Signal transduction and amplification in a circadian oscillator: Interaction between two colored noises

Journal of Theoretical Biology 265 (2010) 565–571

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Signal transduction and amplification in a circadian oscillator: Interaction between two colored noises Shi Jian-Cheng n College of Chemistry and Life Sciences, Guangxi Teachers Education University, MingXiu Road 175, Nanning 530001, People’s Republic of China

a r t i c l e in fo

abstract

Article history: Received 18 February 2010 Received in revised form 12 May 2010 Accepted 31 May 2010 Available online 8 June 2010

The signal transduction and amplification in a Neurospora circadian clock system is studied by using the mechanism of internal signal stochastic resonance (ISSR). Two cases have been investigated: the case of no correlations between multiplicative and additive colored noises and the case of correlations between two noises. The results show that, in both cases, the noise-induced circadian oscillations can be transduced with the phenomenon of internal signal stochastic resonance (ISSR). However, the correlation time and intensity of an additive colored noise play different roles for the ISSR, driven by multiplicative colored noise, while the correlation time and intensity of multiplicative colored noise hardly influence the ISSR driven by additive colored noise. In addition, the ISSR can be amplified or suppressed at an appropriate range of the correlation intensity between two colored noises. The fundamental frequency of noise-induced circadian oscillations is hardly shifted with the increment of the intensity and correlation time of colored noises, which implies that the Neurospora system could be resistant to colored noises, exhibit strong vitality and sustain intrinsic circadian rhythms. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Three-variable Neurospora model Interplay of colored noises Internal signal stochastic resonance Correlation intensity of noises Circadian clock

1. Introduction Noise (fluctuations in temperature, light, and humidity) is ubiquitous in real biological system, the interaction between nonlinear dynamics and noise can lead to stochastic resonance (SR) phenomenon (Benzi et al., 1981; Neiman and Sung, 1996; Volman and Levine, 2009, 2008). In the past years, the constructive role of noise has attracted increasing attraction (Perc et al., 2006; Perc et al., 2007; Gosak et al., 2007; Ozer et al., 2009), and some significant phenomena, such as an internal signal stochastic resonance (ISSR) (Shi and Li, 2007), spatio-temporal stochastic resonance (Jung and Mayer-Kress, 1995) and an internal-noise spatial coherence resonance (Perc, 2005a, 2005b) in biochemical tissue-like media and excitable biochemical media have been found. It is well known that circadian clocks (i.e., circadian rhythms) play an important role in making living organisms entrain to the 24-h cycle of environment, such that it allows an organism to build up an efficient organization of its daily activity. However, some individuals are unable to entrain to environmental cues and find themselves struggling with abnormalities of circadian clock, which may occur later or earlier than the usual circadian clock. Therefore, it is very important for us to find a proper way in the theoretical studies to induce and amplify normal circadian rhythms. Up to now, a lot of theoretical works for it have been reported (Emery et al.,

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1998; Yi and Jia, 2005; Shi and Li, 2007), but these works regarding fluctuations have been on the consideration of systems with just one noise source. However, the various noise sources (e.g., the cases of no correlations and correlations between the internal and external noises) are unavoidable and must always concur in any actual circadian rhythms process. Furthermore, the largest amount of works about the resonant behaviors in circadian system are assumed as white noise, but more realistic models of circadian system require considering the case of colored noise (Buldu´ et al., 2001; Li and Wang, 2004). Yi et al. (2006) studied an enhancement of internal-noise coherence resonance by modulation of external noise in Drosophila circadian oscillator, but they considered the case of white noise and did not consider the correlations between the internal and external noises. To our knowledge, for the circadian clock system, few researches have focused on how the interaction of external and internal colored noises (correlative) influences the circadian rhythms. Our motivation is, therefore, to investigate the interaction of external and internal colored noises (correlative) on the signal transduction and amplification by using the mechanism of internal signal stochastic resonance (ISSR) in Neurospora circadian clock. Note that the multiplicative colored noise (light noise) will be regarded as an external noise, which comes mainly from an external fluctuation outside the cell, while the additive noise will be regarded as an internal noise, which comes mainly from an internal fluctuation inside the cell in the present work (Li and Li, 2004). The remainder of this paper is structured as follows: in Section 2 we introduce the model and mathematical methods presently in use. Results and discussion are presented in Section 3, and in the last section we summarize our results.

