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Synchronization in a multiplex network of gene oscillators
Physics Letters A 383 (2019) 125919
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Synchronization in a multiplex network of gene oscillators Abdul Jalil M. Khalaf a , Fawaz E. Alsaadi b , Fuad E. Alsaadi c , Viet-Thanh Pham d,∗ , Karthikeyan Rajagopal e a
Ministry of Higher Education and Scientific Research, Baghdad, Iraq Department of information Technology, Faculty of Computing and IT, King Abdulaziz University, Jeddah, Saudi Arabia c Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia d Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam e Center for Nonlinear Dynamics, Defence University, Ethiopia b
a r t i c l e
i n f o
Article history: Received 19 July 2019 Received in revised form 23 August 2019 Accepted 23 August 2019 Available online 29 August 2019 Communicated by M. Perc Keywords: Synchronization Multilayer network Gene oscillators
a b s t r a c t Study of the synchronization in the network of gene oscillator network has vital importance in understanding of rhythmicity of molecular and cellular activities. In this paper, we analyze a network of linearly coupled genetic oscillators in a multiplex structure. The coupling strength values are changed and the coupling range is considered to be fixed, but different in the two layers. The analyses are done in two cases of periodic and chaotic oscillations. By computing the statistical measures, the interlayer and intralayer synchronization states are studied. The results show that the layer with higher coupling range has more enhanced synchrony and is less affected by the turbulent behavior of the other layer. On the other hand, the layer with lower coupling range approaches synchronization by strengthening the interlayer and intralayer couplings. The interlayer synchronization is also achieved in high coupling strength values. © 2019 Elsevier B.V. All rights reserved.
1. Introduction In a complex system, the observed behavior is the collection of the activities of the interacting elements [1]. In order to study these systems, the elements can be modeled as a set of nodes, with some links between them, representing the interactions. The links can be a description of the flow of information between the components [2]. The multilayer structure provides a more realistic framework for the study of complex systems. In real systems, the interaction of the components has various types or can vary in time [3]. The use of multiple layers in a network provides the possibility of considering different interaction types. As an example, a social system is a hypernetwork which is composed of a group of dynamical systems such as families, friends, colleagues, . . . , coupled through the connections of distinct networks. Therefore it can be considered as a multilayer network [4]. Biological systems are complex systems comprised of many interacting bio-entities [5]. In the body, all of the cells interact with each other to perform a special function perfectly. Furthermore, the relationships between bio-entities have different attributes. There-
*
Corresponding author. E-mail address: [email protected] (V.-T. Pham).
https://doi.org/10.1016/j.physleta.2019.125919 0375-9601/© 2019 Elsevier B.V. All rights reserved.
fore, a multilayer network, in which the layers represent different relations, seems to be more illustrative. The function of a biological network is characterized by the relations between the fundamental genes, proteins, metabolites, or cells [6]. Gene regulation is mediated by various interactions between different kinds of molecules [7]. The interactions can be either through direct coupling or indirect one. In direct interaction, the coupling is defined by the differences between concentration variables within neighboring cells. While in indirect one, the substances crossing the cellular membrane into the extracellular space are considered [8]. Many factors can effect on these interactions, such as the number of neighboring cells in direct interaction or the rate of the exchange of substances in indirect interaction. The multilayer structure can be helpful in analyzing these different contributions. Synchronization is a ubiquitous phenomenon in nature which has attracted much attention in different fields such as biology, physics and technology [9–12]. Due to the great potential of chaos in many disciplines, synchronization of coupled chaotic systems has been intensively studied [13–16]. In neuroscience, synchronization has been revealed in the olfactory system or the hippocampal region [17]. In addition, it has strong relations with some brain diseases such as the epileptic seizure [18,19]. Therefore synchronization has been a hot topic in neuroscience [20–23].
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Fig. 1. (a) The motif of a repressilator, composed of three gene elements. (b) The motif of a single gene with a delayed self-inhibition [33].
Fig. 3. (a) The motif of a single gene with two delayed self-inhibitions. (b) The bifurcation diagram of the motif concerning time delay. (c) The chaotic time series of the motif for τ1 = 18 min and τ2 = 4.65 min [33]. Fig. 2. (a) The bifurcation diagram of the single gene element with one negative loop and (b) the periodic time series for τ = 12 min [33].
