Combinatorial dynamics of protein synthesis time delay and negative feedback loop in NF‐κ B signalling pathway

Abstract

The transcription factor NF‐κ B links immune response and inflammatory reaction and its different oscillation patterns determine different cell fates. In this study, a mathematical model with Iκ Bα protein synthesis time delay is developed based on the experimental evidences. The results show that time delay has the ability to drive oscillation of NF‐κ B via Hopf bifurcation. Meanwhile, the amplitude and period are sensitive to the time delay. Moreover, the time delay threshold is a function of four parameters characterising the negative feedback loop. Likewise, the parameters also have effects on the amplitude and period of NF‐κ B oscillation induced by time delay. Therefore, the oscillation patterns of NF‐κ B are collaborative results of time delay coupled with the negative feedback loop. These results not only enhance the understanding of NF‐κ B biological oscillation but also provide clues for the development of anti‐inflammatory or anti‐cancer drugs.

1 Introduction

Nuclear factor kappa B (NF‐κ B) is well known for its critical role in regulation of a vast number of cellular processes including immunity, inflammation and tumour development [1 –3 ]. In mammalian cells, NF‐κ B activity is tightly controlled and exists in dimers, often with a predominance of p65/p50 heterodimers [4, 5 ]. In the resting state, NF‐κ B retains transcriptional inactive and is sequestered in the cytoplasm through binding to a member of the inhibitor of NF‐κ B (Iκ Bα ) proteins [5, 6 ]. In response to a variety of extracellular stimuli, such as cytokines (tumour necrosis factor‐α and interleukin‐1β ), viral and bacterial infections, ultraviolet and ionising radiation and diverse stresses (oncogenic stress), Iκ Bα protein is phosphorylated and depredated quickly leading to the release of NF‐κ B from the complex with Iκ Bα [7 –9 ]. In this way, NF‐κ B is liberated and activated, and then translocates to the nucleus and further regulates the transcription of downstream anti‐apoptotic, pro‐apoptotic and pro‐inflammatory target genes [10, 11 ]. It is reported that NF‐κ B regulates more than 500 target genes, including many chemokines, inflammatory factors and stress response genes by binding to the promoter or enhancer regions of target genes [12 ]. Extensive experiments have shown that deregulated NF‐κ B activity occurs in the vast majority of human tumours and inhibited NF‐κ B pathway activity can block cell cycle and induce apoptosis [13 –17 ]. This suggests that unravelling the underlying regulation mechanisms and biological functions of NF‐κ B signalling pathway is indispensable and essential for the treatment of inflammation and cancer.

Interestingly, experiments have revealed that NF‐κ B can exhibit oscillation between an inactive form outside and an active form inside of the nucleus upon external signal stimuli [18, 19 ]. This is mainly because the core module of NF‐κ B signalling pathway is constructed by a negative feedback loop [20 ]. At the same time, due to the newly activated NF‐κ B can be ceaselessly transported to the nucleus via NF‐κ B oscillation, a relatively high level of activated NF‐κ B in the nucleus is maintained. It indicates that the oscillation of NF‐κ B is a hint of the cells suffering the extracellular stimulation or diseases even cancers [13, 18 ]. Moreover, it has been reported that the characters of NF‐κ B oscillation dynamics can influence NF‐κ B‐dependent downstream gene expression patterns [18 –20 ]. Concretely, NF‐κ B may regulate the transcription of different genes via changing the amplitude and frequency of the oscillation [12, 18, 21 –23 ]. Especially, the different oscillation frequency of NF‐κ B achieved by changing the interval of pulsatile TNFα stimulation can result in different downstream gene expression patterns [24 ]. Thus, a deep and comprehensive understanding of the oscillation principles of NF‐κ B pathway is crucial for clarifying its biological functions and providing clues to develop drugs for the treatments of related diseases.

In recent years, theoretical modelling and stability analysis are generally applied to reveal the underlying biophysical mechanism [25 –27 ]. In order to explore the dynamic mechanism of NF‐κ B oscillation, many theoretical models are put forward to describe the chemical reaction in NF‐κ B pathway [24, 28 –31 ]. However, many works on NF‐κ B oscillation are focused on the negative feedback loop and instantaneous transcriptional and translational events, whereas the time needed in these gene expression processes has been poorly noticed. It is well known that gene expression refers to the process of synthesising genetic information from a gene to a functional gene product that is usually a protein or functional RNA such as miRNA and sRNA. Transcription, transcript splicing and processing, and translation are essential steps in this synthesis process [32 ]. Typically, the average transcriptional delay is about 10–20 min and the translation delay is ∼1–3 min [33 –35 ]. Meanwhile, the existing theoretical results suggest that the time delay can usually lead to oscillation and complex dynamic behaviours of gene regulatory networks [36 –39 ]. This shows that the delay of protein synthesis is a common phenomenon, and it is reasonable and necessary to include the delay factor in the model of gene regulation network.

In this paper, motivated by the above consideration, a new delayed model is proposed to explore the combined regulation mechanisms of time delay and the negative feedback loop on NF‐κ B oscillation dynamics. First, the effect of the protein synthesis time delay on the oscillation patterns of NF‐κ B core module is studied. Next, the effects of some rates characterising the negative feedback loop on NF‐κ B oscillation patterns are analysed. Finally, the combinatorial influence including time delay and the strength of the negative feedback loop is further discussed.

