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Internal signal stochastic resonance induced by colored noise in an intracellular calcium oscillations model
Chemical Physics Letters 387 (2004) 383–387 www.elsevier.com/locate/cplett
Internal signal stochastic resonance induced by colored noise in an intracellular calcium oscillations model Qian Shu Li *, Pin Wang School of Science, Beijing Institute of Technology, Beijing 100081, PR China Received 15 October 2003; in final form 10 January 2004 Published online: 10 March 2004
Abstract An intracellular calcium oscillations model subject to external colored noise is investigated. Internal signal stochastic resonance (ISSR) can be induced and significantly influenced by colored noise. The signal-to-noise ratio (SNR) exhibits two maxima with the increment of the correlation time as the noise intensity is fixed, indicating the occurrence of stochastic bi-resonance. Additionally, we find ISSR and explicit internal signal stochastic resonance (EISSR) have quite similar responses to colored noise, which implies there are some commonalities in their mechanisms. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction Noise plays a constructive role in many nonlinear systems and its effect is widely studied. One of the important effects is stochastic resonance (SR), in which a weak external periodic signal can be amplified by additive noise as it passes through a nonlinear system [1,2]. Since 1981 the concept of SR was first proposed to explain the periodic oscillations of the earth’s ice age [3], SR phenomenon has been widely studied theoretically and experimentally in physical [4,5], chemical [6–8] and biological systems [9–12]. In early studies of SR, it was realized that the occurrence of SR needs three essential ingredients: a nonlinear system, a weak external signal and random noise. However, with the development of SR studies, Hu et al. [13] reported SR-like behavior in which the external signal is replaced by an ‘internal signal’: the deterministic oscillations of the system. When the control parameter is located near the saddlenode bifurcation point, where the system is outside the oscillatory region, noise-induced coherent oscillation (NICO) is observed and the strength of which passes through a maximum with the increment of the noise *
Corresponding author. Fax: +86-10-6891-2665. E-mail address: [email protected] (Q.S. Li).
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.02.042
intensity, showing the characteristic of SR, which might be named by internal signal SR (ISSR) [12]. This phenomenon can also be called autonomous SR [13] or coherence resonance (CR) [14]. In 2001, an attractive type of ISSR, explicit internal signal SR (EISSR), was found by simulating a Belousov–Zhabotinsky reaction [8], the internal signal of which comes from the intrinsic period-1 oscillation. EISSR is thought to have different mechanism with common ISSR and the occurrence of EISSR need to avoid crossing bifurcation point. Calcium ion is considered to be one of the most important secondary messengers in the cytosol of most of living cells, and its oscillations can tune the cellular living processes such as egg fertilization and cell secretion [15–18]. In the past years, the mechanism of calcium oscillations has been actively investigated [19–25], but very limited attention is paid to the effect of random environmental fluctuations. In the present Letter, we investigate the effect of environmental fluctuations on intracellular calcium oscillations. We use colored noise to simulate environmental fluctuations in a real system. Colored noise can enhance or weaken SR [2,26–28], which refers to the long-time dynamics of fluctuations. Our numerical simulation shows that calcium oscillations can be amplified and optimized with the increase of noise intensity, indicating the occurrence of ISSR.
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Q.S. Li, P. Wang / Chemical Physics Letters 387 (2004) 383–387
Meanwhile, stochastic bi-resonance is observed under certain condition. What interests us most is that our results are similar to the results on EISSR induced by colored noise [2]. So, we assume that there probably exist some commonalities in the mechanism of ISSR and EISSR. An explanation for this phenomenon is provided.