Study of synchronization in multilayer networks has provided different collective properties and also different synchronization types [24–26]. The synchronization has also been revealed in cellular communications [27]. Many studies have focused on the synchronization analysis of gene networks with the aim of understanding the rhythmic behavior of living organisms [27–32]. In this paper, we study synchronization in a multiplex gene oscillator network. A simple gene motif is selected as the elements of the network. The motif is composed of single gene element with self-inhibition with time delay. Since the relations in the gene network are not fixed, we consider two different coupling ranges in the two layers. The interlayer and intralayer coupling strength values are taken as the control parameter for the emergence of synchronization in the network. Actually, the synchronization is studied by calculating statistical measures. 2. The model The simplest oscillatory circuit motif is the repressilator, which is relevant to the natural gene network. The repressilator is composed of three genes with sequential inhibitions (Fig. 1a) and can produce stable periodic oscillations. Suzuki et al. [33] showed that
Fig. 4. A multiplex framework of the used network. The black lines show the intralayer couplings and the blue ones show the interlayer couplings. The number of coupling neighbors in the first layer is p 1 = 2 and in the second layer is p 2 = 20. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
A.J.M. Khalaf et al. / Physics Letters A 383 (2019) 125919
3
Fig. 5. The time snapshots of the 2-layer network composed of genes with one negative feedback loop (τ = 12 min) and for σ = 0.004. (a) γ = 0, (b) γ = 0.01, (c) γ = 0.05, (d) γ = 0.1, (e) γ = 0.5. Increment of the interlayer coupling increases the first layer synchronization and finally leads to interlayer synchronization. (f) Time series of the first (blue) and the second gene oscillator (red) of layer I and the first gene oscillator (green) of layer II, corresponding to part (e).
the dynamics of this three-gene circuit is equivalent to a selfinhibitory single gene element with a time delay (Fig. 1b). The time dynamics of a self-inhibitory single-gene element with a negative feedback loop was proposed by the following deterministic rate equation:
d A (t ) dt
= F (A, t ) = g A H − A A A (t − τ ) − k A A (t )
(1)
where A refers to the protein concentration (nM). The measurements are time (t) in minutes, the protein production rate (g A = 50) in nM/min, and the degradation rate (k A = 0.2) in 1/min. H A A is an inhibitory Hill function that if gene X is inhibited by gene Y , then the inhibitory Hill function is defined by:
n H− X Y [ Y ] ≡ 1/ 1 + ( Y / Y 0 )
(2)
where Y 0 = 38 nM is the threshold concentration of the Hill function and n = 4 is its rank. Fig. 2a shows the bifurcation diagram of the single inhibitory gene, and Fig. 2b shows the periodic time series of the gene at τ = 12 min.
Suzuki et al. [33] further showed that adding another negative feedback loop to the single gene element can enrich its dynamics and lead to the emergence of quasi-periodic and chaotic oscillations. Fig. 3a shows the gene element with two negative feedback loops. The rate equation of this motif is described by:
d A (t ) dt
= F (A, t ) = g A H 1−A A A (t − τ1 ) H 2−A A A (t − τ2 ) − k A A (t ) (3)
where the parameters are fixed at: g A = 50 nM/min, k A = 0.21/min, n1 A A = 4 and n2 A A = 2. Fig. 3b shows the bifurcation diagram of this motif and Fig. 3c shows a chaotic time series obtained from the gene at τ1 = 18 min and τ2 = 4.65 min. In this case, using the 0-1 test results in k = 0.996, which confirms the existence of chaotic behavior. In order to investigate the gene network, we consider a twolayer multiplexed network with linear coupling, as shown in Fig. 4. The equations of the network are described by:
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Fig. 6. The time snapshots of the 2-layer network composed of genes with one negative feedback loop (τ = 12 min) and for σ = 0.01. (a) (d) γ = 0.5. Increasing the interlayer coupling raises the coherency of the first layer toward being synchronous with the second layer.