2 Model formulation

According to the biological facts of NF‐κ B pathway, Hoffman et al. [20 ] constructed a high dimensional model including chemical reactions among 26 different molecules to model NF‐κ B system. Subsequently, Krishna et al. [28 ] simplified it into a low‐dimensional model including the interactions among three components that constitute the core module of NF‐κ B pathway. The model includes nucleus NF‐κ B and its inhibitor Iκ Bα mRNA and Iκ Bα protein, which formed a negative feedback loop as shown in Fig. 1. In normal cells, NF‐κ B is quarantined in the cytoplasm via forming a complex with Iκ Bα [28, 40 ]. Upon cellular stimuli, Iκ B kinase (IKK) is first activated, which promotes Iκ Bα protein degradation via phosphorylation. After degradation of Iκ Bα protein, NF‐κ B is released and then translocates to the nucleus and activates hundreds of downstream target genes [28, 41 ]. Among them, Iκ Bα is a target of NF‐κ B which binds to the promoter region of the gene Iκ Bα to initiate its transcription and translation [42, 43 ]. Synthesised Iκ Bα enters to the nucleus and forms Iκ Bα ‐NF‐κ B complex by binding to NF‐κ B, then Iκ Bα ‐NF‐κ B exports back to the cytoplasm. The negative feedback loop mediated by Iκ Bα is regarded as the determinant to produce NF‐κ B oscillation.

Fig 1.

Schematic diagram of the core NF‐κB signalling pathway. The core feedback loop of NF‐κB signalling pathway consists of three components: nucleus NF‐κB, cytoplasmic IκBα and IκBα mRNA. NF‐κB dimers activate the production of IκBα mRNA, which translates into IκBα to inhibit nucleus NF‐κB production. This regulation process forms a negative feedback loop. Here, green arrows signify transcription and translation; the blue arrow indicates transportation in nucleus upon signal stimuli; the red barred arrow denotes the inhibition of nucleus NF‐κB by IκBα

Nevertheless, it should be particularly emphasised that the synthesis of Iκ Bα protein is a complex process including transcription, translation and translocation between nucleus and cytoplasm, and all of them need a certain time [18, 34, 35 ]. However, Iκ Bα protein synthesis time delay is not mentioned in the previous models. Generally, transcriptional and translational delays exist objectively during the gene transcription process. Moreover, time delay can induce multiple complex dynamic behaviours of the system [36 –39 ]. Therefore, based on the inspiring research results in [28 ], we developed a delayed dynamical model characterised by the following non‐linear ordinary differential equation (1 ) to comprehensively explore the collaborative dynamic mechanisms of time delay and the negative feedback loop on NF‐κ B signalling pathway

$$\left\{\begin{array}{rl}& \frac{\mathrm{d}{N}_n(t)}{\mathrm{d}t}=a\frac{1-N_n(t)}{\epsilon +I(t)}-bI(t)\frac{N_n(t)}{\delta +N_n(t)},\\ & \frac{\mathrm{d}{I}_m(t)}{\mathrm{d}t}=N_n^2(t-{\tau }_1)-I_m(t),\\ & \frac{\mathrm{d}I(t)}{\mathrm{d}t}=I_m(t-{\tau }_2)-c\frac{(1-N_n(t))I(t)}{\epsilon +I(t)}.\end{array}\right.$$ (1)

Here, all the variables and parameters are dimensionless. $N_n(t)$ , $I_m(t)$ and $I(t)$ , respectively, represent the concentrations at time t of nucleus NF‐κ B, Iκ Bα mRNA and Iκ Bα protein, so all of them are non‐negative. The first term of the first equation in the model indicates that cytoplasm Iκ Bα protein blocks nuclear import of NF‐κ B. The second term of the first equation describes that NF‐κ B in the nucleus exports to the cytoplasm by forming a NF‐κ B‐Iκ Bα complex. The first term in the second equation presents that nucleus NF‐κ B initiates transcription and activation of Iκ Bα via a dimer form, which takes a certain amount of time and is recorded as ${\tau }_1$ . The second term is the degradation of Iκ Bα mRNA. The first term of the third equation is Iκ Bα protein synthesis from Iκ Bα mRNA, which takes some time and is denoted by ${\tau }_2$ . The second term in the third equation is derived from Iκ Bα protein degradation which contains basal degradation and accelerated degradation by IKK upon cellular stimulation, which is directly proportional to the concentration of NF‐κ B‐Iκ Bα complex. Parameter ${\tau }_1$ denotes the transcriptional time required from a gene segment to a mature Iκ Bα mRNA and ${\tau }_2$ refers to the translational time needed from Iκ Bα mRNA to Iκ Bα protein. Parameter a represents NF‐κ B nuclear import rate hindered from Iκ Bα protein. Parameter b corresponds to NF‐κ B nuclear export rate resulted from Iκ Bα nuclear import. Parameter c signifies the degradation rate of Iκ Bα protein. Parameter $\delta$ is the Michaelis constant of NF‐κ B nuclear export enhanced by Iκ Bα protein, in other word, at which half of nucleus Iκ Bα is complexed to NF‐κ B. Parameter $\epsilon$ is the Michaelis constant of Iκ Bα protein degradation accelerated by NF‐κ B in the cytoplasm. According to the biological meanings, all the above parameters including a, b, c, $\delta$ and $\epsilon$ are positive, which make sense of the denominator.

