# Stochastic Simulation of Delay-Induced Circadian Rhythms in *Drosophila*

- Zhouyi Xu
^{1}and - Xiaodong Cai
^{1}Email author

**2009**:386853

https://doi.org/10.1155/2009/386853

© Z. Xu and X. Cai. 2009

**Received: **11 December 2008

**Accepted: **10 May 2009

**Published: **15 June 2009

## Abstract

Circadian rhythms are ubiquitous in all eukaryotes and some prokaryotes. Several computational models with or without time delays have been developed for circadian rhythms. Exact stochastic simulations have been carried out for several models without time delays, but no exact stochastic simulation has been done for models with delays. In this paper, we proposed a detailed and a reduced stochastic model with delays for circadian rhythms in *Drosophila* based on two deterministic models of Smolen et al. and employed exact stochastic simulation to simulate circadian oscillations. Our simulations showed that both models can produce sustained oscillations and that the oscillation is robust to noise in the sense that there is very little variability in oscillation period although there are significant random fluctuations in oscillation peeks. Moreover, although average time delays are essential to simulation of oscillation, random changes in time delays within certain range around fixed average time delay cause little variability in the oscillation period. Our simulation results also showed that both models are robust to parameter variations and that oscillation can be entrained by light/dark circles. Our simulations further demonstrated that within a reasonable range around the experimental result, the rates that *dclock* and *per* promoters switch back and forth between activated and repressed sites have little impact on oscillation period.

## 1. Introduction

Almost all living organisms, including animals, plants, fungi, and cyanobacteria, exhibit daily periodic oscillations in their biochemical or physiological behavior, which are known as circadian rhythms [1–7]. The mechanism of circadian oscillation has been an extensive research topic in the last three decades. It has been found that circadian rhythms in fact are determined by oscillatory expression of certain genes [89]. Specifically, circadian clocks consist of a network of interlocked transcriptional-translational feedback loops formed by a number of genes [2]. In *Drosophila*, transcription of *per* and *tim* genes is activated by a heterodimer consisting of two transcriptional activators dCLOCK and CYCLE [10–13]. The PER protein in turn binds to the dCLOCK-CYCLE heterodimer, which inhibits the DNA binding activity of the dimer, thereby repressing the transcription of *per* and *tim* [11–14]. While this forms a negative feedback loop, there is also a positive feedback loop, in which PER and TIM activate dCLOCK synthesis by binding dCLOCK and relieving dCLOCK's repression of dclock transcription [1516].

Several mathematical models have been proposed for circadian oscillation in *Drosophila* [121417–22]. The models of Smolen et al. [1214] introduce time delays in the expression of *dclock* and *per* genes, while other models do not have such delays. Numerical simulations using ordinary differential equations (ODE) show that all these models can produce circadian oscillations. In particular, times delays were found to be essential for simulation of circadian oscillations with the model of Smolen et al. [1214].

Since there is significant stochasticity in gene expression arising from fluctuations in transcription and translation [23–25], it is desirable to simulate circadian oscillations in the presence of noise. Toward this end, several stochastic models were proposed [426–29], and Gillespie's stochastic simulation algorithm (SSA) [3031] was employed to simulate circadian oscillations. All these stochastic models [426–29] do not include time delays. In order to reflect the noise in gene expression, Smolen et al. used two approximate stochastic simulation methods to simulate circadian oscillation based on their models with delays [1214]. However, their models lumped transcription and translation into one single process and did not model the process that dCLOCK binds to or dissociates with *dclock* and *per* genes to activate or inhibit transcription. Since transcription is a major source of intrinsic noise [2324], the approximate stochastic simulation of Smolen et al. may underestimate the effect of noise. Li and Lang [32] used similar approximate stochastic simulation methods to simulate reduced model of Smolen et al. [14], but with an emphasis on the noise-sustained oscillation in the region of parameter values where the deterministic model predicted no oscillation. Currently, no exact stochastic simulation has been done for circadian rhythm models with random delays, partially due to the fact that Gillespie's SSA cannot handle delays in certain reactions.

Recently, we developed an exact SSA algorithm for systems of chemical reactions with delays [33]. The goal of this paper is to apply this exact SSA to simulate circadian oscillations in *Drosophila* using a model with time delays and to investigate the effects of noise and random time delays on circadian oscillations. We first develop two stochastic models with random delays for circadian oscillations in *Drosophila* based on the two deterministic models of Smolen et al. [1214]. Using our exact SSA, we then simulate free-running circadian oscillation under constant darkness. Our simulations demonstrate that both models can produce sustained oscillations. The variability in oscillation period is very small although the variability in oscillation peaks is considerably large. In particular, although time delays are essential to oscillation, random fluctuations in time delays do not cause significant changes in oscillation period as long as the average delays are fixed. Our simulations also showed that circadian oscillations of both models are robust to parameter variations. The entrainment by light was also simulated for both models, yielding results consistent with experimental observations. To see the effect of transcription noise, we also run simulations with different values for the rate that dCLOCK binds or unbinds to *per* and *dclock* genes.

