Skip Navigation


Bioinformatics Advance Access originally published online on February 10, 2005
Bioinformatics 2005 21(10):2478-2487; doi:10.1093/bioinformatics/bti316
This Article
Right arrow Abstract Freely available
Right arrow FREE Full Text (Print PDF) Freely available
Right arrow All Versions of this Article:
21/10/2478    most recent
bti316v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in ISI Web of Science
Right arrow Similar articles in PubMed
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Ohki, N.
Right arrow Articles by Hagiwara, M.
Right arrow Search for Related Content
PubMed
Right arrow PubMed Citation
Right arrow Articles by Ohki, N.
Right arrow Articles by Hagiwara, M.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The Author 2005. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions{at}oupjournals.org

Bio-Object, a stochastic simulator for post-transcriptional regulation

Nobukazu Ohki 1 and Masatoshi Hagiwara 1,2,*

1Department of Functional Genomics, Medical Research Institute 1-5-45 Yushima, Bunkyo-ku, Tokyo 113-0034, Japan
2Laboratory of Gene Expression, School of Biomedical Science, Tokyo Medical and Dental University 1-5-45 Yushima, Bunkyo-ku, Tokyo 113-0034, Japan

*To whom correspondence should be addressed.


   Abstract
TOP
 Abstract
1 INTRODUCTION
 2 METHODS
 3 RESULTS
 4 DISCUSSION
 REFERENCES
 

Motivation: Recently, biologists learnt that the transport and degradation of transcribed mRNA and protein present critically important steps for the regulation of gene expression through extensive studies of RNA interference, none-sense mediated decay and ubiquitination. However, adequate consideration of these factors has not been done in the past in in silico analysis compared with transcriptional regulations.

Results: We have developed a bio-system simulator ‘Bio-Object’ and assessed the contribution of numerous factors including movements, stability and interactions of both mRNAs and proteins in the virtual cell space to the Drosophila circadian rhythm. The oscillations of period (per), timeless (tim) and Drosophila Clock ({dClk}) mRNAs and proteins predicted by the simulations agreed with the observed data in Drosophila and were lost with the knock-out of either the per or the dClk gene as observed experimentally. Bio-Object predicts that (1) the stability of dClk mRNA, (2) the stability of dCLK and (3) the affinity of the PER–TIM complex are determinants of the circadian duration.

Availability: The source code is available for download from http://www.tmd.ac.jp/mri/mri-end/bio-object/download/

Contact: m.hagiwara.end{at}mri.tmd.ac.jp

Supplementary information: A detailed explanation of Bio-Object is available at http://www.tmd.ac.jp/mri/mri-end/bio-object/


   1 INTRODUCTION
TOP
 Abstract
 1 INTRODUCTION
2 METHODS
 3 RESULTS
 4 DISCUSSION
 REFERENCES
 
A living cell is a system of thousands of chemical reactions. A chemical reaction starts when each movable molecule interacts with its partner. In the case of gene expression, mRNA is produced when RNA polymerase and transcription factors form an appropriate complex on the promoter region of a gene. The transcribed mRNA (precursor mRNA) is processed (capping, polyA addition and splicing) by RNA-binding proteins and changed to a mature mRNA. The mature mRNA is translocated to cytoplasm, and bound by ribosomes. Then, genetic information is translated to a protein. Considering RNA interference (RNAi) (Fire et al., 1998), none-sense mediated decay (NMD) (Cheng and Maquat, 1993) and proteosome-mediated protein degradation (Matthews, 1989), we felt it necessary to provide a new computational framework to simulate such post-transcriptional regulation.

As a model of gene regulation, the production system of Drosophila circadian rhythm is quite attractive because the system is well investigated and operated with a limited number of core-factors. In the previous study, our group constructed a deterministic model on Drosophila circadian rhythms, created some mathematical ‘null mutants’ (e.g. per01, dClkJrk and per01;dClkJrk) and revealed that the interlocked feedback model provides a possible explanation for the robust oscillation of Drosophila circadian rhythms (Ueda et al., 2001). Gonze et al. (2002) adopted a stochastic simulation algorithm to model a simplified regulatory network for the Drosophila circadian rhythm. They changed the number of molecules in the model and assessed the effect of molecular noise on the circadian oscillations. The results of their theoretical experiment show that robust circadian oscillations can occur with a limited number of protein molecules.

In spite of the extensive biochemical and theoretical studies, the identification of determinant(s) for the 24 h duration, which is common to all organisms, is still an open question. Therefore, we applied our newly developed biological objective model (Bio-Object) to the Drosophila circadian rhythm, and tried to identify determinant(s) of the duration.


   2 METHODS
TOP
 Abstract
 1 INTRODUCTION
 2 METHODS
3 RESULTS
 4 DISCUSSION
 REFERENCES
 
2.1 Basic concept of Bio-Object
Bio-Object prepares a virtual cell with an (x,y) coordinate plane on which molecules move around. Biochemical reactions occur when a moving molecule meets its partner molecule. A virtual cell is composed of a square and a circle mimicking the area of a cell and a nucleus, respectively, as shown in Figure 1A. In order to detect access by molecules, the area is separated by latticework into meshes, and each mesh is named ‘Cell’. Multiple molecules exist within the Cell, and the occurrence of reaction is decided for every molecule in the same Cell. Various events occur due to the reaction, and the events of transcription, translation and complex formation are shown respectively in each visual cell of Figure 1B.