Make variable transforms as $\tau ={\tau }_1+{\tau }_2$ and ${\tilde{I}}_m(t)=I_m(t-{\tau }_2)$ . Put them into system (1 ), we obtain a new mathematical model as follows:

$$\left\{\begin{array}{rl}& \frac{\mathrm{d}{N}_n(t)}{\mathrm{d}t}=a\frac{1-N_n(t)}{\epsilon +I(t)}-bI(t)\frac{N_n(t)}{\delta +N_n(t)},\\ & \frac{\mathrm{d}{\tilde{I}}_m(t)}{\mathrm{d}t}=N_n^2(t-\tau )-{\tilde{I}}_m(t),\\ & \frac{\mathrm{d}I(t)}{\mathrm{d}t}={\tilde{I}}_m(t)-c\frac{(1-N_n(t))I(t)}{\epsilon +I(t)}.\end{array}\right.$$ (2)

Replacing ${\tilde{I}}_m(t)$ still with $I_m(t)$ , system (2 ) converts to

$$\left\{\begin{array}{rl}& \frac{\mathrm{d}{N}_n(t)}{\mathrm{d}t}=a\frac{1-N_n(t)}{\epsilon +I(t)}-bI(t)\frac{N_n(t)}{\delta +N_n(t)},\\ & \frac{\mathrm{d}{I}_m(t)}{\mathrm{d}t}=N_n^2(t-\tau )-I_m(t),\\ & \frac{\mathrm{d}I(t)}{\mathrm{d}t}=I_m(t)-c\frac{(1-N_n(t))I(t)}{\epsilon +I(t)},\end{array}\right.$$ (3)

where $\tau$ represents the whole time required for Iκ Bα synthesis from the beginning of transcription to the emergence of a mature functional protein of Iκ Bα. It is easy to see from the above derivation that systems (1 ) and (3 ) are equivalent. Also, theoretical analysis illuminates that the stability of system (1 ) depends only on the total delay. The only difference is that the evolution of new $I_m$ over time is ${\tau }_2$ units faster than original $I_m$ . Next, aiming to understand the combinatorial mechanism of time delay and the negative feedback loop in NF‐κ B pathway, we will systematically study the dynamics of model (3 ). In addition, the model parameter values are listed in Table 1, which are selected as much as possible based on literatures. Among them, parameter value of a, the binding rate of Iκ Bα with cytoplasm NF‐κ B to impede the nuclear import, is estimated via referring to different literatures under biochemical constraints so that the system is stable without delay [24, 28 –31, 35 ]. Then, when the reasonable delay appears, the system can oscillate. In all calculations and numerical simulations, the parameter values are used unless otherwise stated.

Table 1.

Parameters values used in the model

Parameter	Description	Value	Reference
a	NF‐κ B nuclear import hindered from Iκ Bα protein	0.2	estimated
b	NF‐κ B nuclear export resulted from Iκ Bα nuclear import	954.5	[28 ]
$\delta$	concentration at which half of nucleus Iκ Bα is complexed to NF‐κ B	0.029	[28 ]
c	degradation of Iκ Bα protein	0.035	[28 ]
$\epsilon$	concentration at which half of cytoplasmic NF‐κ B is complexed to Iκ Bα	0.00002	[28 ]

3 Main results

It is reported that different oscillation modes of nuclear NF‐κ B will lead to different downstream gene expressions and result in different cell fates including proliferation and apoptosis [18, 44 ]. Therefore, NF‐κ B plays a dual role of carcinogenic and antitumor capabilities due to its ability to promote cancer cells survival and apoptosis. On the one hand, if the downstream pro‐apoptotic factors such as TNF, Bax, c‐myc and so on. are activated, then the damaged cells will be programmed death. Thus, it will avoid inheriting the damage and wrong information to the next generation of daughter cells. In this case, nuclear NF‐κ B oscillation plays a role of suppressing cancer [12, 45 ]. On the other hand, if the downstream anti‐apoptotic genes such as TRAF, c‐iAP and so on are activated, then the damaged cells will survive. Hence its progeny cells are also unhealthy. If they continue to develop in this way, they will easily cause inflammation and even cancer [13 –17 ]. Therefore, it is believed that the modes of nuclear NF‐κ B oscillation determines the downstream genes expressing and cell fate choices and thus plays a crucial role in inflammation and cancer [18 –24 ]. However, little is known about the control of NF‐κ B oscillation and its properties.

3.1 Effect of the time delay of Iκ Bα protein synthesis on NF‐κ B oscillator

In this subsection, we discuss the effect of Iκ Bα protein synthesis time delay on the stability of system (3 ) with a combinational way of theoretical predictions and numerical simulations. Particular attention is the oscillation behaviour generated by Hopf bifurcation, which is usually used to design biochemical oscillators.

3.1.1 Theoretical analysis of the time delay effect

According to the relevant stability and bifurcation theory of delay differential equations, the stability of positive equilibrium and existence of periodic solution resulting from Hopf bifurcation of system (3 ) are studied [46 ].

