A Mathematical Model for the Effect of Low-Dose Radiation on the G2/M Transition

Abstract

We develop a mathematical model to study the immediate effect of low-dose radiation on the G2 checkpoint and the G2/M transition of the cell cycle via a radiation pathway (the ATM–Chk2 pathway) of an individual mammalian cell. The model consists of a system of nonlinear differential equations describing the dynamics of a network of regulatory proteins that play key roles in the G2/M transition, cell cycle oscillations, and the radiation pathway. We simulate the application of a single pulse of low-dose radiation at different intensities ($\sim$ 0–0.4 Gy) and times during the latter part of the G2-phase. We use bifurcation analysis to characterize the effect of radiation on the G2/M transition via the ATM–Chk2 pathway. We show that radiation between 0.1 and 0.3 Gy can delay the G2/M transition, and radiation higher than 0.3 Gy can fully activate the G2 checkpoint. Also, our results show that radiation can be low enough to neither delay the G2/M transition nor activate the G2 checkpoint ($\sim$ 0.1 Gy). Our model supports the idea that the cell response to radiation during G2-phase explains hyper-radiosensitivity and increased radioresistance (HRS/IRR) observed at low dose.

Appendix: Parameter Values

Table 2 Parameter values

The Goldbeter–Koshland kinetics model for protein activation is given by

$$\begin{aligned} \frac{\mathrm{d} x}{\mathrm{d}t} = v_{1}\frac{1-x}{J_{1}+1-x}-v_{2}\frac{x}{J_{2}+x}, \end{aligned}$$

where x is the normalized concentration of the active form of the protein, $1-x$ is the concentration of the inactive form of the protein, $v_{1}$ is the maximum activation rate, $v_{2}$ is the maximum inactivation rate, and $J_{1}$ and $J_{2}$ are the Michaelis constants. The smaller the Michaelis constants, the faster the activation/inactivation switch. It can be shown that the activation of x is obtained when

$$\begin{aligned} \frac{v_{1}}{v_{2}}>1. \end{aligned}$$

Ultrasensitivity refers to a fast switch in activity when ${v_{1}}$ is close to ${v_{2}}$ and the Michaelis constants are small. Parameter values in the radiation pathway module, such as activation/inactivation rate and Michaelis constants, are carefully chosen so that our simulations match experimental results (see Table 2).