## 2. Methods

### 2.1. The Detailed Model of Circadian Oscillation with Time Delays

#### 2.1.1. Model Description

*Drosophila*circadian oscillator based on the deterministic model of Smolen et al. [12], which is depicted in Figure 1. In Smolen's model, transcription of

*dclock*gene is repressed by dCLOCK protein after a time delay of [1113]. dCLOCK activates the synthesis of PER protein with a time delay of . PER is then phosphorylated [34], and unphosphorylated and phosphorylated PER can bind to dCLOCK thereby relieving dCLOCK's repression of

*dclock*transcription. It was reported that PER undergoes multiple and sequential phosphorylation [34], but exact times of phosphorylation are unknown. Following Smolen et al. [12], we assumed that PER can be phosphorylated up to 10 times. Although the TIM gene also plays an important role in circadian rhythm, Smolen et al. [12] used a single "lumped" variable, PER, to represent both PER and TIM, since the time courses of PER and TIM proteins are similar in shape and largely overlap. Smolen et al. [12] characterized the circadian oscillator in

*Drosophila*using 23 ordinary differential equations (ODEs). We first convert these 23 ODEs into 46 chemical reactions. Smolen et al. [12] lumped transcription and translation of

*dclock*and

*per*into one single step. They did not model the process that dCLOCK binds to and dissociates with

*dclock*gene and

*per*gene. Since this binding and unbinding processes, transcription, and translation are major sources of intrinsic noise [35–38], we model these processes explicitly. Our stochastic model, containing 29 molecular species in Table 1, is featured with 54 reactions in Table 2 which include 44 reactions converted from Smolen's ODE and 10 new reactions.

Rid | Reaction | |
---|---|---|

1 | ||

2 | ||

3 | ||

4 | ||

5 | ||

6 | ||

7 | ||

8 | ||

9 | ||

10 | ||

11 | ||

22 | ||

54 |

Molecular species.

Reaction represent transcription of gene, degradation of mRNA, translation of mRNA, and degradation of dCLOCK protein, respectively. Reaction 5 models the process that dCLOCK protein binds to gene and reaction 6 represents dissociation of dCLOCK with . Reaction 7 and 8 specify the event that dCLOCK binds to and dissociates with gene. Reactions 9, 10, and 11 represent transcription of gene after it is activated by dCLOCK, degradation of mRNA, and translation of mRNA, respectively. Reactions represent the phosphorylation of PER and reaction 22 represents the degradation of PER. Reactions represent the association of dCLOCK with PER at different levels of phosphorylation. Reactions describe the phosphorylation of dCLOCK and ( ) heterodimer. Reactions represent the degradation of dCLOCK and ( ) heterodimer.

#### 2.1.2. Parameter Estimation

Each reaction is associated with a reaction probability rate constant, , which determines the probability that a specific reaction occurs in an infinitesimal time interval. The probability rate constant of a specific reaction can be calculated from conventional rate constant as follows: for a monomolecular reaction, for a bimolecular reaction with two different reactants, and for a bimolecular reaction with one reactant [39], where , and is the Avogadro constant and is the system volume. We assume that a lateral neuron in is a sphere of a radius around 6 [1440], which results in a volume L. As many other existing models [121441], we do not separate nuclear and cytoplasmic compartments. We retain most parameter values from Smolen's et al. [12] including and . The remaining 10 parameters, , , , and , are determined in our simulation. In the following, we describe 54 reactions and how each probability rate constant was determined.

Here represents the concentration of the species in the bracket. A time delay is included in reaction 3 accounting for time needed for transcription, translation, and other potential mechanisms for activating the transcription of . Smolen et al. chose to be a deterministic number equal to 5 hours [12]. Taking into account uncertainty in this delay, we choose as a random variable uniformly distributed in the interval (4h–6h).

In reaction 5, dCLOCK binds to the E-box of [1113], but there is no experimental report on the values of and the dissociation rate . However, the dissociation rate of myogenin protein with the E-box of E12 gene was reported to be [44]. Therefore, we choose . The equilibrium constant of reactions 5 and 6 is equal to the Michaelis constant in [12] that describes the regulation of dCLOCK synthesis by dCLOCK and was chosen to be 1 nM [12]. Using this equilibrium constant, we calculate to be . Reactions 7 and 8 specify the event that dCLOCK binds to and dissociates with gene. The equilibrium constant of reactions 7 and 8 is equal to the Michaelis constant in [12], which was 1 nM. This Michaelis constant reflects the regulation of PER synthesis by dCLOCK. After choosing , is calculated from the equilibrium constant as .

The transcription rate of
gene
is chosen to be 20
, and the degradation rate of
mRNA
is calculated as
from the half-life of
mRNA which was estimated to be 2 hours [4243]. Also, we assume that there are two copies of
gene, and thus, the initial value for the number of molecules of *perg* in Table 1 is 2. The synthesis rate of PER protein was chosen to be
nM
in the model of Smolen et al. [12], which equivalently is
molecules per hour. In our model, the average PER synthesis rate is
molecules per hour. Letting
, we got
. Similar to
, we also tested the sensitivity of simulation results to
. Increasing or decreasing
two times while fixing the ratio of
only causes negligible change in the mean and standard error (SE) of period and peaks (data not shown).