For system (3 ), the following conclusions hold:

• (i) If (H1) is satisfied, then the positive equilibrium $E^{\ast }=(N_n^{\ast },I_m^{\ast },I^{\ast })$ of system (3 ) is asymptotically stable for all $\tau >0$.

• (ii) If (H2) and (H3) are true, then there exist ${\delta }_j>0(j=0,1,2,\mathrm{3...})$ such that the positive equilibrium $E^{\ast }=(N_n^{\ast },I_m^{\ast },I^{\ast })$ of system (3 ) is asymptotically stable when time delay $\tau \epsilon [{\delta }_j,{\tau }_i^j)$ , and unstable when $\tau \epsilon ({\tau }_i^j,{\delta }_{j+1})$ . Furthermore, Hopf bifurcation occurs at the positive equilibrium point $E^{\ast }=(N_n^{\ast },I_m^{\ast },I^{\ast })$ when $\tau ={\tau }_i^j$ . Meanwhile, when $\tau$ crosses the threshold ${\tau }_i^j$ , a set of periodic solutions are bifurcated from the equilibrium point.

Assuming that the positive equilibrium point of the system (3 ) is $E^{\ast }=(N_n^{\ast },I_m^{\ast },I^{\ast })$ . Then we can obtain the linearised equations at the equilibrium point as follows:

$$\left\{\begin{array}{rl}& {\dot{N}}_n(t)=C_1{N}_n(t)+C_{2}I(t),\\ & {\dot{I}}_m(t)=C_3{N}_n(t-\tau )-I_m(t),\\ & \dot{I}(t)=C_4{N}_n(t)+I_m(t)+C_{5}I(t),\end{array}\right.$$ (4)

where

$$C_1=-\frac{a}{\epsilon +I^{\ast }}-\frac{b\delta I^{\ast }}{{(\delta +N_n^{\ast })}^2},$$

$$C_2=-\frac{a(1-N_n^{\ast })}{{(\epsilon +I^{\ast })}^2}-\frac{b{N}_n^{\ast }}{\delta +N_n^{\ast }},$$

$$C_3=2N_n^{\ast },$$

$$C_4=\frac{c{I}^{\ast }}{\epsilon +I^{\ast }},$$

$$C_5=-\frac{c(1-N_n^{\ast })\epsilon }{{(\epsilon +I^{\ast })}^2}.$$

Obviously, the characteristic equation of system (4 ) is

$$\begin{array}{c}{\lambda }^3+A_1{\lambda }^2+A_2\lambda +a_0+a_1{\mathrm{e}}^{-\lambda \tau }=0,\end{array}$$ (5)

where

$$A_1=1-C_1-C_5,$$

$$A_2=C_1{C}_5-C_1-C_5-C_2{C}_4,$$

$$a_0=C_1{C}_5-C_2{C}_4,$$

$$a_1=-C_2{C}_3.$$

Suppose that $\pm i\omega (\omega >0)$ is a pair of pure virtual roots of (5 ). Substitute it into (5 ) and separate the real and imaginary parts, which yields

$$\sin (\omega \tau )=\frac{-a_{1}P}{P^2+Q^2},$$

$$\cos (\omega \tau )=\frac{a_{1}Q}{P^2+Q^2},$$

where

$$\begin{array}{c}P=a_0-A_1{\omega }^2,\\ Q=A_2\omega -{\omega }^3.\end{array}$$

According to the identical equation ${\sin }^2(\omega \tau )+{\cos }^2(\omega \tau )=1$ , we can obtain the exponential polynomial equation of $\omega$

$$\begin{array}{c}{b}_0+b_1{\omega }^2+b_2{\omega }^4+b_3{\omega }^6+b_4{\omega }^8+b_5{\omega }^{10}+{\omega }^{12}=0,\end{array}$$ (6)

where

$$b_0=a_0^4-a_0^2{a}_1^2,$$

$$b_1=-4a_0^3{A}_1+2a_0^2{A}_2^2+2a_0{a}_1^{2}A1-a_1^2{A}_2^2,$$

$$b_2=6a_0^2{A}_1^2+A_2^4-4a_0^2{A}_2-4a_0{A}_1{A}_2^2-a_1^2{A}_1^2+2a_1^2{A}_2,$$

$$b_3=-4a_0{A}_1^3-4A_2^3+2a_0^2+2A_1^2{A}_2^2+8a_0{A}_1{A}_2-a_1^2,$$

$$b_4=A_1^4+6A_2^2-4A_1^2{A}_2-4a_0{A}_1,$$

$$b_5=-4A_2+2A_1^2.$$

Here, we make the following assumptions:

(H1) $b_i>0(i=1,2,\cdots ,5)$.

(H2) Equation (6 ) has at least one positive root.

If (H1) is satisfied, (6 ) has no positive roots and all roots have negative real parts. Therefore, the positive equilibrium $E^{\ast }=(N_n^{\ast },I_m^{\ast },I^{\ast })$ of system (3 ) is asymptotically stable for all $\tau >0$ . Thus, the first conclusion (i) is proved.

