS dynamics

(a) – (c) The bifurcation diagram of QS system level versus production rate of AiiA by LacI $(u)$, production rate of LacI by AiiA $(o)$, AI degradation rate induced by AiiA $(r)$, respectively. This HB represents the Hopf bifurcation, the red line represents a steady state, the black line represents an unstable state and the green dots represent the maximum and minimum of the limit cycle

Next, we consider the other two parameters e, f that affect the signal molecule AI. First, the dynamic features of the QS system with response to e are obtained. Fig. 4 a shows a bifurcation diagram of QS versus e.

Fig. 4.

Effects of important model parameters on the QS dynamics

(a) , (b) The bifurcation diagram of QS system level versus The maximum transcription rate in the absence of an inhibitor $(e)$ and the ratio of AI decay rate to mRNA decay rate $(f)$, respectively. This HB represents the Hopf bifurcation, the red line represents a steady state, the black line represents an unstable state, and the green dots represent the maximum and minimum of the limit cycle

At low value of e the QS remains at a low stable level and then rises sharply with the increasing of e. In addition, the Hopf bifurcation point appears at e = 8.198, and when e beyond this bifurcation point, QS level can undergo a supercritical Hopf bifurcation and sustained oscillations in a wide range. Here, the amplitude of the oscillations is increasing with the increasing of e. As the maximum transcription rate in the absence of an inhibitor $(e)$ increased, the TetR concentration also increased, showing a positive correlation. From a biological perspective, it is concluded that increasing the maximum transcription rate in the absence of an inhibitor $(e)$ can greatly change the dose of drug release. Second, the ratio of AI decay rate to mRNA decay rate, f, is focused. Fig. 4 b indicates the bifurcation diagram of QS versus f. It is discernable that when f is small enough (<2.285), QS remains stable. Moreover, with the increasing of f and passes through the supercritical Hopf bifurcation value 2.285, the QS lost its stability and Hopf bifurcation occurs, which lead to a sustained oscillation. Importantly, one can see that the amplitude of oscillation from HB is rising with the increasing of f. From a practical point of view, as mentioned earlier, the dose of drug release depends on the amplitude of the QS system. Therefore, we can conclude that increasing the ratio of AI decay rate to mRNA decay rate $(f)$ helps to continuously increase the dose of drug release.

Above numerical simulation results show that u, o, r, e and f all has the ability to change the QS oscillation property. Moreover, the oscillatory amplitude and period depend on these five parameters. Remarkably, as previously mentioned, when the bacteria get to the target cancer cells, they lyse and release the drug in a suicidal manner. The dose and time of the drug released by the bacteria is determined by the amplitude and period of QS oscillation, respectively. Therefore, parameters u, o, r indicating the intensity of action of the reaction AiiA on the QS system and parameters e, f affecting the concentration of the signal molecule AI can be used to modulate the dose and time of drug released by bacteria.

3.2 Dependence of dynamic on time delay

From the viewpoint of mathematics, it has been proved that sufficient time delay can be used to generate oscillations when certain conditions are met. However, almost all mathematical models ignored the important protein synthesis delays, let alone for the conditions under which the delay induces oscillations in the QS system. In addition, there is no mention in the related research fields of the effect of the protein synthesis time delay on the bacterial release of drugs in a controlled manner. Based on these concepts, in this subsection, we theoretically and numerically studied the dependence of QS system oscillatory behaviour on time delay. In order to investigate whether such a time delay will cause oscillation of the QS system, the key step is to understand the dynamic behaviour of the QS system without time delay. That is, the protein synthesis time delay is considered to be a transient process (i.e. $\tau =0$). It is calculated that if the protein synthesis time is not considered in the system, the QS system is locally asymptotically stable, as shown in Fig. 5. Instead, under the effects of this time delay (i.e. $\tau >0$), we can get the critical value ${\tau }_0=4.6315$ of time delay with theoretical calculation (see the Appendix). The QS system remains stable when $\tau =4$ which is less than the critical value ${\tau }_0=4.6315$, as can be seen Fig. 6. However, when $\tau =4.8$ more than ${\tau }_0=4.6315$, the equilibrium of QS system loses its stability and meanwhile Hopf bifurcation occurs, which indicates that oscillatory behaviour generated from the bifurcation point, as can be seen from Fig. 7. Taking together, the time delay during the process of protein production can result in oscillation of QS system under the critical condition.

Fig. 5.

Temporal evolution of the concentrations of TetR, CI, LacI, AiiA and AI when

$\tau =0$

(a) The temporal evolution of the concentrations of TetR and CI, (b) The temporal evolution of the concentrations of LacI and AiiA, (c) The temporal evolution of the AI concentration, (d) The phase portrait of CI and TetR

Fig. 6.

Temporal evolution of the concentrations of TetR, CI, LacI, AiiA and AI when

$\tau =4<{\tau }_0=4.6315$

(a) The temporal evolution of the concentrations of TetR and CI, (b) The temporal evolution of the concentrations of LacI and AiiA, (c) The temporal evolution of the AI concentration, (d) The phase portrait of CI and TetR

Fig. 7.