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2. Dynamical model

Table 1 Parameter descriptions and values used in Eq. (1).

Herein, in order to investigate the signal transduction and amplification of the noise-induced circadian oscillations, we employ a three-variable Neurospora model proposed by Leloup (Leloup et al., 1999) and Gonze (Gonze and Goldbeter, 2000), which is depicted in Fig. 1. The core mechanism of this circadian rhythmicity relies on the negative regulation exerted by a clock protein (FRQ) on the transcription of clock gene (frq) into the clock gene messenger (mRNA), and the translation of mRNA leads to the synthesis of the clock protein. The time evolution of the three variables is governed by the following kinetic equations:

Parameter Description

Value

Transcription rate of the clock gene control parameter Threshold beyond which the nuclear protein repress the transcription of its gene Hill coefficient characterizing the repression Maximum rate of mRNA degradation Michaelis constant related to mRNA degradation Translation rate of mRNA to protein Maximum rate of protein degradation Michaelis constant related to protein degradation Transport rate of protein into the nucleus Transport rate of protein out of the nucleus

us kl n um km ks ud kd k1 k2

0.2 nM 4 0.3 nM h  1 0.2 nM 2.0 h  1 1.5 nM h  1 0.1 nM 0.2 h  1 0.2 h  1

d½M=dt ¼ us KIn =ðKIn þ ½PN n Þum ½M=ðKM þ ½MÞ

ð1Þ

where [M], [PC], and [PN] denote the concentrations of the clock gene mRNA, of the cytosolic and nuclear forms of the clock protein, respectively. The parameter us denotes the rate of an frq transcription, which is chosen as the light-controlled parameter because it increases with light intensity (Gonze and Goldbeter, 2000). The other parameters relating to the physical properties of the model are listed in Table 1, and the reaction steps and corresponding transition rates involved in the model are listed in Table 2. Further detailed description and meaning of parameters about this model can be found in Gonze and Goldbeter (2000); Gonze et al. (2002) and Hou and Xin (2003). The system undergoes a supercritical Hopf bifurcation at us ¼0.257 nM h  1 as shown in Fig. 2. It means that, if us ous , the system is in a quiescent state, while us 4 us , the system is in an oscillatory state, which is further demonstrated in Figs. 3 and 4. As is seen from Fig. 3, the oscillatory signal does occur when us ¼0.258, 0.260 and 0.262 nM h  1 (us 40.257 nM h  1), not when us ¼0.253 nM h  1 (us o0.257 nM h  1). As Fig. 4 shows, the system remains stable (curve a) in the absence of noise. However, the fluctuation of M increases with the increment of noise intensity, which means that the system can enter deeper oscillatory region with larger noise intensity (curves b and c). It is worth mentioning that us is chosen as 0.256 nM h  1 such that the system is in quiescent state in the present work.

Table 2 Reaction steps and corresponding transition rates involved in the model. Reaction step

Description

Transition rate

G-M + G

Transcription of the clock gene

M-

mRNA degradation

M-PC +M PC-

Translation of mRNA into protein Degradation of cytosolic protein

PC-PN PN-PC

Transport of protein into the nucleus Transport of protein out of the nucleus

us KIn :V KIn þ ½PN n um ½M :V W2 ¼ a2 :V ¼ Km þ ½M W3 ¼ a3.V ¼ Ks[M].V u ½PC  :V W4 ¼ a4 :V ¼ d KI þ ½PC  W5 ¼ a5.V ¼ K1[PC].V W6 ¼ a6.V ¼ K2[PN].V W1 ¼ a1 :V ¼