A delay is introduced in reaction 11. This time delay accounts for the time needed for the transcription and translation of gene. Smolen et al. [12] chose to be 8 hours. However, the total time needed for transcription and translation maybe less than 8 hours [45] and also there may be some fluctuations in . Therefore, we chosen as a random variable with mean 6 hours, uniformly distributed in (4.8h–7.2h).

Reaction 22 represents the degradation of PER and is equal to , where nM and nM [12].

Reactions represent the association of dCLOCK with PER at different levels of phosphorylation. The deterministic rates for all these reactions are 30 n and thus the probability rate constants are , . Reactions describe the phosphorylation of dCLOCK and ( ) heterodimer and we have , . Reactions represent the degradation of dCLOCK and ( ) heterodimer. We have , and , where and are given earlier.

### 2.2. The Reduced Model of Circadian Oscillation with Time Delays

#### 2.2.1. Model Description

Rid | Reaction | |
---|---|---|

1 | ||

2 | ||

3 | ||

4 | ||

5 | ||

6 | ||

7 | ||

8 | ||

9 | ||

10 | ||

11 | ||

12 | ||

13 | ||

14 |

#### 2.2.2. Parameter Estimation

The rate is chosen to be 10 which is slightly lower than that in the detailed model. This is because the synthesis rate of dCLOCK protein in the reduced model of Smolen et al. [14] was nM , which is smaller than in the detailed model. The rate is the same as that in the detailed model. Letting equal to , we calculate . The rate is the same as that in the reduced model of Smolen et al. [14], equal to 0.5 .

The unbinding rate of dCLOCK to gene, , is chosen identical to that in the detailed model. The equilibrium constant , which is equal to the Michaelis constant in [14] that describes the regulation of dCLOCK synthesis by dCLOCK, was chosen to be 0.1 nM [14]. Using this equilibrium constant, we calculate to be 1.44 . Similarly, is the same as that in the detailed model. The equilibrium constant , which is equal to the Michaelis constant in [14] that describes the regulation of PER synthesis by the transcriptional activators dCLOCK, is chosen to be 0.3 nM [14]. Then we calculate to be 0.48 .

The transcription rate of gene is chosen to be 10 , which is lower than that in the detailed model, because the synthesis rate of PER protein in the reduced model of Smolen et al. [14] was nM which is smaller than that in the detailed model. The degradation rate of mRNA is again . Letting equal to , we calculate as . The degradation rate of PER is the same as that in the reduced model of Smolen et al. [14], equal to 0.5 . The degradation rate of dCLOCK and PER complex is , identical to that in the detailed model and we have .

Time delays and are chosen as follows. As the effective delay contributed by PER phosphorylation is incorporated into and , , and should be longer than those in the detailed model. Therefore we chose and uniformly distributed in the time interval (5h–9h) and (7h–11h), respectively.

### 2.3. Stochastic Simulation

Gillespie's SSA [31] is often employed to simulate the stochastic dynamics of genetic networks [2346]. However, Gillespie's SSA cannot deal with delays in certain reactions. Recently, we developed an exact SSA for systems of chemical reactions with delays [33], which can handle both deterministic and random delays. We use this exact SSA to simulate the dynamics of the systems described in Tables 1 and 3.

### 2.4. Data Analysis

Customized MATLAB Software (Mathworks Inc.) was written to analyze data generated from stochastic simulations, for example, to calculate the mean and SE of protein levels, to identify the peaks of dCLOCK and PER during oscillation, and to calculate the peak amplitudes. Oscillation periods were calculated using the short-time Fourier transform (STFT) method [47]. Specifically, Fourier transform was applied to protein levels of dCLOCK and PER within a time window of 70 hours, after the mean level was subtracted. The largest peak at a non-zero frequency was identified as the oscillation frequency within the time window and the period of the oscillation is the inverse of the oscillation frequency. Note that the maximum period that can be identified by the STFT is 35 hours since a time window of 70 hours was used.

## 3. Results

### 3.1. Simulation of Oscillation in the Presence of Noise

Statistics of oscillations for the detailed stochastic model.

Mean | SE | CV | |
---|---|---|---|

Period (h) | 23.93 | 0.78 | 3.26% |

Peak value of total PER | 10149 | 1530.1 | 15.08% |

Peak value of free dCLOCK | 1377.1 | 184.62 | 13.41% |

Peak value of total dCLOCK | 1437.1 | 156.24 | 10.87% |

Peak-to-through amplitude of total dCLOCK | 1016.5 | 191.51 | 18.84% |

Statistics of oscillations for the reduced stochastic model.

Mean | SE | CV | |
---|---|---|---|

Period (h) | 23.60 | 0.80 | 3.39% |

Peak value of Total PER | 196.47 | 40.49 | 20.61% |

Peak value of free dCLOCK | 183.09 | 35.49 | 19.38% |

Peak value of total dCLOCK | 201.51 | 31.05 | 15.41% |

Peak-to-through amplitude of total dCLOCK | 172.47 | 36.10 | 20.93% |

We investigate the effect of the random time delays with fixed average time delays. Since both detailed and reduced models produced similar results, we here only present results for detailed model. Note that time delays and in our simulations are random variables uniformly distributed in , where the standard value of and standard value of , with denoting the average time delay. To test the sensitivity of the range of random delays, we run more simulations using different and but with a fixed . Specifically, when we fix to be 5 hours and 6 hours for and , respectively, if and are uniformly distributed in , the mean period is 23.89 hours and the SE is 0.76 hour; if and are uniformly distributed in , the mean period is 23.81 hours and the SE is 0.79 hour. In both cases, the mean period and SE are very close to the results from standard value of and . Therefore, our simulations show that the random changes in the delays do not cause significant variations in the oscillation period as long as the average delays are fixed.

