Introduction

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. Ca2+ is mostly stored in the endoplasmic reticulum and mitochondria, namely calcium store. 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. This is the well-known calcium-induced calcium release mechanism.

In many studies on intracellular calcium oscillation(ICO) system, there have been a variety of channels showing calcium-induced calcium release and a variety of models1,2,3,4, and some phenomena have been found. Such as stochastic resonance in mesoscopic stochastic model5 and in ICO with Gaussian white noises and time delay6, reverse resonance in ICO with Gaussian colored noises and time delay6 and in ICO with non-Gaussian noises7, coherence resonance in ICO with non-Gaussian noises7 and in ICO with internal and external noises8, oscillatory coherence9 and resonant activation10 in ICO with Gaussian colored noises and time delay. Besides, calcium puffs has been found in neuronal cells11 and in a model with clustered Ca2+ release channels12, where, local waves, abortive waves, global oscillation and tide waves also have been exhibited. For other phenomena, there are stochastic backfiring in stochastic DeYoung-Keizer-model of the inositol 1,4,5-trisphosphate receptor channel13, dispersion gap and localized spiral waves in a bistable three component reaction-diffusion system modeling ICO14, Ca2+ spiral wave based on modified spatially extended Tang-Othmer Ca2+ model15.

In ICO system, it is inevitable to have the effect of stochastic force, i.e., noise. Therefore, Matjaž Perc group16,17,18,19,20,21,22 has studied the effects of noise on ICO. By analyzing experimental data, they have derived directly that noise and other stochastic effects indeed play a central role16,17. At the same time, they have obtained that, noise is good for the stability and robustness of ICO18 and for detecting of weak calcium signals within the cell19, and noise could induce periodic calcium waves20,21. When the cells are coupled, the effects of noise on ICO become lesser22. Then, they have introduced this spatial coherence resonance to excitable media23,24.

Martin Falcke group13,14,25,26,27,28,29,30,31,32,33,34,35 has done much research about ICO. They have studied a discrete stochastic model for calcium dynamics in living cells25, spatial and temporal structures in intracellular Ca2+ dynamics caused by fuctuations26, and key characteristics of Ca2+ puffs in deterministic and stochastic frameworks30.

In our research on ICO6,7,9,10,36,37,38,39,40,41, taking into account time delay in processes of active and passive transport of intracellular Ca2+, the roles of time delay and Gaussian colored noises on ICO system have been studied. In view of non-Gaussian noise in ICO with time delay, we have only studied the role of non-Gaussian noises on ICO, but the effect of time delay on ICO system has not been studied. Thus, studying the effect of time delay on ICO system with non-Gaussian noises, specifically, studying the effect of time delay on autocorrelation property of this ICO system is necessary in this paper.

First, according to ref. 7, the ICO model with non-Gaussian noises and time delay is presented. Then, the time delay, normalized autocorrelation function(NAF), and characteristic correlation time(CCT) of cytosolic and calcium store’s Ca2+ concentration are respectively simulated. Finally, conclusions are drawn.

The model for ICO with non-Gaussian noises

There is a layer of membrane to separate calcium store and cytosol, in the membrane there is some channel cluster. In transport process of intracellular Ca2+ between cytosol and calcium store, it depends that channel cluster opens as transport pathway, in this way Ca2+ could only transport from cytosol into calcium store or from calcium store into cytosol. When Ca2+ transporting in channel, no matter what from cytosol into calcium store, or from calcium store into cytosol, it is sure that this process all takes time. Moreover, both the transport process from cytosol into calcium store and the transport process from calcium store into cytosol, they all contain two processes, i.e., the active transport process and the passive transport process. In order to study easily, taking into account same time delay τ in processes of active and passive transport of Ca2+ in a real cell. In this paper, x and y denote the concentration of free Ca2+ of cytosol and calcium store in a cell, respectively. Based on calcium-induced calcium release, the Langevin equations of ICO system can be read as follows according to our previous result7:

with

and

Here v0 is the steady flow of Ca2+ to the cytosol, v1 is the maximum rate of the stimulus induced influx of Ca2+ from the extracellular medium, β0 is the external control parameter that denotes the degree of extracellular simulation. The rates v2 and v3 refer, respectively, to pumping of Ca2+ into 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. kfy is a diffusional flow of Ca2+ from store to cytosol, kx denotes the uptake from the cytosol, V is the system size. V2 and V3 denote the maximum values of the rates v2 and v3, respectively. The parameters k1, k2, and k3 are threshold constants for pumping, release, and activation of release by Ca2+ and by inositol 1,4,5-trisphosphate. W = W(x; xτ, yτ), xτ = x(t − τ), yτ = y(t − τ). λ denotes cross-correlation degree of internal and external noise before merger38.

By analysising experimental data, Matjaž Perc group16 have established that the nature of ICO is stochastic. Specifically, namely it 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 and non-Gaussian 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 non-Gaussian noise, which also contains Gaussian noise only if the control parameter takes 1 then non-Gaussian noise evolves into Gaussian noise. Importantly, after careful analysising experimental data and comparing with mathematical models of Martin Falcke group’s results35, we conclude, the noise in real ICO is very likely non-Gaussian noise. So that the noises η1(t) and η2(t) in Eqs (1 and 2) are considered as non-Gaussian noises which are characterized by the following Langevin equation42:

Where ξi(t) is a standard Gaussian white noise of zero mean and correlation ξi(t)ξi(t′) = δ(t − t′). Vip(ηi) is given by

and the statistical properties of non-Gaussian noise ηi(t) is defined as

Where τ1 denotes the correlation time of the non-Gaussian noises ηi(t), and D denotes the noise intensity of Gaussian white noise ξi(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 ξ1(t) and ξ2(t) have same strength D, and non-Gaussian noises η1(t) and η2(t) have same p and correlation time τ1.

Time series, NAF and CCT of intracellular Ca2+ concentration

By means of second-order stochastic Runge-Kutta type algorithm43, for the specific simulation algorithm of ICO system, see ref. 7, discretize time in steps of size Δ = 0.001s, 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 100~200 nM in basal state44 and y = 5 μM45, 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. The value of parameters are set as7: v0 = 1 μM/s, v1 = 7.3 μM/s, β0 = 0.287, kf = 1/s, k = 10/s, V2 = 65 μM/s, V3 = 500 μM/s, k1 = 1 μM, k2 = 0.9 μM, k3 = 2 μM, V = 1000 μm3, and λ = 0.1. Additionally, in this paper the parameters of non-Gaussian noises are p = 0.9, D = 0.5, and τ1 = 10 s.

First, the variation of time series for time delay is plotted in Fig. 1. There is also a critical value τc ≃ 0.1 s of τ. As τ → τc, ICO almost disappears(see Fig. 1(b), τ = 0.1s). Either τ < τc, the time coherence is obvious when time delay τ is short(see Fig. 1(a), τ = 0.005s), or τ > τc, the oscillation also strengthens as time delay increases(see Fig. 1(c–d), τ = 1 s and 50 s). In addition, it is clearly seen that Ca2+ concentrations in cytosol x(t) is lower than calcium store y(t), this is also fact in a real cell. By simulating time series, it could obtains many properties of stochastic dynamics system, e.g., in neuronal network system46,47,48.

Figure 1
figure 1

The variation of time series of Ca2+ concentration in cytosol x(t)(solid line) and calcium store y(t)(dotted line) for time delay τ.

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Second, to describe the fluctuation decay of intracellular Ca2+ concentration of cytosol x(t) and calcium store y(t) in the stationary state, one respectively define NAF of state variable x(t) and y(t) as Cx(θ) and Cy(θ)(see refs 7 and 9) as follows

Where θ is autocorrelation time, point bracket 〈〉 denotes statistical average over time.