γ = 0, (b) γ = 0.01, (c) γ = 0.1,
Fig. 7. Synchronization measures concerning coupling strengths, for the 2-layer network composed of genes with one negative feedback loop (τ = 12 min). (a) Interlayer synchronization error, (b) first layer synchronization factor, (c) second layer synchronization factor.
A.J.M. Khalaf et al. / Physics Letters A 383 (2019) 125919
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Fig. 8. The time snapshots of the 2-layer network composed of genes with two negative feedback loops (τ1 = 18 min and τ2 = 4.65 min) and for σ = 0.002. (a) γ = 0, (b) γ = 0.01, (c) γ = 0.05, (d) γ = 0.1, (e) γ = 0.5. By increasing of γ , at first, the coherency of the second layer is disturbed and then is recovered. The first layer also moves from the asynchronous state to synchronous behavior. (f) Time series of the first (blue) and the second gene oscillator (red) of layer I and the first gene oscillator (green) of layer II, corresponding to part (e).
d A i ,1 dt d A i ,2 dt
i+ p1
= F ( A i ,1 , t ) + σ
= F ( A i ,2 , t ) + σ
N
(4)
i+ p2
E =<
( A j ,1 − A i ,1 ) + γ ( A i ,2 − A i ,1 )
j =i − p 1
N 1
( A j ,2 − A i ,2 ) + γ ( A i ,1 − A i ,2 )
j =i − p 2
where A i ,1 and A i ,2 represents the gene elements in layer one and two, respectively. The intralayer coupling strength is represented by σ , and the interlayer coupling strength is represented by γ . p 1 and p 2 are the number of nearest neighbors in coupling in the first and second layers and are considered to be p 1 = 2 and p 2 = 20. The total number of the gene elements in each layer is assumed to be N = 500. The interlayer and intralayer coupling strength values are the control parameters for investigating network behavior. The intralayer synchronization refers to the synchronization of each layer separately. If the two layers become synchronous with each other, it is called interlayer synchronization. We quantify the intralayer synchronization with using a statistical factor and the interlayer synchronization by defining the following error:
j =1
| A 1, j − A 2, j | A 21, j + A 22, j
>t
(5)
where <>t denotes the average over time. The statistical factor for the intralayer synchronization (R) is based on the mean field theory, and R = 1 shows a complete synchronization. This factor can be calculated by:
F=
R=
N 1
N
1 N
xi
i =1
N
(6)
F 2 − F 2
i =1 [<
A 2i > − < A i >2 ]
3. Results We firstly hypothesize that the network is comprised of the genes with one negative feedback loop, and then we analyze the network of genes with two negative feedback loops. In order to
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Fig. 9. The time snapshots of the 2-layer network composed of genes with two negative feedback loops (τ1 = 18 min and τ2 = 4.65 min) and for σ = 0.02. (a) (b) γ = 0.05, (c) γ = 0.2, (d) γ = 0.5. In this case, the increment of γ leads to movement of the first layer from the chimera state to the synchronous state.
solve the network, the 4th order Runge-Kutta method was used, and the initial conditions were chosen randomly in [0, 100]. 3.1. The network of gene elements with one delayed self-inhibition The network patterns are observed by taking the two coupling strength values as the control parameters. In the case of independence of the two layers (γ = 0), for very small intralayer couplings as σ = 0.004, the first layer is asynchronous, and the second layer has complete phase synchronization as shown in Fig. 5a. If the interlayer links connect, then the layers may effect on each other. Fig. 5b shows the layers’ behavior for γ = 0.01. In this case, the asynchronous state of the first layer causes a disruption in the synchronization of the second layer oscillators. When γ increases to γ = 0.05, as it is shown in Fig. 5c, the second layer has more effects on the first layer and change its behavior to chimera state, at which some oscillators are synchronous and some are asynchronous [34,35]. By more increasing of γ , the first layer tends to be synchronous and the interlayer synchronization enhances (as shown in Fig. 5d, e). Fig. 5f shows some of the oscillators time series in the case of complete synchronization (Fig. 5e). Next the network is investigated by considering stronger intralayer coupling strength as σ = 0.01. In this case, when there is no link between the layers, the first layer shows chimera state, and the second one shows phase synchronization (as shown in Fig. 6a). Connecting the interlayer links leads to the influence of the first layer on the second one. Therefore, the second layer loses its complete phase synchronization (as shown in Fig. 6b for γ = 0.01). Further increasing of γ to γ = 0.1, makes the second layer return to its synchronous state. In this case, the first layer shows a sine-like synchronization (as shown in Fig. 6c). Finally growing γ to higher values, enhances the interlayer synchronization, and brings the first layer to phase synchronous state (as shown in Fig. 6d).