Smolen et al. [1214] also investigated the effects of noise using stochastic simulation. There are three major differences between our stochastic simulation and that of Smolen et al.: ( ) we employed exact SSA, whereas they used approximate SSAs, ( ) two delays critical to circadian oscillation are random in our simulation but deterministic in the simulation of Smolen et al., and ( ) we explicitly simulated the transcription process and the binding/unbinding events between dCOLCK and per and dclock promoters, whereas Smolen et al. lumped transcription and translation of dclock and per into a one-step process.

To convert concentration into number of molecules, we used the volume of typical lateral neuron cells, whereas Smolen et al. determined a scale factor by trial. For the detailed model, this resulted in different scale factors and protein levels in our simulation as shown in Figure 2 are approximate 10 times of those in the simulation of Smolen et al. as depicted in Figure 3 of [12]. To make fair comparison, we ran simulations using the same scale factor as Smolen et al. [12]. Our simulation results showed that the mean peak values of PER, free dCLOCK and total dCLOCK are 1205, 176, and 183, respectively, which are comparable to the results of Smolen et al. [12]. The mean period in our simulation is 24 hours and the CV of periods is 3.33%. These results are also comparable to the results of Smolen et al.: a mean period of 23.5 hours and a CV of 5%. The CVs of the peaks of PER, free dCLOCK, and dCLOCK in our simulation are 15.47%, 15.20%, and 12.26%, respectively, which are greater than the CV of PER (9%) in the simulation of Smolen et al. [12]. For the reduced model, it turns out that protein levels in our simulation are similar to those in the simulation of Smolen et al. [14]. The CV of periods in our simulation (3.39%) is slightly smaller than that obtained in simulation of Smolen et al. (4.78%). Since no result about the CV of peak protein levels was reported by Smolen et al., we cannot compare the CV of peak protein levels.

In summary, although the noise in our models may be stronger than that in the models of Smolen et al. due to the random delays, transcription process, and random activation and repression of the promoters of per and dclock, the CV of periods in our simulation is slightly smaller than that in the simulation of Smolen et al. [14]. This result indicates that approximate simulation may have yielded nonnegligible errors. It is difficult to evaluate the effect of such possible errors in the approximate method of Smolen et al. [14], but our simulation method is exact and can correctly capture the stochastic dynamics of the circadian rhythm. It seems that strong noise in our model is reflected in the peak protein levels because the CVs of peak protein levels in our detailed model are larger than those in the detailed model of Smolen et al. [12].

### 3.2. Robustness Test in the Presence of Noise

In living cells, biochemical parameters often vary significantly from cell to cell due to stochastic effects, even if the cells are genetically identical [50]. But circadian oscillations with close period are still withstood in or mammals. Therefore, a model of circadian rhythm should be robust in the sense that small parameter variations should not lead to large period variations. For the deterministic models, Smolen et al. [1214] have shown that circadian rhythm is robust when a parameter changes its value by . Here, we test if circadian rhythm is robust with respect to parameter changes in the presence of intrinsic noise. To test robustness, each parameter is decreased or increased by from the standard value, with all other parameters fixed at the standard values, and then the mean and SE of oscillation periods and peaks are determined from simulation results. Since and are random variables, we decrease or increase their mean values by .

We also tested the robustness of oscillation for the reduced model. The reduced model has 14 probability rate constants and 2 time delays. Therefore, 33 sets of simulations were run including the set with standard parameter values. Figure 6 plots relative change of the mean value of the period and peaks for the parameter sets with one changed parameter comparing with the standard parameter set and Figure 7 plots the CV of the period and peaks for all parameter sets. It is seen that the change is small in period but relatively large in peaks when a parameter changes. It is also seen that CV of the periods is very small for both models, in the interval . Therefore, the system is very robust to the parameter variation in oscillation period. Note that the CV of the peaks of the reduced model is larger than that of the detailed model. This is due to the fact that the reduced model has lower number of molecules in the system so that there is larger internal noise.

As in the detailed model, the period, the peak of the free dCLOCK and the peak of the total PER in the reduced model are most sensitive to , , and , respectively. Specifically, decreasing (increasing) the mean value of by 20% decreased (increased) the mean period by ( ) and the corresponding CV was ( ). Decreasing (increasing) by decreased (increased) the mean peak of free dCLOCK by ( ) and the corresponding CV was ( ). Decreasing (increasing) by decreased (increased) the mean peak of total PER by ( ) and the corresponding CV was ( ). Comparing the results of two models, it appears that both models have small changes in the mean period and relatively large changes in the mean peaks and that the reduced model has slightly larger CVs.