Cx(θ) and Cy(θ) vs. autocorrelation time θ are studied as time delay varies. In Fig. 2, for the case of short time delay, both Cx(θ) and Cy(θ) have similar regular with time delay, i.e., both NAFs show damped motion when time delay is very short(e.g., τ = 0.005s), as time delay increases, however, they become a level line(e.g., τ = 0.1s). In Fig. 3, for the case of long time delay, where NAFs Cx(θ) and Cy(θ) have different regular: in Fig. 3(a), Cx(θ) exhibits underdamped motion with moderate time delay(e.g., τ = 0.5 s), and it is almost periodic motion, but this motion slowly decreases as time delay prolongs(e.g., τ = 5 s), finally it becomes a level line(e.g., τ = 50 s); in Fig. 3(b), Cy(θ) exhibits damped motion with moderate time delay(e.g., τ = 0.5 s), but it finally becomes underdamped motion(e.g., τ = 50 s), and it is almost periodic motion.

Figure 2
figure 2

For the case of short time delay, the role of time delay on NAFs vs. autocorrelation time θ is simulated.

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Figure 3
figure 3

For the case of long time delay, the role of time delay on NAFs vs. autocorrelation time θ is simulated.

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In order to explain the ordering of time series, it is necessary to introduce CCT Λx of cytosolic Ca2+ concentration x(t) and Λy of calcium store’s Ca2+ concentration y(t) to reflect correlation, they can be defined as follows

The larger the CCT, the more pronounced the correlation is, and the more orderly the time series is.

In Fig. 4, Λx and Λy vs. time delay τ is plotted. Here, CCTs Λx and Λy have different regular as time delay varies. In Fig. 4(a), the variation of CCT Λx for time delay shows, it firstly increases and then exhibits a concave structure, which implies that the moderate time delay induces reverse resonance in cytosolic calcium oscillation. In Fig. 4(b), the variation of CCT Λy for time delay shows, it firstly increases and then presents double concave structure, which also implies that the moderate time delay induces reverse resonance in calcium store’s calcium oscillation. Anyway, these manifest a reverse resonance phenomenon with respect to time delay in ICO system with non-Gaussian noises.

Figure 4
figure 4

CCT Λx and Λy vs. time delay are simulated.

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Among our previous research, when we studying the role of time delay on ICO, there is a different phenomena with different type noise. If noise is Gaussian white noise, there is also critical phenomenon of time delay37,39. If noise is Gaussian colored noise, time delay would induce reverse resonance and stochastic resonance6, oscillatory coherence9, resonant activation10, periodic square calcium wave36, stability transition40, calcium spikes41. Comparing above results concerning non-Gaussian noise with previous our works about Gaussian one, there is many differences. In the other stochastic system with time delay49,50,51,52,53, one have also researched about the role of time delay.

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, we have studied the role of time delay on intracellular calcium dynamics. By computing the time series, NAF, and CCT of cytosolic and calcium store’s Ca2+ concentration, some conclusions are obtained.

Firstly, the effects of time delay on time series are analyzed, there is a critical value of 0.1s of time delay: ICO decreases as time delay trends to this value, however, ICO is very obvious as time delay doesn’t trend to this value. Especially, time coherence appears when time delay is very short. Then, the variation of NAF for time delay shows: for the case of short time delay, both NAFs of cytosolic and calcium store’s Ca2+ concentration present damped motion when time delay is very short, but finally they become a level line as time delay increases. For the case of long time delay, NAF of cytosolic Ca2+ concentration presents underdamped motion, but it decreases into a level line as time delay further prolongs; NAF of calcium store’s Ca2+ concentration changes from damped motion to underdamped motion as time delay further prolongs. Finally, the variation of CCT for time delay shows: CCT of cytosolic and calcium store’s Ca2+ concentration respectively present a concave structure and two concave structure as time delay varies, these imply that reverse resonance occurs both in cytosol and calcium store when ICO system is driven non-Gaussian noises and time delay.

Additional Information

How to cite this article: Duan, W.-L. and Zeng, C. Role of time delay on intracellular calcium dynamics driven by non-Gaussian noises. Sci. Rep. 6, 25067; doi: 10.1038/srep25067 (2016).