View larger version (41K):
[in this window]
[in a new window]
  		Fig. 1 Basic concept of Bio-Object. (A) The virtual cell. (B) Various events that occur in the virtual cell. (1) mRNA is produced at the transcription site in the nucleus under the regulation of a transcription factor. (2) The mRNA migrates to the cytoplasm, and then ‘protein A’ is produced when the mRNA meets a ribosome. (3) Protein A moves around the cytoplasm, and then a complex is produced when protein A meets its reaction partner, protein B. And the formation of the complex alters the properties of these proteins. Each molecule in the Bio-Object has its own lifetime and it disappears by that lifetime (Table 2). In the visual cell, the number and distribution of these molecules are changed by the events as time passes. (C) The movement of molecules in the model. The distance that a molecule moves in one unit of time, dA,t, depends on the product of multiplication between the average distance that a molecule moves in one unit time, D, and a random number of the normal distribution, Rg [1]. The direction of movement, scaled in radian, depends only on the uniform random number Ru [2]. From these two parameters, the position of a newly located molecule is calculated [3]. The numerical limitation shown in equation [4] is applied to a newly calculated location of the molecule, since the molecule can move around only inside of the cell. If the molecule is limited to existing only in the nucleus (the radius is defined as r), the numerical limitation shown in Equation [5] must be adopted for this case. On the contrary, if the distribution is restricted only within the cytoplasm, the numerical limitation of [6] is used instead of [5]. The movement of the molecule will be recalculated when the results of calculations are inconsistent with those limitations.

 
We assume that a random walk model is able to emulate the movement of molecules in the diffusion process inside the cell. The distance which the molecule moves in one unit time is obtained by statistical mechanics. Therefore, the equation for calculating the (x,y) coordinates of ‘molecule A’ at the time t(XA(t), YA(t)) can be described as shown in Figure 1C.

The cell is a 3-dimensional (3D) system, so if it is reflected in the Bio-Object, then a 3D latticework should be used for the simulation. However, the 3D latticework needs a large memory and for this reason is not suitable for calculation using a personal computer. Therefore, we have adopted a 2D latticework as the model cell in the Bio-Object. However, this simplification will change the volume available for the movement of molecules in the cell, which in turn may affect the relative rates of the chemical reactions. We recognize this weak point of the current model and intend to improve the model in the next version.

2.2 Fundamental flow chart in Bio-Object
The simulation algorithm is composed of two parts: the ‘initial condition module’ and the ‘main simulation loop’ (Fig. 2). In the former part, a virtual cell space is prepared and the location of the transcription site and the activity of an initial transcription are defined in the cell space. Then, a fixed number of ribosomes and an initial number of molecules are arranged in a specific place in the virtual cell space. The latter part, the main simulation loop, starts to update all data in the first step by using the information in the initial condition module. This main simulation loop consists of a ‘molecule simulation loop’ and a ‘reaction simulation loop’. In the molecule simulation loop, each molecule changes its location and reduces its lifetime. This process is repeated in all the existing molecules. Next, the reaction simulation loop starts by taking account of the updated information. In this step, a simulation of biological reactions is performed at each Cell. When the target Cell is a transcription site, the activity of transcription is evaluated and the transcription rate is varied if it is necessary. Subsequently, the transcription process is started to produce new mRNAs according to the updated transcription rate. When the mRNA comes to the ribosome Cell in the cytoplasm area, the translation process starts to produce new protein molecules. In the last step of the reaction simulation loop, the formation of a new complex is checked in all Cells of the virtual cell. Finally, all molecules in the system are updated and the molecule disappears when the lifetime becomes zero. A detailed explanation of these processes is given in the supplementary information (Section 1 in the Supplementary material).



View larger version (20K):
[in this window]
[in a new window]
  		Fig. 2 The flow chart of a simulation algorithm of the Bio-Object simulator. See text for a detailed description of the flow chart.

 
2.3 Simulation for simple enzyme reaction, E + S {Leftrightarrow} ES {Rightarrow} E + P
To estimate the adequacy of the model and algorithm, we applied the Bio-Object simulation method to a simple enzyme reaction. For the initial condition, the numbers of substrates and enzymes are 100 000 and 1000, respectively, and the others are zero. The probability of ES complex formation is defined as 0.5 when they meet in the same Cell and the probability of each ES complex proceeding to the product is also defined as 0.5. Under these conditions, an equilibrium between the reactants (substrates and enzymes) and the ES complex is maintained and concurrently products are produced from the complex in the simulation (Fig. 3).



View larger version (18K):
[in this window]
[in a new window]
  		Fig. 3 Evaluation of our simulation model in a simple enzyme reaction. Each plot shows the temporal change in the number of molecules for a simple model of enzyme reaction, E + S {Leftrightarrow} ES {Rightarrow} E + P (E: enzyme, S: enzyme-substrate complex and P: product).