### 3.3. Light Entrainment of Oscillation in the Presence of Noise

Models of circadian rhythms must be able to maintain synchrony with environmental cycles to drive behavioral, physiological and metabolic outputs at appropriate time of day [7]. Circadian rhythms can be entrained by external cues, such as daily environmental cycles of light and temperature, but light is generally considered as the strongest and most pervasive factor. Therefore, the responses of the rhythm are often simulated by light pulses or light/dark (L/D) cycles [121851–53]. We first consider the detailed model. In , light induces to enhance the degradation of phosphorylated TIM [125455]. Since there is no separate variable for TIM in our model, the degradation of phosphorylated PER was induced to simulate the effect of light, as done by Smolen et al. [12]. Here the phosphorylated PER includes all unbounded and bounded PERs ( and dCLOCK. dCLOCK. ). The dCLOCK is released after the PER complex with dCLOCK is degraded by light and the degradation rate of all phosphorylated PER is 0.9 [12]. In addition, to keep the oscillation period, the maximum degradation rate of dCLOCK, , was reduced to 1.5 nM [12] and the probability rate constants and , in our stochastic model were reduced correspondingly.

### 3.4. Impact of Transcription Activation Rate

As we mentioned earlier, the rate that dCLOCK binds to and genes is unknown but was estimated in our simulation. The rate that dCLOCK dissociates with and genes is chosen to be equal to the experimentally reported dissociation rate of myogenin protein with the E-box of E12 gene. In a deterministic model, these rates generally do not affect oscillation as long as their ratio is fixed. However, these rates may have significant effects on transcription noise even when their ratio is fixed [232456]. In the following, we change the value of while keep the ratio fixed to see whether the oscillation period changes. Since both detailed model and reduced models yield similar results, we only give results for the detailed model.

## 4. Discussion

We have presented a detailed and a reduced stochastic model for delay-induced circadian rhythm in based on the deterministic models of Smolen et al. [1214], and employed our recently developed exact stochastic simulation algorithm [33] to simulate the circadian rhythm. This work is unique since no exact stochastic simulation has been carried out for circadian rhythms based on a model with random time delays. As discussed in [33], several SSAs have been developed for reaction systems with delays [5758]. However, the algorithm in [57] and two algorithms in [58] are not exact. The work in [33] also proved that another heuristic algorithm in [58] is exact but requires more computation than the exact SSA in [33]. Since both algorithms are exact, they should produce the same statistical results. Another SSA for systems with delays was proposed in [59], but an approach similar to that in [57] was used, and thus, it is not exact either. Smolen et al. [1214], as well as Li and Lang [32], also simulated delay-induced circadian oscillation, but they used approximate stochastic simulation methods.

Our simulation results demonstrated that the intrinsic noise causes large fluctuations in oscillation peaks but very small fluctuations in oscillation period. This observation is seen in all simulations under different conditions, such as constant darkness and L/D cycles. Deterministic simulation cannot reveal this phenomena, since both period and peaks are constant. Our stochastic simulations also showed that circadian oscillation is robust in the presence of noise in the sense that noise has little effect on oscillation period although it can change oscillation peaks significantly. We also showed that random delays within certain range do not cause significant variations in the oscillation period as long as the average delays are fixed. To the best of our knowledge, these two results have not been observed in previous stochastic simulation of circadian rhythms. These two observations imply that circadian oscillation is robust in the presence of noise and random delays and that the randomness inherent to the oscillation circuit may not have much biological impact on the organism. As discussed in [50], when a protein regulates its targets, it often operates on a Hill curve. Once the level of the regulating protein is higher or lower than certain value, the protein operates at the top or bottom of the curve and the fluctuation of its level to certain extend does not affect much the regulating effect on its targets. Therefore, the relatively large variations in the peak values of PER and dCLOCK proteins observed in our simulation may not have a strong biological impact.

Similar to previous deterministic simulations and approximate stochastic simulations [1214], our stochastic simulation shows that both detailed and reduced stochastic models can provide sustained oscillations under darkness and L/D cycles. Our results also show right phase of all the components in the system, correct phase and anti-phase relationship of mRNAs and proteins, and also the appropriate lags between mRNAs and proteins. Our stochastic simulation further demonstrated that circadian rhythm is robust to parameter variations in the presence of noise. Increasing or decreasing each parameter by of its standard value changes the mean period by less than and causes negligible changes in the CV of oscillation periods. The model is not sensitive to the time delay during the mRNA translation, but it is most sensitive to the average time delay during mRNA translation, which shows that time delay is essential to circadian oscillation in the two models. However, random fluctuations in these two time delays have little effect on the oscillation period as long as the average delays are fixed. We also found that the binding and unbinding rates of dCLOCK to and genes within a reasonable range have little impact on the circadian oscillation. Increasing or decreasing the binding and unbinding rates by 10 times relative to an experimentally reported rate while keeping their ratio fixed does not cause significant changes in the period and peaks of oscillation.