If (H2) holds, then (5 ) has a pair of pure virtual roots $\pm i\omega (\omega >0)$ at the critical value of $\tau$ . Then corresponding critical value of $\tau$ can be expressed as

$$\begin{array}{c}{\tau }_i^j=\frac{2j\pi }{{\omega }_i}+\frac{1}{{\omega }_i}\arccos \left[\frac{-a_1{P}_i}{P_i^2+Q_i^2}\right],1\le i\le 12,j=0,1,2,\cdots ,\end{array}$$ (7)

where

$$\begin{array}{c}{P}_i=a_0-A_1{\omega }_i^2,\\ Q_i=A_2{\omega }_i-{\omega }_i^3.\end{array}$$

Furthermore, by calculating the derivative of (5 ) with respect to $\tau$ , we have

$${\left(\frac{d\lambda }{d\tau }\right)}^{-1}=\frac{(3{\lambda }^2+2A_1\lambda +A_2)}{a_1\lambda {\mathrm{e}}^{-\lambda \tau }}-\frac{\tau }{\lambda }.$$

Thus, it satisfies

$${\left.\mathrm{Re}{\left(\frac{d\lambda }{d\tau }\right)}^{-1}\right|}_{\tau ={\tau }_i^j}=\frac{{\omega }_i^2(D_1+D_2)+D_3}{a_1^2{\omega }_i^2},$$ (8)

where

$$\begin{array}{rl}& D_1=2a_1{A}_1\cos ({\omega }_i{\tau }_i^j),\\ & D_2=3a_1{\omega }_i\sin ({\omega }_i{\tau }_i^j),\\ & D_3=-a_1{A}_2{\omega }_i\sin ({\omega }_i{\tau }_i^j).\end{array}$$

Next, we make another assumption.

(H3) ${\omega }_i^2(D_1+D_2)+D_3>0$.

It is evident that if (H3) holds, then $\mathrm{sign}\mathrm{Re}\left(\mathrm{d}\lambda /\mathrm{d}\tau \right){|}_{\tau ={\tau }_i^j}=\mathrm{sign}\mathrm{Re}{\left(\mathrm{d}\lambda /\mathrm{d}\tau \right)}^{-1}{|}_{\tau ={\tau }_i^j}>0$ . It indicates that the roots of (5 ) cross the imaginary axis from left to right. That is to say, the state of system (3 ) changes from a stable state to an unstable state at the equilibrium point $E^{\ast }=(N_n^{\ast },I_m^{\ast },I^{\ast })$ . At the critical point $\tau ={\tau }_i^j$ , the Hopf bifurcation of the system (3 ) occurs. Taken together, the conclusion (ii) holds. □

3.1.2 Numerical analysis of the time delay effect

In order to give a visual understanding and verify the above theoretical results about the effects of time delay on NF‐κ B oscillation, numerical simulations are performed as shown in Figs. 2 and 3. The parameter values in system (3 ) are displayed in Table 1 unless otherwise specified. Substituting these parameters into system (3 ), it can be calculated that the positive equilibrium point of system (2 ) is $E^{\ast }=(0.170291,0.0289991,0.0142539)$ . Moreover, the time delay thresholds are calculated according to (7 ). Without loss of generality, we only pay attention to the minimal time delay threshold, ${\tau }_0=min({\tau }_i^j)=0.11$ , and other values can be analysed in a similar way. The bifurcation diagram of NF‐κ B system versus the time delay $\tau$ is obtained in Fig. 2, which shows that Hopf bifurcation occurs at time delay $\tau =0.11$ . Moreover, the amplitude of the bifurcated oscillation increases with the time delay $\tau$ . The evolutions of the system (3 ) over time are depicted in Fig. 3. As Fig. 3 a, NF‐ $\kappa$ B signalling system is locally asymptotically stable at $E^{\ast }$ if $\tau =0<{\tau }_0$ , i.e. the Iκ Bα protein synthesis time is regarded as an immediate process. However, if $\tau =0.15>{\tau }_0$ , the system keeps sustained oscillation as described in Fig. 3 b. These simulation results are completely consistent with Theorem 1. Taken together, the time delay during the process of Iκ Bα protein production can result in the oscillation of NF‐κ B core module.

Fig 2.

Bifurcation diagram of NF‐κB system with time delay

$\tau$

(a) Bifurcation diagram of NF‐κ B with time delay $\tau$ , (b) Bifurcation diagram of the Iκ Bα mRNA and Iκ Bα protein with time delay $\tau$

Fig 3.

Evolutions of NF‐κB system over time when $\tau$ located upon both sides of the critical value ${\tau }_0=0.11$ . Red, blue and purple lines are corresponding to the concentration of nucleus NF‐κB, mRNA of IκBα and protein of IκBα, respectively

(a) Evolutions of NF‐κ B system over time at $\tau =0<{\tau }_0=0.11$ , (b) Evolutions of NF‐κ B system over time at $\tau =0.15>{\tau }_0=0.11$

To further elucidate the effect of time delay on the oscillation behaviour of NF‐κ B system, the changing trends of oscillation period and amplitude as time delay $\tau$ varies are captured in Figs. 4 a and b, respectively. It can be seen from Fig. 4 a that the oscillatory period of all the three substances is almost a monotone increasing function of time delay $\tau$ . As in Fig. 4 b, the oscillatory amplitude also increases with time delay, while the speed of NF‐κ B levels off after the initial ramp‐up. As above, the oscillatory patterns of NF‐κ B system strongly rely on the time delay $\tau$.

