  We develop a kinetic reaction model for cells having irradiated DNA molecules
due to ionizing radiation exposure. Our theory simultaneously accounts for the
time-dependent reactions of the DNA damage, the DNA mutation, the DNA repair,
and the proliferation and apoptosis of cells in a tissue with a minimal set of
model parameters. In contrast to existing theories for radiation exposition, we
do not assume the relationships between the total dose and the induced mutation
frequency. Our theory provides a universal scaling function that reasonably
explains the mega-mouse experiments in Ref.[W. L. Russell and E. M. Kelly,
Proc. Natl. Acad. Sci. USA. {\bf 79} (1982) 542.] with different dose rates.
Furthermore, we have estimated the effective dose rate, which is biologically
equivalent to the ionizing effects other than those caused by artificial
irradiation. This value is $ 1.11 \times 10^{-3} ~\rm{[Gy/hr]}$, which is
significantly larger than the effect caused by natural background radiation.