Temporal evolution of the concentrations of TetR, CI, LacI, AiiA and AI when

$\tau =4.8>{\tau }_0=4.6315$

(a) The temporal evolution of the concentrations of TetR, CI and LacI, (b) The temporal evolution of the AiiA concentration, (c) The temporal evolution of the AI concentration, (d) The phase portrait of CI and TetR

To further study the effect of time delay on the oscillatory behaviour of QS system, we will analyse the effect of the change in the parameter $\tau$ on the oscillation period and amplitude through numerical simulation. As a particular example, Figs. 8 and 9 show that the temporal process of the QS system when the parameters taken as $\tau =7$, $\tau =15$ and $\tau =23$. The result indicated that the amplitude and period of QS system oscillations increases with increasing the time delay $\tau$. Collectively, these results indicate that both of the oscillatory amplitudes and the periods are sensitive and critically rely on the time delay $\tau$, which reveal that the time delay provide a highly tunable period and amplitude of QS oscillation. However, the dose and time of the drug released by the bacteria are determined by the amplitude and period of the QS oscillation, respectively. Thus, the dose and timing of the drug released by the bacteria can be flexibly designed by adjusting the length of the time delay of protein synthesis. An important question arises how to actually change the length of the protein synthesis time delay. Experiments have shown that the protein synthesis time delay can be changed by the transcription inhibitors rifampicin (Rif), and translation inhibitors chloramphenicol (Cam) and kasugamycin (Ksg) [42–45].

Fig. 8.

Change of amplitude and period of oscillations as the length of the time delay varies. Blue, green and cyan lines, respectively show the temporal process with $\tau =7$, $\tau =15$ and $\tau =23$

(a) The comparation diagram of the TetR oscillation, (b) The comparation diagram of the CI oscillation, (c) The comparation diagram of the LacI oscillation

Fig. 9.

Change of amplitude and period of oscillations as the length of the time delay varies. Blue, green and cyan lines, respectively, shown the temporal process with $\tau =7$, $\tau =15$ and $\tau =23$

(a) The comparation diagram of the AiiA oscillation, (b) The comparation diagram of the AI oscillation

3.3 Dependence of dynamics on the combinatorial regulation of AiiA, AI and time delay

As above, under the critical condition, time delay can cause larger oscillations and play a pivotal role in controlling the dynamics of QS system. Moreover, the time delay can be adjust to a desired value via modern biotechnology. An issue comes naturally that when AiiA and AI are constantly changing, how does the critical value of the time delay change, which decides the efficiency of drug release by bacteria. Here, this issue is reflected by the dependence of time delay threshold ${\tau }_0$ on the above five important parameters. Therefore, our next investigation is about the effects of different important model parameters on the critical value ${\tau }_0$ of the time delay $\tau$ with combination dynamics simulation approaches. Fig. 10 a shows how the variation trend of the critical value ${\tau }_0$ of the time delay $\tau$ depends on the gradual increase of u. It can be observed that the overall trend of the critical value ${\tau }_0$ is still increasing as the value of u is gradually increased. When u is between 0.2 and 0.5, the growth trend of this critical value ${\tau }_0$ is relatively flat. When u is between 0.5 and 1, the growth trend of this critical value ${\tau }_0$ is more prominent. When u exceeds 1, the growth trend of the critical value ${\tau }_0$ restored a relatively flat trend. It can be concluded that when the production rate of AiiA by LacI $(u)$ increases, a longer time delay is required to generate the QS system oscillation, reducing the efficiency of starting bacteria to release drugs. Fig. 10 b shows the trend of the critical value ${\tau }_0$ of the time delay $\tau$ when the parameter o is constantly changed. It can be seen that with the increasing value of o continuously, the upward trend and degree of this critical value ${\tau }_0$ become more and more obvious. With the increase in the production rate of LacI by AiiA $(o)$, a larger time delay is required to cause the oscillation of QS system, thereby starting the bacterial population to release the drug. Fig. 10 c and 11 a shows the variation trend of the critical value ${\tau }_0$ when the parameters r and e are gradually increased, respectively. The trend graphs that change the two parameters r and e affecting ${\tau }_0$ are very similar, but compared to parameters u and o are opposite. As depicted, the critical value ${\tau }_0$ shows a downward trend with the gradual increase of r and e, and the decline is more and more gentle. With the increase of the maximum transcription rate in the absence of an inhibitor $(e)$ and AI degradation rate induced by AiiA $(r)$, the QS system needs a shorter time delay to generate oscillation, which can increase the efficiency of the bacterial population to release drugs. Fig. 11 b shows the trend diagram of critical value ${\tau }_0$ versus f. As drawn, the overall variation trend of critical value ${\tau }_0$ presents a declining state with the gradual increase of f. In addition, the downward trend is becoming more and more pronounced, and there is a tendency to continue to fall substantially. With the increase of the ratio of AI decay rate to mRNA decay rate $(f)$, the QS system needs a short time delay to generate oscillations, which can make bacteria release drugs more efficiently.

Fig. 10.

This trend graph shows the effect of important parameters on the time delay critical value ${\tau }_0$. The number next to the red point represents a different thresholds ${\tau }_0$ for the different parameters

(a) The changing trend of time delay critical value 0 with respect to the production rate of AiiA by LacI, (b) The changing trend of time delay critical value 0 with respect to the production rate of LacI by AiiA, (c) The changing trend of time delay critical value 0 with respect to the maximum transcription rate in the absence of an inhibitor

Fig. 11.

