Theoretical study on the regulation of circadian rhythms by RNA methylation.

Messenger RNAs are often destabilized by methylation, suggesting that mRNA methylation alters mRNA and protein dynamics. This may indicate that the gene regulatory system is reflected by the metabolic system through mRNA methylation because methylation substrates are components of the metabolic system. Elucidating the mechanisms by which mRNA methylation regulates gene regulatory systems has posed considerable challenges due to the numerous targets of mRNA methylation. Recent studies have demonstrated that inhibition of mRNA N6-methyladenosine methylation elongates circadian periods. The aim of this study was to understand the mechanisms by which mRNA methylation regulates circadian rhythms. Using a detailed realistic model and a simple model, we demonstrated that period elongation of circadian rhythms by decreasing mRNA methylation can be achieved by two possibilities, i.e., decreasing mRNA methylation stabilizes nonoscillatory mRNAs such as Ck1δ and/or stabilizes oscillatory mRNAs of clock genes such as Per and Cry. In addition, we predicted that period elongation by stabilizing nonoscillatory mRNA (Ck1δ) should always be accompanied by the distortion of the circadian waveform. Finally, we discuss the validity of the two possible mechanisms on the regulation of circadian rhythms by mRNA methylation by quantifying waveform distortion of circadian gene activity data with or without mRNA methylation inhibitors.

Non-sinusoidal Waveform in Temperature-Compensated Circadian Oscillations.

Time series of biological rhythms are of various shapes. Here, we investigated the waveforms of circadian rhythms in gene-protein dynamics using a newly developed, to our knowledge, index to quantify the degree of distortion from a sinusoidal waveform. In general, most biochemical reactions accelerate with increasing temperature, but the period of circadian rhythms remains relatively stable with temperature change, a phenomenon known as "temperature compensation." Despite extensive research, the mechanism underlying this remains unclear. To understand the mechanism, we used transcriptional-translational oscillator models for circadian rhythms in the fruit fly Drosophila and mammals. Given the assumption that reaction rates increase with temperature, mathematical analyses revealed that temperature compensation required waveforms that are more nonsinusoidal at higher temperatures. We then analyzed a post-translational oscillator (PTO) model of cyanobacteria circadian rhythms. Because the structure of the PTO is different from that of the transcriptional-translational oscillator, the condition for temperature compensation would be expected to differ. Unexpectedly, the computational analysis again showed that temperature compensation in the PTO model required a more nonsinusoidal waveform at higher temperatures. This finding held for both models even with a milder assumption that some reaction rates do not change with temperature, which is consistent with experimental evidence. Together, our theoretical analyses predict that the waveform of circadian gene-activity and/or protein phosphorylation rhythms would be more nonsinusoidal at higher temperatures, even when there are differences in the network structures.

Two Ck1δ transcripts regulated by m6A methylation code for two antagonistic kinases in the control of the circadian clock.

The N6-methylation of internal adenosines (m6A) in mRNA has been quantified and localized throughout the transcriptome. However, the physiological significance of m6A in most highly methylated mRNAs is unknown. It was demonstrated previously that the circadian clock, based on transcription-translation negative feedback loops, is sensitive to the general inhibition of m6A. Here, we show that the Casein Kinase 1 Delta mRNA (Ck1δ), coding for a critical kinase in the control of circadian rhythms, cellular growth, and survival, is negatively regulated by m6A. Inhibition of Ck1δ mRNA methylation leads to increased translation of two alternatively spliced CK1δ isoforms, CK1δ1 and CK1δ2, uncharacterized until now. The expression ratio between these isoforms is tissue-specific, CK1δ1 and CK1δ2 have different kinase activities, and they cooperate in the phosphorylation of the circadian clock protein PER2. While CK1δ1 accelerates the circadian clock by promoting the decay of PER2 proteins, CK1δ2 slows it down by stabilizing PER2 via increased phosphorylation at a key residue on PER2 protein. These observations challenge the previously established model of PER2 phosphorylation and, given the multiple functions and targets of CK1δ, the existence of two isoforms calls for a re-evaluation of past research when CK1δ1 and CK1δ2 were simply CK1δ.

Discrete and ultradiscrete models for biological rhythms comprising a simple negative feedback loop.

Many biological rhythms are generated by negative feedback regulation. Griffith (1968) proved that a negative feedback model with two variables expressed by ordinary differential equations do not generate self-sustained oscillations. Kurosawa et al. (2002) expanded Griffith׳s result to the general type of negative feedback model with two variables. In this paper, we propose discrete and ultradiscrete feedback models with two variables that exhibit self-sustained oscillations. To obtain the model, we applied tropical discretization and ultradiscretization to a continuous model with two variables and then investigated its bifurcation structures and the conditions of parameters for oscillations. We found that when the degradation rate of the variables is lower than their synthesis rate, the proposed models generate oscillations by Neimark-Sacker bifurcation. We further demonstrate that the ultradiscrete model can be reduced to a Boolean system under some conditions.