A mathematical model of the mutation process in bacteria Escherichia coli induced by ultraviolet radiation is developed. Our model is based on the experimental data characterizing the main processes of the bacterial SOS response. Here we have modeled a whole sequence of the events leading to the fixation of the primary DNA lesion as a point mutation. A quantitative analysis of the key ways of the SOS mutagenesis was performed in terms of modern system biology. The dynamic changes of the basic SOS protein concentrations and the process of the translesion synthesis by the modified replication complex are described quantitatively. We have also demonstrated the applicability of the developed model to the description of the mutagenesis in individual genes. As an example, an estimation of the mutation frequency in E. coli's lacI gene is performed.