This trend graph shows the effect of important parameters on the time delay critical value ${\tau }_0$. The number next to the red point represents a different thresholds ${\tau }_0$ for the different parameters

(a) The changing trend of time delay critical value 0 with respect to the AI degradation rate induced by AiiA, (b) The changing trend of time delay critical value 0 with respect to the ratio of AI decay rate to mRNA decay rate

4 Conclusion

In this study, we have developed a mathematical model that includes AiiA and a protein synthesis time delay, for purpose to understand the oscillatory mechanism of the QS system and how bacteria can controllably release drugs. However, it is difficult to detect oscillations in practice, which requires experiments to observe amount of spatiotemporal data [37]. Instead, numerous mathematical methods can be used to predict the presence of these oscillations [46, 47]. It prompted us to apply mathematical methods to predict the dynamics of QS systems and its practical application. The main work of this article is to study how the oscillation behaviour of the QS system changes when AiiA, AI and time delay are changed, respectively. Furthermore, the effect of the dynamics behaviour of QS system on the dose and time of drug release by bacteria is analysed. The combinatorial regulation of AiiA, AI and time delay is discussed, i.e. what is the trend of the critical value of time delay when AiiA, AI concentration changes. In other words, when the concentrations of AiiA and AI change, the QS system requires a longer or shorter time delay to initiate the release drug by bacteria. Specifically, first, how the dynamic behaviours of QS system depend on AiiA and AI are studied, which is reflected by five parameters u (production rate of AiiA by LacI), o (production rate of LacI by AiiA), e (the maximum transcription rate in the absence of an inhibitor), r (AI degradation rate induced by AiiA) and f (the ratio of AI decay rate to mRNA decay rate). The research demonstrated that these five parameters can drive the QS oscillation. The parameters u and o have similar dynamics properties, and the Hopf bifurcation critical values of u and o can be found, which drive QS system from oscillations to stability. Furthermore, the parameters r, e and f have similar dynamic properties, and these three parameters also have the capacity to drive Hopf bifurcation, which can make the system transit from a steady state to an oscillatory state. In general, these three parameters u, o, r reflect the important role of AiiA protein and the two parameters e and f reflect the important role of AI protein on QS system. Second, the oscillatory behaviour of QS system is analysed by taking time delay as a control parameter in two cases of attendance and absence of time delay. The results revealed that the critical value ${\tau }_0$ of the time delay $\tau$ plays a crucial role in inducing the oscillation of QS system. Moreover, the length of the delay can regulate the oscillatory amplitude and period. In addition, how the dynamics of QS system depend on the combinatorial regulation of AiiA, AI and time delay are further discussed, which is shown via the changes of the above five important parameters affect the critical value ${\tau }_0$ of the time delay $\tau$. The results show that the trend and extent of critical value ${\tau }_0$ continue to be changed with the increase of different parameters.

According to the above theoretical analysis and numerical simulation, we can obtain the following biological conclusions. As mentioned previously, the dose and time at which the bacteria release the drug is determined by the amplitude and period of the QS oscillation. Our results show that larger time delays can result in bigger amplitude and period of QS oscillations. Therefore, in order to meet the desired drug dosage and release time required to treat cancer, the period and amplitude of the oscillation can be further adjusted by changing the length of the time delay, thereby indirectly adjusting the dose and time of drug release. This involves a problem, how to adjust the length of the time delay in the actual biological system. In particular, the time delay herein refers to the time required to synthesise the four proteins TetR, CI, LacI and AiiA (i.e. the gene fragment is transcribed and translated into a mature protein). Here, it is assumed that the synthesis time of these three proteins is identical. Consequently, in order to change the length of the time delay, it is necessary to regulate the time required for transcription and translation involved in gene expression. To our knowledge, both transcription and translation process contains three stages, namely initiation, elongation and termination [42]. Recently, experiments have shown that Rif, a transcriptional inhibitor, has the ability to block transcription initiation and inhibit bacterial RNA polymerase entry into the prolonged phase of mRNA [43, 44]. Therefore, the length of time delay can be increased by using Rif via inhibiting the initiation and elongation of transcription. Furthermore, both Cam and Ksg can act as translation inhibitors [43]. The main difference between Cam and Ksg is that the former inhibits the elongation of translation [45], while the latter inhibits the initiation of translation [42]. Therefore, Cam and Ksg can increase the length of time delay by inhibiting the elongation and initiation of translation. In summary, Rif, Cam and Ksg can be used to modulate the time delay required for protein synthesis.

In comparison with other scholars' research content in the relevant area, the uniqueness of this study lies in three aspects. (i) Comparing the works [14, 16–20], there are two key differences in our research. The first difference is that all of the models ignore the role of time delay on the QS pathway, which is considered as a key consideration in this study. The second difference is that the important element AiiA is rarely considered, which suppresses the signal molecule AI in the QS network. The new model in this study including the above two factors, which is closer to the actual biological significance. The present research can be viewed as a complement and advancement of the previous research, and it also clearly reveals the important role of time delay in the dynamic behaviour of the QS system. (ii) In the absence of time delay, the dynamic effects of QS system are elucidated when varying AiiA and AI via changing some important parameters. In addition, how critical value ${\tau }_0$ of time delay $\tau$ depends on these important model parameters is also analysed. (iii) In the study, the dynamic behaviour of QS system generated by time delay and important model parameters is studied via a combination way of quantitative and qualitative analysis. Theoretical analysis shows that the system has a time delay threshold value ${\tau }_0$, which is the Hopf bifurcation point. When the time delay gradually increases and passes through the critical value, the Hopf bifurcation occurs, and meanwhile oscillation appears. In addition, the period and amplitude of the oscillation of the QS system is largely affected by the length of the time delay. The numerical results are consistent with the theoretical predictions.

