PROBLEM SET 7
POPULATION GENETICS
The goal of this problem set is to add to your understanding of the regulation of
phenotypic variation in populations. By reviewing some ideas in quantitative population
genetics, you will explore the influence of various factors (dominance, allele frequency,
and genotype values) on the frequency distribution of quantitative traits. The
fundamental “theorem” of quantitative genetics is
Vp = Vg + Ve,
or phenotypic variance is the sum of genotypic and environmental variances.
The exercise will consist of two parts. First, you will review theoretical
expectations for distribution of phenotypes under various assumptions. Second, you will
generate samples from populations using a Monte-Carlo technique. First, however some
preliminaries.
Assume the following genetic model for a quantitative trait:
Y=G+E
Genotype
A1A1
A1A2
A2A2
Frequency
p2
2pq
q2
Mean Value
a
b
c
where Y is the phenotypic value, G is the genotype at an autosomal locus, and E is the
random environmental effect; A1 and A2 are the allelles at the locus with gene frequencies
p and q (p + q = 1). The quantities a, b, and c are mean values (e.g. size) of genotypes
A1A1, A1A2, A2A2, having frequencies p2, 2pq and q2 respectively. Random
environmental effects, E, are assumed to have a normal distribution with mean zero and
an environmental variance, Ve. The probability density function of a phenotype Y would
be:
f(Y)= p[A1A1}*f(Y|A1A1) + p[A1A2]*f(Y|A1A2) + p[A2A2]*f(Y|A2A2)
= p2*g(Y,a,Ve) + 2pq*g(Y,b,Ve) + q2*g(Y,c,Ve)
1
where g(Y,m,V) is a probability density function of a variable (Y) with mean m and
variance V. The formula for the probability density function of a normal distribution,
g(Y,a,Ve), is:
  Y  a 2 
1


gY , a ,Ve  
 * exp 

.
* Ve 
 2 * 31416
 2 * Ve 
The parameters of this distribution are p, a, b, c and Ve. The following
relationships may be useful for the evaluating the dependence of phenotypic distribution
on its parameters:
Overall genotypic mean: G = ap2 + 2bpq + cq2
Total genetic variance:
Vg = a2p2 + 2b2pq + c2q2 - (G)2
= 2pq[p(a - b) + q(b - c)]2 + p2q2[a - 2b + c]2
= Va + Vd
where Va is called the additive variance and Vd is called the variance due to
dominance.
Heritability: hg 
2
Vg
Vp
and,
Degree of dominance: x 
ha2 
Va
Vp
a  2b  c
ac
Part I: Set up a spread-sheet to calculate Vg, Va, Vd, Vp, ha, hg and x given a set of
parameters values for p, a, b, c, and Ve. The spreadsheet should also contain a graph of
f(Y) vs Y from at least 20 pairs of data. The range of Y should by plus or minus 2
standard deviations ( Ve ) beyond the maximum genotypic range. Evaluate the
probability density function by varying parameters values to obtain three types of
phenotypic distributions (trimodal, bimodal, and unimodal).
Part II: Create a Visual Basic module to generate three sets of 500 phenotypes using
these three sets of parameter values. Use our standard function to generate standard
normal random variable to add environmental variability and random numbers to assign
genotype of each of the 500 individuals. Remember our standard function is:
R = 2* RND - 1
2
ZR = log ((1+R)/(1-R))/1.82
From each of three samples of 500 individuals, generate a histogram of frequency vs Y.
The histograms should contain a minimum of 15 bins.
SUBMIT:
1. An EXCEL file with
a. Part I worksheet(s) and Part II module (3 points)
b. Plots of 3 phenotypic distributions in Part I (5 points)
c. Three histograms in Part II (5 points)
2. A document file with
a. A flow chart of your Visual Basic module in Part II (3 points)
b. A discussion comparing theoretical and simulated samples of frequencies of
phenotypes. You should include insights you have gained about the influence
of genotype values, dominance, and environmental variability on the
observability of underlying genotypic frequencies in populations. You must
elaborate on the values of Heritability and Dominance. Use your text book
from Population Biology to discuss them. (24 points).
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