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Statistics for anti-synchronization of intracellular calcium dynamics
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Statistics for anti-synchronization of intracellular calcium dynamicsR Wei-Long Duan a,∗, Chunhua Zeng b a
City College, Kunming University of Science and Technology, Kunming 650051, China State Key Laboratory of Complex Nonferrous Metal Resources Clean Utilization/Faculty of Science, Kunming University of Science and Technology, Kunming 650093, China
b
a r t i c l e
i n f o
PACS: 05.45.Xt 87.18.Tt 87.16.Xa Keywords: Synchronization Anti-synchronization Intracellular calcium oscillation Non-Gaussian noise
a b s t r a c t The anti-synchronization of calcium oscillation between cytosol and calcium store in intracellular calcium oscillation system with non-Gaussian noises and time delay are studied by means of second-order stochastic Runge–Kutta type algorithm. Basic on statistic, the normalized global synchronization error of Ca2+ concentration in cytosol and calcium store is simulated, the results exhibit, it shifts into single trough from multi troughs structure as parameter p (which is used to control the degree of the departure from the non-Gaussian noise and Gaussian noise) or density of non-Gaussian noises increases, however, it shifts into multi troughs from single trough structure as correlation time of non-Gaussian noises increases. In addition, the short, moderate, and long time delay respectively induce antisynchronously quasi-periodic, synchronously quasi-periodic, and chaotic oscillation. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Ca2+ is an ubiquitous and versatile second messenger that transmits information through changes of the cytosolic Ca2+ concentration, namely, Ca2+ signaling pathway translates external signals into intracellular responses by increasing the cytosolic Ca2+ concentration in a stimulus dependent pattern. Specifically, the increasing of concentration can be caused either by Ca2+ entry from the extracellular medium through plasma membrane channels, or by Ca2+ release from the internal calcium store. There are a variety of channels showing calcium-induced calcium release and a variety of models [1–4] so far. In many studies on intracellular calcium oscillation(ICO) system, some phenomena have been found such as stochastic resonance [5,6], reverse resonance [6–8], oscillatory coherence [9], coherence resonance [7], resonant activation [10], bistability solutions with hysteresis [11,12], calcium puffs [13], various spontaneous Ca2+ patterns [14], stochastic backfiring [15], dispersion gap and localized spiral waves [16], stability transition [17], calcium wave instability [18], colored noise optimized calcium wave [19], periodic square calcium wave [20], etc. Importantly, ICO has been intensively studied by Perc group [21–24] and Falcke group [15,16,25–29]. Among, Perc group have found that noise and other stochastic effects indeed play a central role [21,22], in the transmission processes of intraR This project was supported by the National Natural Science Foundation of China (grant nos. 11305079 and 11347014), the Candidate Talents Training Fund of Yunnan Province (Project No. 2015HB025) and Introduction of talent capital group fund project of Kunming University of Science and Technology (grant no. KKZ3201407030). ∗ Corresponding author. E-mail address: [email protected] (W.-L. Duan).
http://dx.doi.org/10.1016/j.amc.2016.07.041 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: W.-L. Duan, C. Zeng, Statistics for anti-synchronization of intracellular calcium dynamics, Applied Mathematics and Computation (2016), http://dx.doi.org/10.1016/j.amc.2016.07.041
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cellular Ca2+ , there may be non-Gaussian noise [29]. Recently, we have studied ICO system driven by non-Gaussian noises [7,8], then found that ICO exhibits synchronous or anti-synchronous oscillation between cytosol and calcium store [7]. Interestingly, this kind of synchronization and anti-synchronization oscillation can regulate the calcium concentration in cytosol and calcium store [30], then can change the calcium wave across cells, so that can adjust finally the signal transmission across cells. Thus, in this paper, we compute the normalized global synchronization error of Ca2+ concentration in cytosol and calcium store to study anti-synchronous ICO. First, according to Refs. [7,8], the ICO model with non-Gaussian noises and time delay is presented. Then, the normalized global synchronization error is computed. Finally, conclusions are obtained. 2. The model for ICO with non-Gaussian noises and time delay Taking into account same time delay τ in processes of active and passive transport of Ca2+ in a real cell, the Langevin equations of ICO system read as follows [7,8]:
