Modeling periodic transcriptions in tandem gene systems

We study periodic gene transcription in a tandem gene system, where an external periodic signal modulates the expression of an upstream gene, whose products subsequently regulate the expression of a downstream gene. For the external signal characterized by cosine-wave modulation of the transcription rate, we derive analytical formulas for the mean transcription levels, delays, amplitudes, and noise intensities. The results reveal that the mean transcription levels of both genes exhibit periodic behavior, where the transcription of the downstream gene shows a phase delay with the amplitude proportional to that of the upstream gene. Numerical simulations confirm these findings, demonstrating that the noise increases with the signal strength and is consistently higher for the downstream gene. The delays are determined by the signal's angular frequency and mRNA degradation rates. This work provides insights into the propagation of periodic signals in gene networks, highlighting the coupling roles of stochastic fluctuations and gene regulations in transcriptional dynamics. The findings have important implications for understanding the regulation of circadian rhythms and for the rational design of synthetic gene circuits.