2. Model This model was proposed by Borghans et al. [29] to account for complex intracellular calcium oscillations, based on the mechanism of calcium-induced calcium release (CICR) [23,30]. In 1999, Houart et al. [31] investigated in detail the various dynamic behaviors of the model. The dynamical evolution equations are given by dx ¼ m0 þ bm1 m2 þ m3 þ kf y kx; dt dy ¼ m2 m3 kf y; dt dz ¼ bm4 m5 ez; dt where m2 ¼ mM2
x2 ; k22 þ x2
m3 ¼ mM3
xm y2 z4 ; kxm þ xm ky2 þ y 2 kz4 þ z4
m5 ¼ mM5
k5p
ð1Þ
zp xn : n p þ z kd þ xn
nored when the effect of colored noise on the dynamics of nonlinear system is considered. CðtÞ is exponential Gaussian colored noise, and s0 is the correlation time of colored noise. The system is modulated in the following two ways: (i) varying noise intensity with constant correlation time; (ii) varying correlation time with constant noise intensity. We choose three correlation times s0 ¼ 0:5, 0.7 and 1 s and three noise intensities B ¼ 2:0, 3.0, 4.0 to study the system, respectively. We take p ¼ 0:6 s and D ¼ 0:005 for Np ðtÞ. Eqs. (1)–(3) are integrated by a Euler algorithm with a fixed time step of 0.06 s. The time evolution of the system lasts 66 000 s. To quantify the ISSR effect, 16 384 points are used to obtain frequency spectra by fast Fourier transformation. The SNR is picked as the parameter evaluating the effect of ISSR. The most commonly used definition of the SNR for external periodic signals is the ratio of the height of the first harmonic peak in the power spectral density at the signal frequency and the level of the background noise at the same frequency [1]. In this 1 Letter, SNR is defined as SNR ¼ H ðDx=xp Þ [13], where H is the peak height, xp is the frequency where the peak occurs and Dx is the half width of the peak. Thus SNR so defined depends on two factors. H represents the height of output peak of system, which changes with the increment of noise intensity; Dx=xp reasonably corresponds to the relative width of the peak, which is in fact the familiar quality factor of a signal. Each plot of SNR is obtained by averaging 20 runs.
3. Results and discussion
x, y and z represent the concentration of cytosolic Ca2þ , Ca2þ sequestered in internal store and inositol 1,4,5trisphosphate (IP3 ), respectively. b denotes the degree of extracellular simulation and varies from 0 to 1. The detailed description of this model can be found in [31]. To study the effect of environmental fluctuations on this system, we apply colored noise to the parameter b as
A linear stability analysis for this three-variable system is carried out. The system has two supercritical Hopf bifurcation points, occurring at b 0:0289 and 0.7714, respectively. See Fig. 1 for the bifurcation diagram. We choose b0 ¼ 0:772, which is a stable state near the second Hopf bifurcation point.
b ¼ b0 ½1 þ BCðtÞ;
3.1. Varying noise intensity with constant correlation time
ð2Þ
where b0 is a constant value, B is the noise intensity of colored noise CðtÞ. We can introduce another equation as follows: dCðtÞ ¼ CðtÞ=s0 þ Np ðtÞ; dt
ð3Þ
where Np ðtÞ is Gaussian white noise with hNp ðtÞi ¼ 0 and hNp ðtÞNp ðt0 Þi ¼ 2Ddðt t0 Þ. p is the pulse length of white noise (the interval of noise added on the system), which has been found to be an important factor to affect ISSR [12]. But its effects are smeared out by integration in formula (3). Therefore, the influence of p is usually ig-
Fig. 2 shows the SNR versus noise intensity with correlation time s0 ¼ 0:5, 0.7 and 1 s. The appearance of a maximum in the output SNR at an optimal noise level indicates that ISSR occurs. Now, we take the first case of s0 ¼ 0:5 s as an example to show the formation of resonant peaks in Fig. 2. In Fig. 3, we have plotted the time series of x with s0 ¼ 0:5 s and B ¼ 0, 1, 4.2 and 5.4, respectively. Without noise, the system will remain stable (Fig. 3a). The fluctuation of x increases with the increment of noise intensity, which means that the system can enter deeper oscillatory region with larger noise intensity.