1.0 mRNA Concentration[M](nM)

d½PC =dt ¼ Ks ½Mud ½PC =ðKd þ½PC ÞK1 ½PC  þ K2 ½PN  d½PN =dt ¼ K1 ½PC K2 ½PN 

0.9 0.8 0.7 0.6 0.5 0.4 0.3

Figure and Figure captions LIGHT

0.2 0.25

0.26

0.27

0.28

0.29

0.30

Control Parameter vs (nM.h-1) +

frq transcription

ks

3. Results and discussion

nuclear FRQ(PN) k1

υs0

frq mRNA(M)

Fig. 2. Bifurcation diagram for the deterministic Eq. (1). The Hopf bifurcation value is about 0.257 nM h  1.

-

FRQ(PC)

3.1. The case of no correlations between additive and multiplicative colored noises

k2

υd

υm Fig. 1. Scheme of the model for circadian oscillations in Neurospora.

When the system is subjected to the multiplicative and additive colored noises, Eq. (1) becomes d½M=dt ¼ ðu0s þ u0s B1 C1 ðtÞÞKIn =ðKIn þ ½PN n Þum ½M=ðKM þ½MÞþ B2 C2 ðtÞ d½PC =dt ¼ Ks ½Mud ½PC =ðKd þ ½PC ÞK1 ½PC  þ K2 ½PN  d½PN =dt ¼ K1 ½PC K2 ½PN 

ð2Þ

S. Jian-Cheng / Journal of Theoretical Biology 265 (2010) 565–571

567

M (nM)

0.8

0.6

0.4 1.0

M (nM)

0.8

0.6

0.4 600

800

1000

600

t(h)

800

1000 t(h)

Fig. 3. The time series of M at various control parameters us. (a) us ¼ 0.256 nM h  1; (b) us ¼ 0.258 nM h  1; (c) us ¼ 0.260 nM h  1; (d) uh ¼0.262 nM h  1. Other parameters are shown in Table 1.

M (nM)

0.8

0.6

0.4 t(h)

M (nM)

0.8

0.6

0.4 600

700

800 t(h)

900

1000

Fig. 4. The time series of M at various intensity of multiplicative colored noise (B1). (a) B1 ¼0 nM h  1/2; (b) B1 ¼0.02 nM h  1/2; (c) B1 ¼0.08 nM h  1/2. us ¼ 0.256 nM h  1, correlation time (m)¼ 5 h. Other parameters are shown in Table 1.

where C1(t) and C2(t) are the multiplicative and additive colored noises, B1 and B2 are the intensity of multiplicative colored noise and additive colored noise, respectively. They are determined by the following equations m1 dC1 =dt ¼ C1 þ xðtÞ,

and m2 dC2 =dt ¼ C2 þ ZðtÞ

ð3Þ

m1 and m2 are the correlation time of multiplicative colored noise and additive colored noise, respectively. x(t) and Z(t) are the Gaussion white noises. For the case of no correlations between the additive and multiplicative colored noises /C1 ðtÞC2 ðt þ tÞS ¼ /C2 ðtÞC1 ðt þ pÞS ¼ 0

ð4Þ

Eqs. (2) and (3) are numerically solved by using Euler’s method and the time evolution of the system lasts 19,000 h, the last 16,384 data points are used to obtain frequency spectra by fast Fourier transform. Based on the power spectrum, signal-to-noise ratio (SNR) is defined as H(Do/of)  1 (Hu et al., 1993), where H is the maximum peak height of the spectrum, of is the principal peak frequency, and Do is the width of the peak at its half height. Each data is obtained by averaging 20 runs. In what follows, we mainly investigate the interaction of the external and internal colored noises on the signal transduction and amplification by using the mechanism of an internal signal stochastic resonance (ISSR). To address the problem, we will

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investigate the effect of the correlation time of colored noise on an ISSR and the effect of the intensity of colored noise on an ISSR.

