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MTH 428/528 Spring 2020 Assignment #3
发布时间：2022-02-09
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MTH 428/528
Spring 2020
Assignment #3
Consider a population of N diploid individuals (with two copies of their genetic material in each cell). We study two linked loci, where the first locus can be occupied by either one of two alleles A and a that have the same fitness, and the second locus can be occupied by either one of two alleles B and b of the same fitness. Thus, we have ten different genotypes:
|
a a b b |
Here the first row is the first locus and the second row is the second locus, while the columns
represent the two homologous chromosomes in the diploid population. Also, the chromosome ordering is irrelevant: e.g. and are indistinguishable.
We enumerate these genotypes by the integers from 0 to 9. The population in generation t can be thought as a ten-dimensional vector
X(t) = ╱X0 (t), X1 (t), X2 (t), X3 (t), X4 (t), X5 (t), X6 (t), X7 (t), X8 (t), X9 (t)、,
where Xk (t) is the respective population of genotype k. We assume the population is of fixed size:
X0 (t) + X1 (t) + X2 (t) + X3 (t) + X4 (t) + X5 (t) + X6 (t) + X7 (t) + X8 (t) + X9 (t) = N.
In the process of formation of the next generation t + 1, the individuals in generation t
produce (segregate) gametes. Two gametes will then join to form an individual of the next generation. For example, or will merge into .
|
A a b B |
|
a B |
|
A a b B |
in or , coming from different columns. Let r be the probability of a recombination
|
A a b B |
, , ,
with respective probabilities
(1 - r),
(1 - r),
r,
and
r.
We extend the Wright-Fisher model to the case of two loci. Here, each individual in gener- ation t + 1 is produced by randomly selecting two individuals in generation t. Each of them segregates a gamete, and the gametes are joined to form a new individual. This is done until we have N individuals in generation t + 1. And so on, from one generation to the next generation.
Problem 1. Suppose X(t) = i, where
i = (i0 , i1 , i2 , i3 , i4 , i5 , i6 , i7 , i8 , i9 ), where i0 + i1 + i2 + i3 + i4 + i5 + i6 + i7 + i8 + i9 = N.
Find the (conditional) probabilities pk (i) (k = 0, . . . , 9) for the new individual in population t + 1 to be of genetic type k .
Problem 2. Find the general form for the transition probabilities in this modified Wright- Fisher model
p(i, j) = P╱X(t + 1) = j |X(t) = i、
for
i = (i0 , i1 , i2 , i3 , i4 , i5 , i6 , i7 , i8 , i9 ), where i0 + i1 + i2 + i3 + i4 + i5 + i6 + i7 + i8 + i9 = N,
and
j = (j0 , j1 , j2 , j3 , j4 , j5 , j6 , j7 , j8 , j9 ), where j0 + j1 + j2 + j3 + j4 + j5 + j6 + j7 + j8 + j9 = N.
Hint: use pk (i) in Problem 1 and multinomial coefficients (and multinomial distribution). This should be similar to the single locus Wright-Fisher model, where binomial coefficients and binomial distribution were used.