γ = 0,
Fig. 7 shows the synchronization measures concerning intralayer and interlayer couplings. Fig. 7a shows that increases in both intralayer and interlayer coupling strength causes decrement of interlayer synchronization error (E). Fig. 7b shows the synchronization factor of the first layer, which is defined by R 1 in the figure. According to the obtained factor, increasing of σ helps in synchronization of the first layer in low interlayer couplings. However, when γ is large enough, the first layer is synchronous for all σ values. Fig. 7c shows the synchronization factor of the second layer, defined by R 2 , and confirms that except for σ = 0, the second layer has high synchronization factor. 3.2. The network of gene elements with two delayed self-inhibitions In this subsection, the gene oscillators exhibit chaotic behavior. Similar to the previous case, the effects of coupling strengths are investigated. Staring from very low intralayer coupling as σ = 0.002, and no interlayer links (γ = 0), the first layer is asynchronous, and the second layer is phase synchronized (as shown in Fig. 8a). When the interlayer links connect, the asynchrony of the first layer disrupts the phase synchronization of the second layer (as shown in Fig. 8b for γ = 0.01). If γ increases, both layers have disturbed phase synchronization state (as shown in Fig. 8c, d) and further increasing of it leads to interlayer synchronization and intralayer phase synchronization (as shown in Fig. 8e). Fig. 8f shows some of the oscillators time series in the case of complete synchronization (Fig. 8e). In the next step, the intralayer coupling strength is fixed at stronger values as σ = 0.02. In this case, when γ = 0, the first layer exhibits chimera state, and the second layer is phase synchronized (as shown in Fig. 9a). Increasing of γ to γ = 0.05 changes the behavior of the first layer to a sine-like synchronization (as shown in Fig. 9b). More increasing of interlayer coupling strength changes the behavior of the layers to have interlayer synchroniza-
A.J.M. Khalaf et al. / Physics Letters A 383 (2019) 125919
Fig. 10. Synchronization measures concerning coupling strengths, for the 2-layer network composed of genes with two negative feedback loops (τ1 = 18 min and 4.65 min). (a) Interlayer synchronization error, (b) first layer synchronization factor, (c) second layer synchronization factor.
tion and intralayer phase synchronization (as shown in Fig. 9c, d). The synchronization factors, in the case of using gene elements with two delayed self-inhibitions, are shown in Fig. 10. Fig. 10a shows that at each value of γ , by increasing the intralayer coupling, the interlayer synchronization error decays to zero. Fig. 10b demonstrates that increasing of both γ and σ is necessary for intralayer synchronization of the first layer. Subsequently, Fig. 10c shows that irrespective of σ value, the second layer’s synchronization enhances by growing of γ .
7
τ2 =
interlayer coupling strength. Calculating the interlayer synchronization error confirmed that by increasing both coupling strengths, the error decreases and comes to zero. However, when the gene elements were in chaotic oscillation, the error in low intralayer couplings was more than when the gene elements were in periodic oscillation. Declaration of competing interest None.