Currently, our research model is based on biological facts to improve and perfect the model proposed by predecessors. Next, we will use big data and statistical methods for modelling and breaking through the current limitations. In addition, our current research results are still limited to the theoretical level. If we can combine experiments and prove their practical application in the clinic, which will probably be more meaningful.

This section discusses the stability of the equilibrium and the existence of local Hopf bifurcations of the system (1). In particular, since the system has specific biological significance, only the positive equilibrium is concerned here. Suppose the system (1) has a positive equilibrium denoted by $E^{\ast }=(X^{\ast },Y^{\ast },Z^{\ast },V^{\ast },P^{\ast })$, then it satisfies the following equation:

$$\left\{\begin{array}{rl}& a\left[-X^{\ast }+\frac{e}{1+{(Z^{\ast })}^n}\right]=0,\\ & b\left[-Y^{\ast }+\frac{e}{1+{(X^{\ast })}^n}\right]=0,\\ & c\left[-Z^{\ast }+\frac{e}{1+{(Y^{\ast })}^n}+\frac{o{(V^{\ast })}^n}{j^n+{(V^{\ast })}^n}+\frac{k{P}^{\ast }}{1+P^{\ast }}\right]=0,\\ & m\left[-V^{\ast }+\frac{u{(Z^{\ast })}^n}{q^n+{(Z^{\ast })}^n}\right]=0,\\ & -f{P}^{\ast }+g{Y}^{\ast }-\frac{r{V}^{\ast }{P}^{\ast }}{h+P^{\ast }}-l(P^{\ast }-U)=0.\end{array}\right.$$ (2)

Through the variable transforms $\overline{x}(t)=x(t-\tau )-X^{\ast }$, $\overline{y}(t)=y(t-\tau )-Y^{\ast }$, $\overline{z}(t)=z(t-\tau )-Z^{\ast }$, $\overline{v}(t)=v(t-\tau )-V^{\ast }$ and $\overline{p}(t)=p(t)-P^{\ast }$ and still use x, y, z, v, p $\overline{x}$ to present $\overline{y}$, $\overline{z}$, $\overline{v}$, $\overline{p}$, then the system [1] can be transformed into the following form:

$$\left\{\begin{array}{rl}& \dot{x}=-ax(t)+b_{11}z(t-\tau )+f_1,\\ & \dot{y}=-by(t)+b_{12}x(t-\tau )+f_2,\\ & \dot{z}=-cz(t)+b_{13}y(t-\tau )+b_{14}v(t-\tau )+b_{15}p(t)+f_3,\\ & \dot{v}=-mv(t)+b_{16}z(t-\tau )+f_4,\\ & \dot{p}=-(f+l+b_{17})p(t)+gy(t-\tau )p(t)-b_{18}v(t-\tau )+f_5,\end{array}\right.$$ (3)

where

$$b_{11}=-\frac{2e{Z}^{\ast }}{{(1+{(Z^{\ast })}^2)}^2},b_{12}=-\frac{2e{X}^{\ast }}{{(1+{(X^{\ast })}^2)}^2},$$

$$b_{13}=-\frac{2e{Y}^{\ast }}{{(1+{(Y^{\ast })}^2)}^2},b_{14}=\frac{2o{j}^2{V}^{\ast }}{{(j^2+V^{\ast 2})}^2},$$

$$b_{15}=\frac{k}{{(1+P^{\ast })}^2},b_{16}=\frac{2u{q}^2{Z}^{\ast }}{{(q^2+Z^{\ast 2})}^2},$$

$$b_{17}=\frac{hr{V}^{\ast }}{{(h+P^{\ast })}^2},b_{18}=\frac{r{P}^{\ast }}{h+P^{\ast }},$$

$$f_1=c_{11}{z}^2(t-\tau )+c_{12}{z}^3(t-\tau ),$$

$$f_2=c_{13}{x}^2(t-\tau )+c_{14}{x}^3(t-\tau ),$$

$$\begin{array}{rl}{f}_3& =c_{15}{y}^2(t-\tau )+c_{16}{y}^3(t-\tau )+c_{17}{v}^2(t-\tau )\\ & +c_{18}{v}^3(t-\tau )+c_{19}{p}^2(t)+c_{20}{p}^3(t),\end{array}$$

$$f_4=c_{21}{z}^2(t-\tau )+c_{22}{z}^3(t-\tau ),$$

$$\begin{array}{rl}{f}_5& =c_{23}{p}^2(t)+c_{24}v(t-\tau )p(t)+c_{25}{p}^3(t)\\ & +c_{26}v(t-\tau )p^2(t),\end{array}$$

and

$$c_{11}=\frac{2e(3Z^{\ast 2}-1)}{{(1+Z^{\ast 2})}^3},c_{12}=-\frac{24e(Z^{\ast 3}-Z^{\ast })}{{(1+Z^{\ast 2})}^4},$$

