Novel effect of interplay of internal and external noise on the dynamics of calcium oscillations

Chemical Physics 377 (2010) 132–135

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Novel effect of interplay of internal and external noise on the dynamics of calcium oscillations Hongying Li a,⇑, Juan Ma b a b

Hefei Normal University, Hefei, Anhui 230601, China China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

a r t i c l e

i n f o

Article history: Received 23 June 2010 In final form 8 September 2010 Available online 17 September 2010 Keywords: External noise Internal noise Calcium oscillation

a b s t r a c t Using a mesoscopic stochastic model, the effect of interplay of external and internal noise on the dynamics of calcium oscillations was studied. When the system was tuned near a Hopf bifurcation point and driven by external noise or internal noise only, the existence of external noise coherence resonance (ENCR) or internal-noise stochastic resonance (INSR) was found, respectively. When both of the noises were considered, it was found that ENCR could be suppressed by internal noise, while INSR could be enhanced by external noise in a certain range of external noise intensity. It was also interesting to note that the optimal system size can be regulated by the external noise when the INSR occurs. The cell system may adapt to adjust the optimal size according to the external noise, indicating some kind of self-tuning mechanism involved in stochastic calcium dynamics. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction In the last two decades, the constructive effects of external noise in nonlinear systems have gained much attention. It was demonstrated that there existed an optimal external noise intensity, at which the response of a system to a periodic signal was maximally ordered, which was well-known as stochastic resonance (SR) [1]. Many previous works have also demonstrated that nonlinear systems in the presence of external noise can also display SR-like behavior, even without an external signal, this phenomenon was being called external noise coherence resonance (ENCR) [2]. SR and ENCR have been widely studied in various systems [3–12]. In recent years, the effects of internal noise resulting from the stochastic reaction events in small scale systems have been paid much attention in biological and chemical systems, such as ion channels [13], circadian clock [14], intracellular calcium signaling [15–17], genetic regulation [18], surface catalytic reaction system [19,20], neuron system [21] and so on. It was found that stochastic oscillations could be observed in a region sub-threshold to deterministic oscillatory dynamics, and there existed an optimal internal noise intensity at which the stochastic oscillations showed the best performance, which was well-known as internal-noise stochastic resonance (INSR), or system size resonance.

⇑ Corresponding author. Tel.: +86 0551 4416789. E-mail address: [email protected] (H. Li). 0301-0104/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2010.09.004

So far, most works regarding noise have focused on the influence to systems with just one noise, either internal noise or external noise. However, these two kinds of noise are both unavoidable in small scale systems, so one must take into account the interplay of internal and external noise on a system’s dynamics. Only recently, the interplay of external noise and internal noise has been studied in a circadian oscillator system [22], NO reduction system [23,24] and CO oxidation system [25], and so on. But to our knowledge, few works have been carried out so far on the influence of interplay of internal and external noise on the dynamics of calcium oscillations in cell systems. In this paper, based on the mesoscopic stochastic model of a calcium oscillation system, we have studied the interplay effect of external and internal noise on the dynamics of calcium oscillations. 2. Model and equations The mesoscopic stochastic model we used here is identical with the stochastic model in Ref. [17], except for the modulation of external noise on the degree of cell stimulation by agonist. For completeness, we will briefly describe the model below. The original model used here accounting for the intracellular calcium oscillations was proposed by Shen and Larter [26]. The functioning of the model system is based on the mechanisms of inositol 1,4,5-trisphosphate cross-coupling (ICC) and the calciuminduced calcium release (CICR). If the internal noise is ignored, the dynamics can be described by a simple three-variable system of equations:

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H. Li, J. Ma / Chemical Physics 377 (2010) 132–135

d½Cacyt  ¼ J ch þ J leak  J pump þ kin1  r þ kin2  J out ; dt d½Caer  ¼ J pump  J ch  J leak ; dt d½IP 3  ¼ Jþ  J ; dt

ð1Þ

where

½IP3 4

Jch ¼ kch 

!