Fig 4.

Dependence of oscillatory patterns including period and amplitude on time delay

$\tau$

(a) Dependence of oscillatory period on time delay $\tau$ , (b) Dependence of oscillatory amplitude on time delay $\tau$

To sum up, Iκ Bα protein synthesis time delay can not only provide a way to induce or eliminate NF‐κ B oscillation but also can alter the period and amplitude of the oscillation. Experiments have shown that the period and amplitude of NF‐κ B oscillation can lead to bipolar cell fates including survival and apoptosis of cancer cells [12, 18, 21 –23 ]. The above research results provide a method for regulating NF‐κ B oscillation dynamics through time delay, which may provide a new insight for the treatment of cancer and other diseases. Therefore, Iκ Bα protein synthesis time delay may become an important potential target for treating immunological diseases and cancer. To our knowledge, transcription and splicing of intron sequences increase the time required for the extension and splicing of genes to mature RNA [33, 47, 48 ]. Thus, an effective way to vary the time delay is changing the number of introns within the gene. Takashima et al. [49 ] previously showed that deletion of all three introns within Hes7 gene reduces the time delay by 19 min and completely abolishes oscillatory expression. Based on the conclusion that each consecutive intron splicing in mammalian cells required about 0.4–7.5 min [50, 51 ], the number of introns within the genes of Iκ Bα can be calculated in order to achieve the desired length of Iκ Bα protein synthesis time delay $\tau$.

3.2 Combinatorial regulation of time delay and the negative feedback loop on NF‐κ B oscillator

The negative feedback loop is a common way to design biological oscillator. Multiple researchers suggested that the occurrence of oscillation dynamics of NF‐κ B pathway should be attributed to Iκ Bα ‐mediated negative feedback loop. On the other hand, the results obtained in the front subsection suggested that time delay can also drive oscillation of NF‐κ B pathway even in the parameter region with non‐oscillation. Therefore, we believe that the oscillation dynamics of NF‐κ B pathway should be collaboratively controlled by time delay coupled with the negative feedback loop. In fact, the assertion that the functional consequences of NF‐κ B signalling pathway depend on the period and amplitude of oscillation is confirmed as early as 2004 [18 ]. However, how the characteristics of the oscillation, such as the periodic and amplitude, are regulated via a cooperative way by the negative feedback loop and time delay is received poor attention so far. Generally, the effect of the negative feedback loop on oscillation is reflected through the parameter rates involved in the feedback loop on the oscillation patterns. Moreover, the expression of nucleus NF‐κ B is a hallmark in NF‐κ B signalling pathway because that the qualitative stable properties of NF‐κ B, Iκ Bα mRNA and Iκ Bα protein are consistent. Based on these considerations, only the oscillating properties of NF‐κ B need to be studied in detail, and the properties of Iκ Bα mRNA and Iκ Bα protein can be similarly derived.

In this section, we are committed to reveal this collaborative regulation mechanism. Firstly, four parameters characterising the negative feedback loop are, respectively, discussed with and without time delay $\tau$ . Secondly, the function relation among time delay threshold ${\tau }_0$ and the above four important parameters are intuitively displayed.

3.2.1 Combinatorial regulation of time delay and the nuclear import of NF‐κ B

Parameter a refers to the rate at which NF‐κ B is imported from the cytoplasm to the nucleus, and this process is repressed by Iκ Bα protein via forming a NF‐κ B‐Iκ Bα complex. In other words, parameter a characterises the discourage strength of Iκ Bα protein to NF‐κ B nuclear import. The corresponding dynamic behaviours of system (3 ) are compared under the combinatorial regulation of time delay $\tau$ and the discourage strength a, which are exhibited in Fig. 5. As shown in Fig. 5 a, the qualitative and quantitative behaviour of nucleus NF‐κ B are almost fixing regardless of the discourage strength a takes at time delay $\tau =0$ . In contrast, as in Fig. 5 b, when time delay $\tau =0.15>0$ and the inhibitory strength a of Iκ Bα protein to nuclear NF‐κ B varies from weak to strong, the dynamic of NF‐κ B gradually changes from a sustained oscillation through a damping oscillation to a monostable state with non‐oscillation, which indicates that Hopf bifurcation occurs. The bifurcation diagram (Fig. 6 ) further shows that the system undergoes a Hopf bifurcation at $a=0.46$ . Moreover, the amplitude is radically decreased and close to 0 at enough discourage strength a as displayed in Figs. 5 b, 6 and 7 a. At the same time, the period is also sharply decreased, but the speed is gradually slowed down, and even the varying direction changed from a decrease trend to an increase trend. It corresponds that the system gradually tends to a stable state through slowly frequency and low amplitude fluctuation as illustrated in Figs. 5 b and 7 b. In summary, the inhibitory strength of Iκ Bα protein to NF‐κ B nuclear import cannot change the stability of system (3 ) without time delay. Moreover, if there is a certain time delay, the varied suppression strength from Iκ Bα protein to NF‐κ B nuclear import can lead to essential and qualitative changes about the stability of system (3 ). Concretely, the weaker the discourage strength, the more favourable to NF‐κ B oscillation production, and the larger to the amplitude and period of the oscillation. It is because that the larger discourage strength may lead to the weaker negative feedback loop. In view of this, we predict that appropriate enhancement of the discourage strength of Iκ Bα protein to NF‐κ B nuclear import may be an effective way to repress nucleus NF‐κ B oscillation.