$$c_{13}=\frac{2e(3X^{\ast 2}-1)}{{(1+X^{\ast 2})}^3},c_{14}=-\frac{24e(X^{\ast 3}-X^{\ast })}{{(1+X^{\ast 2})}^4},$$

$$c_{15}=\frac{2e(3Y^{\ast 2}-1)}{{(1+Y^{\ast 2})}^3},c_{16}=-\frac{24e(Y^{\ast 3}-Y^{\ast })}{{(1+Y^{\ast 2})}^4},$$

$$c_{17}=-\frac{2j^{2}o(-j^2+3V^{\ast 2})}{{(j^2+V^{\ast 2})}^3},c_{18}=\frac{24o{V}^{\ast }(-j^4+j^2{V}^{\ast 2})}{{(j^2+V^{\ast 2})}^4},$$

$$c_{19}=-\frac{2k}{{(1+P^{\ast })}^3},c_{20}=\frac{6k}{{(1+P^{\ast })}^4},$$

$$c_{21}=-\frac{2q^{2}u(-q^2+3Z^{\ast 2})}{{(q^2+Z^{\ast 2})}^3},c_{22}=\frac{24u{Z}^{\ast }(-q^4+q^2{Z}^{\ast 2})}{{(q^2+Z^{\ast 2})}^4},$$

$$c_{23}=-\frac{2hr{V}^{\ast }}{{(h+P^{\ast })}^3},c_{24}=\frac{rh}{{(h+P^{\ast })}^2},$$

$$c_{25}=\frac{6hr{V}^{\ast }}{{(h+P^{\ast })}^4},c_{26}=-\frac{2hr}{{(h+P^{\ast })}^3}.$$

The characteristic equation of the linearised system corresponding to the system (3) is

$$\begin{array}{cc}\text{det}& \left(\begin{array}{ccccc}\lambda +a& 0& -b_{11}{e}^{-\lambda \tau }& 0& 0\\ -b_{12}{\mathrm{e}}^{-\lambda \tau }& \lambda +b& 0& 0& 0\\ 0& -b_{13}{\mathrm{e}}^{-\lambda \tau }& \lambda +c& -b_{14}{\mathrm{e}}^{-\lambda \tau }& -b_{15}\\ 0& 0& -b_{16}{\mathrm{e}}^{-\lambda \tau }& \lambda +m& 0\\ 0& -g{\mathrm{e}}^{-\lambda \tau }& 0& b_{18}{\mathrm{e}}^{-\lambda \tau }& \lambda +U\end{array}\right)=0,\end{array}$$ (4)

where

$$U=(f+l+b_{17}).$$

The above characteristic equation can be reduced to the following exponential polynomial equation:

$$\begin{array}{rl}& -{\lambda }^5+p_1{\lambda }^4+p_2{\lambda }^3+p_3{\lambda }^2+p_4\lambda +p_5\\ & +(q_1{\lambda }^2+q_2\lambda +q_3){\mathrm{e}}^{-3\lambda \tau }+(h_1{\lambda }^2+h_2\lambda +h_3){\mathrm{e}}^{-2\lambda \tau }\\ & & +(l_1{\lambda }^3+l_2{\lambda }^2+l_3\lambda +l_4){\mathrm{e}}^{-\lambda \tau }=0,\end{array}$$ (5)

where

$$p_1=-a-b-c-m-U,$$

$$p_2=-ab-ac-bc-am-bm-cm-aU-bU-cU-mU,$$

$$\begin{array}{rl}{p}_3& =-abc-abm-acm-bcm-abU-acU-bcU-amU\\ & -bmU-cmU,\end{array}$$

$$p_4=-abcm-abcU-abmU-acmU-bcmU,$$

$$p_5=-abcmU,q_1=b_{11}{b}_{12}{b}_{13},$$

$$q_2=m{b}_{11}{b}_{12}{b}_{13}+U{b}_{11}{b}_{12}{b}_{13}+g{b}_{11}{b}_{12}{b}_{15},$$

$$q_3=mU{b}_{11}{b}_{12}{b}_{13}+gm{b}_{11}{b}_{12}{b}_{15},h_1=-(b_{15}{b}_{16}{b}_{18}),$$

$$h_2=-(a{b}_{15}{b}_{16}{b}_{18}+b{b}_{15}{b}_{16}{b}_{18}),h_3=-ab{b}_{15}{b}_{16}{b}_{18},$$

$$l_1=b_{14}{b}_{16},l_2=a{b}_{14}{b}_{16}+b{b}_{14}{b}_{16}+U{b}_{14}{b}_{16},$$

$$l_3=ab{b}_{14}{b}_{16}+aU{b}_{14}{b}_{16}+bU{b}_{14}{b}_{16},l_4=abU{b}_{14}{b}_{16}.$$

As we all known, if all the roots of the characteristic (5) have a strict negative real part, then the equilibrium point $E^{\ast }=(X^{\ast },Y^{\ast },Z^{\ast },V^{\ast },P^{\ast })$ is locally asymptotically stable. However, if a pair of pure imaginary roots appears, the equilibrium will lose the stability.

To discuss the distribution of the roots of the characteristic (5), we introduce the following lemma.