dt x = A1 (x; xτ , yτ ) + B1 (x; xτ , yτ )η1 (t ),
(1)
dt y = A2 (x, y; xτ ) + B2 (x, y; xτ )η2 (t ),
(2)
A1 (x; xτ , yτ ) = v0 + v1 β0 − v2 + v3τ + k f yτ − kx,
(3)
A2 (x, y; xτ ) = v2τ − v3 − k f y,
(4)
with
B1 ( x; xτ , yτ ) =
B2 (x, y; xτ ) = W ( x; xτ , yτ ) =
v21 β02 + 2v1 β0 λW + W 2,
v2τ + v3 + k f y
V
(5)
,
(6)
v0 + v1 β0 + v2 + v3τ + k f yτ + kx V
,
(7)
and
V2 x2 V3 x4 y2 , v3 = 4 , 2 + k1 (x + k42 )(y2 + k23 ) V x2 V3 x4τ y2τ v2τ = 2 2 τ 2 , v3τ = 4 . xτ + k1 (xτ + k42 )(y2τ + k23 )
v2 =
x2
(8) (9)
Here, x and y denote concentration of free Ca2+ of cytosol and calcium store in a cell, respectively. The rate v2 and v3 refer, respectively, to pumping of Ca2+ into the calcium store and to release of Ca2+ from store into cytosol in a process activated by cytosolic Ca2+ . v2τ is v2 with time delay, and v3τ is v3 with time delay. W = W (x; xτ , yτ ), xτ = x(t − τ ), yτ = y(t − τ ). λ denotes cross-correlation degree of internal and external noise before merger [12]. The noises η1 (t) and η2 (t) in Eqs. (1) and (2) are considered as non-Gaussian noises [8] which are characterized by the following Langevin equation [31]:
1 d dηi (t ) =− V (η ) + dt τ1 dηi ip i
√ 2D
τ1
ξi (t ), i = 1, 2.
(10)
Where ξ i (t) is a standard Gaussian white noise of zero mean and correlation ξi (t )ξi (t ) = δ (t − t ).Vip (ηi ) is given by
Vip (ηi ) =
η2 D τ1 ln[1 + ( p − 1 ) i ], τ1 ( p − 1 ) D 2
(11)
and the statistical properties of non-Gaussian noise ηi (t) is defined as
ηi (t ) = 0, ⎧ 2D ⎨ , p ∈ (−∞, 53 ), ηi2 (t ) = τ1 (5 − 3 p) ⎩ ∞, p ∈ [ 35 , 3 ),
(12)
(13)
where τ 1 denotes the correlation time of the non-Gaussian noises ηi (t), and D denotes the noise intensity of Gaussian white noise ξ i (t). The parameter p is used to control the degree of the departure from the non-Gaussian noise to Gaussian noise. The distribution of the noise is Gaussian for p = 1, non-Gaussian with long tail for p > 1, and characterized by a “more than Gaussian” cutoff for p < 1. Here, in order to study easily, supposing noises ξ 1 (t) and ξ 2 (t) have same strength D, and non-Gaussian noises η1 (t) and η2 (t) have same p and correlation time τ 1 . Please cite this article as: W.-L. Duan, C. Zeng, Statistics for anti-synchronization of intracellular calcium dynamics, Applied Mathematics and Computation (2016), http://dx.doi.org/10.1016/j.amc.2016.07.041
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3
3. The normalized global synchronization error By means of second-order stochastic Runge–Kutta type algorithm [32], for the specific simulation algorithm of ICO system, see Ref. [7], discretize time in steps of size = 0.001 s, one can stochastically simulate the time evolution of intracellular Ca2+ concentration in the cytosol x(t) and calcium store y(t). Experimentally, x is in the order of 100 ∼ 200nM in basal state [33] and y = 5 μM [34], so that the initial values x(0) and y(0) independently take uniformly random from 0.1 ∼ 0.2 μM and 4−5 μM. In the condition of time delay, it is rational to let x(t − τ ) = x(0 ) and y(t − τ ) = y(0 ) as t < τ . In this paper, the other parameters are set as Ref. [7]. Due to x(t) and y(t) would be all more than 1, in order to study further, we first normalize time series x(t) and y(t) as follows
X (t ) =
x(t ) y(t ) , Y (t ) = , max(x(t )) max(y(t ))
(14)
in this way the normalized time series of intracellular Ca2+ concentration of cytosol X(t) and calcium store Y(t) are obtained. Here, we aim to study anti-synchronous oscillation of intracellular Ca2+ between cytosol and calcium store, because antisynchronous states is opposite to synchronous states, like studying synchronous states, one define the normalized global synchronization error [35] as follow which distinguish synchronous states from anti-synchronous states
σ=
X (t ) − Y (t )2 . X 2 (t ) + Y 2 (t )
(15)
According to Ref. [36], for globally synchronous states σ ideally approaches 0, while for desynchronous states σ is nonzero, namely for anti-synchronous states σ ideally approaches 1. In practice, due to experimental mismatch and noise, the σ of synchronous states is small but nonzero, at the same time, the σ of anti-synchronous states is high but less than 1. In a word, the higher the σ is, the more anti-synchronous the states are; the lower σ the σ is, the more synchronous the states are; these correspond quasi-periodic oscillation. The other correspond chaotic oscillation.
