Theoretical study on the regulation of circadian rhythms by RNA methylation

Unstable eigenvectors and reduced amplitude spaces specifying limit cycles of coupled oscillators with simultaneously diagonalizable matrices: with applications from electric circuits to gene regulation

ABSTRACT

A fascinating phenomenon is the self-organization of coupled systems to a whole. This phenomenon is studied for a particular class of coupled oscillatory systems exhibiting so-called simultaneously diagonalizable matrices. For three exemplary systems, namely, an electric circuit, a coupled system of oscillatory neurons, and a system of coupled oscillatory gene regulatory pathways, eigenvectors and amplitude equations are derived. It is shown that for all three systems, only the unstable eigenvectors and their amplitudes matter for the dynamics of the systems on their respective limit cycle attractors. A general class of coupled second-order dynamical oscillators is presented in which stable limit cycles emerging via Hopf bifurcations are solely specified by appropriately defined unstable eigenvectors and their amplitudes. While the eigenvectors determine the orientation of limit cycles in state spaces, the amplitudes determine the evolution of states along those limit cycles. In doing so, it is shown that the unstable eigenvectors define reduced amplitude spaces in which the relevant long-term dynamics of the systems under consideration takes place. Several generalizations are discussed. First, if stable and unstable system parts exhibit a slow-fast dynamics, the fast variables may be eliminated and approximative descriptions of the emerging limit cycle dynamics in reduced amplitude spaces may be again obtained. Second, the principle of reduced amplitude spaces holds not only for coupled second-order oscillators, but can be applied to coupled third-order and higher order oscillators. Third, the possibility to apply the approach to multifrequency limit cycle attractors and other types of attractors is discussed.

Post-transcriptional modulators and mediators of the circadian clock

The endogenous circadian timekeeping system drives ~24-h rhythms in gene expression and rhythmically coordinates the physiology, metabolism and behavior in a wide range of organisms. Regulation at various levels is important for the accurate functioning of this circadian timing system. The core circadian oscillator consists of an interlocked transcriptional-translational negative feedback loop (TTFL) that imposes a substantial delay between the accumulation of clock gene mRNA and its protein to generate 24-h oscillations. This TTFL mediated daily oscillation of clock proteins is further fine-tuned by post-translational modifications that regulate the clock protein stability, interaction with other proteins and subcellular localization. Emerging evidence from various studies indicates that besides TTFL and post-translational modifications, post-transcriptional regulation plays a key role in shaping the rhythmicity of mRNAs and to delay the accumulation of clock proteins in relation to their mRNAs. In this review, we summarize the current knowledge on the importance of post-transcriptional regulatory mechanisms such as splicing, polyadenylation, the role of RNA-binding proteins, RNA methylation and microRNAs in the context of shaping the circadian rhythmicity in Drosophila and mammals. In particular, we discuss microRNAs, an important player in post-transcriptional regulation of core-clock machinery, circadian neural circuit, clock input, and output pathways. Furthermore, we provide an overview of the microRNAs that exhibit diurnal rhythm in expression and their role in mediating rhythmic physiological processes.