For the following transcendental equation

$$\begin{array}{rl}& p(\lambda ,{\mathrm{e}}^{-\lambda {\tau }_1},\dots ,{\mathrm{e}}^{-\lambda {\tau }_m})\\ & ={\lambda }^n+p_1^{(0)}{\lambda }^{n-1}+\cdots +p_{n-1}^{(0)}\lambda +p_n^{(0)}\\ & +[p_1^{(1)}{\lambda }^{n-1}+\cdots +p_{n-1}^{(1)}\lambda +p_n^{(1)}]{\mathrm{e}}^{-\lambda {\tau }_1}\\ & +\cdots +[p_1^{(m)}{\lambda }^{n-1}+\cdots +p_{n-1}^{(m)}\lambda +p_n^{(m)}]{\mathrm{e}}^{-\lambda {\tau }_m},\end{array}$$

where ${\tau }_i\ge (i=1,2,\dots ,m)$ and $p_j^{(i)}(i=1,2,\dots ,m;j=1,2,\dots ,n)$ are constants. Then as $({\tau }_1,{\tau }_2,\dots ,{\tau }_m)$ vary, the sum of orders of the zeros of $p(\lambda ,{\mathrm{e}}^{-\lambda {\tau }_1},\dots ,{\mathrm{e}}^{-\lambda {\tau }_m})$ on the right half open plane can change only when a zero root or a pair of pure virtual roots appears [48].

In order to study the distribution of the roots of the characteristic (5), we discuss it in two cases including with and without time delay.

Case I: If there is no time delay, i.e. $\tau =0$, then the characteristic (5) can be reduced to

$$\begin{array}{rl}& -{\lambda }^5+p_1{\lambda }^4+(p_2+l_1){\lambda }^3+(p_3+q_1+l_2+h_1){\lambda }^2\\ & +(p_4+q_2+h_2+l_3)\lambda +(p_5+q_3+h_3+l_4)=0.\end{array}$$ (6)

According to the famous Routh–Hurwitz criterion, the necessary and sufficient conditions for all roots of (6) to have a strict negative real part are

$${\mathrm{\Delta }}_1=p_1>0,$$

$${\mathrm{\Delta }}_2=\left|\begin{array}{cc}{p}_1& -1\\ A_1& p_2+l_1\end{array}\right|>0,$$

$${\mathrm{\Delta }}_3=\left|\begin{array}{ccc}{p}_1& -1& 0\\ A_1& p_2+l_1& p_1\\ A_3& A_2& A_1\end{array}\right|>0,$$

$${\mathrm{\Delta }}_4=\left|\begin{array}{cccc}{p}_1& -1& 0& 0\\ A_1& p_2+l_1& p_1& -1\\ A_3& A_2& A_1& p_2+l_1\\ 0& 0& A_3& A_2\end{array}\right|>0,$$

$${\mathrm{\Delta }}_5=\left|\begin{array}{ccccc}{p}_1& -1& 0& 0& 0\\ A_1& p_2+l_1& p_1& -1& 0\\ A_3& A_2& A_1& p_2+l_1& p_1\\ 0& 0& A_3& A_2& A_1\\ 0& 0& 0& 0& A_3\end{array}\right|>0,$$

where $A_1=p_3+q_1+l_2+h_1,A_2=p_4+q_2+l_3+h_2,A_3=p_5+q_3+l_4+h_3$

when the assumption (H1) is established, i.e. ${\mathrm{\Delta }}_i>0(i=1,2,3,4,5)$, then the equilibrium $E^{\ast }=(X^{\ast },Y^{\ast },Z^{\ast },V^{\ast },P^{\ast })$ is locally asymptotically stable.

Case II: If the time delay exists, i.e. $\tau >0$

$$\begin{array}{rl}& -{\lambda }^5+p_1{\lambda }^4+p_2{\lambda }^3+p_3{\lambda }^2+p_4\lambda +p_5\\ & +(q_1{\lambda }^2+q_2\lambda +q_3)e^{-3\lambda \tau }+(h_1{\lambda }^2+h_2\lambda +h_3)e^{-2\lambda \tau }\\ & +(l_1{\lambda }^3+l_2{\lambda }^2+l_3\lambda +l_4)e^{-\lambda \tau }=0,\end{array}$$ (7)

Suppose $\pm i\omega (\omega >0)$ is a pair of pure virtual roots of (7), then $\omega$ satisfies the following conditions

$$\begin{array}{rl}& p_5+i{p}_4\omega -p_3{\omega }^2-i{p}_2{\omega }^3+p_1{\omega }^4-i{\omega }^5\\ & +(l_4+i{l}_3\omega -l_2{\omega }^2-i{l}_1{\omega }^3)(\cos (\omega \tau )-i\sin (\omega \tau ))\\ & +(h_3+i{h}_2\omega -h_1{\omega }^2)(\cos (2\omega \tau )-i\sin (2\omega \tau ))\\ & +(q_3+i{q}_2\omega -q_1{\omega }^2)(\cos (3\omega \tau )-i\sin (3\omega \tau ))=0.\end{array}$$ (8)