3.1.1. The effect of the correlation time of colored noise on an ISSR Let now consider the effect of the correlation time on an ISSR. Fig. 5a shows the SNR as a function of multiplicative colored noise intensity (B1) at different correlation time (m2) of an additive colored noise. Two interesting features are clearly revealed: (i) the ISSR is suppressed with increasing the m2 from 1 to 7 h. However, when m2 is further increased from 7 to 9 h, an ISSR hardly change. It means that m2 plays a negative role for the ISSR, and there exists a critical value of m2 to make the ISSR to be robust for the additive colored noise. (ii) When the intensity of multiplicative colored noise (B1) is 0.081 nM h  1/2, ISSR curves overlap completely, indicating there exists a threshold of the multiplicative colored noise for destroying the effect of m2, i.e., critical intensity of the multiplicative colored noise (B1c). That is to say, when B1 oB1c, the m2 plays a suppressive role for an ISSR, while B1 4 B1c, B1 is so strong that the effect of m2 is destroyed. In the above considerations, the circadian rhythms (noiseinduced circadian oscillations) can be achieved and transduced with the phenomenon of an internal signal stochastic resonance (ISSR), and ISSR can be amplified or suppressed by modulating the

3.2x10-4 3.1x10-4

SNR

3.0x10-4 2.9x10-4 m2=1 h m2=3 h m2=5 h m2=7 h m2=9 h

2.8x10-4 2.7x10-4

correlation time of the additive colored noise (m2). For human being, the disorganized circadian oscillations could lead to circadian clock disorders and various syndromes, such as, phase-advanced or phase-delayed sleep syndromes (Mahowald et al., 1991). The results obtained here might yield insights into the process of curing the circadian rhythms syndromes, e.g., some studies have shown that bright light exposure might correct the circadian rhythms syndromes (Dawson and Campbell, 1991; Rosenthal et al., 1990). Up to now, many studies have demonstrated that distinct behavioral or perceptual brain states, including sensory perception, motor function, memory, and consciousness, are associated with different brain rhythms that reflect synchronization of neurons and their entrainment in a regular firing pattern (Baruchi et al., 2008; Raichman and Ben-Jacob, 2008; Walter, 2008). We expect that the results could be useful for further understanding other rhythms, such as cardiac, respiratory rhythms. As already mentioned in Fig. 5a, additive colored noise can strongly influence the ISSR driven by multiplicative colored noise. Intuitively, the ISSR driven by one colored noise could be influenced by another colored noise. To further confirm the speculation, we proceed to investigate how multiplicative colored noise influences the ISSR driven by additive colored noise. It is easy to see from Fig. 6a that the ISSR hardly change with the increment of the correlation time (m1) of multiplicative colored noise, which means that the multiplicative colored noise hardly influence the ISSR driven by an additive colored noise. Furthermore, by comparing Fig. 5a with Fig. 6a, it is seen that the optimal noise intensity for an ISSR is about 0.036 nM h  1/2 (see Fig. 5a), while 0.0025 nM h  1/2 (see Fig. 6a). Taken together, for additive and multiplicative colored noises, the system might be more sensitive to the additive colored noise. It is known that different noise sources may affect the system’s stochastic dynamics in different ways (Li and Li, 2004). Here, the additive colored noise term is added directly to the deterministic equation, while the multiplicative colored noise introduced in the system is related to the control parameter values. Therefore, the mechanisms of the internal signal stochastic resonance (ISSR) phenomenon induced by multiplicative colored noise and additive colored noise may be different (Yi et al., 2006).

2.6x10-4 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 B1 (nM.h-1/2) 0.060 m2=1 h m2=3 h m2=5 h m2=7 h m2=9 h

0.055

Frequency (Hz)

0.050 0.045 0.040 0.035 0.030 0.025 0.020

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 B1 (nM.h-1/2) Fig. 5. (a) SNR for the clock gene mRNA vs. multiplicative colored noise intensity (B1) at various correlation time of the additive colored noise (m2); (b) the frequency for the clock gene mRNA vs. multiplicative colored noise intensity (B1) at various correlation time of the additive colored noise (m2). Other parameters are shown in Table 1.