Q.S. Li, P. Wang / Chemical Physics Letters 387 (2004) 383–387
385
1.0 0.8
x [µM]
0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
β
Fig. 1. The bifurcation diagram of the model. Parameters: m0 ¼ m1 ¼ 2 lM min1 , m4 ¼ 2 lM min1 , k ¼ 10 min1 , kf ¼ 1 min1 , e ¼ 0:1 min1 , mM2 ¼ 6 lM min1 , mM3 ¼ 20 lM min1 , mM5 ¼ 5 lM min1 , k2 ¼ 0:1 lM, k5 ¼ 1 lM, kx ¼ 0:5 lM, ky ¼ 0:2 lM, kz ¼ 0:2 lM, kd ¼ 0:4 lM, m ¼ p ¼ 2, n ¼ 4. (These parameters remain unchanged.)
Fig. 2. SNR versus noise intensity. The SNR passes through a maximum at B 4:2 with s0 ¼ 0:5 s (a), B 2:8 with s0 ¼ 0:7 s (b) and B 2:1 with s0 ¼ 1 s (c).
Smaller oscillation occurs at B ¼ 1 (Fig. 3b). More orderly spikes appear at B ¼ 4:2 (Fig. 3c), which results in a peak in the SNR (Fig. 2a). When noise intensity is
further increased, the regulation of spikes is destroyed (Fig. 3d), which results in the decrease in the SNR. We can draw the same conclusion from the corresponding
(a)
(b) 0.6
x [µM]
x [µM]
0.6
0.4
0.2
0.4
0.2 65000
65500
66000
66500
64500
65000
Time [min]
66000
66500
66000
66500
Time [min]
(c)
(d) 0.6
x [µM]
0.6
x [µM]
65500
0.4
0.2 64500
0.4
0.2 65000
65500
Time [min]
66000
66500
64500
65000
65500
Time [min]
Fig. 3. The time series of x at s0 ¼ 0:5 s with noise intensity of 0 (a), 1 (b), 4.2 (c) and 5.4 (d), respectively.
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Q.S. Li, P. Wang / Chemical Physics Letters 387 (2004) 383–387 0.03
B=2 B=3 B=4
c
c b
600
a
b SNR
Amplitude
0.02
0.01
a
400
0.00 0.02
0.04
Frequency (Hz) Fig. 4. The smoothed Fourier spectra of noise-induced oscillations at s0 ¼ 0:5 s with B ¼ 1 (a), 4.2 (b) and 5.4 (c).
frequency spectra (see Fig. 4). We note that the height of peak H and the width of peak x increase at the same time as noise intensity is increased. However, at the beginning, the increment of H is remarkable, with the increase of noise intensity, the increment of x becomes dominant. It implies SNR increases as noise intensity is increased for low values of noise intensity until a maximum is reached and then SNR decreases, which indicates the occurrence of ISSR. Fig. 2 displays that the maximum of SNR is shifted to lower noise intensity when the correlation time is increased: s0 ¼ 0:5 s, B 4:2 (Fig. 2a); s0 ¼ 0:7 s, B 2:8 (Fig. 2b); and s0 ¼ 1 s, B 2:1 (Fig. 2c). It means that the system is more sensitive to noise intensity when the longer correlation time is chosen. Moreover, the maximum of SNR decreases with the increment of correlation time, indicating that colored noise can weaken ISSR. The experimental verification was obtained by Misono [32] in an optical bistable system. It suggests there are some commonalities in chemical and physical nonlinear systems. 3.2. Varying correlation time with constant noise intensity In Fig. 5, we have depicted the SNR versus correlation time with noise intensity B ¼ 2:0 (a), 3.0 (b) and 4.0 (c). The SNR clearly displays two maxima, the first one more pronounced than the second. This phenomenon, referred to as stochastic bi-resonance, indicates there are multiple values of correlation time at which the response of the system is enhanced. Furthermore, Fig. 5 manifests that the first maximum is shifted to lower correlation time with the increment of noise intensity: s0 1:1 s with B ¼ 2:0 (a); s0 0:7 s with B ¼ 3:0 (b) and s0 0:5 s with B ¼ 4:0 (c), respectively. Compared with Fig. 2, it seems that s0 plays the same role as noise intensity.