3 0.7 0.6
0.6 0.5
0.5
2
0.4
σ
0.4
2.5
0.3 0.3 0.2 0.1
1.5 0.2 0.1
1
0
p 1
10
0
10
0.5
−1
10
τ(s)
−2
10
Fig. 1. The normalized global synchronization error σ as a function of time delay τ and parameter p of non-Gaussian noises.
Please cite this article as: W.-L. Duan, C. Zeng, Statistics for anti-synchronization of intracellular calcium dynamics, Applied Mathematics and Computation (2016), http://dx.doi.org/10.1016/j.amc.2016.07.041
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0.7 0.6 0.6 4
0.5 0.5
3.5 0.4
3
σ
0.4
0.3
2.5
0.3
2
0.2
1.5
0.2 0.1 0.1
D
1
0
0.5 1
10
0
10
−1
10
0 −2
10
τ(s) Fig. 2. The normalized global synchronization error σ as a function of time delay τ and intensity D of non-Gaussian noises.
In the following, the effect of time delay and non-Gaussian noises on normalized global synchronization error σ are simulated. In Fig. 1, the normalized global synchronization error σ vs. time delay τ and parameter p of non-Gaussian noises is simulated, here D = 1, τ1 = 10 s. It exhibits that, the short time delay induces anti-synchronously quasi-periodic calcium oscillation between cytosol and calcium store. However, the other time delay destroys this quasi-periodic oscillation and makes it become chaotic oscillation, even turns to synchronously quasi-periodic oscillation when parameter p is small. As parameter p increases, σ shifts into single trough from multi troughs structure. In Fig. 2, the normalized global synchronization error σ vs. time delay τ and intensity D of non-Gaussian noises is simulated, here p = 0.9, τ1 = 10s. It is clearly seen that, the short time delay induces anti-synchronously quasi-periodic oscillation, however, the moderate time delay induces synchronously quasi-periodic oscillation and the long time delay induces chaotic oscillation; as non-Gaussian noises increase, σ shifts into single trough from multi troughs structure. In Fig. 3, the normalized global synchronization error σ vs. time delay τ and correlation time τ 1 of non-Gaussian noises is simulated, here p = 0.9, D = 1. As stated in figure, the short time delay also induces anti-synchronously quasi-periodic oscillation, the moderate time delay also induces synchronously quasi-periodic oscillation, and the long time delay also induces chaotic oscillation; however, as correlation time τ 1 increases, σ shifts into multi troughs from single trough structure. Where, the more the troughs show, the easier the synchronously quasi-periodic oscillation occurs; oppositely, the less the troughs show, the more difficult the synchronously quasi-periodic oscillation occurs. There is important role for signal transmission across cells to research anti-synchronous ICO, because calcium wave can change the electric activities of neurons, i.e., calcium wave plays important role in neurodynamics. Recently, many researchers have studied neurodynamics [37–41] and obtained outstanding results. For the role of noise on some stochastic systems, there are also some research [42–45]. 4. Conclusions In view of non-Gaussian noises and time delay in transmission processes of intracellular Ca2+ , by means of second-order stochastic Runge–Kutta type algorithm, in order to research on anti-synchronization of Ca2+ oscillation between cytosol and Please cite this article as: W.-L. Duan, C. Zeng, Statistics for anti-synchronization of intracellular calcium dynamics, Applied Mathematics and Computation (2016), http://dx.doi.org/10.1016/j.amc.2016.07.041
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0.6
5
0.7 0.6
0.5 0.5 100 0.4
σ
0.4
0.3
80
0.3 60 0.2
0.2
40
τ (s)
0.1 20 0.1
1
0 0
0
10
−1
τ(s)
10
Fig. 3. The normalized global synchronization error σ as a function of time delay τ and correlation time τ 1 of non-Gaussian noises.
