A Mathematical Model for the Detection Mechanism of DNA Double-Strand Breaks Depending on Autophosphorylation of ATM

Background: After IR stress, DNA double-strand breaks (DSBs) occur and repair proteins (RPs) bind to them, generating DSB-RP complexes (DSBCs), which results in repaired DSBs (RDSBs). In recent experimental studies, it is suggested that the ATM proteins detect these DNA lesions depending on the autophosphorylation of ATM which exists as a dimer before phosphorylation. Interestingly, the ATM proteins can work as a sensor for a small number of DSBs (approximately 18 DSBs in a cell after exposure to IR). Thus the ATM proteins amplify the small input signals based on the phosphorylation of the ATM dimer proteins. The true DSB-detection mechanism depending on ATM autophosphorylation has yet to be clarified. Methodology/Principal Findings: We propose a mathematical model for the detection mechanism of DSBs by ATM. Our model includes both a DSB-repair mechanism and an ATM-phosphorylation mechanism. We model the former mechanism as a stochastic process, and obtain theoretical mean values of DSBs and DSBCs. In the latter mechanism, it is known that ATM autophosphorylates itself, and we find that the autophosphorylation induces bifurcation of the phosphorylated ATM (ATM*). The bifurcation diagram depends on the total concentration of ATM, which makes three types of steady state diagrams of ATM*: monostable, reversible bistable, and irreversible bistable. Bistability exists depending on the Hill coefficient in the equation of ATM autophosphorylation, and it emerges as the total concentration of ATM increases. Combining these two mechanisms, we find that ATM* exhibits switch-like behaviour in the presence of bistability, and the detection time after DNA damage decreases when the total concentration of ATM increases. Conclusions/Significance: This work provides a mathematical model that explains the DSB-detection mechanism depending on ATM autophosphorylation. These results indicate that positive auto-regulation works both as a sensor and amplifier of small input signals.