0
1
2
3
4
Correlation Time τ0 [arb.units] Fig. 5. SNR versus correlation time. The first maximum appears at s0 1:1 s with B ¼ 2:0 (a), s0 0:7 s with B ¼ 3:0 (b) and s0 0:5 s with B ¼ 4:0 (c).
4. Conclusion The above results show that ISSR can be induced and significantly influenced by colored noise. ISSR can be weakened by colored noise, i.e., the maximum of SNR decreases with the increment of the correlation time. Moreover, the SNR goes through two maxima with the increment of the correlation time as the noise intensity is fixed, which implies that internal intrinsic signal may be enhanced at multiple values of correlation time. Our group have done the same work on EISSR [2]. Two sets of results are very similar: the maximum of SNR is shifted to lower noise intensity as the correlation time is increased; the SNR shows SRlike behavior with the variation of correlation time as the noise intensity is fixed and stochastic multiresonance is observed; EISSR can be weakened by colored noise. These facts suggest there probably exist some commonalities in the mechanism of ISSR and EISSR. We assume noise plays a twofold role in ISSR. On the one hand, noise can induced the system to enter the oscillatory region. On the other hand, noise can enhance internal intrinsic order of oscillations. In EISSR, the system is already inside the oscillatory region so noise only can enhance internal order. Based on our assumption, smaller noise may induce EISSR. Xin group investigated the phenomena of ISSR and EISSR in a model of molecular machinery [33], and they found EISSR can occur at lower noise intensity. Their results show that our assumption may be reasonable. Further theoretical and experimental work is needed. It is well known that many processes in excitable and non-excitable cells are regulated by the oscillatory changes of cytosolic calcium concentration. Calcium oscillations can control the birth, life, death of cell [15].
Q.S. Li, P. Wang / Chemical Physics Letters 387 (2004) 383–387
So, the study on the effect of external fluctuations on calcium oscillations is significant to reveal the essence of cellular processes. The finding of the present Letter would be helpful to understand further the beneficial role of noise at the cellular level.
Acknowledgements This work was supported by specialized research fund for the doctoral program of higher education.
References [1] L. Gammaitoni, P. H€anggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70 (1998) 223. [2] R. Zhu, Q.S. Li, Z.C. Liu, Chem. Phys. Lett. 351 (2002) 410. [3] R. Benzi, S. Sutera, A. Vulpiani, J. Phys. A 14 (1981) L453. [4] M.M. Alibegov, Phys. Rev. E 59 (1999) 4841. [5] S.M. Bezrukov, I. Vodyanoy, Nature 385 (1997) 319. [6] L.F. Yang, Z.H. Hou, B.J. Zhou, H.W. Xin, J. Chem. Phys. 109 (1998) 6456. [7] S. Zhong, H.W. Xin, J. Phys. Chem. A 104 (2000) 297. [8] Q.S. Li, R. Zhu, J. Chem. Phys. 115 (2001) 6590. [9] I. Hidaka, D. Nozaki, Y. Yamamoto, Phys. Rev. Lett. 85 (2000) 3740. [10] G. Bal azsi, L.B. Kish, Phys. Lett. A 265 (2000) 304. [11] D.F. Russell, L.A. Wilkens, F. Moss, Nature 402 (1999) 291.
