Non-Gaussian noises induce transitions in intracellular calcium dynamics

Chaos, Solitons and Fractals 94 (2017) 63–67

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Non-Gaussian noises induce transitions in intracellular calcium dynamicsR Ling Lin a, Wei-Long Duan b,∗ a b

Science and technology college, Puer University, Puer 665000, China City college, Kunming University of Science and Technology, Kunming 650051, China

a r t i c l e

i n f o

Article history: Received 14 November 2016 Revised 26 November 2016 Accepted 29 November 2016 Available online 2 December 2016 Keywords: Intracellular calcium oscillation Non-Gaussian noise Stationary probability distribution Time series

a b s t r a c t The effect of non-Gaussian noises on time series and stability of intracellular calcium oscillation is researched by means of second-order stochastic Runge–Kutta type algorithm. By simulating time series and stationary probability distribution of Ca2+ concentration in cytosol and calcium store, the results show: (i) the bigger parameter p(which is used to control the degree of the departure from the non-Gaussian noise and Gaussian noise.) would induce anti-synchronous quasi-periodic oscillation; (ii) the stability weakens as p increases, but strengthens as correlation time of non-Gaussian noises prolongs; (iii) the strong nonGaussian noises induce transitions.

1. Introduction Intracellular calcium oscillation(ICO) can regulate the calcium concentration in cytosol and calcium store [1], and can change the calcium wave across cells. Furthermore, calcium wave can change the electric activities of neurons, so that signal transmission can be adjusted. Thus, Ca2+ is an ubiquitous and versatile second messenger. In many studies on ICO, there are a variety of models [2–5]. Importantly, some phenomena have been found such as stochastic resonance [6,7], reverse resonance [7–9], coherence resonance [8], oscillatory coherence [10], resonant activation [11], bitransitions solutions with hysteresis [12,13], calcium puffs [14], various spontaneous Ca2+ patterns [15], stochastic backfiring [16], dispersion gap and localized spiral waves [17], transitions [18], calcium wave instability [19], colored noise optimized calcium wave [20], periodic square calcium wave [21] etc. ICO has also been intensively studied by Perc group [22–25] and Falcke group [16,17,26–30]. Here, Perc group has found that noise and other stochastic effects indeed play a central role [22,23]. In R This project was supported by the Scientific research project of Puer College(Grant No. 201322), the Science Research Fund of Yunnan Provincial Education Department(Grant No. 2016ZZX047), the National Natural Science Foundation of China(Grant No. 11305079 and Grant No. 11665014), 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.chaos.2016.11.017 0960-0779/© 2016 Published by Elsevier Ltd.

© 2016 Published by Elsevier Ltd.

the transmission processes of intracellular Ca2+ , it may be nonGaussian noise [30]. Specifically, namely there is noise to act on the transport process of intracellular Ca2+ between cytosol and calcium store. Based on stochastic dynamics, the noise is divided into Gaussian noise, non-Gaussian noise, and bounded noise. Among the Gaussian noise only exists in stochastic dynamics system under the ideal process, but the noise in the really stochastic dynamics system, including biological system, is basically all nonGaussian noise, which also contains Gaussian noise only if the control parameter takes 1 then non-Gaussian noise evolves into Gaussian noise. Importantly, after careful analysing experimental data and comparing with mathematical models of Martin Falcke group’s results [30], we conclude, the noise in real ICO is very likely non-Gaussian noise. Thus, we have studied ICO system driven by non-Gaussian noises [8,9,31,32], then found that ICO exhibits synchronous or anti-synchronous oscillation between cytosol and calcium store [8]. However, the role of non-Gaussian noises on time series and transitions of ICO haven’t been researched so far. Thus, in this paper, we mainly research the role of non-Gaussian noises on time series and transitions of ICO. In Section 2, the ICO model with non-Gaussian noises and time delay is presented. Then the time series and stationary probability distribution(SPD) of intracellular Ca2+ concentration are simulated in Section 3. Finally, conclusions are obtained in Section 4. 2. The model for ICO with non-Gaussian noises and time delay In view of same time delay τ in processes of active and passive transport of Ca2+ in a real cell, the Langevin equations of ICO

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(a) p=0.8

x(μM), y(μM)

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x(t) y(t)