4. Conclusion References In this paper, we studied the synchronization in a network of coupled gene oscillators. As the elements of the network, we used a simple gene motif consisting of a single gene with inhibitory delayed feedback, in two cases of one loop which showed periodic behavior and two loops which could show chaotic behavior. The multiplex framework was used to be able to consider different interactions. The number of elements in the coupling was fixed differently in the two layers. The strength of the inter and intralayer links was considered as the control parameter for synchronization, similar to the using of the coupled controller in monolayer networks [36]. Even by increasing the number of layers, this coupled controller is still applicable for synchronization of the hypernetwork. The intralayer synchronization was quantified with using a synchronization factor, and the interlayer synchronization was measured by computing an error. The results showed that the intralayer synchronization of the first layer was enhanced by increasing both coupling strengths. Furthermore, the intralayer synchronization of the second layer was enhanced by increasing the
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Synchronization in a multiplex network of gene oscillators Abdul Jalil M. Khalaf a , Fawaz E. Alsaadi b , Fuad E. Alsaadi c , Viet-Thanh Pham d,∗ , Karthikeyan Rajagopal e a
Ministry of Higher Education and Scientific Research, Baghdad, Iraq Department of information Technology, Faculty of Computing and IT, King Abdulaziz University, Jeddah, Saudi Arabia c Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia d Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam e Center for Nonlinear Dynamics, Defence University, Ethiopia b
a r t i c l e
i n f o
Article history: Received 19 July 2019 Received in revised form 23 August 2019 Accepted 23 August 2019 Available online 29 August 2019 Communicated by M. Perc Keywords: Synchronization Multilayer network Gene oscillators
a b s t r a c t Study of the synchronization in the network of gene oscillator network has vital importance in understanding of rhythmicity of molecular and cellular activities. In this paper, we analyze a network of linearly coupled genetic oscillators in a multiplex structure. The coupling strength values are changed and the coupling range is considered to be fixed, but different in the two layers. The analyses are done in two cases of periodic and chaotic oscillations. By computing the statistical measures, the interlayer and intralayer synchronization states are studied. The results show that the layer with higher coupling range has more enhanced synchrony and is less affected by the turbulent behavior of the other layer. On the other hand, the layer with lower coupling range approaches synchronization by strengthening the interlayer and intralayer couplings. The interlayer synchronization is also achieved in high coupling strength values. © 2019 Elsevier B.V. All rights reserved.
1. Introduction In a complex system, the observed behavior is the collection of the activities of the interacting elements [1]. In order to study these systems, the elements can be modeled as a set of nodes, with some links between them, representing the interactions. The links can be a description of the flow of information between the components [2]. The multilayer structure provides a more realistic framework for the study of complex systems. In real systems, the interaction of the components has various types or can vary in time [3]. The use of multiple layers in a network provides the possibility of considering different interaction types. As an example, a social system is a hypernetwork which is composed of a group of dynamical systems such as families, friends, colleagues, . . . , coupled through the connections of distinct networks. Therefore it can be considered as a multilayer network [4]. Biological systems are complex systems comprised of many interacting bio-entities [5]. In the body, all of the cells interact with each other to perform a special function perfectly. Furthermore, the relationships between bio-entities have different attributes. There-
*
Corresponding author. E-mail address: [email protected] (V.-T. Pham).
https://doi.org/10.1016/j.physleta.2019.125919 0375-9601/© 2019 Elsevier B.V. All rights reserved.
fore, a multilayer network, in which the layers represent different relations, seems to be more illustrative. The function of a biological network is characterized by the relations between the fundamental genes, proteins, metabolites, or cells [6]. Gene regulation is mediated by various interactions between different kinds of molecules [7]. The interactions can be either through direct coupling or indirect one. In direct interaction, the coupling is defined by the differences between concentration variables within neighboring cells. While in indirect one, the substances crossing the cellular membrane into the extracellular space are considered [8]. Many factors can effect on these interactions, such as the number of neighboring cells in direct interaction or the rate of the exchange of substances in indirect interaction. The multilayer structure can be helpful in analyzing these different contributions. Synchronization is a ubiquitous phenomenon in nature which has attracted much attention in different fields such as biology, physics and technology [9–12]. Due to the great potential of chaos in many disciplines, synchronization of coupled chaotic systems has been intensively studied [13–16]. In neuroscience, synchronization has been revealed in the olfactory system or the hippocampal region [17]. In addition, it has strong relations with some brain diseases such as the epileptic seizure [18,19]. Therefore synchronization has been a hot topic in neuroscience [20–23].
2
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Fig. 1. (a) The motif of a repressilator, composed of three gene elements. (b) The motif of a single gene with a delayed self-inhibition [33].
Fig. 3. (a) The motif of a single gene with two delayed self-inhibitions. (b) The bifurcation diagram of the motif concerning time delay. (c) The chaotic time series of the motif for τ1 = 18 min and τ2 = 4.65 min [33]. Fig. 2. (a) The bifurcation diagram of the single gene element with one negative loop and (b) the periodic time series for τ = 12 min [33].