K 4 ½Cacyt  ð½Cacyt  þ K 4 Þð½Cacyt  þ K 5 Þ



½IP3 4 þ K 41

3  ½Caer :

Jleak ¼ kleak  ½Caer : Jpump ¼ kpump 

½Cacyt 2 ½Cacyt 2 þ K 22

:

Jout ¼ kout  ½Cacyt : J þ ¼ kþ  r 

½Cacyt  ½Cacyt  þ K 3

J ¼ k  ½IP 3 : ð2Þ [Cacyt], [Caer] and [IP3] represent the concentration of free Ca2+ in the cytosol, free Ca2+ in the endoplasmic reticulum (ER) and the inositol 1,4,5-trisphosphate in the cytosol, respectively. The parameter r measures the degree of cell stimulation by agonist and is selected as the control parameter. The meanings and values of the other parameters have been explained in detail in Ref. [17]. And hence will not be stated here again. See Fig. 1 for a simple description of

the mechanism. The system exhibits two Hopf bifurcation (HB) points at r1 = 0.2345 and r2 = 0.6859, respectively [26,27]. However, for a typical cell system, the number of reaction molecules is often low [28–30], and we must consider the internal noise which results from the random fluctuations of the stochastic reaction events. The reactions in the cell can be grouped into eight elementary processes. See Fig. 1 for a simple description of the six processes, and Table 1 for the stochastic processes and the corresponding transition rates. In Table 1, X and Y represent the number of calcium ions in the cytosol and in the ER, respectively. So X = [Cacyt]V, Y = [Caer]V and Z = [IP3]V, where V is the cell volume. Then the Chemical Langevin Equation (CLE) for the current model is as follows:

pffiffiffiffiffi pffiffiffiffiffi d½Cacyt  1 ¼ ½ða1 þ a2  a3 þ a4 þ a5  a6 Þ þ a1 n1 ðtÞ þ a2 n2 ðtÞ dt V pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi  a3 n3 ðtÞ þ a4 n4 ðtÞ þ a5 n5 ðtÞ  a6 n6 ðtÞ;  pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi d½Caer  1  ¼ ða1  a2 þ a3 Þ  a1 n1 ðtÞ  a2 n2 ðtÞ þ a3 n3 ðtÞ ; dt V  pffiffiffiffiffi pffiffiffiffiffi d½IP3  1  ¼ ða7  a8 Þ þ a7 n7 ðtÞ  a8 n8 ðtÞ ; dt V ð3Þ Where ni(t) (i = 1, 2, 3, 4, 5, 6, 7, 8) are Gaussian white noises with hni(t)i = 0 and hni(t)nj(t0 )i = dij(t  t0 ) .The reaction rates ai are proporpffiffiffiffi tional to V, so the internal noise item in the CLE scales as 1= V . To study the influence of external noise, we consider the degree of cell stimulation by agonist is perturbed by an external noise:

r ¼ r 0 ½1 þ DnðtÞ;

ð4Þ

where n(t) is the Gaussian white noise with hn(t)i = 0 and hn(t)n(t0 )i = d(t  t0 ); D denotes the intensity of external noise. The dimension of D is sec1/2. We tune r0 = 0.22, which is slightly smaller than the left Hopf bifurcation (HB) value r1, so that the cell is at a stable state in the absence of noise. In the following parts, we will use the CLE (3) and Eq. (4) to study the interplay effect of the external noise and internal noise on the dynamics of calcium oscillations. 3. Results and discussion 3.1. Enhancement of INSR by external noise

Fig. 1. Schematic representation of the model proposed by Shen and Larter based on the interplay between CICR and ICC.