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[12] S. Zhong, F. Qi, H.W. Xin, Chem. Phys. Lett. 342 (2001) 583. [13] G. Hu, T. Ditzinger, C.Z. Ning, H. Haken, Phys. Rev. Lett. 71 (1993) 807. [14] A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78 (1997) 775. [15] M.J. Berridge, M.D. Bootman, P. Lipp, Nature 395 (1998) 645. [16] V. Grubelnik, A.Z. Larsen, U. Kummer, Biopys. Chem. 94 (2001) 59. [17] M.J. Berridge, Nature 361 (1993) 315. [18] A. Ghosh, M.E. Greenberg, Science 268 (1995) 239. [19] A. Goldbeter, G. Dupont, M.J. Berridge, Proc. Natl. Acad. Sci. USA 87 (1990) 1461. [20] G. Dupont et al., FASEB J. 14 (2000) 279. [21] A. Goldbeter, Nature 420 (2002) 238. [22] G. Balazsi, A.H. Cornell-Bell, F. Moss, Chaos 13 (2003) 515. [23] G. Dupont, A. Goldbeter, Cell Calcium 14 (1993) 311. [24] L. L€aer, M. Kloppstech, C. Sch€ ofl, Biophys. Chem. 91 (2001) 157. [25] J.Q. Zhang, F. Qi, H.W. Xin, Biophys. Chem. 94 (2001) 201. [26] P. H€anggi, P. Jung, C. Zerbe, F. Moss, J. Stat. Phys. 70 (1993) 25. [27] D. Nozaki, Y. Yamamoto, Phys. Lett. A 243 (1998) 281. [28] D. Nozaki, D.J. Mar, P. Grigg, J.J. Collins, Phys. Rev. Lett. 82 (1999) 2402. [29] J.A.M. Borghans, G. Dupont, A. Goldbeter, Biophys. Chem. 66 (1997) 25. [30] A. Goldbeter, Biochemical Oscillations and Cellular Rhythms, Cambridge University Press, Cambridge, 1996. [31] G. Houart, G. Dupont, A. Goldbeter, Bull. Math. Biol. 61 (1999) 507. [32] M. Misono, T. Kohmoto, Y. Fukuda, M. Kunitomo, Opt. Commun. 152 (1998) 255. [33] Z.W. Wang, Z.H. Hou, H.W. X, Chem. Phys. Lett. 51 (2002) 362.
Internal signal stochastic resonance induced by colored noise in an intracellular calcium oscillations model Qian Shu Li *, Pin Wang School of Science, Beijing Institute of Technology, Beijing 100081, PR China Received 15 October 2003; in final form 10 January 2004 Published online: 10 March 2004
Abstract An intracellular calcium oscillations model subject to external colored noise is investigated. Internal signal stochastic resonance (ISSR) can be induced and significantly influenced by colored noise. The signal-to-noise ratio (SNR) exhibits two maxima with the increment of the correlation time as the noise intensity is fixed, indicating the occurrence of stochastic bi-resonance. Additionally, we find ISSR and explicit internal signal stochastic resonance (EISSR) have quite similar responses to colored noise, which implies there are some commonalities in their mechanisms. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction Noise plays a constructive role in many nonlinear systems and its effect is widely studied. One of the important effects is stochastic resonance (SR), in which a weak external periodic signal can be amplified by additive noise as it passes through a nonlinear system [1,2]. Since 1981 the concept of SR was first proposed to explain the periodic oscillations of the earth’s ice age [3], SR phenomenon has been widely studied theoretically and experimentally in physical [4,5], chemical [6–8] and biological systems [9–12]. In early studies of SR, it was realized that the occurrence of SR needs three essential ingredients: a nonlinear system, a weak external signal and random noise. However, with the development of SR studies, Hu et al. [13] reported SR-like behavior in which the external signal is replaced by an ‘internal signal’: the deterministic oscillations of the system. When the control parameter is located near the saddlenode bifurcation point, where the system is outside the oscillatory region, noise-induced coherent oscillation (NICO) is observed and the strength of which passes through a maximum with the increment of the noise *
Corresponding author. Fax: +86-10-6891-2665. E-mail address: [email protected] (Q.S. Li).