Study of synchronization in multilayer networks has provided different collective properties and also different synchronization types [24–26]. The synchronization has also been revealed in cellular communications [27]. Many studies have focused on the synchronization analysis of gene networks with the aim of understanding the rhythmic behavior of living organisms [27–32]. In this paper, we study synchronization in a multiplex gene oscillator network. A simple gene motif is selected as the elements of the network. The motif is composed of single gene element with self-inhibition with time delay. Since the relations in the gene network are not fixed, we consider two different coupling ranges in the two layers. The interlayer and intralayer coupling strength values are taken as the control parameter for the emergence of synchronization in the network. Actually, the synchronization is studied by calculating statistical measures. 2. The model The simplest oscillatory circuit motif is the repressilator, which is relevant to the natural gene network. The repressilator is composed of three genes with sequential inhibitions (Fig. 1a) and can produce stable periodic oscillations. Suzuki et al. [33] showed that
Fig. 4. A multiplex framework of the used network. The black lines show the intralayer couplings and the blue ones show the interlayer couplings. The number of coupling neighbors in the first layer is p 1 = 2 and in the second layer is p 2 = 20. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
A.J.M. Khalaf et al. / Physics Letters A 383 (2019) 125919
3
Fig. 5. The time snapshots of the 2-layer network composed of genes with one negative feedback loop (τ = 12 min) and for σ = 0.004. (a) γ = 0, (b) γ = 0.01, (c) γ = 0.05, (d) γ = 0.1, (e) γ = 0.5. Increment of the interlayer coupling increases the first layer synchronization and finally leads to interlayer synchronization. (f) Time series of the first (blue) and the second gene oscillator (red) of layer I and the first gene oscillator (green) of layer II, corresponding to part (e).
the dynamics of this three-gene circuit is equivalent to a selfinhibitory single gene element with a time delay (Fig. 1b). The time dynamics of a self-inhibitory single-gene element with a negative feedback loop was proposed by the following deterministic rate equation:
d A (t ) dt
= F (A, t ) = g A H − A A A (t − τ ) − k A A (t )
(1)
where A refers to the protein concentration (nM). The measurements are time (t) in minutes, the protein production rate (g A = 50) in nM/min, and the degradation rate (k A = 0.2) in 1/min. H A A is an inhibitory Hill function that if gene X is inhibited by gene Y , then the inhibitory Hill function is defined by:
n H− X Y [ Y ] ≡ 1/ 1 + ( Y / Y 0 )
(2)
where Y 0 = 38 nM is the threshold concentration of the Hill function and n = 4 is its rank. Fig. 2a shows the bifurcation diagram of the single inhibitory gene, and Fig. 2b shows the periodic time series of the gene at τ = 12 min.
Suzuki et al. [33] further showed that adding another negative feedback loop to the single gene element can enrich its dynamics and lead to the emergence of quasi-periodic and chaotic oscillations. Fig. 3a shows the gene element with two negative feedback loops. The rate equation of this motif is described by:
d A (t ) dt
= F (A, t ) = g A H 1−A A A (t − τ1 ) H 2−A A A (t − τ2 ) − k A A (t ) (3)
where the parameters are fixed at: g A = 50 nM/min, k A = 0.21/min, n1 A A = 4 and n2 A A = 2. Fig. 3b shows the bifurcation diagram of this motif and Fig. 3c shows a chaotic time series obtained from the gene at τ1 = 18 min and τ2 = 4.65 min. In this case, using the 0-1 test results in k = 0.996, which confirms the existence of chaotic behavior. In order to investigate the gene network, we consider a twolayer multiplexed network with linear coupling, as shown in Fig. 4. The equations of the network are described by:
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Fig. 6. The time snapshots of the 2-layer network composed of genes with one negative feedback loop (τ = 12 min) and for σ = 0.01. (a) (d) γ = 0.5. Increasing the interlayer coupling raises the coherency of the first layer toward being synchronous with the second layer.