The effect of only internal noise on the calcium oscillations has been investigated in Ref. [17].We found that when the internal noise was considered, the internal noise induced calcium oscillations occurred, and there existed an optimal system size V at which the regularity of the calcium oscillations was the best. From the CLE, pffiffiffiwe ffi can see that the internal noise item in the CLE scales as 1= V , so an optimal system size implies an optimal internal noise

Table 1 Stochastic transition processes and corresponding rates. Transition processes (1)

Description

Transition rates 2+



The release of Ca

(4) (5) (6) (7)

X !Xþ1 Y !Y 1 X !Xþ1 Y !Y 1 X !X1 Y !Y þ1 X !Xþ1 X !Xþ1 X !X1 Z !Zþ1

The agonist-depended influx into the cytosol The Constant Ca2+ influx into the cell Transport of cytosolic Ca2+ into the extracellular medium The production of IP3 activated by cytosolic ca2+

a4 ¼ V  kin1  r a5 ¼ V  kin2 a6 ¼ V  Jout ¼ V  kout  ½Cacyt 

(8)

Z !Z1

The degradation of IP3 which stimulates the release of Ca2+ from the ER into the cytosol

a8 ¼ V  J ¼ V  k  ½IP 3 

(2) (3)

from the ER into the cytosol in a process induced by IP3

Leaky transport of Ca2+ from the ER to the cytosol The pump of Ca2+ from the cytosol into the ER

a1 ¼ V  Jch ¼ V  kch 

½IP3 4 ½IP3 4 þK 41

 

a2 ¼ V  Jleak ¼ V  kleak  ½Caer  a3 ¼ V  Jpump ¼ V  kpump 

½Cacyt 2 ½Cacyt 2 þK 22

½Ca



a7 ¼ V  Jþ ¼ V  kþ  r  ½Cacyt cyt þK 3



3 K 4 ½Cacyt  ð½Cacyt þK 4 Þð½Cacyt þK 5 Þ

 ½Caer 

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H. Li, J. Ma / Chemical Physics 377 (2010) 132–135

8

(b)

10

8

R

10

R

(a)

D=0.0 D=0.5 D=0.7 D=0.8 D=1.0 D=1.5 D=2.0 D=2.5

12

6 4

6

4

2 2.0

2.5

3.0 3.5 Log(V)

4.0

4.5

2

0.0

0.5

1.0

1.5

2.0

2.5

1/2

D(sec )

Fig. 2. (a) The dependence of R on system size V for different external noise intensity D; (b) the dependence of the maximal R of each curve in (a) on the external noise intensity D.

level, this phenomenon is being called ‘‘system size resonance” or ‘‘internal-noise stochastic resonance (INSR)”. In the following part, we will investigate the effect of the external noise on the INSR. In Eq. (4), we set r0 = 0.22, which is slightly smaller than the left Hopf bifurcation (HB) value r1, so that no oscillations occur in the absence of external and internal noise. When both of the noises are considered, calcium oscillations occur. To measure the regularity of the calcium oscillations, we calculated the coherence ffi (see Ref. [31] for measure R, which was defined as R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2

the definition of R). The dependence of R on system size V for different external noise intensity D is shown in Fig. 2(a). For convenience of comparison, the R curve without external noise is also displayed (solid line with empty circles). At V = 102 lm3, the internal noise is so large that the system is insensitive to the change of external noise intensity, therefore, the R values for different values of D are very close. When the internal noise intensity decreases, for example, at V = 104 lm3, R increases at first and then declines with increasing of D, which reasonably illustrates the occurrence of external noise coherence resonance (ENCR). From Fig. 2(a), we can also see that when the external noise is small (D 6 0.8), R goes through a maximum at an optimal system size V, indicating the occurring of internal-noise stochastic resonance (INSR). And the optimal system size for the INSR is affected by external noise and becomes larger with increasing D in a range of D 6 0.8. In addition, an interesting result is that the curves of

12 10

R

8 6 4 2 0.3

0.6

0.9

1.2

1.5

1.8

2.1

1/2

D(sec ) Fig. 3. The dependence of R on the external noise intensity D when the internal noise is ignored.