 
However, in the latter term of the reaction, the time scale of the molecular number shows a sigmoid curve in a decreasing or increasing pattern, because the amount of substrates is reduced to break the equilibrium. These results are consistent with both the Haldane–Briggs model in which the ES complex amount is assumed as constant and the Michealis–Menten model in which the amount of ES complex is not fixed. Moreover, a stochastic fluctuation is observed in our model when the number of molecules becomes very small. Thus, this model is suitable for the simulation of phenomena that depend on a fluctuation in the system.

2.4 Parameter set for a Drosophila circadian rhythm
As a target system for theoretical analysis by Bio-Object, we have chosen Drosophila circadian rhythms (Dunlap, 1999; Young and Kay, 2001; Stanewsky, 2002). The Drosophila clock component can maintain its circadian rhythm for a week in a constant darkness condition (Emery et al., 1997). This maintenance of rhythms without Zeitgeber depends on the gene regulatory system which produces robust gene expression oscillations (Ueda et al., 2001; Leloup and Goldbeter, 1999). The model with two interlocked feedback loops reported by Glossop et al. (1999) is one of the most likely candidates to explain the Drosophila circadian oscillator and its property, and it is also supported by theoretical verification (Ueda et al., 2001). In one loop, per and tim are negatively regulated by their own products, PER and TIM proteins. PER protein binds the TIM protein, and the PER–TIM complex enters the nucleus. The PER–TIM complex then indirectly represses the transcription of a per and a tim gene through interaction with their transcriptional activator dCLK–CYC complex (Lee et al., 1999). In another loop, dClk is negatively regulated by its own product, dCLK protein, and the CYC protein which is continuously expressed at a high level from the Cycle (cyc) gene (Glossop et al., 1999; Bae et al., 2000). These two proteins also form a complex and indirectly repress the transcription of dClk by activating vrille (vri) which is a transcriptional repressor of dClk (Glossop et al., 2003). But the loss of function of the dCLK–CYC complex through the interaction with the PER–TIM complex causes an activating effect on the dClk expression indirectly (Glossop et al., 1999; Lee et al., 1999). Apparently, this structure of gene regulation network is essential, but post-transcriptional regulation (e.g. temporal change of mRNA stability) is also important for sustaining the overt rhythm (So and Rosbash, 1997). Furthermore, the temporal and spatial change in the localization of PER and TIM proteins from cytoplasm to nucleus is an essential process for the maintenance of the circadian rhythm (Gekakis et al., 1995).

We applied Bio-Object to the model with two interlocked feedback loops, and focused on the model based on the regulation by four clock genes: per, tim, dClk and cyc. The reaction rules of this gene regulation model are listed in Table 1. To estimate the influence of post-transcriptional regulation (e.g. mRNA stability, protein stability, translation rate and complex formation rate) on the circadian oscillation, we tuned the parameter sets which were consistent with biological data, and could produce a period close to 24 h in constant darkness (Table 2).


View this table:
[in this window]
[in a new window]
  		Table 1 Definition of reaction partner and localization of molecule

 

View this table:
[in this window]
[in a new window]
  		Table 2 Parameter values of the model

 
In general, the size of the eukaryotic cell ranges from 10 to 100 µm, and the size of the nucleus is about 10 µm (Alberts et al., 2003). Because of the limitation of the memory size of a personal computer for complex calculations, we have assumed that the size of the cell and the nucleus are 20 and 12 µm, respectively. The number of ribosomes in a yeast cell is estimated to be 200 000 (Warner, 1999). In the present simulation we limited the number of genes to four (per, tim, cyc and dClk); therefore, 20,000 ribosomes are sufficient for the translation in the model. And we used 7.3 µm/min as the diffusion rate of every molecule because every molecule can reach any place in the cell at a diffusion rate within the time resolution of 2 min in this simulation condition (Nomura et al., 2001). There are several reports related to the half-lives of mRNAs and proteins (Nishinokubi et al., 2003; Bae et al., 2000; Gerner et al., 2002; Cao and Parker, 2001). Considering these reports and experiments for the temporal change of the amount of clock protein and clock mRNA, we have assumed that the half-life of the mRNAs and that of the proteins are 2 and 10 h, respectively. In the case of the protein dimer (PER–TIM and dCLK–CYC), the half-life of the complex is assumed to be 20 h because of the stabilization effect of complex formation (Bae et al., 2000; Kloss et al., 1998; Price et al., 1998). However, the protein tetramer (PER–TIM–dCLK–CYC) does not play a significant role in the system, so we set the half-life of the tetramer to 1 h in order to remove the complex from the system quickly (Lee et al., 1999). There are certain reports that have examined relative changes in the transcription rate and translation rate of clock genes (Shu and Hong-Hui, 2004; So and Rosbash, 1997). But there is no experimental report that has quantitatively examined those rates. Therefore, in order to assume the transcription rates and translation rates for our simulation model, we have referred to some theoretical models about transcription and translation (Thattai and van Oudenaarden, 2001; McAdams and Arkin, 1997). Since we do not have any information on either the probability of complex formation among the clock proteins or the probability of the complex binding to the transcription sites, we tuned those two kinds of probabilities so that they could produce a period close to 24 h for the Drosophila circadian rhythm. We called this parameter set that could produce sustained oscillations the ‘standard condition’.