γ = 0, (b) γ = 0.01, (c) γ = 0.1,
Fig. 7. Synchronization measures concerning coupling strengths, for the 2-layer network composed of genes with one negative feedback loop (τ = 12 min). (a) Interlayer synchronization error, (b) first layer synchronization factor, (c) second layer synchronization factor.
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Fig. 8. The time snapshots of the 2-layer network composed of genes with two negative feedback loops (τ1 = 18 min and τ2 = 4.65 min) and for σ = 0.002. (a) γ = 0, (b) γ = 0.01, (c) γ = 0.05, (d) γ = 0.1, (e) γ = 0.5. By increasing of γ , at first, the coherency of the second layer is disturbed and then is recovered. The first layer also moves from the asynchronous state to synchronous behavior. (f) Time series of the first (blue) and the second gene oscillator (red) of layer I and the first gene oscillator (green) of layer II, corresponding to part (e).
d A i ,1 dt d A i ,2 dt
i+ p1
= F ( A i ,1 , t ) + σ
= F ( A i ,2 , t ) + σ
N
(4)
i+ p2
E =<
( A j ,1 − A i ,1 ) + γ ( A i ,2 − A i ,1 )
j =i − p 1
N 1
( A j ,2 − A i ,2 ) + γ ( A i ,1 − A i ,2 )
j =i − p 2
where A i ,1 and A i ,2 represents the gene elements in layer one and two, respectively. The intralayer coupling strength is represented by σ , and the interlayer coupling strength is represented by γ . p 1 and p 2 are the number of nearest neighbors in coupling in the first and second layers and are considered to be p 1 = 2 and p 2 = 20. The total number of the gene elements in each layer is assumed to be N = 500. The interlayer and intralayer coupling strength values are the control parameters for investigating network behavior. The intralayer synchronization refers to the synchronization of each layer separately. If the two layers become synchronous with each other, it is called interlayer synchronization. We quantify the intralayer synchronization with using a statistical factor and the interlayer synchronization by defining the following error:
j =1
| A 1, j − A 2, j | A 21, j + A 22, j
>t
(5)
where <>t denotes the average over time. The statistical factor for the intralayer synchronization (R) is based on the mean field theory, and R = 1 shows a complete synchronization. This factor can be calculated by:
F=
R=
N 1
N
1 N
xi
i =1
N
(6)
F 2 − F 2
i =1 [<
A 2i > − < A i >2 ]
3. Results We firstly hypothesize that the network is comprised of the genes with one negative feedback loop, and then we analyze the network of genes with two negative feedback loops. In order to
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Fig. 9. The time snapshots of the 2-layer network composed of genes with two negative feedback loops (τ1 = 18 min and τ2 = 4.65 min) and for σ = 0.02. (a) (b) γ = 0.05, (c) γ = 0.2, (d) γ = 0.5. In this case, the increment of γ leads to movement of the first layer from the chimera state to the synchronous state.
solve the network, the 4th order Runge-Kutta method was used, and the initial conditions were chosen randomly in [0, 100]. 3.1. The network of gene elements with one delayed self-inhibition The network patterns are observed by taking the two coupling strength values as the control parameters. In the case of independence of the two layers (γ = 0), for very small intralayer couplings as σ = 0.004, the first layer is asynchronous, and the second layer has complete phase synchronization as shown in Fig. 5a. If the interlayer links connect, then the layers may effect on each other. Fig. 5b shows the layers’ behavior for γ = 0.01. In this case, the asynchronous state of the first layer causes a disruption in the synchronization of the second layer oscillators. When γ increases to γ = 0.05, as it is shown in Fig. 5c, the second layer has more effects on the first layer and change its behavior to chimera state, at which some oscillators are synchronous and some are asynchronous [34,35]. By more increasing of γ , the first layer tends to be synchronous and the interlayer synchronization enhances (as shown in Fig. 5d, e). Fig. 5f shows some of the oscillators time series in the case of complete synchronization (Fig. 5e). Next the network is investigated by considering stronger intralayer coupling strength as σ = 0.01. In this case, when there is no link between the layers, the first layer shows chimera state, and the second one shows phase synchronization (as shown in Fig. 6a). Connecting the interlayer links leads to the influence of the first layer on the second one. Therefore, the second layer loses its complete phase synchronization (as shown in Fig. 6b for γ = 0.01). Further increasing of γ to γ = 0.1, makes the second layer return to its synchronous state. In this case, the first layer shows a sine-like synchronization (as shown in Fig. 6c). Finally growing γ to higher values, enhances the interlayer synchronization, and brings the first layer to phase synchronous state (as shown in Fig. 6d).