3 2 1 0

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t(s) Fig. 1. The time series of Ca2+ concentration in cytosol x(t)(solid line) and in calcium store y(t)(dotted line) vs. parameter p of non-Gaussian noises, here D = 0.1 and τ1 = 10s. As p increases, ICO shows anti-synchronous quasi-periodic oscillation.

system read [8,9,31]:

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)

with

A1 (x; xτ , yτ ) = v0 + v1 β0 − v2 + v3τ + k f yτ − kx,

(3)

The noises η1 (t) and η2 (t) in Eqs. (1) and (2) are considered as non-Gaussian noises [9] which are characterized by the following Langevin equation [33]:

1 d dηi (t ) =− V (η ) + dt τ1 dηi ip i

B1 ( x; xτ , yτ ) =



 B2 (x, y; xτ ) =

v21 β02 + 2v1 β0 λW + W 2 ,

v2τ + v3 + k f y V

 W ( x; xτ , yτ ) =

(4)

(5)

,

(6)

v0 + v1 β0 + v2 + v3τ + k f yτ + kx V

,

(7)

and

v2 =

V2 x2 V3 x4 y2 , v3 = 4 , 2 + k1 (x + k42 )(y2 + k23 )

x2

v2τ =

V2 x2τ V3 x4τ y2τ , v3τ = 4 . 2 (xτ + k42 )(y2τ + k23 ) τ + k1

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 [13].

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 ) = A2 (x, y; xτ ) = v2τ − v3 − k f y,

√ 2D



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,

< η (t ) >= 2 i

(12)

⎧ ⎨ ⎩

2D

τ1 ( 5 − 3 p )

, p ∈ (−∞, 53 ),

(13)

∞, p ∈ [ 35 , 3 ).

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 . 3. Time series and SPD of intracellular Ca2+ concentration By means of second-order stochastic Runge–Kutta type algorithm [34], for the specific simulation algorithm of ICO system, see Ref. [8], discretize time in steps of size  = 0.001 s, one can stochastically simulate the time evolutions of intracellular Ca2+ concentration in the cytosol x(t) and calcium store y(t). Experimentally, x is in the order of 100∼ 200 nM in basal state [35] and y =

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t(s) Fig. 2. The time series of Ca2+ concentration in cytosol x(t)(solid line) and in calcium store y(t)(dotted line) vs. intensity D of non-Gaussian noises, here p = 0.9 and τ1 = 10s. As non-Gaussian noises increase, ICO strengthens. 5

(a) τ1=5s

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t(s) Fig. 3. The time series of Ca2+ concentration in cytosol x(t)(solid line) and in calcium store y(t)(dotted line) vs. correlation time τ 1 of non-Gaussian noises, here p = 0.9 and D = 0.1. As τ 1 prolongs, ICO weakens.

5 μM [36], so that the initial values x(0) and y(0) independently take uniformly random from 0.1 ∼ 0.2 μ M and 4 ∼ 5 μ M. 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 . The parameters are set as: v0 = 1 μM/s, v1 = 7.3 μM/s, β0 = 0.287, k f = 1/s, k = 10 /s, V2 = 65 μM/s, V3 = 500 μM/s, k1 = 1 μM, k2 = 0.9 μM, k3 = 2 μM, V = 10 0 0 μm3 , λ = 0.1, and τ = 0.01 s. The role of non-Gaussian noises on time series of intracellular Ca2+ concentration in cytosol x(t) and in calcium store y(t) is simulated and analyzed. Fig. 1 shows the variation for time series with parameter p of non-Gaussian noises, it is clearly seen that cytosolic and calcium store’s Ca2+ all exhibit anti-synchronous oscillation. As p increases, the time coherence of oscillation strengthens, and it shifts into anti-synchronous quasi-periodic oscillation(see Fig. 1(b),

p = 2.5) from chaotic oscillation(see Fig. 1(a), p = 0.8). In Fig. 2, the effect of intensity D of non-Gaussian noises on time series is simulated. When non-Gaussian noises are weak(see Fig. 2(a), D = 0.01), the calcium oscillation in cytosol and calcium store almost stops, as expected, the calcium oscillation between cytosol and calcium store strengthens and exhibits time coherence as non-Gaussian noises increase(see Fig. 2(b), D = 2). Besides, the oscillation is also anti-synchronous. In Fig. 3, the effect of correlation time τ 1 of nonGaussian noises on time series is plotted. When τ 1 is short(see Fig. 3(a), τ1 = 5 s), the calcium oscillation between cytosol and calcium store exhibits anti-synchronous oscillation, as correlation time increases, it almost disappears(see Fig. 3(b), τ1 = 100 s). Recently, the role of non-Gaussian noises on time series of stochastic systems [37–42] has been introduced too.