calcium store, the normalized global synchronization error of Ca2+ concentration in cytosol and calcium store is simulated. The results exhibit, the short time delay induces anti-synchronously quasi-periodic calcium oscillation between cytosol and calcium store, the moderate time delay induces synchronously quasi-periodic oscillation, and the long time delay induces chaotic oscillation. As parameter p of non-Gaussian noises increases, the normalized global synchronization error shifts into single trough from multi troughs. As non-Gaussian noises increase, the normalized global synchronization error shifts into single trough from multi troughs structure. However, as correlation time of non-Gaussian noises increases, the normalized global synchronization error shifts into multi troughs from single trough structure. References [1] A. Goldbeter, G. Dupont, M.J. Berridge, Minimal model for signal-induced Ca2+ oscillations and for their frequency encoding through protein phosphorylation, PNAS 87 (1990) 1461–1465. [2] Y. Shiferaw, D. Sato, A. Karma, Coupled dynamics of voltage and calcium in paced cardiac cells, Phys. Rev. E 71 (2005) 021903. [3] A.C. Ventura, L. Bruno, S.P. Dawson, Simple data-driven models of intracellular calcium dynamics with predictive power, Phys. Rev. E 74 (2006) 011917. [4] R. Thul, M. Falcke, Frequency of elemental events of intracellular Ca2+ dynamics, Phys. Rev. E 73 (2006) 061923. [5] H. Li, Z. Hou, H. Xin, Internal noise stochastic resonance for intracellular calcium oscillations in a cell system, Phys. Rev. E 71 (2005) 061916. [6] W.L. Duan, F. Long, C. Li, Reverse resonance and stochastic resonance in intracellular calcium oscillations, Physica A 401 (2014) 52–57. [7] L. Lin, W.L. Duan, The phenomena of an intracellular calcium oscillation system with non-Gaussian noises, Chaos, Solitons Fractals 77 (2015) 132–137. [8] W.L. Duan, C.H. Zeng, Role of time delay on intracellular calcium dynamics driven by non-Gaussian noises, Sci. Rep. 6 (2016) 25067. [9] W.L. Duan, Time delay induces oscillatory coherence in intracellular calcium oscillation system, Physica A 405 (2014) 10–16. [10] W.L. Duan, P.F. Duan, Time delay induces resonant activation in intracellular calcium oscillations, Chin. J. Phys. 52 (2014) 1059–1068. [11] J.M.A.M. Kusters, J.M. Cortes, W.P.M. van Meerwijk, D.L. Ypey, A.P.R. Theuvenet, C.C.A.M. Gielen, Hysteresis and bistability in a realistic cell model for calcium oscillations and action potential firing, Phys. Rev. Lett. 98 (2007) 098107. [12] W.L. Duan, L.J. Yang, D.C. Mei, Simulation of time delay effects in the intracellular calcium oscillation of cells, Phys. Scr. 83 (2011) 015004. [13] S. Rüdiger, J.W. Shuai, I.M. Sokolov, Law of mass action, detailed balance, and the modeling of calcium puffs, Phys. Rev. Lett. 105 (2010) 048103. [14] J.W. Shuai, P. Jung, Selection of intracellular calcium patterns in a model with clustered Ca2+ release channels, Phys. Rev. E 67 (2003) 031905. [15] M. Falcke, L. Tsimring, H. Levine, Stochastic spreading of intracellular Ca2+ release, Phys. Rev. E 62 (20 0 0) 2636. [16] M. Falcke, M. Or-Guil, M. Bär, Dispersion gap and localized spiral waves in a model for intracellular Ca2+ dynamics, Phys. Rev. Lett. 84 (20 0 0) 4753. [17] L. Lin, W.L. Duan, Extrinsic periodic information interpolates between monostable and bistable states in intracellular calcium dynamics, Physica A 427 (2015) 155–161. [18] C.B. Tabi, I. Maïna, A. Mohamadou, H.P.E. Fouda, T.C. Kofané, Wave instability of intercellular Ca2+ oscillations, EPL 106 (2014) 18005.