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.02.042
intensity, showing the characteristic of SR, which might be named by internal signal SR (ISSR) [12]. This phenomenon can also be called autonomous SR [13] or coherence resonance (CR) [14]. In 2001, an attractive type of ISSR, explicit internal signal SR (EISSR), was found by simulating a Belousov–Zhabotinsky reaction [8], the internal signal of which comes from the intrinsic period-1 oscillation. EISSR is thought to have different mechanism with common ISSR and the occurrence of EISSR need to avoid crossing bifurcation point. Calcium ion is considered to be one of the most important secondary messengers in the cytosol of most of living cells, and its oscillations can tune the cellular living processes such as egg fertilization and cell secretion [15–18]. In the past years, the mechanism of calcium oscillations has been actively investigated [19–25], but very limited attention is paid to the effect of random environmental fluctuations. In the present Letter, we investigate the effect of environmental fluctuations on intracellular calcium oscillations. We use colored noise to simulate environmental fluctuations in a real system. Colored noise can enhance or weaken SR [2,26–28], which refers to the long-time dynamics of fluctuations. Our numerical simulation shows that calcium oscillations can be amplified and optimized with the increase of noise intensity, indicating the occurrence of ISSR.
384
Q.S. Li, P. Wang / Chemical Physics Letters 387 (2004) 383–387
Meanwhile, stochastic bi-resonance is observed under certain condition. What interests us most is that our results are similar to the results on EISSR induced by colored noise [2]. So, we assume that there probably exist some commonalities in the mechanism of ISSR and EISSR. An explanation for this phenomenon is provided.
2. Model This model was proposed by Borghans et al. [29] to account for complex intracellular calcium oscillations, based on the mechanism of calcium-induced calcium release (CICR) [23,30]. In 1999, Houart et al. [31] investigated in detail the various dynamic behaviors of the model. The dynamical evolution equations are given by dx ¼ m0 þ bm1 m2 þ m3 þ kf y kx; dt dy ¼ m2 m3 kf y; dt dz ¼ bm4 m5 ez; dt where m2 ¼ mM2
x2 ; k22 þ x2
m3 ¼ mM3
xm y2 z4 ; kxm þ xm ky2 þ y 2 kz4 þ z4
m5 ¼ mM5
k5p
ð1Þ
zp xn : n p þ z kd þ xn
nored when the effect of colored noise on the dynamics of nonlinear system is considered. CðtÞ is exponential Gaussian colored noise, and s0 is the correlation time of colored noise. The system is modulated in the following two ways: (i) varying noise intensity with constant correlation time; (ii) varying correlation time with constant noise intensity. We choose three correlation times s0 ¼ 0:5, 0.7 and 1 s and three noise intensities B ¼ 2:0, 3.0, 4.0 to study the system, respectively. We take p ¼ 0:6 s and D ¼ 0:005 for Np ðtÞ. Eqs. (1)–(3) are integrated by a Euler algorithm with a fixed time step of 0.06 s. The time evolution of the system lasts 66 000 s. To quantify the ISSR effect, 16 384 points are used to obtain frequency spectra by fast Fourier transformation. The SNR is picked as the parameter evaluating the effect of ISSR. The most commonly used definition of the SNR for external periodic signals is the ratio of the height of the first harmonic peak in the power spectral density at the signal frequency and the level of the background noise at the same frequency [1]. In this 1 Letter, SNR is defined as SNR ¼ H ðDx=xp Þ [13], where H is the peak height, xp is the frequency where the peak occurs and Dx is the half width of the peak. Thus SNR so defined depends on two factors. H represents the height of output peak of system, which changes with the increment of noise intensity; Dx=xp reasonably corresponds to the relative width of the peak, which is in fact the familiar quality factor of a signal. Each plot of SNR is obtained by averaging 20 runs.