3.1.2. The effect of the intensity of colored noise on an ISSR In Figs. 5a and 6a, we show the effect of correlation time on an ISSR. Now, we are interested in investigating how the intensity of colored noise affects the ISSR. As is shown in Fig. 7a, some important features are reflected: (i) the ISSR driven by multiplicative colored noise is enhanced with increasing the intensity of an additive colored noise (B2), which suggests B2 plays an active role for ISSR. (ii) The SNR at various B2 is all increased firstly, reaches ‘‘plateau’’ and then do not change as the intensity of multiplicative colored noise increases, which might be called ISSR without tuning (Collins et al., 1995). The phenomenon suggests that the noise-induced circadian rhythms in the Neurospora system can be resistant to multiplicative colored noise (the high light fluctuation) above the critical value (B1 ¼0.061 nM h  1/2). (iii) When the intensity of multiplicative colored noise is increased to critical value (B1 ¼0.061 nM h  1/2), all resonance curves overlap completely, implying the critical multiplicative colored noise is so strong that the effect of additive colored noise disappears. It might be the reason that, when the ISSR driven by multiplicative colored noise occurs, the additive colored noise can enhance the performance of stochastic oscillations by introducing extra dynamics. The extra dynamics may play a crucial role as an energy source, which makes an SNR increase with increasing the intensity of additive colored noise, until the total noise loses its constructive role (Yi et al., 2006).

S. Jian-Cheng / Journal of Theoretical Biology 265 (2010) 565–571

3.2x10-4

3.3x10-4

3.1x10-4 SNR

3.0x10-4

3.0x10-4 2.8x10-4 SNR

m1=1 h m1=3 h m1=5 h m1=7 h m1=9 h

3.2x10-4

2.6x10-4 2.4x10-4

B2=0.0001 nM.h-1/2

2.8x10-4

2.2x10-4

B2=0.0005 nM.h-1/2

2.7x10-4

2.0x10-4

2.6x10-4

1.8x10-4

2.9x10-4

B2=0.0003 nM.h-1/2 B2=0.0007 nM.h-1/2 B2=0.0009 nM.h-1/2

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

2.5x10-4

B1 (nM.h-1/2) 0.000

0.003

0.006

0.009

0.012

0.015

B2 (nM.h-1/2)

m1=1 h m1=3 h m1=5 h m1=7 h m1=9 h

0.055 0.050 0.045

Frequency (Hz)

0.060

0.060

Frequency (Hz)

569

0.055

B2=0.0001 (nM.h-1/2)

0.050

B2=0.0005 (nM.h-1/2)

0.045 0.040

B2=0.0003 (nM.h-1/2) B2=0.0007 (nM.h-1/2) B2=0.0009 (nM.h-1/2)

0.035 0.030

0.040

0.025

0.035

0.020 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.030

B1 (nM.h-1/2)

0.025 0.020 0.000

0.003

0.006

0.009

0.012

0.015

Fig. 7. (a) SNR for the clock gene mRNA vs. multiplicative colored noise intensity (B1) at various intensity of the additive colored noise (B2); (b) the frequency for the clock gene mRNA vs. multiplicative colored noise intensity at (B1) at various intensity of the additive colored noise (B2). Other parameters are shown in Table 1.

B2 (nM.h-1/2) Fig. 6. (a) SNR for the clock gene mRNA vs. additive colored noise intensity (B2) at various correlation time of the multiplicative colored noise (m1); (b) the frequency for the clock gene mRNA vs. an additive colored noise intensity (B2) at various correlation time of the multiplicative colored noise (m1). Other parameters are shown in Table 1.

