
Smolen (2002), Circadian Oscillator
November 2006, model of the month by Enuo He
Original model: BIOMD0000000025
Almost all living organisms ranging from bacteria to humans display rhythmic activities coinciding with the daynight cycle, which are maintained by a circadian clock. That is why circadian oscillators became some of the favourite systems tackled by modellers.
Starting from the pioneering work of Goldbeter 1995 [1] (BIOMD0000000016), mathematical models of circadian oscillators became more and more complex and comprehensive, including more components as people gathered more knowledge of the biological system (e.g. [2], BIOMD0000000036; [3], BIOMD0000000021; [4], BIOMD0000000022; [5], BIOMD0000000055; for a review, see [6]). However, because of the high level of detail incorporated in those models, an intuitive understanding of the oscillation mechanisms is difficult to acquire.
Some authors therefore decided to develop simpler models, including as few differential equations as possible, in order to examine minimal but biologically realistic requirements for getting circadian oscillations (e.g. [7], BIOMD0000000024, which only contained a negative feedback).
Figure 1: Simulation of circadian oscillations by a reduced model (from [8]) A) Schematic of model. The dCLOCK protein activates the synthesis of PER. PER represses its own synthesis indirectly, by binding and inactivating dCLOCK. dCLOCK also represses its own synthesis. A time delay 1 is included between changes in dCLOCK concentration and in PER synthesis,and a delay 2 is included between changes in dCLOCK concentration and in dCLOCK synthesis. (B) Simulated oscillations. Blue, dark green, and light green traces are, respectively, for [PER], [dCLOCK], and [dCLOCK]free.

The model by Smolen et al 2002 [8] (BIOMD0000000025) is one of those minimal models, which in addition displayed all the controls (both positive and negative feedback) that were known at that time. The authors presented a reduced mathematical model that reflected the essential dynamical features of the circadian oscillator systems.
This model was reduced from an earlier detailed model [9] neglecting the dynamics of two oscillations in Drosophila: TIM and CYCLE. The model consisted of two differential equations, each one comprising a time delay. These delayed differential equations described the evolution of PER and dCLOCK concentrations. The reduced model succeeded in simulating circadian oscillations of [PER] and [dCLOCK]. Moreover the oscillation amplitude and period were robust to parameter variation.
This model is described in Figure 1A. dCLOCK activates per transcription and thus PER synthesis. PER represses per transcription (and thus PER synthesis) by binding to dCLOCK. PER also activates dCLOCK synthesis by binding to dCLOCK and relieving dCLOCK's repression of dclock transcription. Thus, the model contains both a negative feedback loop, in which PER binds dCLOCK and thereby deactivates per transcription, and a positive feedback loop, in which activation of per transcription by dCLOCK results in binding of dCLOCK by PER and derepression of dclock.
The Drosophila model readily simulated largeamplitude circadian oscillations in the level of PER and dCLOCK (Figure 1B). When [PER] rises at the beginning of oscillation (at t=12h), free dCLOCK is rapidly eliminated. The loss of free dCLOCK also increases its synthesis. Degradation of PER along with new dCLOCK synthesis rapidly regenerates free dCLOCK (at t=24h). After another delay, the free dCLOCK activates PER synthesis, beginning the next oscillation (at t=34h). However, because of the time delay function, positive feedback does not suppress the oscillation of the system. With [dCLOCK] fixed, the negative feedback loop still operates. PER still inhibits its own synthesis by binding to dCLOCK and blocking transcriptional activation.

Bibliographic References

A. Goldbeter. A model for circadian oscillations in the Drosophila period protein (PER). Proc Biol Sci., 261(1362):31924, Sep 1995. [PubMed]

J. J. Tyson, C. I. Hong, C. D. Thron, and B. Novak. A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM. Biophys J.,77:24112417, 1999. [PubMed]

J. C. Leloup and A. Goldbeter. Chaos and birhythmicity in a model for circadian oscillations of the PER and TIM proteins in drosophila. J Theor Biol., 198(3):44559, Jun1999. [PubMed]

H. R. Ueda, M. Hagiwara, and H. Kitano. Robust oscillations within the interlocked feedback model of Drosophila circadian rhythm. J Theor Biol., 210:401406, 2001. [PubMed]

J. C. W. Locke, M. M. Southern, L. KozmaBognár, V. Hibberd, P. E. Brown, M. S. Turner, and A. J. Millar. Extension of a genetic network model by iterative experimentation and mathematical analysis. Mol Syst Biol., 1:E1E9, 2005. [PubMed]

A. Goldbeter. Computational approaches to cellular rhythms. Nature, 420:238245, 2002. [PubMed]

T. Scheper, D. Klinkenberg, C. Pennartz, and J. van Pelt. A mathematical model for the intracellular circadian rhythm generator. J Neurosci., 19:4047, 1999. [PubMed]

P. Smolen, D. A. Baxter, and J. H. Byrne. A reduced model clarifies the role of feedback loops and time delays in the Drosophila circadian oscillator. Biophys J., 83(5):234959, Nov 2002. [PubMed]

P. Smolen, D. A. Baxter, and J. H. Byrne. Modeling circadian oscillations with interlocking positive and negative feedback loops. J Neurosci., 21:66446656, 2001. [PubMed]