We have compared our exact simulations with approximate simulations of Smolen et al. [1214] in Section 3.1. Another work by Li and Lang [32] also employed approximate SSAs to simulate the reduced model of Smolen et al. [14]. Like Smolen et al. [1214], Li and Lang [32] used deterministic delays, whereas we employed random delays which are more appropriate to reflect the delays in transcription, translation, and other chemical process. Li and Lang emphasized on the noise induced oscillation and showed that noise can sustain oscillation in the parameter region where no oscillation is predicted by the deterministic model, whereas we here focused on the robustness of oscillation in the presence of intrinsic noise and the effect of random delays. We showed that the oscillation is robust in the presence of noise since there is very little variability in oscillation period in spite of large random variability in peaks, and that random changes in delays within a large interval around the fixed average delay cause little variability in the oscillation period.

## Declarations

### Acknowledgments

This work was supported by the National Science Foundation (NSF) under NSF CAREER Award no. 0746882.

## Authors’ Affiliations

## References

- Moore-Ede M, Sulzman FM, Fuller CA:
*The Clocks That Time Us Physiology of the Circadian Timing System*. Harvard University Press, Cambridge, Mass, USA; 1982.Google Scholar - Young MW, Kay SA: Time zones: a comparative genetics of circadian clocks.
*Nature Reviews Genetics*2001, 2(9):702-715. 10.1038/35088576View ArticleGoogle Scholar - Yu W, Hardin PE: Circadian oscillators of
*Drosophila*and mammals.*Journal of Cell Science*2006, 119(23):4793-4795. 10.1242/jcs.03174View ArticleGoogle Scholar - Gonze D, Halloy J, Goldbeter A: Deterministic versus stochastic models for circadian rhythms.
*Journal of Biological Physics*2002, 28(4):637-653. 10.1023/A:1021286607354View ArticleGoogle Scholar - Williams JA, Sehgal A: Molecular components of the circadian system in
*Drosophila*.*Annual Review of Physiology*2001, 63: 729-755. 10.1146/annurev.physiol.63.1.729View ArticleGoogle Scholar - Reppert SM, Weaver DR: Molecular analysis of mammalian circadian rhythms.
*Annual Review of Physiology*2001, 63: 647-676. 10.1146/annurev.physiol.63.1.647View ArticleGoogle Scholar - Hardin PE: The circadian timekeeping system of
*Drosophila*.*Current Biology*2005, 15(17):R714-R722. 10.1016/j.cub.2005.08.019View ArticleGoogle Scholar - Dunlap JC: Molecular bases for circadian clocks.
*Cell*1999, 96(2):271-290. 10.1016/S0092-8674(00)80566-8View ArticleGoogle Scholar - Edmunds LN:
*Cellular and Molecular Bases of Biological Clocks*. Springer, New York, NY, USA; 1988.Google Scholar - Darlington TK, Wager-Smith K, Ceriani MF,
*et al*.: Closing the circadian loop: CLOCK-induced transcription of its own inhibitors PER and TIM.*Science*1998, 280(5369):1599-1603. 10.1126/science.280.5369.1599View ArticleGoogle Scholar - Lee C, Bae K, Edery I: PER and TIM inhibit the DNA binding activity of a
*Drosophila*CLOCK- CYC/dBMAL1 heterodimer without disrupting formation of the heterodimer: a basis for circadian transcription.*Molecular and Cellular Biology*1999, 19(8):5316-5325.View ArticleGoogle Scholar - Smolen P, Baxter DA, Byrne JH: Modeling circadian oscillations with interlocking positive and negative feedback loops.
*Journal of Neuroscience*2001, 21(17):6644-6656.Google Scholar - Bae K, Lee C, Hardin PE, Edery I: dCLOCK is present in limiting amounts and likely mediates daily interactions between the dCLOCK-CYC transcription factor and the PER-TIM complex.
*Journal of Neuroscience*2000, 20(5):1746-1753.Google Scholar - Smolen P, Baxter DA, Byrne JH: A reduced model clarifies the role of feedback loops and time delays in the
*Drosophila*circadian oscillator.*Biophysical Journal*2002, 83(5):2349-2359. 10.1016/S0006-3495(02)75249-1View ArticleGoogle Scholar - Lee C, Bae K, Edery I: The
*Drosophila*CLOCK protein undergoes daily rhythms in abundance, phosphorylation, and interactions with the PER-TIM complex.*Neuron*1998, 21(4):857-867. 10.1016/S0896-6273(00)80601-7View ArticleGoogle Scholar - Glossop NRJ, Lyons LC, Hardin PE: Interlocked feedback loops within the
*Drosophila*circadian oscillator.*Science*1999, 286(5440):766-768. 10.1126/science.286.5440.766View ArticleGoogle Scholar - Goldbeter A: A model for circadian oscillations in the