Let me have a brief review of the fundamental frequency of noise-induced circadian oscillations. It is found from Figs. 5b, 6b and 7b that the fundamental frequency is hardly shifted with increasing the intensity and correlation time of colored noise, i.e., their values are almost 0.036 Hz. The reason may lie in the fact that the Neurospora system could be resistant to the colored noises, exhibit strong vitality and sustain intrinsic circadian rhythms. Mizuno (2004) has pointed out that higher plants employ their plant response regulators for their own signal transduction and eliciting plant hormone responses, which are used to modulate sophisticated biological processes, including circadian rhythms and other light signal responses. So we believe that the result here is helpful for us to further understand other phenomena such as biological rhythm and signal detection and transduction in biological system (Hou and Xin, 2003; Gonze et al., 2002; Li and He 2005).

3.2. The case of correlations between additive colored noise and multiplicative colored noise We have argued that the external colored noise (light noise) can drive the Neurospora circadian oscillator from its quiescent

state to an oscillatory state, the synthetic rate of mRNA might be affected. Accordingly, the internal noise which is generated by the stochastic properties of the biochemical reaction might also be influenced. Therefore, the importance of the interactions of internal noise (additive colored noise) and external noise (multiplicative colored noise) cannot be ignored. One notes that the interactions of internal and external noises are termed as the correlation between two noises, and its strength is defined as correlation intensity (l). In this section, our aim is to assess how the correlation intensity (l) influences the internal signal stochastic resonance (ISSR). Note that the correlations between C1(t) (multiplicative colored noise) and C2(t) (additive colored noise) pffiffiffiffiffiffiffiffiffiffican be described as: /C1 ðtÞC2 ðt þ tÞS ¼ /C2 ðtÞC1 ðt þ pÞS ¼ l B1 B2 dðtÞ (Jia et al., 2000). The other parameters relating to the physical properties of the model are the same as Eqs. (2) and (3). When the correlation intensity (l) is 1 h1/2 nM  1, Fig. 8 displays the SNR vs. the intensity of multiplicative colored noise (B1) at different correlation time (m2) of the additive colored. As one can see that the ISSR does not occur when m2 ¼ 1 and 3 h, occurs when m2 ¼ 5, 7 and 8 h. The result indicates that the m2 plays an important role for the occurrence of an ISSR. To investigate the effect of correlation intensity (l) on an ISSR, curves of m2 ¼5, 7 and 8 h in Figs. 5a and 8 are compared in Fig. 9a, b and c, respectively. It is seen from Fig. 9 that the ISSR in Fig. 5a (no correlations between two colored noises) is weaker than that in Fig. 8 (l ¼1 h1/2 nM  1). The result implies that the correlation intensity (l) plays a suppressive role for an ISSR. One notes that the result is further confirmed in Fig. 10. It can be

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m2=1 h m2=3 h m2=5 h m2=7 h m2=9 h

3.1x10-4

SNR

3.0x10-4

no correlation between two colored noises correlation intensity=1 (h1/2.nM-1) 3.1x10-4

3.0x10-4 SNR

2.9x10-4

2.8x10-4

2.9x10-4

2.8x10-4

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 B1 (nM.h-1/2)

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Fig. 8. SNR for the clock gene mRNA vs. multiplicative colored noise intensity (B1) at various correlation time of the additive colored noise (m2). Other parameters are shown in Table 1.

4. Conclusions The signal transduction and amplification in a Neurospora circadian clock system is studied by using the mechanism of an internal signal stochastic resonance (ISSR). Two cases have been investigated: the case of no correlations between multiplicative and additive colored noises and the case of correlations between two noises. The results show that, whether the multiplicative and additive noises are correlative, the noise-induced circadian oscillations can be transduced with the phenomenon of an internal signal stochastic resonance (ISSR). However, the correlation time of an additive colored noise plays a suppressive role for the ISSR driven by multiplicative colored noise, while the

no correlation between two colored noises correlation intensity=1 (h1/2.nM-1) 3.1x10-4

