Curtis (1986) proposed a model for the formation and repair of cellular DNA damage (called lesions) due to radiation. In Radiation damage to cells occurs, for example, in the treatment of cancer, or when a nuclear power station explodes and pours radiation into the atmosphere. Lethal lesions kill the cell, and so are bad (unless it’s a cancer cell, in which case they’re good).

their model there are two kinds of lesions; lethal and potentially lethal. Let P denote the number of potentially lethal lesions, and let L denote the number of lethal lesions. Potentially lethal lesions can repair themselves, at a rate k1, but they can also combine with another potentially lethal lesion to form a lethal lesion, in a process called binary misrepair. Binary misrepair occurs at the rate k2P 2

. If R is the rate at which radiationWhy the P 2? Well, it’s because 2 P have to combine to form an L, and so the law of mass action says that the rate will be proportional to P 2

.

induced lesions form, and if we assume that all lesions are only potentially lethal, not lethal, this leads to the pair of differential equations R will be closely related to the size of the radiation dose, although we don’t specify this relationship exactly, as that’s not an easy thing to work out.

dP dt

= R − k1P − k2P 2

, dL dt

= k2P 2

.

a. Show that, for any scientifically valid value of R, there is a unique scientifically valid steady state value of P, and that this steady state is stable.

b. Is there a steady state for L? Explain.

Now suppose that a huge, short, dose of radiation was applied just before t = 0. This dose suffices to increase P very quickly