*Drosophila*period protein (PER).*Proceedings of the Royal Society B*1995, 261(1362):319-324. 10.1098/rspb.1995.0153View ArticleGoogle Scholar - Scheper TO, Klinkenberg D, van Pelt J, Pennartz C: A model of molecular circadian clocks: multiple mechanisms for phase shifting and a requirement for strong nonlinear interactions.
*Journal of Biological Rhythms*1999, 14(3):213-220. 10.1177/074873099129000623View ArticleGoogle Scholar - Tyson JJ, Hong CI, Thron CD, Novak B: A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM.
*Biophysical Journal*1999, 77(5):2411-2417. 10.1016/S0006-3495(99)77078-5View ArticleGoogle Scholar - Leloup J-C, Goldbeter A: Chaos and birhythmicity in a model for circadian oscillations of the PER and TIM proteins in
*Drosophila*.*Journal of Theoretical Biology*1999, 198(3):445-459. 10.1006/jtbi.1999.0924View ArticleGoogle Scholar - Petri B, Stengl M: Phase response curves of a molecular model oscillator: implications for mutual coupling of paired oscillators.
*Journal of Biological Rhythms*2001, 16(2):125-141. 10.1177/074873001129001836View ArticleGoogle Scholar - Ueda HR, Hagiwara M, Kitano H: Robust oscillations within the interlocked feedback model of
*Drosophila*circadian rhythm.*Journal of Theoretical Biology*2001, 210(4):401-406. 10.1006/jtbi.2000.2226View ArticleGoogle Scholar - Kaern M, Elston TC, Blake WJ, Collins JJ: Stochasticity in gene expression: from theories to phenotypes.
*Nature Reviews Genetics*2005, 6(6):451-464. 10.1038/nrg1615View ArticleGoogle Scholar - Raser JM, O'Shea EK: Noise in gene expression: origins, consequences, and control.
*Science*2005, 309(5743):2010-2013. 10.1126/science.1105891View ArticleGoogle Scholar - Rao CV, Wolf DM, Arkin AP: Control, exploitation and tolerance of intracellular noise.
*Nature*2002, 420(6912):231-237. 10.1038/nature01258View ArticleGoogle Scholar - Gonze D, Halloy J, Goldbeter A: Robustness of circadian rhythms with respect to molecular noise.
*Proceedings of the National Academy of Sciences of the United States of America*2002, 99(2):673-678. 10.1073/pnas.022628299View ArticleGoogle Scholar - Gonze D, Halloy J, Goldbeter A: Stochastic models for circadian oscillations: emergence of a biological rhythm.
*International Journal of Quantum Chemistry*2004, 98(2):228-238. 10.1002/qua.10875View ArticleGoogle Scholar - Calander N: Propensity of a circadian clock to adjust to the 24 h day-night light cycle and its sensitivity to molecular noise.
*Journal of Theoretical Biology*2006, 241(4):716-724.View ArticleMathSciNetGoogle Scholar - Miura S, Shimokawa T, Nomura T: Stochastic simulations on a model of circadian rhythm generation.
*BioSystems*2008, 93(1-2):133-140. 10.1016/j.biosystems.2008.05.002View ArticleGoogle Scholar - Gillepsie DT: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions.
*Journal of Computational Physics*1976, 22(4):403-434. 10.1016/0021-9991(76)90041-3View ArticleMathSciNetGoogle Scholar - Gillespie DT: Exact stochastic simulation of coupled chemical reactions.
*Journal of Physical Chemistry*1977, 81(25):2340-2361. 10.1021/j100540a008View ArticleGoogle Scholar - Li Q, Lang X: Internal noise-sustained circadian rhythms in a
*Drosophila*model.*Biophysical Journal*2008, 94(6):1983-1994. 10.1529/biophysj.107.109611View ArticleGoogle Scholar - Cai X: Exact stochastic simulation of coupled chemical reactions with delays. Journal of Chemical Physics 2007., 126(12):Google Scholar
- Edery I, Zwiebel LJ, Dembinska ME, Rosbash M: Temporal phosphorylation of the
*Drosophila*period protein.*Proceedings of the National Academy of Sciences of the United States of America*1994, 91(6):2260-2264. 10.1073/pnas.91.6.2260View ArticleGoogle Scholar - McAdams HH, Arkin A: It's a noisy business! Genetic regulation at the nanomolar scale.
*Trends in Genetics*1999, 15(2):65-69. 10.1016/S0168-9525(98)01659-XView ArticleGoogle Scholar - Elowitz MB, Leibier S: A synthetic oscillatory network of transcriptional regulators.
*Nature*2000, 403(6767):335-338. 10.1038/35002125View ArticleGoogle Scholar - Blake WJ, Kaern M, Cantor CR, Collins JJ: Noise in eukaryotic gene expression.
*Nature*2003, 422(6932):633-637. 10.1038/nature01546View ArticleGoogle Scholar - Fraser HB, Hirsh AE, Giaever G, Kumm J, Eisen MB: Noise minimization in eukaryotic gene expression.
*PLoS Biology*2004., 2(6): article e137Google Scholar - Gillespie DT: Stochastic simulation of chemical kinetics.
*Annual Review of Physical Chemistry*2007, 58: 35-55. 10.1146/annurev.physchem.58.032806.104637View ArticleGoogle Scholar - Ewer J, Frisch B, Hamblen-Coyle MJ, Rosbash M, Hall JC: Expression of the period clock gene within different cell types in the brain of