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Please cite this article as: W.-L. Duan, C. Zeng, Statistics for anti-synchronization of intracellular calcium dynamics, Applied Mathematics and Computation (2016), http://dx.doi.org/10.1016/j.amc.2016.07.041
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Applied Mathematics and Computation 0 0 0 (2016) 1–6
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Statistics for anti-synchronization of intracellular calcium dynamicsR Wei-Long Duan a,∗, Chunhua Zeng b a
City College, Kunming University of Science and Technology, Kunming 650051, China State Key Laboratory of Complex Nonferrous Metal Resources Clean Utilization/Faculty of Science, Kunming University of Science and Technology, Kunming 650093, China
b
a r t i c l e
i n f o
PACS: 05.45.Xt 87.18.Tt 87.16.Xa Keywords: Synchronization Anti-synchronization Intracellular calcium oscillation Non-Gaussian noise
a b s t r a c t The anti-synchronization of calcium oscillation between cytosol and calcium store in intracellular calcium oscillation system with non-Gaussian noises and time delay are studied by means of second-order stochastic Runge–Kutta type algorithm. Basic on statistic, the normalized global synchronization error of Ca2+ concentration in cytosol and calcium store is simulated, the results exhibit, it shifts into single trough from multi troughs structure as parameter p (which is used to control the degree of the departure from the non-Gaussian noise and Gaussian noise) or density of non-Gaussian noises increases, however, it shifts into multi troughs from single trough structure as correlation time of non-Gaussian noises increases. In addition, the short, moderate, and long time delay respectively induce antisynchronously quasi-periodic, synchronously quasi-periodic, and chaotic oscillation. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Ca2+ is an ubiquitous and versatile second messenger that transmits information through changes of the cytosolic Ca2+ concentration, namely, Ca2+ signaling pathway translates external signals into intracellular responses by increasing the cytosolic Ca2+ concentration in a stimulus dependent pattern. Specifically, the increasing of concentration can be caused either by Ca2+ entry from the extracellular medium through plasma membrane channels, or by Ca2+ release from the internal calcium store. There are a variety of channels showing calcium-induced calcium release and a variety of models [1–4] so far. In many studies on intracellular calcium oscillation(ICO) system, some phenomena have been found such as stochastic resonance [5,6], reverse resonance [6–8], oscillatory coherence [9], coherence resonance [7], resonant activation [10], bistability solutions with hysteresis [11,12], calcium puffs [13], various spontaneous Ca2+ patterns [14], stochastic backfiring [15], dispersion gap and localized spiral waves [16], stability transition [17], calcium wave instability [18], colored noise optimized calcium wave [19], periodic square calcium wave [20], etc. Importantly, ICO has been intensively studied by Perc group [21–24] and Falcke group [15,16,25–29]. Among, Perc group have found that noise and other stochastic effects indeed play a central role [21,22], in the transmission processes of intraR This project was supported by the National Natural Science Foundation of China (grant nos. 11305079 and 11347014), the Candidate Talents Training Fund of Yunnan Province (Project No. 2015HB025) and Introduction of talent capital group fund project of Kunming University of Science and Technology (grant no. KKZ3201407030). ∗ Corresponding author. E-mail address: [email protected] (W.-L. Duan).
http://dx.doi.org/10.1016/j.amc.2016.07.041 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: W.-L. Duan, C. Zeng, Statistics for anti-synchronization of intracellular calcium dynamics, Applied Mathematics and Computation (2016), http://dx.doi.org/10.1016/j.amc.2016.07.041
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cellular Ca2+ , there may be non-Gaussian noise [29]. Recently, we have studied ICO system driven by non-Gaussian noises [7,8], then found that ICO exhibits synchronous or anti-synchronous oscillation between cytosol and calcium store [7]. Interestingly, this kind of synchronization and anti-synchronization oscillation can regulate the calcium concentration in cytosol and calcium store [30], then can change the calcium wave across cells, so that can adjust finally the signal transmission across cells. Thus, in this paper, we compute the normalized global synchronization error of Ca2+ concentration in cytosol and calcium store to study anti-synchronous ICO. First, according to Refs. [7,8], the ICO model with non-Gaussian noises and time delay is presented. Then, the normalized global synchronization error is computed. Finally, conclusions are obtained. 2. The model for ICO with non-Gaussian noises and time delay Taking into account same time delay τ in processes of active and passive transport of Ca2+ in a real cell, the Langevin equations of ICO system read as follows [7,8]:
dt x = A1 (x; xτ , yτ ) + B1 (x; xτ , yτ )η1 (t ),
(1)
dt y = A2 (x, y; xτ ) + B2 (x, y; xτ )η2 (t ),
(2)
A1 (x; xτ , yτ ) = v0 + v1 β0 − v2 + v3τ + k f yτ − kx,
(3)
A2 (x, y; xτ ) = v2τ − v3 − k f y,
(4)
with
B1 ( x; xτ , yτ ) =
B2 (x, y; xτ ) = W ( x; xτ , yτ ) =
v21 β02 + 2v1 β0 λW + W 2,
v2τ + v3 + k f y
V
(5)
,
(6)
v0 + v1 β0 + v2 + v3τ + k f yτ + kx V
,
(7)
and
V2 x2 V3 x4 y2 , v3 = 4 , 2 + k1 (x + k42 )(y2 + k23 ) V x2 V3 x4τ y2τ v2τ = 2 2 τ 2 , v3τ = 4 . xτ + k1 (xτ + k42 )(y2τ + k23 )
v2 =
x2
(8) (9)
Here, x and y denote concentration of free Ca2+ of cytosol and calcium store in a cell, respectively. The rate v2 and v3 refer, respectively, to pumping of Ca2+ into the calcium store and to release of Ca2+ from store into cytosol in a process activated by cytosolic Ca2+ . v2τ is v2 with time delay, and v3τ is v3 with time delay. W = W (x; xτ , yτ ), xτ = x(t − τ ), yτ = y(t − τ ). λ denotes cross-correlation degree of internal and external noise before merger [12]. The noises η1 (t) and η2 (t) in Eqs. (1) and (2) are considered as non-Gaussian noises [8] which are characterized by the following Langevin equation [31]:
1 d dηi (t ) =− V (η ) + dt τ1 dηi ip i
√ 2D
τ1
ξi (t ), i = 1, 2.
(10)
Where ξ i (t) is a standard Gaussian white noise of zero mean and correlation ξi (t )ξi (t ) = δ (t − t ).Vip (ηi ) is given by
Vip (ηi ) =
η2 D τ1 ln[1 + ( p − 1 ) i ], τ1 ( p − 1 ) D 2
(11)
and the statistical properties of non-Gaussian noise ηi (t) is defined as
ηi (t ) = 0, ⎧ 2D ⎨ , p ∈ (−∞, 53 ), ηi2 (t ) = τ1 (5 − 3 p) ⎩ ∞, p ∈ [ 35 , 3 ),
(12)
(13)
where τ 1 denotes the correlation time of the non-Gaussian noises ηi (t), and D denotes the noise intensity of Gaussian white noise ξ i (t). The parameter p is used to control the degree of the departure from the non-Gaussian noise to Gaussian noise. The distribution of the noise is Gaussian for p = 1, non-Gaussian with long tail for p > 1, and characterized by a “more than Gaussian” cutoff for p < 1. Here, in order to study easily, supposing noises ξ 1 (t) and ξ 2 (t) have same strength D, and non-Gaussian noises η1 (t) and η2 (t) have same p and correlation time τ 1 . Please cite this article as: W.-L. Duan, C. Zeng, Statistics for anti-synchronization of intracellular calcium dynamics, Applied Mathematics and Computation (2016), http://dx.doi.org/10.1016/j.amc.2016.07.041
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3. The normalized global synchronization error By means of second-order stochastic Runge–Kutta type algorithm [32], for the specific simulation algorithm of ICO system, see Ref. [7], discretize time in steps of size = 0.001 s, one can stochastically simulate the time evolution of intracellular Ca2+ concentration in the cytosol x(t) and calcium store y(t). Experimentally, x is in the order of 100 ∼ 200nM in basal state [33] and y = 5 μM [34], so that the initial values x(0) and y(0) independently take uniformly random from 0.1 ∼ 0.2 μM and 4−5 μM. In the condition of time delay, it is rational to let x(t − τ ) = x(0 ) and y(t − τ ) = y(0 ) as t < τ . In this paper, the other parameters are set as Ref. [7]. Due to x(t) and y(t) would be all more than 1, in order to study further, we first normalize time series x(t) and y(t) as follows
X (t ) =
x(t ) y(t ) , Y (t ) = , max(x(t )) max(y(t ))
(14)
in this way the normalized time series of intracellular Ca2+ concentration of cytosol X(t) and calcium store Y(t) are obtained. Here, we aim to study anti-synchronous oscillation of intracellular Ca2+ between cytosol and calcium store, because antisynchronous states is opposite to synchronous states, like studying synchronous states, one define the normalized global synchronization error [35] as follow which distinguish synchronous states from anti-synchronous states
σ=
X (t ) − Y (t )2 . X 2 (t ) + Y 2 (t )
(15)
According to Ref. [36], for globally synchronous states σ ideally approaches 0, while for desynchronous states σ is nonzero, namely for anti-synchronous states σ ideally approaches 1. In practice, due to experimental mismatch and noise, the σ of synchronous states is small but nonzero, at the same time, the σ of anti-synchronous states is high but less than 1. In a word, the higher the σ is, the more anti-synchronous the states are; the lower σ the σ is, the more synchronous the states are; these correspond quasi-periodic oscillation. The other correspond chaotic oscillation.