   3 RESULTS
TOP
 Abstract
 1 INTRODUCTION
 2 METHODS
 3 RESULTS
4 DISCUSSION
 REFERENCES
 
3.1 Sustained oscillations in wild-type (standard condition) model
Sustained oscillations in the amount of per mRNA, tim mRNA, dClk mRNA, PER, TIM and dCLK are obtained from a simulation using the parameter sets shown in Table 2 (Fig. 4A and B). The oscillations of per mRNA and tim mRNA are in phase (Fig. 4C) and their levels peak in Zeitgeber time (ZT) 10–14 and subsequently bottom in ZT 22–24. The protein level of PER and TIM peaks in ZT 16–18 after a delay of about 4–6 h from their mRNA. On the other hand, the oscillations of per mRNA and dClk mRNA are in anti-phase (Fig. 4D). The oscillation of dClk mRNA peaks in ZT 22–24 and subsequently bottoms in ZT 10–14, and the protein level of dCLK peaks in ZT 4–6. The cyc mRNA and CYC levels are continuously maintained at a high level, and thus no oscillation pattern is observed (Fig. 4B). These temporal properties of oscillations are consistent with experimental observations (Dunlap, 1999; Young and Kay, 2001; Stanewsky, 2002). The sustained oscillation continues for at least 10 days (Fig. 4E). The robustness of the oscillation is confirmed by calculating an autocorrelation function that oscillates with a near constant amplitude (Fig. 4F). Using these results as the standard in the theoretical experiments on the interlocked two feedback loops model of Drosophila circadian rhythm, we have comprehensively analyzed a large number of variations by changing parameters.



View larger version (29K):
[in this window]
[in a new window]
  		Fig. 4 Simulation of circadian oscillation in the constant darkness condition. Parameter values are shown in Table 2. Time course of (A) per and tim expression and (B) dClk expression are simulated. Average of 10 independent simulation results are plotted at each 2-h interval. (C) tim and (D) dClk mRNA levels are plotted against per mRNA levels. tim and per mRNA reach peak or bottom levels almost simultaneously (in phase), whereas dClk mRNA approaches the peak level when the per mRNA reaches the bottom level (in anti-phase). (E) The oscillation in the number of molecules continues for at least 240 h (10 days). Only the simulated time courses of the PER protein are shown. (F) Time evolution of the autocorrelation function for the oscillation of the number of PER protein.

 
3.2 In silico knockout study of Drosophila circadian rhythm
Several mutants that lost the circadian rhythms are reported in Drosophila. Particularly in a mutant lacking PER (per01), the characteristic changes of the dClk mRNA level are observed by analysis for the double mutant per01;dClkJrk (Glossop et al., 1999). Though the dClk mRNA level is decreased in the per01 single mutant, the decreased level of dClk is restored to the wild-type level in the per01;dClkJrk double mutant. The increase or the decrease in the level of dClk mRNA is explained by the regulatory mechanism, indicated in the two interlocked feedback loops model, in which functional dCLK–CYC represses the transcription of dClk mRNA and functional PER–TIM represses the function of dCLK–CYC. Then, in order to examine whether the increase or the decrease in the dClk mRNA level can be also observed in the theoretical experiment by our model, we executed an in silico knockout study of Drosophila circadian rhythm by the application of Bio-Object. Two types of in silico knockout mutants which correspond to per01 single mutants and per01;dClkJrk double mutants are created. In per01 mutants, the sustaining oscillations are abolished and the dClk mRNA level is decreased to about 15% of the peak level in the wild type. On the other hand, in per01;dClkJrk double mutants, the sustaining oscillations are also abolished, but the dClk mRNA level is restored to 110% of the peak level in the wild type (Section 2 in Supplementary information). These results are consistent with reported observations that show the decreased level of dClk mRNA level in per01 single mutants and the increased level of that in per01; dClkJrk double mutants. Our theoretical model also ascertains the adequacy of the interlocked feedback model that accounts for the observed experimental results (Glossop et al., 1999) as confirmed by our previous model based on the deterministic equations (Ueda et al., 2001).

3.3 Differential roles of the stability of per mRNA and dClk mRNA
The effect of per mRNA stability on the circadian oscillation has been well studied in comparison with that of dClk mRNA. So and Rosbash (1997) reported that per mRNA levels were regulated at a post-transcriptional level, which contributed to the Drosophila clock gene mRNA cycling. Chen et al. (1998) have shown the possibility that the stabilized per mRNA shortens the behavior rhythm by 3 h from that generated by a wild type. In order to examine whether per mRNA stability can affect the circadian rhythms, a virtual mutant cell model was created and examined. In the short half-life per mRNA mutant model, oscillations were abolished, but in the long half-life per mRNA mutant model, oscillations were maintained with an emphasized amplitude (Fig. 5A). However, no evidence could be found that the stability of per mRNA affected the duration, as shown in the experimental reports (So and Rosbash, 1997; Chen et al., 1998).