Separating the real and imaginary parts, we get

$$\left\{\begin{array}{rl}& (l_4-l_2{\omega }^2)\cos (\omega \tau )+(l_3\omega -l_1{\omega }^3)\sin (\omega \tau )\\ & =H\omega +R{\omega }^2-p_1{\omega }^4+J,\\ & (l_3\omega -l_1{\omega }^3)\cos (\omega \tau )+(-l_4\omega +l_2{\omega }^2)\sin (\omega \tau )\\ & =L\omega +T{\omega }^2+p_2{\omega }^3+{\omega }^5+Q.\end{array}\right.$$ (9)

where

$$H=-h_2\sin (2\omega \tau )-q_2\sin (3\omega \tau ),$$

$$R=p_3+h_1\cos (2\omega \tau )+q_1\cos (3\omega \tau ),$$

$$J=-p_5-h_3\cos (2\omega \tau )-q_3\cos (3\omega \tau ),$$

$$L=-h_2\cos (2\omega \tau )-q_2\cos (3\omega \tau ),$$

$$T=-h_1\sin (2\omega \tau )-q_1\sin (3\omega \tau ),$$

$$Q=h_3\sin (2\omega \tau )+q_3\sin (3\omega \tau )-p_4.$$

Simplify the above formula and get the following equation

$$\left\{\begin{array}{c}\sin (\omega \tau )=\frac{-l_{4}Q+e_9\omega +e_{10}{\omega }^2+e_{11}{\omega }^3+e_{12}{\omega }^4+e_{13}{\omega }^5+e_{14}{\omega }^7}{{(l_4)}^2+e_7{\omega }^2+e_8{\omega }^4+{(l_1)}^2{\omega }^6},\\ \cos (\omega \tau )=\frac{l_{4}J+e_1\omega +e_2{\omega }^2+e_3{\omega }^3+e_4{\omega }^4+e_5{\omega }^5+e_6{\omega }^6-l_1{\omega }^8}{{(l_4)}^2+e_7{\omega }^2+e_8{\omega }^4+{(l_1)}^2{\omega }^6},\end{array}\right.,$$ (10)

where

$$e_1=H{l}_4+Q{l}_3,e_2=-J{l}_2+L{l}_3+R{l}_4,$$

$$e_3=-H{l}_2-Q{l}_1+T{l}_3,e_4=-L{l}_1-l_4{p}_1+l_3{p}_2-l_{2}R,$$

$$e_5=-l_{1}T,e_6=l_3+p_1{l}_2-l_1{p}_2,e_7=(l_3){}^2-2l_2{l}_4,$$

$$e_8=(l_2){}^2-2l_1{l}_3,e_9=J{l}_3-L{l}_4,e_{10}=H{l}_3+Q{l}_2-T{l}_4,$$

$$e_{11}=-J{l}_1+L{l}_2-l_4{p}_2+R{l}_3,e_{12}=-H{l}_1+T{l}_2,$$

$$e_{13}=-l_4-l_3{p}_1+l_2{p}_0-R{l}_1,e_{14}=l_2+l_1{p}_1.$$

Employing ${\cos }^2(\omega \tau )+{\sin }^2(\omega \tau )=1$, one can obtain the following equation

$$\begin{array}{rl}& h_0+h_1\omega +h_2{\omega }^2+h_3{\omega }^3+h_4{\omega }^4+h_5{\omega }^5+h_6{\omega }^6+h_7{\omega }^7\\ & +h_8{\omega }^8+h_9{\omega }^9+h_{10}{\omega }^{10}+h_{11}{\omega }^{11}+h_{12}{\omega }^{12}-2e_5{l}_1{\omega }^{13}\\ & +h_{13}{\omega }^{14}+{(l_1)}^2{\omega }^{16}=0,\end{array}$$ (11)

where

$$h_0=J^2{l}_4^2-l_4^4+l_4^4{Q}^2,$$

$$h_1=2e_{1}J{l}_4-2e_9{l}_{4}Q,$$

$$h_2=e_1^2+e_9^2+2e_{2}J{l}_4-2e_7{l}_4^2-2e_{10}{l}_{4}Q,$$

$$h_3=2e_1{e}_2+2e_9{e}_{10}+2e_{3}J{l}_4-2e_{11}{l}_{4}Q,$$

$$\begin{array}{rl}{h}_4& =e_{10}^2+e_2^2+2e_1{e}_3-e_7^2+2e_9{e}_{11}+2e_{4}J{l}_4-2e_8{l}_4^2\\ & -2e_{12}{l}_{4}Q,\end{array}$$

$$h_5=2e_{10}{e}_{11}+2e_2{e}_3+2e_1{e}_4+2e_{12}{e}_9+2e_{5}J{l}_4-2e_{13}{l}_{4}Q,$$

$$\begin{array}{rl}{h}_6& =e_{11}^2+2e_{10}{e}_{12}+e_3^2+2e_2{e}_4+2e_1{e}_5-2e_7{e}_8+2e_{13}{e}_9\\ & +2e_{6}J{l}_4-2l_1^2{l}_4^{2}Q,\end{array}$$

$$h_7=2e_{11}{e}_{12}+2e_{10}{e}_{13}+2e_3{e}_4+2e_2{e}_5+2e_1{e}_6-2e_{14}{l}_{4}Q,$$

$$\begin{array}{rl}{h}_8& =e_{12}^2+2e_{11}{e}_{13}+e_4^2+2e_3{e}_5+2e_2{e}_6-e_8^2+2e_{14}{e}_9\\ & -2e_7{l}_1^2-2l_{1}J{l}_4,\end{array}$$