3 0.7 0.6
0.6 0.5
0.5
2
0.4
σ
0.4
2.5
0.3 0.3 0.2 0.1
1.5 0.2 0.1
1
0
p 1
10
0
10
0.5
−1
10
τ(s)
−2
10
Fig. 1. The normalized global synchronization error σ as a function of time delay τ and parameter p of non-Gaussian noises.
Please cite this article as: W.-L. Duan, C. Zeng, Statistics for anti-synchronization of intracellular calcium dynamics, Applied Mathematics and Computation (2016), http://dx.doi.org/10.1016/j.amc.2016.07.041
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0.7 0.6 0.6 4
0.5 0.5
3.5 0.4
3
σ
0.4
0.3
2.5
0.3
2
0.2
1.5
0.2 0.1 0.1
D
1
0
0.5 1
10
0
10
−1
10
0 −2
10
τ(s) Fig. 2. The normalized global synchronization error σ as a function of time delay τ and intensity D of non-Gaussian noises.
In the following, the effect of time delay and non-Gaussian noises on normalized global synchronization error σ are simulated. In Fig. 1, the normalized global synchronization error σ vs. time delay τ and parameter p of non-Gaussian noises is simulated, here D = 1, τ1 = 10 s. It exhibits that, the short time delay induces anti-synchronously quasi-periodic calcium oscillation between cytosol and calcium store. However, the other time delay destroys this quasi-periodic oscillation and makes it become chaotic oscillation, even turns to synchronously quasi-periodic oscillation when parameter p is small. As parameter p increases, σ shifts into single trough from multi troughs structure. In Fig. 2, the normalized global synchronization error σ vs. time delay τ and intensity D of non-Gaussian noises is simulated, here p = 0.9, τ1 = 10s. It is clearly seen that, the short time delay induces anti-synchronously quasi-periodic oscillation, however, the moderate time delay induces synchronously quasi-periodic oscillation and the long time delay induces chaotic oscillation; as non-Gaussian noises increase, σ shifts into single trough from multi troughs structure. In Fig. 3, the normalized global synchronization error σ vs. time delay τ and correlation time τ 1 of non-Gaussian noises is simulated, here p = 0.9, D = 1. As stated in figure, the short time delay also induces anti-synchronously quasi-periodic oscillation, the moderate time delay also induces synchronously quasi-periodic oscillation, and the long time delay also induces chaotic oscillation; however, as correlation time τ 1 increases, σ shifts into multi troughs from single trough structure. Where, the more the troughs show, the easier the synchronously quasi-periodic oscillation occurs; oppositely, the less the troughs show, the more difficult the synchronously quasi-periodic oscillation occurs. There is important role for signal transmission across cells to research anti-synchronous ICO, because calcium wave can change the electric activities of neurons, i.e., calcium wave plays important role in neurodynamics. Recently, many researchers have studied neurodynamics [37–41] and obtained outstanding results. For the role of noise on some stochastic systems, there are also some research [42–45]. 4. Conclusions In view of non-Gaussian noises and time delay in transmission processes of intracellular Ca2+ , by means of second-order stochastic Runge–Kutta type algorithm, in order to research on anti-synchronization of Ca2+ oscillation between cytosol and Please cite this article as: W.-L. Duan, C. Zeng, Statistics for anti-synchronization of intracellular calcium dynamics, Applied Mathematics and Computation (2016), http://dx.doi.org/10.1016/j.amc.2016.07.041
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0.6
5
0.7 0.6
0.5 0.5 100 0.4
σ
0.4
0.3
80
0.3 60 0.2
0.2
40
τ (s)
0.1 20 0.1
1
0 0
0
10
−1
τ(s)
10
Fig. 3. The normalized global synchronization error σ as a function of time delay τ and correlation time τ 1 of non-Gaussian noises.
