Mathematical Modeling of Circadian Gene Expression in Mammalian Cells

Circadian rhythms in mammals are self-sustained repeating activities driven by the circadian gene expression in cells, which is regulated at both transcriptional and posttranscriptional stages. In this work, we first used mathematical modeling to investigate the transcriptional regulation of circadian gene expression, with a focus on the mechanisms of robust genetic oscillations in the mammalian circadian core clock. Secondly, we built a coarse-grained model to study the post-transcriptional regulation of the rhythmicities of poly(A) tail length observed in hundreds of mRNAs in mouse liver. Lastly, we examined the application of Sobol indices, which is a global sensitivity analysis method, to mathematical models of biological oscillation systems, and proposed two methods tailored for the calculation of circular Sobol indices. In the first project, we modified the core negative feedback loop in a mathematical model of the mammalian genetic oscillator so that the unrealistic tight binding between the repressor PER and the activator BMAL1 is relaxed for robust oscillations. By studying the modified extended models, we found that the auxiliary positive feedback loop, rather than the auxiliary negative feedback loop, makes the oscillations more robust, yet they are similar when accounting for circadian rhythms (~24h period). In the second project, we investigated the regulation of rhythmicities in poly(A) tail length by four coupled rhythmic processes, which are transcription, deadenylation, polyadenylation, and degradation. We found that rhythmic deadenylation is the strongest contributor to the rhythmicity in poly(A) tail length and the rhythmicity in the abundance of the mRNA subpopulation with long poly(A) tails. In line with this finding, the model further showed that the experimentally observed distinct peak phases in the expression of deadenylases, regardless of other rhythmic controls, can robustly cluster the rhythmic mRNAs by their peak phases in poly(A) tail length and abundance of the long-tailed subpopulation. In the last project, we reviewed the theoretical basis of Sobol indices and identified potential problems when it is applied to mathematical models of biological oscillation systems. Based on circular statistics, we proposed two methods for the calculation of circular Sobol indices and compared their performance with the original Sobol indices in several models. We found that though the relative rankings of the contribution from parameters are the same across three methods, circular Sobol indices can better quantitatively distinguish the contribution of individual parameters. Through this work, we showed that mathematical modeling combined with sensitivity analysis can help us understand the mechanisms underlying the circadian gene expression in mammalian cells. Also, testable predictions are made for future experiments and new ideas are provided that can enable potential chronopharmacology research.