View larger version (52K):
[in this window]
[in a new window]
  		Fig. 5 Effect of changed parameters related to post-transcriptional regulation on circadian oscillations. Parameter values (not noticed here) used in these simulations are shown in Table 2. Time courses of average number of per mRNAs at each time point (every 2 h) are obtained from 10 independent simulations. The average half-life of per (A) and dClk (B) mRNA are changed in the simulation. ‘Short half-life mRNA’ has an average half-life of 1.0 h while ‘long half-life mRNA’ has an average half-life of 6.0 h. The translation rate of PER(C) and dCLK(D) was set as 1 molecule at every translation process for ‘low translation rate’ and as 4 molecules at every translation process for ‘high translation rate’, respectively. The average half-life of PER (E) and dCLK (F) proteins are changed in the simulation. In the ‘short half-life’ conditions, proteins have an average half-life of 5.0 h while in the ‘long half-life’ conditions, proteins have an average half-life of 20.0 h. (G) Both parameter values of ‘reaction probability of PER–TIM formation’ and ‘reaction probability of dCLK–CYC formation’ are increased to 0.005 molecules/attempt. (H) The parameter value of reaction probability of PER–TIM–dCLK–CYC tetramer formation is decreased to 0.1 molecule/attempt.

 
We next constructed a virtual mutant which gained an altered half-life for dClk mRNA and examined its effect. Surprisingly, this case produced different results from the per mRNA case. In the short half-life, the dClk mRNA mutant still maintained 24 h oscillations. In the long half-life, however, the dClk mRNA mutant elongated the circadian period from 24 to about 30 h (Fig. 5B). These results indicate that the oscillatory system is robust against the abundance of dClk mRNA, because the short half-life dClk mRNA mutant shows a low-level mRNA amount (data not shown). The above results suggest another property of the circadian oscillator. A sufficiently large amount of dClk mRNA (data not shown), which results from the increased stability of dClk mRNA, can elongate the circadian period. Unfortunately, we could not compare these theoretically supported suggestions with experimental data because no investigation into this post-transcriptional regulation point has been reported yet.

3.4 Contrary effects of translation rates of PER and dCLK on the oscillation
The effects of the translation rates of clock proteins have not been well investigated. Chen et al. (1998) investigated whether the translation rate of PER protein changed during the day and showed it was kept constant all day long. However, a possibility that mutated sequences in the non-coding region affect the translation control of clock proteins cannot be completely denied (Chen et al., 1998; Jackson, 1993). Therefore, we have created a mutant model for two cases: an altered translation rate for PER and an altered translation rate for dCLK. The daily 24 h oscillations do not change in the model for which the translation rate of PER is doubled (Fig. 5C). On the other hand, the oscillations completely disappeared when the translation rate of PER was reduced to half. However, the reduced translation rate of dCLK seemed to have no effect on the rhythm, even though a doubled dCLK translation rate forces the oscillations to become unstable, and the rhythms are attenuated and damped (Fig. 5D).

3.5 Stability of dCLK determines the circadian duration
The post-transcriptional regulations on clock proteins have been extensively studied and some regulatory factors were recently identified in the oscillation system. DOUBLETIME (DBT), which is closely related to human casein kinase 1{varepsilon} mediates PER protein phosphorylation to reduce the PER stability (Kloss et al., 1998; Price et al., 1998). However, a control of dCLK stability is not yet known. As a clue to this concealed mechanism, a daily pattern of change in the phosphorylation state of dCLK has been already observed (Lee et al., 1998). To see how these controls of the PER and dCLK stability can affect the circadian rhythm, we have constructed four simulation models with PER and dCLK proteins of different stabilities.

The circadian oscillation preserves the rhythm with a 24 h duration, but a shortened duration (20 h) was observed in the condition that dCLK became unstable. In contrast, when we stabilized the dCLK proteins, the duration was elongated to 32 h (Fig. 5F). As observed in altered dClk mRNA stability models, the stability of the dCLK protein could determine the circadian duration, suggesting that the amount of dCLK molecules in a cell may be the determinant of the circadian duration.

The rhythm was immediately lost when we reduced the stability of the PER protein, indicating that the stability of the PER protein exerts effects on the maintenance of the circadian rhythm (Fig. 5E; see Section 3 of Supplementary material for further information). However, no evidence was found to show that the stability of the PER protein had an influence on the oscillatory duration as observed in dbts and dbtl mutant (Price et al., 1998). It can be still assumed that the DBT controls not only the stability of PER but also the interaction between PER and TIM by regulating a phosphorylation state of PER. As the alternation of this interaction could cause changes in the amount of the PER–TIM complex, we next examined whether this affects the circadian duration.

3.6 Complex formation of PER–TIM and dCLK–CYC
Monomer proteins of PER and TIM enter the nucleus only when they interact with each other through the PAS domain and form a heterodimer complex PER–TIM which regulates the activity of the dCLK–CYC complex. The dCLK and CYC also interact through the PAS domain and form a heterodimer complex. This complex formation will enable their transcriptional regulation by binding DNA through the dHLH domain (Lee et al., 1999). Thus PER–TIM and dCLK–CYC complexes are key factors for the circadian oscillation in Drosophila (Dunlap, 1999; Young and Kay, 2001; Stanewsky, 2002).