calcium store, the normalized global synchronization error of Ca2+ concentration in cytosol and calcium store is simulated. The results exhibit, the short time delay induces anti-synchronously quasi-periodic calcium oscillation between cytosol and calcium store, the moderate time delay induces synchronously quasi-periodic oscillation, and the long time delay induces chaotic oscillation. As parameter p of non-Gaussian noises increases, the normalized global synchronization error shifts into single trough from multi troughs. As non-Gaussian noises increase, the normalized global synchronization error shifts into single trough from multi troughs structure. However, as correlation time of non-Gaussian noises increases, the normalized global synchronization error shifts into multi troughs from single trough structure. References [1] A. Goldbeter, G. Dupont, M.J. Berridge, Minimal model for signal-induced Ca2+ oscillations and for their frequency encoding through protein phosphorylation, PNAS 87 (1990) 1461–1465. [2] Y. Shiferaw, D. Sato, A. Karma, Coupled dynamics of voltage and calcium in paced cardiac cells, Phys. Rev. E 71 (2005) 021903. [3] A.C. Ventura, L. Bruno, S.P. Dawson, Simple data-driven models of intracellular calcium dynamics with predictive power, Phys. Rev. E 74 (2006) 011917. [4] R. Thul, M. Falcke, Frequency of elemental events of intracellular Ca2+ dynamics, Phys. Rev. E 73 (2006) 061923. [5] H. Li, Z. Hou, H. Xin, Internal noise stochastic resonance for intracellular calcium oscillations in a cell system, Phys. Rev. E 71 (2005) 061916. [6] W.L. Duan, F. Long, C. Li, Reverse resonance and stochastic resonance in intracellular calcium oscillations, Physica A 401 (2014) 52–57. [7] L. Lin, W.L. Duan, The phenomena of an intracellular calcium oscillation system with non-Gaussian noises, Chaos, Solitons Fractals 77 (2015) 132–137. [8] W.L. Duan, C.H. Zeng, Role of time delay on intracellular calcium dynamics driven by non-Gaussian noises, Sci. Rep. 6 (2016) 25067. [9] W.L. Duan, Time delay induces oscillatory coherence in intracellular calcium oscillation system, Physica A 405 (2014) 10–16. [10] W.L. Duan, P.F. Duan, Time delay induces resonant activation in intracellular calcium oscillations, Chin. J. Phys. 52 (2014) 1059–1068. [11] J.M.A.M. Kusters, J.M. Cortes, W.P.M. van Meerwijk, D.L. Ypey, A.P.R. Theuvenet, C.C.A.M. Gielen, Hysteresis and bistability in a realistic cell model for calcium oscillations and action potential firing, Phys. Rev. Lett. 98 (2007) 098107. [12] W.L. Duan, L.J. Yang, D.C. Mei, Simulation of time delay effects in the intracellular calcium oscillation of cells, Phys. Scr. 83 (2011) 015004. [13] S. Rüdiger, J.W. Shuai, I.M. Sokolov, Law of mass action, detailed balance, and the modeling of calcium puffs, Phys. Rev. Lett. 105 (2010) 048103. [14] J.W. Shuai, P. Jung, Selection of intracellular calcium patterns in a model with clustered Ca2+ release channels, Phys. Rev. E 67 (2003) 031905. [15] M. Falcke, L. Tsimring, H. Levine, Stochastic spreading of intracellular Ca2+ release, Phys. Rev. E 62 (20 0 0) 2636. [16] M. Falcke, M. Or-Guil, M. Bär, Dispersion gap and localized spiral waves in a model for intracellular Ca2+ dynamics, Phys. Rev. Lett. 84 (20 0 0) 4753. [17] L. Lin, W.L. Duan, Extrinsic periodic information interpolates between monostable and bistable states in intracellular calcium dynamics, Physica A 427 (2015) 155–161. [18] C.B. Tabi, I. Maïna, A. Mohamadou, H.P.E. Fouda, T.C. Kofané, Wave instability of intercellular Ca2+ oscillations, EPL 106 (2014) 18005.
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Please cite this article as: W.-L. Duan, C. Zeng, Statistics for anti-synchronization of intracellular calcium dynamics, Applied Mathematics and Computation (2016), http://dx.doi.org/10.1016/j.amc.2016.07.041