The Wright-Fisher Model of Genetic Drift solved by 470

Hardy-Weinberg Revisited

The principle of genetic equilibrium is an idealistic model for population genetics that simply cannot hold for all genes in practice. For one, evolution has proven too powerful for equilibrium to possibly hold. At the same time, evolution works on the scale of eons, and at any given moment in time, most populations are essentially stable.

Yet we could overlook the inevitable effects of simple random chance in disrupting the allelic frequency for a given gene, a phenomenon called genetic drift.

In this problem, we would like to obtain a simple mathematical model of genetic drift, and so we will need to make a number of simplifying assumptions. First, assume that individuals from different generations do not mate with each other, so that generations exist as discrete, non-overlapping quantities. Second, rather than selecting pairs of mating organisms, we simply randomly select the alleles for the individuals of the next generation based on the allelic frequency in the present generation. Third, the population size is stable, so that we do not need to take into account the population growing or shrinking between generations. Taken together, these three assumptions make up the Wright-Fisher model of genetic drift.

Problem

Consider flipping a weighted coin that gives "heads" with some fixed probability $p$ (i.e., $p$ is not necessarily equal to 1/2).

We generalize the notion of binomial random variable from “Independent Segregation of Chromosomes” to quantify the sum of the weighted coin flips. Such a random variable $X$ takes a value of $k$ if a sequence of $n$ independent "weighted coin flips" yields $k$ "heads" and $n-k$ "tails." We write that $X \in \mathrm{Bin}(n, p)$.

To quantify the Wright-Fisher Model of genetic drift, consider a population of $N$ diploid individuals, whose $2N$ chromosomes possess $m$ copies of the dominant allele. As in “Counting Disease Carriers”, set $p = \frac{m}{2N}$. Next, recall that the next generation must contain exactly $N$ individuals. These individuals' $2N$ alleles are selected independently: a dominant allele is chosen with probability $p$, and a recessive allele is chosen with probability $1-p$.

Given: Positive integers $N$ ($N \leq 7$), $m$ ($m \leq 2N$), $g$ ($g \leq 6$) and $k$ ($k \leq 2N$).

Return: The probability that in a population of $N$ diploid individuals initially possessing $m$ copies of a dominant allele, we will observe after $g$ generations at least $k$ copies of a recessive allele. Assume the Wright-Fisher model.

Sample Dataset

4 6 2 1

Sample Output

0.772