When you take a single dose of an antibiotic, or other kind of medication, the concentration, C (in µg/mL), of drug in your blood stream is often well described by the equation C(t) = C0te−kt

, for some constants C0 and k. Here, t usually has the units of hours.

Furthermore, the rate, r, at which that drug reacts can often be well modelled by a Michaelis–Menten equation r(C) = Vmax C

Km + C

, for some constants Vmax > 0 and Km > 0.

a. Pick values of C0 and k and plot C(t). From trying a few different values, infer the physical interpretation of the constant C0 and k. Why is this curve a reasonable model of the concentration of drug in your blood?

cises 311

b. Similarly, what is the physical interpretation of the constants Vmax and Km?

c. Calculate how fast the reaction rate is changing as a function of time.

d. Set Vmax = 2 µg mL−1 hour−1 , Km = 1.5 µg mL−1 ,C0 = 1 µg mL−1 , k = 1 hour−1 , and plotr and dr/dt as functions of time.

e. Why would we care about how fast the reaction rate is changing?