Monotonic Dynamics of Mrna Degradation by Two Pathways∗

mRNA degradation plays an important role in gene regulation. However, a defect in mRNA decay is expected to result in an increase in mRNA levels. In this paper, we will first establish a model of mRNA regulation by two pathways denoted by 5′ → 3′ and 3′ → 5′ for short, where there are two degradation rates δ1, δ2 on 5 ′ → 3′ pathway and the degradation rate on 3′ → 5′ pathway is δ3. The advantage of this model is that it captures fundamental biochemical reactions in the gene expression process in eukaryotic cells. Then we obtain several basic principles on the monotonicity of the mean level of newly accumulated mRNAs. It is proved that (1) the newly mean level is strictly increasing in p and κ, but is strictly decreasing in γ, where p, κ and γ are the initial activation frequency, the activation rate, and the inactivation rate, respectively; (2) the newly mean level is strictly decreasing in both δ2 and δ3, remarkably, is strictly increasing in δ1 when δ2 < δ3 and decreasing when δ2 > δ3 and; (3) the newly mean level is strictly increasing in time t when p < κ/(κ+ γ). These conclusions not only provide a better understanding on gene expression dynamics but also would be helpful to design reasonable gene expression modules.