*Drosophila*adults and mosaic analysis of these cells' influence on circadian behavioral rhythms.*Journal of Neuroscience*1992, 12(9):3321-3349.Google Scholar - Xie Z, Kulasiri D: Modelling of circadian rhythms in
*Drosophila*incorporating the interlocked PER/TIM and VRI/PDP1 feedback loops.*Journal of Theoretical Biology*2007, 245(2):290-304. 10.1016/j.jtbi.2006.10.028View ArticleMathSciNetGoogle Scholar - Hardin PE, Hall JC, Rosbash M: Circadian oscillations in period gene mRNA levels are transcriptionally regulated.
*Proceedings of the National Academy of Sciences of the United States of America*1992, 89(24):11711-11715. 10.1073/pnas.89.24.11711View ArticleGoogle Scholar - Lin Y, Han M, Shimada B,
*et al*.: Influence of the period-dependent circadian clock on diurnal, circadian, and aperiodic gene expression in*Drosophila*melanogaster.*Proceedings of the National Academy of Sciences of the United States of America*2002, 99(14):9562-9567. 10.1073/pnas.132269699View ArticleGoogle Scholar - Spinner DS, Liu S, Wang S-W, Schmidt J: Interaction of the myogenic determination factor myogenin with E12 and a DNA target: mechanism and kinetics.
*Journal of Molecular Biology*2002, 317(3):431-445. 10.1006/jmbi.2002.5440View ArticleGoogle Scholar - So WV, Rosbash M: Post-transcriptional regulation contributes to
*Drosophila*clock gene mRNA cycling.*EMBO Journal*1997, 16(23):7146-7155. 10.1093/emboj/16.23.7146View ArticleGoogle Scholar - McAdams HH, Arkin A: Stochastic mechanisms in gene expression.
*Proceedings of the National Academy of Sciences of the United States of America*1997, 94(3):814-819. 10.1073/pnas.94.3.814View ArticleGoogle Scholar - Vetterli M, Kovacevic J:
*Wavelets and Subband Coding*. Prentice-Hall, Upper Saddle River, NJ, USA; 1995.MATHGoogle Scholar - Sehgal A, Rothenfluh-Hilfiker A, Hunter-Ensor M, Chen Y, Myers MP, Young MW: Rhythmic expression of timeless: a basis for promoting circadian cycles in period gene autoregulation.
*Science*1995, 270(5237):808-810. 10.1126/science.270.5237.808View ArticleGoogle Scholar - Bae K, Lee C, Sidote D, Chuang K-Y, Edery I: Circadian regulation of a
*Drosophila*homolog of the mammalian Clock gene: PER and TIM function as positive regulators.*Molecular and Cellular Biology*1998, 18(10):6142-6151.View ArticleGoogle Scholar - Alon U:
*An Introduction to Systems Biology: Design Principles of Biological Circuits*. Chapman & Hall/CRC Press, Boston, Mass, USA; 2006.Google Scholar - Leloup J-C, Goldbeter A: A model for circadian rhythms in
*Drosophila*incorporating the formation of a complex between the PER and TIM proteins.*Journal of Biological Rhythms*1998, 13(1):70-87. 10.1177/074873098128999934View ArticleGoogle Scholar - Leloup J-C, Goldbeter A: Modeling the molecular regulatory mechanism of circadian rhythms in
*Drosophila*.*BioEssays*2000, 22(1):84-93. 10.1002/(SICI)1521-1878(200001)22:1<84::AID-BIES13>3.0.CO;2-IView ArticleGoogle Scholar - Gonze D, Leloup J-C, Goldbeter A: Theoretical models for circadian rhythms in
*Neurospora*and*Drosophila*.*Comptes Rendus de l'Academie des Sciences. Series III*2000, 323(1):57-67. 10.1016/S0764-4469(00)00111-6View ArticleGoogle Scholar - Myers MP, Wager-Smith K, Rothenfluh-Hilfiker A, Young MW: Light-induced degradation of TIMELESS and entrainment of the
*Drosophila*circadian clock.*Science*1996, 271(5256):1736-1740. 10.1126/science.271.5256.1736View ArticleGoogle Scholar - Zeng H, Qian Z, Myers MP, Rosbash M: A light-entrainment mechanism for the
*Drosophila*circadian clock.*Nature*1996, 380(6570):129-135. 10.1038/380129a0View ArticleGoogle Scholar - Forger DB, Peskin CS: Stochastic simulation of the mammalian circadian clock.
*Proceedings of the National Academy of Sciences of the United States of America*2005, 102(2):321-324. 10.1073/pnas.0408465102View ArticleGoogle Scholar - Bratsun D, Volfson D, Tsimring LS, Hasty J: Delay-induced stochastic oscillations in gene regulation.
*Proceedings of the National Academy of Sciences of the United States of America*2005, 102(41):14593-14598. 10.1073/pnas.0503858102View ArticleGoogle Scholar - Barrio M, Burrage K, Leier A, Tian T: Oscillatory regulation of Hes1: discrete stochastic delay modelling and simulation.
*PLoS Computational Biology*2006., 2(9): article e117Google Scholar - Ribeiro A, Zhu R, Kauffman SA: A general modeling strategy for gene regulatory networks with stochastic dynamics.
*Journal of Computational Biology*2006, 13(9):1630-1639. 10.1089/cmb.2006.13.1630View ArticleMathSciNetGoogle Scholar

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