INSR become higher with the increases of D, which shows that the external noise can enhance the INSR in a certain range of external noise intensity (D 6 0.8). The cell system may adapt to adjust the optimal size according to the external noise. If D > 0.8, R monotonically increases and the peak disappears. It is because the external noise is large enough that the constructive role of internal noise is suppressed. When D increases further, the R curve becomes lower. The maximal R of each curve is displayed in Fig. 2(b). It is clearly shown that the maximal R rises at first and then falls with the increases of external noise intensity D, indicating the existence of optimal external noise intensity for the regularity of calcium oscillations. 3.2. Suppressions of ENCR by internal noise Firstly, we will investigate the effect of only external noise on the calcium oscillations. When only the external noise is considered, the external noise induced calcium oscillations occur. We also use R to measure the regularity of calcium oscillations. The dependence of R on the external noise intensity D is shown in Fig. 3. We can see that R increases at first and then declines with the increases of D, which illustrates the occurrence of external noise coherence resonance (ENCR). Interestingly, we find that the maximal values of R in Fig. 2(b) are very close to the values of R in Fig. 3 except for the region where D 6 0.8. We think the reason is as follows: When D 6 0.8, internal-noise stochastic resonance (INSR) occurs, the maximal values of R appear at an optimal system size. However, when D > 0.8, R monotonically increases, the maximal values of R appear at the end of the curves corresponding to large system size. From the CLE, pffiffiffiffi we can see that the internal noise item in the CLE scales as 1= V , when the system size is large enough, the internal noise level is small enough. So in this case, the external noise plays a major role and the effect of internal noise can be ignored, which leading that the performance of the calcium oscillations is very similar to those only considering the external noise (see Fig. 3). It is for this reason that led to the interesting phenomenon we mentioned. Secondly, we will investigate the effect of the internal noise on the ENCR. Fig. 4 displays the dependence of R on the external noise intensity D for different system sizes. The R curve without considering internal noise is also displayed, corresponding the system size V = 1. We can see that when the system size is large, for example, V = 105 lm3, the intensity of internal noise is so small that the R curve is obviously close to our previous result without considering the internal noise, showing the occurrence of the

H. Li, J. Ma / Chemical Physics 377 (2010) 132–135

14

V=∞ (internal noise=0) 5 V=10

12

V=10 V=3000 V=1000 V=300

R

10

4

8 6 4

135

optimal size according to the external noise, indicating some kind of self-tuning mechanism involved in stochastic calcium dynamics. Although it is not clear at present whether cell systems use the above regulatory mechanism to play functional roles in cellular process, our results may provide an insight into the understanding of the interplay between the internal and external noises on the dynamics of calcium oscillations. We hope our findings may induce further perspectives on the study of interplay of internal noise and external noise in mesoscopic systems. Acknowledgement

2 0.3 0.6 0.9 1.2 1.5 1.8 2.1

This work is supported by the Natural Science Foundation of Higher Education of Anhui Province (KJ2009B066Z).

1/2

D(sec ) References Fig. 4. The dependence of R on external noise intensity D for different system sizes.

ENCR. With the decreases of system size, the internal noise intensity increases, the ENCR curve becomes lower. When system size decreases further, for example V = 300 lm3, the R curve monotonically decreases with the increases of D and the peak disappears. It is because when the system size is very small, the internal noise is large enough that the constructive role of external noise is suppressed. 4. Conclusion In conclusion, we have studied the effect of interplay of external noise and internal noise on the dynamics of calcium oscillations using a mesoscopic stochastic model. When the system was tuned near a Hopf bifurcation point and driven by external noise or internal noise only, the existence of ENCR or INSR was found, respectively. When both of the noises are considered, it was found that ENCR could be suppressed by internal noise, while INSR could be enhanced by external noise in a certain range of external noise intensity. Our results may be of relevance to calcium signaling. Due to the existence of unavoidable external and internal noise, living cell systems may have learned to exploit them to enhance the calcium oscillation performance via the mechanism of ENCR and INSR, instead of trying to resist them. It is also interesting to note that the optimal system size can be regulated by the external noise when the INSR occurs. The cell system may adapt to adjust the