3. Results and discussion
x, y and z represent the concentration of cytosolic Ca2þ , Ca2þ sequestered in internal store and inositol 1,4,5trisphosphate (IP3 ), respectively. b denotes the degree of extracellular simulation and varies from 0 to 1. The detailed description of this model can be found in [31]. To study the effect of environmental fluctuations on this system, we apply colored noise to the parameter b as
A linear stability analysis for this three-variable system is carried out. The system has two supercritical Hopf bifurcation points, occurring at b 0:0289 and 0.7714, respectively. See Fig. 1 for the bifurcation diagram. We choose b0 ¼ 0:772, which is a stable state near the second Hopf bifurcation point.
b ¼ b0 ½1 þ BCðtÞ;
3.1. Varying noise intensity with constant correlation time
ð2Þ
where b0 is a constant value, B is the noise intensity of colored noise CðtÞ. We can introduce another equation as follows: dCðtÞ ¼ CðtÞ=s0 þ Np ðtÞ; dt
ð3Þ
where Np ðtÞ is Gaussian white noise with hNp ðtÞi ¼ 0 and hNp ðtÞNp ðt0 Þi ¼ 2Ddðt t0 Þ. p is the pulse length of white noise (the interval of noise added on the system), which has been found to be an important factor to affect ISSR [12]. But its effects are smeared out by integration in formula (3). Therefore, the influence of p is usually ig-
Fig. 2 shows the SNR versus noise intensity with correlation time s0 ¼ 0:5, 0.7 and 1 s. The appearance of a maximum in the output SNR at an optimal noise level indicates that ISSR occurs. Now, we take the first case of s0 ¼ 0:5 s as an example to show the formation of resonant peaks in Fig. 2. In Fig. 3, we have plotted the time series of x with s0 ¼ 0:5 s and B ¼ 0, 1, 4.2 and 5.4, respectively. Without noise, the system will remain stable (Fig. 3a). The fluctuation of x increases with the increment of noise intensity, which means that the system can enter deeper oscillatory region with larger noise intensity.
Q.S. Li, P. Wang / Chemical Physics Letters 387 (2004) 383–387
385
1.0 0.8
x [µM]
0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
β
Fig. 1. The bifurcation diagram of the model. Parameters: m0 ¼ m1 ¼ 2 lM min1 , m4 ¼ 2 lM min1 , k ¼ 10 min1 , kf ¼ 1 min1 , e ¼ 0:1 min1 , mM2 ¼ 6 lM min1 , mM3 ¼ 20 lM min1 , mM5 ¼ 5 lM min1 , k2 ¼ 0:1 lM, k5 ¼ 1 lM, kx ¼ 0:5 lM, ky ¼ 0:2 lM, kz ¼ 0:2 lM, kd ¼ 0:4 lM, m ¼ p ¼ 2, n ¼ 4. (These parameters remain unchanged.)
Fig. 2. SNR versus noise intensity. The SNR passes through a maximum at B 4:2 with s0 ¼ 0:5 s (a), B 2:8 with s0 ¼ 0:7 s (b) and B 2:1 with s0 ¼ 1 s (c).
Smaller oscillation occurs at B ¼ 1 (Fig. 3b). More orderly spikes appear at B ¼ 4:2 (Fig. 3c), which results in a peak in the SNR (Fig. 2a). When noise intensity is
further increased, the regulation of spikes is destroyed (Fig. 3d), which results in the decrease in the SNR. We can draw the same conclusion from the corresponding
(a)
(b) 0.6
x [µM]
x [µM]
0.6
0.4
0.2
0.4
0.2 65000
65500
66000
66500
64500
65000
Time [min]
66000
66500
66000
66500
Time [min]
(c)
(d) 0.6
x [µM]
0.6
x [µM]
65500
0.4
0.2 64500
0.4
0.2 65000
65500
Time [min]
66000
66500
64500
65000
65500
Time [min]
Fig. 3. The time series of x at s0 ¼ 0:5 s with noise intensity of 0 (a), 1 (b), 4.2 (c) and 5.4 (d), respectively.