To investigate the role of these dimer complexes in the production and maintenance of circadian oscillation, we manipulated the production rate of the dimer by controlling the parameter of reaction probability related to the complex formation reaction. When the reaction probability of PER–TIM complex formations was increased by five times, oscillations with a 22.5 h short period were observed (Fig. 5G). This result supports the hypothesis that the long oscillation period observed in the PERL mutant is due to weakened interactions between PER and TIM (Gekakis et al., 1995; Rutila et al., 1996). In contrast, the rhythms were lost when the reaction probability of dimer formation between dCLK and CYC was increased to five times that of the standard condition.

The amount of the dCLK–CYC complex, which has an activity that regulates the clock genes, is also regulated by the interaction of the PER–TIM complex (Lee et al., 1999). To estimate the influence of this interaction between PER–TIM and dCLK–CYC, the parameter which represents the reaction probability to a PER–TIM–dCLK–CYC tetramer complex was changed. When the parameter was reduced to one-fifth, the circadian oscillation was damped (Fig. 5H). Taken together, the amount of dCLK–CYC complex also becomes an important regulatory point for maintaining the circadian oscillation.


   4 DISCUSSION
TOP
 Abstract
 1 INTRODUCTION
 2 METHODS
 3 RESULTS
 4 DISCUSSION
REFERENCES
 
4.1 Determinants for maintenance of circadian oscillation
As summarized in Figure 6 (see Section 4 of Supplementary material for further information), the maintenance of stable circadian oscillation seems to be mainly regulated in the per–tim loop. If one of the three conditions: (1) unstable per mRNA, (2) low PER translation rate or (3) unstable PER protein, is satisfied, then the sustained oscillation becomes unstable or completely abolished. This instability of the circadian oscillation in such conditions seems to be caused by an excessive reduction in the amount of the PER–TIM complex which inactivates the key regulatory factor, dCLK–CYC. In the conditions with stable per mRNA, high translation rate or stable PER protein, the model precisely preserves the circadian oscillation. In these conditions, even if an excess amount of PER proteins is produced, the maximum amount of the PER–TIM complex never changes, because the amount of PER–TIM complex depends on the amount of TIM protein. In the per–tim gene regulation loops, the post-transcriptional regulations determine the amplitude of oscillations, and the amplified oscillations seem to contribute to the stability of the whole system. Actually, there are specific sequences within the transcribed portion of per gene and the upstream of the tim transcription start site which increase the expression level of each gene (Stanewsky et al., 1997; Wang et al., 2001).



View larger version (72K):
[in this window]
[in a new window]
  		Fig. 6 Summary of the effect of post-transcriptional regulation on the circadian oscillation of Drosophila. (A) The regulation mechanism in the interlocked feedback loop model of Drosophila circadian rhythm is illustrated. The per, tim, dClk and cyc mRNA (represented by a square) are transcribed in the nucleus and move to the cytoplasm. In the cytoplasm, the parts of these mRNA that meet the ribosome before their lifetime is exhausted start translation to produce proteins. But, if an mRNA fails to meet a ribosome, it will be degraded (represented by a dashed-line square). The translated dCLK and CYC, or PER and TIM, can form a dimer complex and move into the nucleus. The dCLK–CYC complex can activate the transcription of per and tim, and can also repress the transcription of dClk in the model. The PER–TIM complex can form a tetramer complex with the dCLK–CYC complex and can also inactivate the transcriptional activity of the dCLK–CYC complex. The monomer proteins in the cytoplasm whose lifetime becomes zero are degraded (represented by a dashed-line oval). (B) A summary of the effect of post-transcriptional regulation on the circadian oscillation is shown in this table. In this model, per and tim are regulated in the same manner. Thus, the results obtained from a theoretical experiment focusing on per can be also applied to tim. Each alphabet in the parenthesis corresponds to the regulation point represented by the same alphabet in the upper figure. In addition, the difference in the type of regulation is distinguished by a different color: red for the maintenance of robust oscillation; and blue for the regulation of the 24 h period.

 
4.2 Determinants for duration of circadian rhythm
The duration of the circadian rhythm is likely to be regulated by the dClk–cyc feedback loop. As the amount of CYC is maintained constant in this model, the oscillation of dCLK should play the main role in circadian oscillations in the dClk–cyc loop. However, no dClk mutant which has an altered circadian rhythm has been created or discovered, although an arrhythmic rhythm mutant dClkJrk has been reported (Allada et al., 1998). This may suggest that a certain level of dCLK is required for rhythm generation. Actually, the oscillatory amount of dCLK is strictly kept within the fixed range in our simulation compared with factors of per–tim loop, and this is in good accordance with the experimental observation of Bae et al. (2000). The duration of circadian rhythm is also affected in part by the per–tim loop that regulates the interaction between TIM and PER. In fact, the reported mutants such as perL, perS, timrit and timUL have an altered circadian period (Konopka and Benzer, 1971; Matsumoto et al., 1999; Rothenfluh et al., 2000). Mutants may have some conformational changes in the PAS domain or in a particular interaction point where a PER–TIM dimer is formed.