Fig 5.

Influence of parameter a on the dynamic features of system (3 ). Red, blue, cyan, green, orange and purple lines, respectively, denote the evolutions of NF‐κB over time with $a=0.2,a=0.3,a=0.4,a=0.5,a=0.6$ and $a=0.7$

(a) Evolutions of NF‐κ B over time with variation of a at $\tau =0$ , (b) Evolutions of NF‐κ B over time with variation of a at $\tau =0.15$

Fig 6.

Bifurcation diagram of NF‐κB with parameter a at

$\tau =0.15$

Fig 7.

Dependence of oscillatory patterns including amplitude and period on parameter a at

$\tau =0.15$

(a) Dependence of oscillatory amplitude on a, (b) Dependence of oscillatory period on a

3.2.2 Combinatorial regulation of time delay and nuclear export of NF‐κ B

Parameter b refers to the rate at which nucleus NF‐κ B exports from the nucleus to cytoplasm. The exportation process is enhanced by Iκ Bα protein via binding to NF‐κ B and forming a NF‐κ B‐Iκ Bα complex which is the required form of NF‐κ B translocation from the inside to the outside of nucleus. That is to say, parameter b represents the promotion strength of Iκ Bα protein to NF‐κ B nuclear export. The numerical results suggest that the varied promotion strength b can change the expression patterns of nuclear NF‐κ B only in the presence of time delay $\tau$ , which is similar to the function of discourage strength from Iκ Bα protein to NF‐κ B nuclear import. However, the bifurcation direction is opposite as seen in Fig. 8, which leads to an opposite changing trend of the amplitude as shown in Figs. 8 and 9 a. At the same time, the period is sharply decreased as addressed in Fig. 9 b, which is almost homologous to Fig. 7 b. In view of this, we predict that appropriate decrease of NF‐κ B nuclear export rate may be an effective way to reduce the amplitude and eventually eliminate the oscillation.

Fig 8.

Bifurcation diagram of NF‐κB with parameter b at

$\tau =0.15$

Fig 9.

Dependence of oscillatory patterns including period and amplitude on parameter b

(a) Dependence of amplitude on b, (b) Dependence of period on b

3.2.3 Combinatorial regulation of time delay and the degradation rate of Iκ Bα protein

Parameter c refers to the degradation rate of Iκ Bα protein, which is proportional to NFκ B‐Iκ Bα complex and includes IKK induced degradation. Different from the discourage strength a and the promotion strength b, it is obvious from Fig. 10 that the concentration of nucleus NF‐κ B increases with Iκ Bα protein degradation rate c even when the protein synthesis time delay of Iκ Bα is zero. Moreover, the changing trends of the equilibrium point of NF‐κ B, Iκ Bα mRNA and Iκ Bα protein are depicted in Fig. 11. It is easy to observe that the steady states of NF‐κ B and Iκ Bα mRNA increase with parameter c. Different from the former, the Iκ Bα protein shows the opposite trend as addressed in Fig. 11 a and which is clearly visible in the enlarged picture in Fig. 11 b. Fig. 12 gives the bifurcation diagram of NF‐κ B system with respect to degradation rate c at $\tau =0.15$ , where the circle region represents an oscillatory state and the left and right parts mean that the system is stable. Moreover, it is clear from Fig. 12 that the steady state on the right part is higher than the left part, which is because that the activity of the nucleus NF‐κ B increases with the degradation of Iκ Bα protein. Additionally, the amplitude and period increase gradually in the initial stage and then decrease continuously as portrayed in Fig. 13. In sum, the degradation rate of Iκ Bα protein has the ability to induce and eliminate NF‐κ B oscillation more flexible, and also can nimbly change the amplitude and period. This may be due to the complex implications represented by parameter c which integrated the non‐linear regulation of IKK and NF‐κ B on Iκ Bα protein.

Fig 10.

Evolutions of NF‐κB over time with variation of c at

$\tau =0$

Fig 11.

Dependence of equilibrium point on parameter c

(a) Evolutions of equilibrium point depending on c, (b) Enlarged picture of equilibrium state of I depending on c

Fig 12.

Bifurcation diagram of NF‐κB with respect to parameter c at

$\tau =0.15$

Fig 13.

Dependence of oscillatory patterns including period and amplitude on parameter c

(a) Dependence of amplitude on c, (b) Dependence of period on c

3.2.4 Combinatorial regulation of time delay and Michaelis constant $\delta$

The Michaelis constant $\delta$ refers to the concentration at which half of nucleus Iκ Bα is binding to NF‐κ B in the nucleus. Biologically, the bigger the Michaelis constant, the stronger affinity between the substrate and enzyme. The combinatorial regulation of time delay and the Michaelis constant $\delta$ on NF‐κ B oscillator is similar to the regulatory effect of time delay and discourage strength a as pictured in Figs. 14 and 15. Accordingly, it indicates that appropriate increase of the affinity $\delta$ between Iκ Bα and nuclear NF‐κ B may be an effective way to repress the nuclear NF‐κ B oscillation.

Fig 14.

