Bifurcation Analysis of the Regulatory Modules of the Mammalian <i>M/G₁</i>Phase with Time Delay

G0/G1 “gaps” joint the S phase and M phase to form the cell cycle. The dynamics of enzyme reaction to drive the target protein production in M phase is analyzed mathematically. Time delay is introduced since the signal transmission need time in G0/G1 “gaps” phase. Hopf bifurcation of DDEs model is analyzed by applying geometrical analytical method. The instability oscillating periodic solutions arise as subcritical Hopf bifurcation occurs. The Hysteresis phenomena of the limit cycle are also observed underlying the saddle-node bifurcation of the limit cycle. Due to stability switching, interestingly, the bifurcating periodical solution dies out near the vicinity of Hopf lines. By Lyapunov-Schmidt reduction scheme, the normal form is computed on the center manifold. Finally, it is verified that the theory analytical results are in coincidence with the numerical simulation.

Integration of rhythmic metabolome and transcriptome provides insights into the transmission of rhythmic fluctuations and temporal diversity of metabolism in rice

Various aspects of the organisms adapt to cyclically changing environmental conditions via transcriptional regulation.However,the role of rhythmicity in altering the global aspects of metabolism is poorly characterized.Here,we subjected four rice(Oryza sativa)varieties to a range of metabolic profiles and RNA-seq to investigate the temporal relationships of rhythm between transcription and metabolism.More than 40%of the rhythmic genes and a quarter of metabolites conservatively oscillated across four rice accessions.Compared with the metabolome,the transcriptome was more strongly regulated by rhythm;however,the rhythm of metabolites had an obvious opposite trend between day and night.Through association analysis,the time delay of rhythmic transmission from the transcript to the metabolite level was~4 h under long-day conditions,although the transmission was nearly synchronous for carbohydrate and nucleotide metabolism.The rhythmic accumulation of metabolites maintained highly coordinated temporal relationships in the metabolic network,whereas the correlation of some rhythmic metabolites,such as branched-chain amino acids(BCAAs),was significantly different intervariety.We further demonstrated that the cumulative diversity of BCAAs was due to the differential expression of branched-chain aminotransferase 2 at dawn.Our research reveals the flexible pattern of rice metabolic rhythm existing with conservation and diversity.

Bifurcation analysis in a stage-structured predator-prey model with maturation delay

In this paper, we investigate the impact of maturation delay on the positive equilibrium solutions in a stage-structured predator prey system. By analyzing the characteristic equation we derive the conditions for the emergence of Hopf bifurcation. By applying the normal form and the center manifold argument, the direction as well as the sta- bility of periodic solutions bifurcating from Hopf bifurcation is explored. Results show that maturation delay can change the nature of the positive equilibrium solutions, and the loss of equilibrium stability occurs as a consequence of Hopf bifurcation. When Hopf bifurcation takes place, periodic solution arises and is further demonstrated to be asymptotically stable. In addition, the periodic solutions appear only for intermediate maturation delay, that is, there exists a delay window, outside of which the positive equilibrium is locally stable. Furthermore, numerical analysis shows that Hopf bifur- cation is favored by a superior competition for adult predators to juveniles, a smaller mortality on juvenile and/or adult predators, and a higher resource carrying capacity. Interestingly, increasing food carrying capacity can lead to the emergence of irregular chaotic dynamics and regular limit cycles.