The oscillatory system of Drosophila, which has been obtained in the process of its evolution, has two main regulation mechanisms, ‘the maintenance of duration’ and ‘the control of the cycle’. The maintenance of the 24 h period of rhythm provides living organisms with the advantage of adapting their lives to daily changes of the environment. The control of the cycle is necessary to adjust the time lag which is caused by seasonal and geometrical changes in the environment. Our theoretical analysis of the Drosophila circadian rhythm has revealed the separate regulation points of ‘the maintenance of duration’ and ‘the control of the cycle’ in the two interlocked feedback loops as summarized in Figure 6. Our suggestions will provide new insights into both the Drosophila circadian rhythm and instructive information for defining a focusing point for experiments by biological researchers.

4.3 Advantageous features of Bio-Object
As shown in the analysis of Drosophila circadian rhythm, Bio-Object has four advantageous features. First, biological properties experimentally determined can be implemented into the parameters. Therefore, users can easily apply the experimental data to their own simulation. Second, each biologist can produce optional mutants including gain-of-function mutants by setting up parameters in the parameter file according to their own experiment plans. Third, a biologist can execute his own simulation in a lap-top computer on the bench (e.g. on a computer with Pentium IV 3.06 GHz processor and 512 MB RAM, the simulation on Drosophila circadian rhythms processing 1 000 000 molecules and 3000 calculation steps took about 2 h to complete). Fourth, Bio-Object provides researchers with simulation results which can be directly compared with experimental data and/or help them to form a hypothesis that can be tested by further experimentation. Actually, such virtual experiments with this stochastic simulator led us to the conclusion which we did not recognize with our previous model (Ueda et al., 2001). Thus Bio-Object can be used by biologists at bench side to examine their own working hypothesis in in silico analysis before experiments and to help the planning of experiments to test their predictions.


   Acknowledgments
 
We thank Masaru Tomita for useful information and Kazuo Ohki for comments on the draft manuscript. This work was partially supported by a Grant-in-Aid for Scientific Research on Priority Areas (C) Genome Information Science from Ministry of Education, Culture, Sport, Science and Technology.

Received on October 13, 2004; revised on February 4, 2005; accepted on February 7, 2005

   REFERENCES
TOP
 Abstract
 1 INTRODUCTION
 2 METHODS
 3 RESULTS
 4 DISCUSSION
 REFERENCES
 

    Alberts, B., et al. Essential Cell Biology, Second edition, (2003) Garland Publishing, Inc.

    Allada, R., et al. (1998) A mutant Drosophila homolog of mammalian Clock disrupts circadian rhythms and transcription of period and timeless. Cell, 93, 791–804[CrossRef][ISI][Medline].

    Bae, K., et al. (2000) dCLOCK is present in limiting amounts and likely mediates daily interactions between the dCLOCK-CYC transcription factor and the PER-TIM complex. J. Neurosci., 20, 1746–1753[Abstract/Free Full Text].

    Cao, D. and Parker, R. (2001) Computational modeling of eukaryotic mRNA turnover. RNA, 7, 1192–1212[Abstract].

    Chen, Y., et al. (1998) Alterations of per RNA in noncoding regions affect periodicity of circadian behavioral rhythms. J. Biol. Rhythm., 13, 364–379[Abstract].

    Cheng, J. and Maquat, L.E. (1993) Nonsense codons can reduce the abundance of nuclear mRNA without affecting the abundance of pre-mRNA or half-life of cytoplasmic mRNA. Mol. Cell. Biol., 13, 1892–1902[Abstract/Free Full Text].

    Cyran, S.A., et al. (2003) vrille, Pdp1, and dClock form a second feedback loop in the Drosophila circadian clock. Cell, 112, 329–341[CrossRef][ISI][Medline].

    Dunlap, J.C. (1999) Molecular bases for circadian clocks. Cell, 96, 271–290[CrossRef][ISI][Medline].

    Emery, I.F., et al. (1997) Rhythms of Drosophila period gene expression in culture. Proc. Natl Acad. Sci. USA, 94, 4092–4096[Abstract/Free Full Text].

    Fire, A., et al. (1998) Potent and specific genetic interference by double-stranded RNA in Caenorhabditis elegans. Nature, 391, 806–811[CrossRef][Medline].

    Gekakis, N., et al. (1995) Isolation of timeless by PER protein interaction: defective interaction between timeless protein and long-period mutant PERL. Science, 270, 811–815[Abstract/Free Full Text].

    Gerner, C., et al. (2002) Concomitant determination of absolute values of cellular protein amounts, synthesis rates, and turnover rates by quantitative proteome profiling. Mol. Cell. Proteomics., 1, 528–537[Abstract/Free Full Text].

    Glossop, N.R., et al. (1999) Interlocked feedback loops within the Drosophila circadian oscillator. Science, 286, 766–768[Abstract/Free Full Text].

    Glossop, N.R., et al. (2003) VRILLE feeds back to control circadian transcription of clock in the Drosophila circadian oscillator. Neuron, 37, 249–261[CrossRef][ISI][Medline].

    Gonze, D., et al. (2002) Robustness of circadian rhythms with respect to molecular noise. Proc. Natl Acad. Sci. USA, 99, 673–678[Abstract/Free Full Text].