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Q.S. Li, P. Wang / Chemical Physics Letters 387 (2004) 383–387 0.03
B=2 B=3 B=4
c
c b
600
a
b SNR
Amplitude
0.02
0.01
a
400
0.00 0.02
0.04
Frequency (Hz) Fig. 4. The smoothed Fourier spectra of noise-induced oscillations at s0 ¼ 0:5 s with B ¼ 1 (a), 4.2 (b) and 5.4 (c).
frequency spectra (see Fig. 4). We note that the height of peak H and the width of peak x increase at the same time as noise intensity is increased. However, at the beginning, the increment of H is remarkable, with the increase of noise intensity, the increment of x becomes dominant. It implies SNR increases as noise intensity is increased for low values of noise intensity until a maximum is reached and then SNR decreases, which indicates the occurrence of ISSR. Fig. 2 displays that the maximum of SNR is shifted to lower noise intensity when the correlation time is increased: s0 ¼ 0:5 s, B 4:2 (Fig. 2a); s0 ¼ 0:7 s, B 2:8 (Fig. 2b); and s0 ¼ 1 s, B 2:1 (Fig. 2c). It means that the system is more sensitive to noise intensity when the longer correlation time is chosen. Moreover, the maximum of SNR decreases with the increment of correlation time, indicating that colored noise can weaken ISSR. The experimental verification was obtained by Misono [32] in an optical bistable system. It suggests there are some commonalities in chemical and physical nonlinear systems. 3.2. Varying correlation time with constant noise intensity In Fig. 5, we have depicted the SNR versus correlation time with noise intensity B ¼ 2:0 (a), 3.0 (b) and 4.0 (c). The SNR clearly displays two maxima, the first one more pronounced than the second. This phenomenon, referred to as stochastic bi-resonance, indicates there are multiple values of correlation time at which the response of the system is enhanced. Furthermore, Fig. 5 manifests that the first maximum is shifted to lower correlation time with the increment of noise intensity: s0 1:1 s with B ¼ 2:0 (a); s0 0:7 s with B ¼ 3:0 (b) and s0 0:5 s with B ¼ 4:0 (c), respectively. Compared with Fig. 2, it seems that s0 plays the same role as noise intensity.
0
1
2
3
4
Correlation Time τ0 [arb.units] Fig. 5. SNR versus correlation time. The first maximum appears at s0 1:1 s with B ¼ 2:0 (a), s0 0:7 s with B ¼ 3:0 (b) and s0 0:5 s with B ¼ 4:0 (c).
4. Conclusion The above results show that ISSR can be induced and significantly influenced by colored noise. ISSR can be weakened by colored noise, i.e., the maximum of SNR decreases with the increment of the correlation time. Moreover, the SNR goes through two maxima with the increment of the correlation time as the noise intensity is fixed, which implies that internal intrinsic signal may be enhanced at multiple values of correlation time. Our group have done the same work on EISSR [2]. Two sets of results are very similar: the maximum of SNR is shifted to lower noise intensity as the correlation time is increased; the SNR shows SRlike behavior with the variation of correlation time as the noise intensity is fixed and stochastic multiresonance is observed; EISSR can be weakened by colored noise. These facts suggest there probably exist some commonalities in the mechanism of ISSR and EISSR. We assume noise plays a twofold role in ISSR. On the one hand, noise can induced the system to enter the oscillatory region. On the other hand, noise can enhance internal intrinsic order of oscillations. In EISSR, the system is already inside the oscillatory region so noise only can enhance internal order. Based on our assumption, smaller noise may induce EISSR. Xin group investigated the phenomena of ISSR and EISSR in a model of molecular machinery [33], and they found EISSR can occur at lower noise intensity. Their results show that our assumption may be reasonable. Further theoretical and experimental work is needed. It is well known that many processes in excitable and non-excitable cells are regulated by the oscillatory changes of cytosolic calcium concentration. Calcium oscillations can control the birth, life, death of cell [15].
Q.S. Li, P. Wang / Chemical Physics Letters 387 (2004) 383–387
So, the study on the effect of external fluctuations on calcium oscillations is significant to reveal the essence of cellular processes. The finding of the present Letter would be helpful to understand further the beneficial role of noise at the cellular level.
Acknowledgements This work was supported by specialized research fund for the doctoral program of higher education.
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