Bifurcation diagram of NF‐κB with parameter

$\delta$

at

$\tau =0.15$

Fig 15.

Dependence of oscillatory patterns including period and amplitude on parameter

$\delta$

(a) Dependence of amplitude on $\delta$ , (b) Dependence of period on $\delta$

3.2.5 Dependence of time delay threshold ${\tau }_0$ on the four important parameters

The combinatorial regulation of the time delay and the negative feedback loop on NF‐κ B oscillator also can be reflected by the dependence of Hopf bifurcation point (time delay threshold ${\tau }_0$ ) on the above four important parameters. Therefore, when the four important parameters take different values, the corresponding threshold time delay ${\tau }_0$ is calculated according to formula (7 ), and which is displayed in Fig. 16. As shown in Fig. 16 a, the critical value ${\tau }_0$ continuously rises as discourage strength a. From Fig. 16 b, the critical value ${\tau }_0$ monotonously decreases as parameter b increases. In detail, when the promotion strength b increases from 10 to 100, the critical value ${\tau }_0$ falls rapidly. Conversely, when the promotion strength b exceeds 100, the downward trend of ${\tau }_0$ becomes slow and then turns rising. As depicted in Fig. 16 c, the critical value ${\tau }_0$ decreases with the increasing of degradation rate c. But when the degradation rate c increases pass through a certain value about 0.025, the trend of the critical value ${\tau }_0$ changes from decrease to increase. This interesting phenomenon verifies that the degradation rate of Iκ Bα protein owns the more flexible ability to induce the system oscillation. In Fig. 16 d, the critical value ${\tau }_0$ almost linearly increases with the affinity $\delta$ between Iκ Bα and nuclear NF‐κ B. In short, the stronger the discourage strength a and the affinity $\delta$ , the more time delay of Iκ Bα protein synthesis is needed to active the oscillation of nuclear NF‐κ B. By contrast, the weaker the promotion strength b, the longer time delay of Iκ Bα protein synthesis is needed. Furthermore, a relative short time delay is needed for producing NF‐κ B oscillation when the promotion strength is enough strong. However, as degradation rate c of Iκ Bα protein varies from small to big, the time delay needed for NF‐κ B oscillation looks like a parabola.

Fig 16.

Effects of the four parameters characterising the negative feedback loop on the time delay threshold

${\tau }_0$

(a) Dependence of ${\tau }_0$ on a, (b) Dependence of ${\tau }_0$ on b, (c) Dependence of ${\tau }_0$ on c, (d) Dependence of ${\tau }_0$ on $\delta$

4 Conclusion

In this paper, we developed a delayed mathematical model to understand the regulation of the oscillation behaviour in NF‐κ B core module. It is revealed that the time delay can drive the oscillation of NF‐κ B pathway even in the non‐oscillation parameter region. Furthermore, the length of time delay of Iκ Bα protein synthesis can change the amplitude and period of NF‐κ B oscillator. Moreover, the negative feedback loop represented by the four important parameters can also switch the system state between a stable steady state and a sustained oscillatory state, and further regulate the oscillatory amplitude and period. In addition, it is clarified that the critical value ${\tau }_0$ is sensitive to the four important parameters.

Multiple experimental results suggest that the dynamics of NF‐κ B, including oscillatory state, non‐oscillatory state and the oscillatory frequency/period and amplitude, provide critical information to trigger downstream gene expression in response to diverse stimuli, which will lead to different cell fate decision, such as proliferation, differentiation and apoptosis [12, 18 –24 ]. The results obtained in this paper demonstrated that Iκ Bα protein synthesis time delay coupled with the intrinsic negative feedback loop can flexibly induce and eliminate the oscillatory expression and also neatly regulate their period and amplitude. It might shed light to develop related drug target from five aspects, including Iκ Bα protein synthesis, the discourage strength of Iκ Bα protein to NF‐κ B nuclear import, the promotion strength of Iκ Bα protein to NF‐κ B nuclear export, the degradation rate of Iκ Bα protein and the affinity $\delta$ between Iκ Bα and nuclear NF‐κ B, to control the cell fate decision and achieve disease treatment.

As above, although we have given some new ideas to find drug targets, their feasibility need to be verified by further experiment. In fact, the networks involved in inflammation and cancers are complicated with NF‐κ B core modules appearing and often being coupled with several signalling pathways, for example, p53‐mdm2, Wnt and Notch signal pathways, and so on [52 –55 ]. Moreover, in different cancer cells, NF‐κ B signalling pathway exhibits different or even opposite biological functions, such as promoting survival and promoting apoptosis [12 –18, 44, 45 ]. The feature brings greater opportunities and challenges to clarify the internal mechanism of the related disease system. Therefore, if we can deduce the core network including all the important pathways and obtain the potential collaborator mechanism via constructing the corresponding kinetics model, it would be probably more actual and meaningful. On the other hand, how the different dynamic patterns of NF‐κ B pathway trigger different downstream gene expression and then lead to different responses is mysterious and urgent to be further explored. In addition, when the drug is developed and applied to the system, how the pharmacokinetics is and which cell fate will happen, are also matters of concern.

5 Acknowledgments

The authors are grateful to the anonymous referees and editors for their valuable comments and helpful suggestions, and partial support from the National Natural Science Foundation of China (11932003, 11772019 and 11762022).