γ = 0,
Fig. 7 shows the synchronization measures concerning intralayer and interlayer couplings. Fig. 7a shows that increases in both intralayer and interlayer coupling strength causes decrement of interlayer synchronization error (E). Fig. 7b shows the synchronization factor of the first layer, which is defined by R 1 in the figure. According to the obtained factor, increasing of σ helps in synchronization of the first layer in low interlayer couplings. However, when γ is large enough, the first layer is synchronous for all σ values. Fig. 7c shows the synchronization factor of the second layer, defined by R 2 , and confirms that except for σ = 0, the second layer has high synchronization factor. 3.2. The network of gene elements with two delayed self-inhibitions In this subsection, the gene oscillators exhibit chaotic behavior. Similar to the previous case, the effects of coupling strengths are investigated. Staring from very low intralayer coupling as σ = 0.002, and no interlayer links (γ = 0), the first layer is asynchronous, and the second layer is phase synchronized (as shown in Fig. 8a). When the interlayer links connect, the asynchrony of the first layer disrupts the phase synchronization of the second layer (as shown in Fig. 8b for γ = 0.01). If γ increases, both layers have disturbed phase synchronization state (as shown in Fig. 8c, d) and further increasing of it leads to interlayer synchronization and intralayer phase synchronization (as shown in Fig. 8e). Fig. 8f shows some of the oscillators time series in the case of complete synchronization (Fig. 8e). In the next step, the intralayer coupling strength is fixed at stronger values as σ = 0.02. In this case, when γ = 0, the first layer exhibits chimera state, and the second layer is phase synchronized (as shown in Fig. 9a). Increasing of γ to γ = 0.05 changes the behavior of the first layer to a sine-like synchronization (as shown in Fig. 9b). More increasing of interlayer coupling strength changes the behavior of the layers to have interlayer synchroniza-
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Fig. 10. Synchronization measures concerning coupling strengths, for the 2-layer network composed of genes with two negative feedback loops (τ1 = 18 min and 4.65 min). (a) Interlayer synchronization error, (b) first layer synchronization factor, (c) second layer synchronization factor.
tion and intralayer phase synchronization (as shown in Fig. 9c, d). The synchronization factors, in the case of using gene elements with two delayed self-inhibitions, are shown in Fig. 10. Fig. 10a shows that at each value of γ , by increasing the intralayer coupling, the interlayer synchronization error decays to zero. Fig. 10b demonstrates that increasing of both γ and σ is necessary for intralayer synchronization of the first layer. Subsequently, Fig. 10c shows that irrespective of σ value, the second layer’s synchronization enhances by growing of γ .
7
τ2 =
interlayer coupling strength. Calculating the interlayer synchronization error confirmed that by increasing both coupling strengths, the error decreases and comes to zero. However, when the gene elements were in chaotic oscillation, the error in low intralayer couplings was more than when the gene elements were in periodic oscillation. Declaration of competing interest None.
4. Conclusion References In this paper, we studied the synchronization in a network of coupled gene oscillators. As the elements of the network, we used a simple gene motif consisting of a single gene with inhibitory delayed feedback, in two cases of one loop which showed periodic behavior and two loops which could show chaotic behavior. The multiplex framework was used to be able to consider different interactions. The number of elements in the coupling was fixed differently in the two layers. The strength of the inter and intralayer links was considered as the control parameter for synchronization, similar to the using of the coupled controller in monolayer networks [36]. Even by increasing the number of layers, this coupled controller is still applicable for synchronization of the hypernetwork. The intralayer synchronization was quantified with using a synchronization factor, and the interlayer synchronization was measured by computing an error. The results showed that the intralayer synchronization of the first layer was enhanced by increasing both coupling strengths. Furthermore, the intralayer synchronization of the second layer was enhanced by increasing the
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