You are a very talented graduate student in mathematics. You work as a research assistant to Louis, a microbiologist. Louis has determined an “exponential growth model” for a new type of infectious bacterium. If P(t) is the population of the bacterium at discrete time periods (measured in seconds) t = 0, 1, 2,…, then P(t+1) = P(t) + P(0) Exp[-P(t)/P(0)]. So P(1) = 137% P(0), P(2) = 162% P(0). Although the initial growth is explosive, the growth rate decays downward rapidly: After one minute P(60) = 416%P(0); after an hour P(3600) = 819%P(0), which is less than double P(60); after one day P(86400) = 1137% P(0) which is less than 40 percent growth from P(3600). Louis observes that soon the growth rate will be trivial and the population will “level off at an equilibrium level.” He asks you to determine several things: (1) the “cap” or “equilibrium level” for the maximum population (as a multiple of P(0)); (2) at what time t does the population reach 50 percent of this cap; and (3) hypothetically, what the population (as a multiple of P(0)) would be after a trillion (1012) years of growth. What answers do you give Louis?
DNA Sequencing
In this puzzle, Craig is trying to sequence a DNA sample from an ancient dinosaur. Craig consults his brother Gary to determine if the sequencing can be completely finished within a year. Craig’s DNA sequencer can read a single linear segment that is 5,000,000 base pairs long for each hour it is running. The sequencer can only match the linear segments together if they overlap by at least 2,000,000 base pairs. Segments are randomly sampled (with replacement) and the DNA strand is circular. Gary says there is a 99 percent chance the sequencer will finish within a year. How many base pairs long is the strand of dinosaur DNA?
The overlap requirement can be ignored by reducing the length of each segment to its central 1,000,000 base pairs, since the first and last 2,000,000 base pairs must always be used for matching together with other segments. The question then becomes effectively what is the circumference of a circle if there is a 99 percent probability that 8,760 (corresponding to the number of hours in a year) random arcs of length 1,000,000 will cover the entire circle.
The key mathematics to efficiently answer this question were developed in the 1939 article by W. L. Stevens, “Solution to a Geometrical Problem in Probability” (Annals of Eugenics 9: 315–320). We will only describe key formulas from this paper without detailed derivations.
Let X be the ratio of the arc length to the circumference and n be the number of random arcs. If k is a positive integer less than 1/X, then the probability, f(k), that there are uncovered “gap” arcs (not necessarily the same length as the randomly selected arcs) on the circle (not covered by any of the arcs and each immediately to the counterclockwise endpoint of one of k-specified random arcs) is f(k) = (1- kx)n-1. These k arcs must be disjointed from each other, which gives some intuitive insight into the (1- kx) in the formula for f(k).
Now let f(k,t) be the probability that there are gaps to the counterclockwise of k-specified random arcs, that there are no gaps to the counterclockwise of t other than specified random arcs, and that the n–k–t other random arcs may or may not have counterclockwise gaps. Note that f(k) = f(k,0). Therefore f(k,t+1) = f(k,t) –f(k+1,t). Consequently, by repeated iteration f(h,t) = f(h,0) – t f(h+1,0) + (t(t-1)/2) f(h+2,0) -… (-1)t f(h+t,0).
The probability that there will be no gaps between the n random arcs is then f(0,n) = 1 – n (1-x)n-1 + (n(n-1)/2)(1-2x)n-1-…(-1)k (n!/(k! (n–k)!)) (1-kx)n-1, where k is the largest integer not greater than 1/X.
For the stated problem n = 8760 and f(0, 8760) = 99 percent. Once we solve for X, the length of circular strand is 1,000,000/X. Let g(X) = f(0, 8760) for a given X. We can check some by orders of magnitude for X (using a good numerical calculation program, of course): g(0.1) = 100%, g(0.01) = 100%, g(0.001) = 25.2%. Now, start searching by “bisecting” the values of X from 0.001 to 0.01: g(0.005) = 100%, g(0.0025) = 100%, g(0.00125) = 86%,…, g(0.0015604) = 99%. So, X = 0.0015604 and the length of the dinosaur DNA strand is about 1,000,000/0.0015604 = 640,861,318 base pairs.
Bob Conger submitted a solution.
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AR Puzzle Editor Jon Evans is president of Convergent Actuarial Services, Inc. in Delray Beach, Florida.
