Answer Happy

4.2 Binomial distribution The genome of the HIV-1 virus, like any genome, is a string of "letters" (base pairs) in an “alphabet” containing only four letters. The message for HIV is rather short, just n = 104 letters in all. Since any of the letters can mutate to any of the three other choices, there's a total of 30 000 possible distinct one-letter mutations. In 1995, A. Perelson and D. Ho estimated that every day about 1010 new virus particles are formed in an asymptomatic HIV patient. They further estimated that about 1% of these virus particles proceed to infect new white blood cells. It was already known that the error rate in duplicating the HIV genome was about one error for every 3 · 104 "letters” copied. Thus the number of newly infected white cells receiving a copy of the viral genome with one mutation is roughly 1010 x 0.01 (104/(3 - 104)) 3. 107 per day. This number is much larger than the total 30 000 possible 1-letter mutations, so every possible mutation will be generated several times per day. a. How many distinct two-base mutations are there? b. You can work out the probability P, that a given viral particle has two bases copied inaccurately from the previous generation using the sum and product rules of probability. Let P=1/(3.104) be the probability that any given base is copied incorrectly. Then the probability of exactly two errors is P2, times the probability that the remaining 9998 letters don't get copied inaccurately, times the number of distinct ways to choose which two letters get copied inaccurately. Find P2. c. Find the expected number of two-letter mutant viruses infecting new white cells per day and compare to your answer to (a). d. Repeat (a–c) for three independent mutations. e. Suppose an antiviral drug attacks some part of HIV, but that the virus can evade the drug's effects by making one particular, single-base mutation. According to the information above, the virus will very quickly stumble upon the right mutation—the drug isn't effective for very long. Why do you suppose current HIV therapy involves a combination of three different antiviral drugs administered simultaneously?