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(a) p=0.5

400

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200

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150 100 50 0 2.5 2 1.5 1

y(μM) Fig. 4. Stationary probability distribution(SPD) as a function of Ca and τ1 = 10s. As p increases, SPD all exhibits monostable state.

0.5 2+

0.1

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concentration in cytosol x and in calcium store y vs. parameter p of non-Gaussian noises, here D = 0.1

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Fig. 5. Stationary probability distribution(SPD) as a function of Ca2+ concentration in cytosol x and in calcium store y vs. intensity D of non-Gaussian noises, here p = 0.9 and τ1 = 10s. As non-Gaussian noises increase, SPD shifts into bistable state from monostable state.

By using the statistics of the number ratios of data belong to different value zones of the variables and normalizing them, then the effect of non-Gaussian noises on SPD is researched. In Fig. 4, the role of parameter p of non-Gaussian noises is plotted. It shows that, SPD all exhibits monostable state. As p increases, the peak of SPD become lower(see Fig. 4(b), p = 2.5, comparing with Fig. 4(a), p = 0.5), namely the stability of ICO weakens. Then, Fig. 5 shows the variation for SPD with intensity D of non-Gaussian noises. It changes into bistable state(see Fig. 5(b), D = 2) from monostable state(see Fig. 5(a), D = 0.01) as non-Gaussian noises strengthen, although one peak is very small and the peak of SPD rapid becomes low. Namely non-Gaussian noises induce transitions. Finally, in Fig. 6, SPD vs. correlation time τ 1 of non-Gaussian noises is plotted. Here, SPD all exhibits monostable state, but the peak of

SPD rapid becomes high as τ 1 prolongs. It is clearly seen by comparing Fig. 6(a)(τ1 = 5 s) with Fig. 6(b)(τ1 = 100 s), i.e., the stability of ICO strengthens. Moreover, whether monostable state or bistable state, every peak of them all corresponds to the Ca2+ concentration in calcium store being much higher than cytosolic. This also reasonably explains the case of Ca2+ concentration of calcium store being much higher than cytosolic in a real cell forever. Recently, some researchers have studied the stability of the other stochastic systems [43–47].

4. Conclusions In this paper, the effect of non-Gaussian noises on time series and stability of intracellular Ca2+ concentration in ICO system with

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(a) τ1=5s

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Fig. 6. Stationary probability distribution(SPD) as a function of Ca concentration in cytosol x and in calcium store y vs. correlation time τ 1 of non-Gaussian noises, here p = 0.9 and D = 0.1. As τ 1 increases, SPD all exhibits monostable state. 2+

colored noises and time delay is investigated by means of secondorder stochastic Runge–Kutta type algorithm. The time series of cytosolic and calcium store’s Ca2+ concentration show that: the bigger parameter p of non-Gaussian noises would induce anti-synchronous quasi-periodic oscillation; the strong intensity or the short correlation time of non-Gaussian noises would induce anti-synchronous calcium oscillation between cytosol and calcium store; the weak intensity or the long correlation time would eliminate calcium oscillation in cytosol and calcium store. SPD of ICO system all exhibits monostable state as parameter p or correlation time of non-Gaussian noises increases, but the strong non-Gaussian noises induce transitions: SPD shifts into bistable state from monostable state. As parameter p or intensity of non-Gaussian noises increases, the stability of ICO weakens, but the stability strengthens as correlation time of non-Gaussian noises prolongs. Ca2+ is an ubiquitous and versatile second messenger that transmits information through changing 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. Our work reveals that in view of non-Gaussian noises in ICO system appears new phenomena, these are found in real ICO. It demonstrates that nonGaussian noises exist in intracellular calcium dynamics, which has important mean for signal transmission to study ICO. Furthermore, this will lead to important new insights in cell system. For the role of noise on some stochastic systems, there are also some research [48–51]. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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