$$h_9=2e_{12}{e}_{13}+2e_{10}{e}_{14}+2e_4{e}_5+2e_3{e}_6-2e_1{l}_1,$$

$$h_{10}=e_{13}^2+2e_{11}{e}_{14}+e_5^2+2e_4{e}_6-2e_2{l}_1-2e_8{l}_1^2,$$

$$h_{11}=2e_{12}{e}_{14}+2e_5{e}_6-2e_3{l}_1,$$

$$h_{12}=2e_{13}{e}_{14}+e_6^2-2e_4{l}_1-l_1^4,$$

$$h_{13}=e_{14}^2-2e_6{l}_1.$$

denote

$$\begin{array}{rl}H(\omega )& =h_0+h_1\omega +h_2{\omega }^2+h_3{\omega }^3+h_4{\omega }^4+h_5{\omega }^5+h_6{\omega }^6\\ & +h_7{\omega }^7+h_8{\omega }^8+h_9{\omega }^9+h_{10}{\omega }^{10}+h_{11}{\omega }^{11}+h_{12}{\omega }^{12}\\ & -2e_5{l}_1{\omega }^{13}+h_{13}{\omega }^{14}+(l_1){}^2{\omega }^{16},\end{array}$$ (12)

Therefore, (12) has a unique positive root ${\omega }_0$.

Define

$${\tau }_0=\frac{1}{{\omega }_0}\arccos \left[\frac{l_{4}J+e_1{\omega }_0+e_2{\omega }_0^2+e_3{\omega }_0^3+e_4{\omega }_0^4+e_5{\omega }_0^5+e_6{\omega }_0^6-l_1{\omega }_0^8}{{(l_4)}^2+e_7{\omega }_0^2+e_8{\omega }_0^4+{(l_1)}^2{\omega }_0^6}\right],$$ (13)

thus, when $\tau ={\tau }_0$, (11) has a pair of purely imaginary roots $\pm \mathrm{i}{\omega }_0$ and all of its other roots have negative real parts.

Finding derivative of (6) with respect to $\tau$, we can get

$${\left(\frac{\mathrm{d}\lambda }{\mathrm{d}\tau }\right)}^{-1}=\frac{{\mathrm{e}}^{\lambda \tau }{h}_2+{\mathrm{e}}^{2\lambda \tau }{B}_1+{\mathrm{e}}^{3\lambda \tau }{B}_2+2q_1\lambda }{\lambda ({\mathrm{e}}^{\lambda \tau }{B}_3+{\mathrm{e}}^{2\lambda \tau }{B}_4+B_5)}-\frac{\tau }{\lambda },$$ (14)

where $B_1=l_3+2l_2\lambda +3l_1{\lambda }^2,B_2=p_4+q_2+2p_3\lambda +3p_2{\lambda }^2+4p_1{\lambda }^3-5{\lambda }^4,B_3=2h_3+2h_2\lambda +2h_1{\lambda }^2,B_4=l_4+l_3\lambda +l_2{\lambda }^2+l_1{\lambda }^3,B_5=3q_3+3q_2\lambda +3q_1{\lambda }^2$.

Then we can easily find

$$\text{Re}{\left[\frac{\mathrm{d}\lambda (\tau )}{\mathrm{d}\tau }\right]}^{-1}{|}_{\tau ={\tau }_0}=\frac{aA+bB}{A^2+B^2},$$ (15)

where $a=\cos ({\omega }_0{\tau }_0)h_2+\cos (2{\omega }_0{\tau }_0)B_1+\cos (3{\omega }_0{\tau }_0)B_2,b=-\sin ({\omega }_0{\tau }_0)h_2-\sin (2{\omega }_0{\tau }_0)B_1-\sin (3{\omega }_0{\tau }_0)B_2+2q_1{\omega }_0,A=\sin ({\omega }_0{\tau }_0)B_3{\omega }_0+\sin (2{\omega }_0{\tau }_0)B_4{\omega }_0,B=\cos ({\omega }_0{\tau }_0)B_3{\omega }_0+\cos (2{\omega }_0{\tau }_0)B_4{\omega }_0+B_5{\omega }_0$.

Obviously, if condition (H2) $aA+bB>0$ holds, then

$\text{sign}\{[\mathrm{d}(\text{Re}\lambda )/\mathrm{d}\tau ]{|}_{\tau ={\tau }_0}\}=\text{sign}\{[\mathrm{d}(\text{Re}\lambda )/\mathrm{d}\tau ]{}^{-1}{|}_{\tau ={\tau }_0}\}>0$. That is to say the real part of the eigenvalues is increasing monotonously with respect to the parameter, which signifies that the system undergo supercritical Hopf bifurcation. Moreover, according to the data in Table 2, (H2) is satisfied in system (1).

Therefore, the result shows that the root of the characteristic (11) appears the pure imaginary number $\pm \mathrm{i}{\omega }_0$ when $\lambda ({\tau }_0)$. In particular, when $\tau ={\tau }_0$, at the equilibrium $E^{\ast }=(X^{\ast },Y^{\ast },Z^{\ast },V^{\ast },P^{\ast })$, the system (1) has a Hopf bifurcation. In addition, when the time delay $\tau$ is bigger than the threshold ${\tau }_0$, the QS system will oscillate. Therefore, the critical value ${\tau }_0$ is critical to the stability and oscillation behaviour of the QS system.