Bacteria in nature often suffer sudden stresses, such as ultraviolet (UV) irradiation, nutrient deprivation, and chemotoxins that would cause DNA damage and DNA replication failure, which in turn trigger SOS response. According to the strength and duration of the stress, the SOS system not only repairs DNA damage but also induces mutagenesis, so as to adapt to the changing environment. The key proteins in charge of mutagenesis are UmuD and UmuD'. In this paper, we quantitatively measure the growth rate and cellular levels of proteins UmuD and UmuD' in Escherichia coli after various fluences of UV irradiation. To compare with the experimental observations, an ordinary differential equation model is built to describe the SOS response. Considering the fact that the DNA lesions affect cellular protein production and replication origination, the simulation results fit well with the experimental data. Our results show how the fluence of UV irradiation determines the dynamics of the inducing signal and the mutation frequency of the cell.