3.2 Post-Transcriptional Regulation of Clock Genes
The circadian core clock is a network of transcription-translational feedback loops. With its tuned inhibition and activation of gene expression the clock components oscillate with a period of about 24 hours. They are the molecular basis for any circadian regulation, including rhythmic PTR. However, core clock genes themselves are targets for rhythmic degradation, both at the transcript [159, 160, 161] and the protein level [231, 232]. How does a rhythmic degradation of a core clock component effect its oscillation properties, amplitude and phase? Is rhythmic PTR of core clock components able to change the period, a key feature of the circadian clock?
Rhythmic degradation within the core clock is not captured by the model presented in Sec-tion 2.5. Up to now I investigated only genes which do not feed back into a gene network system, see Figure 3.2A. A basic description of a network system which can produce oscillations is pro-vided by the Goodwin model [57]. It provides a phenomenological description of a protein which suppresses the transcription of its own gene. It features the three important components that generate oscillations: negative feedback, delay and nonlinearity [56, 233]. The circadian clock is modeled by the Goodwin oscillator itself [234, 235] or by closely related models [236, 237, 238].
Consequently, the Goodwin model provides a good starting point to characterize the influence of rhythmic PTR on core clock components.
The Goodwin model is a 3-variable model with two activations and one inhibition. The synthesis and degradation rates are linear except of the inhibition rate, which takes the form of a sigmoidal curve, a Hill-curve. To simulate rhythmic PTR in the core clock I let the degradation rate of either one of the variables be rhythmic. This is achieved by introducing a cosine function with phase and amplitude which does not affect the average degradation rate, see Figure 3.2B, Appendix E.1. For each degradation phase and degradation amplitude I simulate the system and determine the oscillation properties, period, magnitude, relative amplitude and phase, of each system variable, see Appendix E.1 for details. In Figure 3.2C an example for a Goodwin oscillator with rhythmic degradation rate is plotted together with the original Goodwin oscillator with constant degradation rates.
Since it is well established that longer half-lives prolong the period of the Goodwin oscilla-tor [234], I expected the period to be affected also by periodic half-lives. This could not be confirmed, instead the period remained stable for almost all parameters, see Figure 3.3. One exception was observed: If the degradation rate of the repressing species oscillated with an am-plitude larger than 0.7 I observed a period doubling. All species then oscillated with twice the period, however this oscillation still contained the original period, see Figure 3.2D.
Apart from this exception, introducing a rhythmic degradation to the Goodwin model had a very stable effect, irrespective which variable was affected by rhythmic degradation. An increasing relative amplitude in the degradation rate resulted in an stronger overall oscillation of the whole system regardless of the degradation phase. This amplitude increase was very large, with up to a 2 fold increase for strongly oscillating degradation rates. This degradation phase independent amplitude increase contrasts with the results of clock-controlled genes, where only a specific phase range produced an amplitude boost, see Section 2.5. Increasing the degradation amplitude also increased the magnitude of all variables, see Figure 3.3. The degradation phase affected only the phase of the oscillating system. The relationship between degradation phase
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3.2 Post-Transcriptional Regulation of Clock Genes
and phase of the system is linear and the phase range of the system spans over all possible phases.
These results suggest that rhythmic degradation of core clock genes is beneficial to the core clock as it increases the overall amplitude and, hence, the biological impact of oscillations. Fur-thermore, rhythmic degradation could serve to easliy shift the phase of all core clock genes since there is a linear relationship between degradation phase and system phase. It seems, that oscil-lating degradation rates in this model system contribute to the overall energy of the osciloscil-lating system. However, the Goodwin oscillator is a toy-model which only consists of one negative feedback loop. The circadian clock network, on the other hand, contains several intertwined negative and positive feedback loops. Conclusions drawn from the Goodwin oscillator’s prop-erties are not necessary true for the circadian clock itself. For example, the fact that longer half-lives cause generally longer periods is true for the Goodwin model [234] but could not be confirmed for a more complex model of the circadian network [239]. Experimental evidence underlines this. A longer period of the core clock was observed either with a longer Cry1-mRNA half-life [161] or with a shortened Cry1-protein half-life [232].
Interestingly, downregulation or upregulation of degradation rate influencing factors - common experimental tools to investigate biological systems - primarily changes the average degradation rate, but not necessarily its rhythmic properties, relative amplitude and phase. To investigate only the rhythmicity of degradation one would need a tightly controlled down or upregulation at specific times of the day in a system which forgets quickly relative to circadian time scales.
To my knowledge, such an experimental system has not been established. At present, it is not possible to disturb and, with that, investigate experimentally the rhythmicity of degradation rates and their contribution to the clock properties. Consequently, this problem can only be tackled by mathematical modeling.
In summary, for a rather simple model I find that when introducing oscillating degradation the system’s period remains stable for most of the cases. Any rhythmicity in the degradation rate contributed to a stronger overall oscillation in all system’s variables, which might be desirable for biological function. For more complex systems, these very general findings must be reviewed.
In order to do that, one could introduce rhythmic degradation to an already established math-ematical model of the circadian clock, for example in the model of Relogio et al.[239] or Woller et al. [240].
However, the proposed model, a Goodwin oscillator with modulated rates, should be discussed a little further. In this model I actually modulated the degradation rate externally. With this, the model describes an oscillating system driven by an oscillating degradation rate with the system’s period. This means we are looking at an oscillator driven by an external force in resonance and this easily explains the contribution of oscillating degradation rates to the overall system’s oscillation strength. Another implementation of this system is realized when the core-clock itself modulates the degradation rate. Hence, the degradation rates are modulated by one of the variables, for example dx =dx(y).
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