    Jackson, R.J. (1993) Cytoplasmic regulation of mRNA function: the importance of the 3 untranslated region. Cell, 74, 9–14[CrossRef][ISI][Medline].

    Kloss, B., et al. (1998) The Drosophila clock gene double-time encodes a protein closely related to human casein kinase Iepsilon. Cell, 94, 97–107[CrossRef][ISI][Medline].

    Konopka, R.J. and Benzer, S. (1971) Clock mutants of Drosophila melanogaster. Proc. Natl Acad. Sci. USA, 68, 2112–2116[Abstract/Free Full Text].

    Lee, C., et al. (1998) The Drosophila CLOCK protein undergoes daily rhythms in abundance, phosphorylation, and interactions with the PER-TIM complex. Neuron, 21, 857–867[CrossRef][ISI][Medline].

    Lee, C., et al. (1999) 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. Mol. Cell. Biol., 19, 5316–5325[Abstract/Free Full Text].

    Leloup, J.-C. and Goldbeter, A. (1999) Chaos and birhythmicity in a model for circadian oscillations of the PER and TIM proteins in Drosophila.. J. Theor. Biol., 198, 445–459[CrossRef][ISI][Medline].

    Matsumoto, A., et al. (1999) timrit Lengthens circadian period in a temperature-dependent manner through suppression of PERIOD protein cycling and nuclear localization. Mol. Cell. Biol., 19, 4343–4354[Abstract/Free Full Text].

    Matthews, K.A., et al. (1989) Developmental distribution of RNA and protein products of the Drosophila-tubulin gene family. Dev. Biol., 132, 45–61[CrossRef][ISI][Medline].

    McAdams, H.H. and Arkin, A. (1997) Stochastic mechanisms in gene expression. Proc. Natl Acad. Sci. USA, 94, 814–819[Abstract/Free Full Text].

    Nishinokubi, I., et al. (2003) Highly conserved Drosophila ananassae timeless gene functions as a clock component in Drosophila melanogaster. Gene, 307, 183–190[CrossRef][ISI][Medline].

    Nomura, Y., et al. (2001) Monitoring of in vitro and in vitro translation of green fluorescent protein and its fusion proteins by fluorescence correlation spectroscopy. Cytometry, 44, 1–6[CrossRef][ISI][Medline].

    Price, J.L., et al. (1998) Double-time is a novel Drosophila clock gene that regulates PERIOD protein accumulation. Cell, 94, 83–95[CrossRef][ISI][Medline].

    Rothenfluh, A., et al. (2000) A TIMELESS-independent function for PERIOD proteins in the Drosophila clock. Neuron, 26, 505–514[CrossRef][ISI][Medline].

    Rutila, J.E., et al. (1996) The timSL mutant of the Drosophila rhythm gene timeless manifests allele-specific interactions with period gene mutants. Neuron, 17, 979–990[CrossRef][ISI][Medline].

    Shu, Y. and Hong-Hui, L. (2004) Transcription, translation, degradation, and circadian clock. Biochem. Biophys. Res. Commun., 321, 1–6[CrossRef][ISI][Medline].

    So, W.V. and Rosbash, M. (1997) Post-transcriptional regulation contributes to Drosophila clock gene mRNA cycling. EMBO J., 16, 7146–7155[CrossRef][ISI][Medline].

    Stanewsky, R. (2002) Clock mechanisms in Drosophila. Cell Tissue Res., 309, 11–26[CrossRef][ISI][Medline].

    Stanewsky, R., et al. (1997) Multiple circadian-regulated elements contribute to cycling period gene expression in Drosophila. EMBO J., 16, 5006–5018[CrossRef][ISI][Medline].

    Thattai, M and van Oudenaarden, A. (2001) Intrinsic noise in gene regulatory networks. Proc. Natl Acad. Sci. USA, 98, 8614–8619[Abstract/Free Full Text].

    Ueda, H.R., et al. (2001) Robust oscillations within the interlocked feedback model of Drosophila circadian rhythm. J. Theor. Biol., 210, 401–406[CrossRef][ISI][Medline].

    Wang, G.K., et al. (2001) Regulation of the cycling of timeless (tim) RNA. J. Neurobiol., 47, 161–175[CrossRef][ISI][Medline].

    Warner, J.R. (1999) The economics of ribosome biosynthesis in yeast. Trends Biochem. Sci., 24, 437–440[CrossRef][ISI][Medline].

    Young, M.W. and Kay, S.A. (2001) Time zones: a comparative genetics of circadian clocks. Nat. Rev. Genet., 2, 702–715[CrossRef][ISI][Medline].


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?



This Article
Right arrow Abstract Freely available
Right arrow FREE Full Text (Print PDF) Freely available
Right arrow All Versions of this Article:
21/10/2478    most recent
bti316v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in ISI Web of Science
Right arrow Similar articles in PubMed
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Ohki, N.
Right arrow Articles by Hagiwara, M.
Right arrow Search for Related Content
PubMed
Right arrow PubMed Citation
Right arrow Articles by Ohki, N.
Right arrow Articles by Hagiwara, M.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?