SNR

3.0x10-4

2.9x10-4

2.8x10-4

2.7x10-4 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 B1 (nM.h-1/2) no correlation between two colored noises correlation intensity=1 (h1/2.nM-1) 3.1x10-4 3.0x10-4 SNR

observed nicely that the ISSR is decreased gradually when the correlation intensity (l) is increased from 0.1 to 0.9 h1/2 nM  1. In particular, Fig. 10 also shows that there are two sets of curves divided by curve of l ¼1 h1/2 nM  1. The upper set corresponds to curves of l ¼0, 0.1, 0.3 and 0.5 h1/2 nM  1; the lower set corresponds to curves of l ¼0.7 and 0.9 h1/2 nM  1. The ISSR for the upper set curves is stronger than that of l ¼1 h1/2 nM  1, while the ISSR for lower set curves is weaker than that of l ¼1 h1/2 nM  1. It illustrates that the enhancement or suppression of an ISSR depends on the correlation intensity (l) of two colored noises. Biological systems, especially cell systems, which have evolved for billions of years, may have learned to exploit noises to enhance the ability to detect weak signals, instead of trying to resist them. Now that all natural processes are accompanied by fluctuations, those noises that are inside or outside the cell systems may play an active role in improving the ability of response of the system in biological information processing (Zhang et al., 2005). In studying an internal signal stochastic resonance (ISSR), different clues to amplify it have been found, such as noise delay (Hou and Xin, 1999), the couplings of elements (Shinohara et al., 2002), and the distance to the bifurcation point (Li and Li, 2004). Here, a clue to amplify the strength of an ISSR is the modulation of the intensity and correlation time of the additive colored noise in a certain range, and the correlation intensity between the additive and multiplicative colored noises. Though it is not easy for us to directly control the intensity, correlation time and correlation intensity of colored noise to obtain the maximum SNR, we expect that biological systems may use the above regulatory mechanism to play functional roles in cellular process (Yi and Jia, 2005).

B1 (nM.h-1/2)

2.9x10-4 2.8x10-4 2.7x10-4 2.6x10-4

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 B1 (nM.h-1/2)

Fig. 9. SNR for the clock gene mRNA vs. multiplicative colored noise intensity (B1) at various correlation time of the additive colored noise (m2). (a) m2 ¼ 5 h; (b) m2 ¼7 h; (c) m2 ¼9 h. Other parameters are shown in Table 1.

intensity of an additive colored noise plays an active role for it. However, the correlation time and intensity of multiplicative colored noise hardly influence the ISSR driven by an additive

S. Jian-Cheng / Journal of Theoretical Biology 265 (2010) 565–571

correlation intensity=0.1 h1/2.nM-1) correlation intensity=0.3 h1/2.nM-1) correlation intensity=0.5 h1/2.nM-1) correlation intensity=0.7 h1/2.nM-1)

3.1x10-4

correlation intensity=0.9 h1/2.nM-1) correlation intensity=1 h1/2.nM-1)

SNR

3.0x10-4 2.9x10-4 2.8x10-4 2.7x10-4 2.6x10-4 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 B1 (nM.h-1/2) Fig. 10. SNR for the clock gene mRNA vs. multiplicative colored noise intensity (B1) at various correlation intensity (l). m1 ¼1 h, m2 ¼ 9 h, B2 ¼0.006 nM h  1/2. Other parameters are shown in Table 1.

colored noise. In addition, the ISSR can be amplified or suppressed by modulating the correlation intensity of colored noises. The fundamental frequency of noise-induced circadian oscillations is hardly shifted with the increment of the intensity and correlation time of colored noise, which implies that the Neurospora system could be resistant to the colored noises, exhibit strong vitality and sustain intrinsic circadian rhythms. We believe that the present results are a representation for a class of mechanisms based on autoregulatory negative feedback loops on gene express in vivo and are useful for inducement and control the circadian rhythms in other complex biological systems.

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