[1] R. Benzi, A. Sutera, A. Vulpiani, J. Phys. A 14 (1981) L453. [2] A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78 (1997) 775. [3] A. Guderian, G. Dechert, K.P. Zeyer, F.W. Schneider, J. Phys. Chem. 100 (1996) 4437. [4] W. Hohman, J. Müller, F.W. Schneider, J. Phys. Chem. 100 (1996) 5388. [5] L.F. Yang, Z.H. Hou, H.W. Xin, J. Chem. Phys. 109 (1998) 2002. [6] A. Nieman, P. Saparin, L. Stone, Phys. Rev. E. 56 (1) (1997) 270. [7] A.S. Pikovsky, J. Kurths, Phys. Rev. Lett 78 (1997) 775. [8] Z.H. Hou, H.W. Xin, Phys. Rev. E 60 (1999) 6329. [9] S. Zhong, H.W. Xin, Chem. Phys. Lett. 321 (2000) 309. [10] O. Kortluke, V.N. Kuzovkov, W. Vou Niessen, Phys. Rev. E 66 (2002) 036139. [11] J.R. Pradines, J.J. Collins, Phys. Rev. E 60 (1999) 6407. [12] A.A. Priplata, J.B. Niemi, J.D. Harry, L.A. Lipsitz, J.J. Collins, Lancet 362 (2003) 1123. [13] G. Schmid, I. Goychuk, P. Hänggi, Europhys. Lett. 56 (2001) 22. [14] Z.H. Hou, H.W. Xin, J. Chem. Phys. 119 (2003) 11508. [15] H.Y. Li, Z.H. Hou, H.W. Xin, Phys. Rev. E 71 (2005) 061916. [16] H.Y. Li, Z.H. Hou, H.W. Xin, Chem. Phys. Lett. 402 (2005) 444. [17] H.Y. Li, J.H. Bi, J. Ma, Acta Phys. -Chim. Sin. 25 (7) (2009) 327. [18] Z.W. Wang, Z.H. Hou, H.W. Xin, Chem. Phys. Lett. 401 (2005) 307. [19] Y.B. Gong, Z.H. Hou, H.W. Xin, J. Phys. Chem. B 108 (2004) 17796. [20] Y.B. Gong, Z.H. Hou, H.W. Xin, J. Phys. Chem. A 109 (2005) 2741. [21] M.S. Wang, Z.H. Hou, H.W. Xin, ChemPhysChem 5 (2004) 1602. [22] M. Yi, Y. Jia, Q. Liu, J.R. Li, C.L. Zhu, Phys. Rev. E 73 (2006) 041923. [23] Y.B. Gong, B. Xu, X.G. Ma, Y.M. Dong, C.L. Yang, J. Phys. Chem. C 111 (2007) 4264. [24] Y.B. Gong, Y.H. Xie, B. Xu, X.G. Ma, J. Chem. Phys. 128 (2008) 124707. [25] Y.B. Gong, B. Xu, X.G. Ma, Chem. Phys. Lett. 458 (2008) 351. [26] P. Shen, R. Larter, Cell Calcium 17 (1995) 225. [27] M. Perc, M. Marhl, Biophys. Chem. 104 (2003) 509. [28] M.B. Elowitz, A.J. Levine, E.D. Siggia, P.S. Swain, Science 297 (2002) 1183. [29] W.J. Blake, M. Kærn, C.R. Cantor, J.J. Collins, Nature 422 (2003) 633. [30] F.J. Isaacs, J. Hasty, C.R. Cantor, J.J. Collins, Proc. Natl. Acad. Sci. 100 (2003) 7714. [31] K. Miyakawa, T.T. anaka, H. Isikawa, Phys. Rev. E 67 (2003) 066206.