Cross- pollinated crops are highly heterozygous due to the free intermating among their plants. They are often referred to as random mating populations because each individual of the population has equal opportunity of mating with any other individual of that population. Such a population is also known as Mendelian population or panmictic population. A population, in this case, consists of all such individuals that share the same gene pool, i.e., have an opportunity to intermate with each other and contribute to the next generation of the population. To understand the genetic make - up of such populations a sophisticated field of study, population genetics, has been developed. The Hardy Weinberg law states that in a large random mating population gene and genotype frequency remain constant generation after generation unless there is selection, mutation, migration or random drift. This is the fundamental law of population genetics and provides the basis for studying Mendelian populations. The law is proposed independently by G. H. Hardy (a mathematician) and W. Weinberg (a physician).

1. Fundamentals of Plant Breeding (PB-212) 1 | P a g e
Dr. Asit Prasad Dash (Associate Professor), Department of PB & G, IAS, SOA (DU)
(dashpraasit@gmail.com)
POPULATION GENETICS AND HARDY WEINBERG LAW
Cross- pollinated crops are highly heterozygous due to the free intermating among their
plants. They are often referred to as random mating populations because each individual of the
population has equal opportunity of mating with any other individual of that population. Such a
population is also known as Mendelian population or panmictic population. A population, in this case,
consists of all such individuals that share the same gene pool, i.e., have an opportunity to intermate
with each other and contribute to the next generation of the population. To understand the genetic
make - up of such populations a sophisticated field of study, population genetics, has been
developed.
Gene Frequency (allelic frequency): The proportion of different alleles of a gene present in a
Mendelian population is known as gene frequency.
Genotype Frequency/Zygotic frequency: The proportion of different genotypes for a gene present in
a Mendelian population is known as genotype frequency/zygotic frequency.
Hardy Weinberg law (Hardy Weinberg Equilibrium): The Hardy Weinberg law states that in a
large random mating population gene and genotype frequency remain constant generation after
generation unless there is selection, mutation, migration or random drift.
This is the fundamental law of population genetics and provides the basis for studying
Mendelian populations. The law is proposed independently by G. H. Hardy (a mathematician) and W.
Weinberg (a physician). The frequencies of the three genotypes for a locus with two alleles, say A and
a, would be p2
AA, 2pq Aa, and q2
aa ; where p and q represents the frequency of alleles A and a in the
population, and the sum of p and q is one, i.e., p+q=1. Hardy-Weinberg law describes a theoretic
situation in which a population is undergoing no evolutionary changes.
The validity of can be demonstrated by assuming either random union of gametes or random
mating among genotypes.
Random union of gametes
Let us consider two alleles of a gene, A and a, their frequencies are p and q respectively and
those two gametes unite randomly to yield zygotes: AA, Aa, aa. Since the union of male and female
gametes is random, the frequency of union of a male gamete with a female gamete will be the
product of their individual frequencies in the population.

2. Fundamentals of Plant Breeding (PB-212) 2 | P a g e
Dr. Asit Prasad Dash (Associate Professor), Department of PB & G, IAS, SOA (DU)
(dashpraasit@gmail.com)
Table: Gene and genotype frequencies for a gene obtained from a random union of gametes in
a Mendelian population.
Suppose, Genotype of gamete A a
Frequency of gamete p q
Random union of gametes will yield
Female gamete Male gametes
A
(p)
a
(q)
A
(p)
AA
(p!
)
Aa
(pq)
a
(q)
Aa
(pq)
aa
(q!
)
Thus, Genotype of zygote AA Aa aa
Frequency of zygote P2
2pq q2
Thus, frequency of zygotes/ genotypes produced through random mating among the gametes will
be: AA=p!
; Aa=2pq; aa= q!
. The 3 genotypes will again produce A and a types gamete in the next
generation. This can be estimated as follow:
• AA individuals will produce only ‘A’ type gametes.
• Aa individuals will produce ½ ‘A’ and ½ ‘a’ type gametes.
• aa individuals will produce only ‘a’ type gametes.
Thus the frequency of A gametes produced by the population will be
= all the gametes from AA individuals + ½ gametes from Aa individuals
= p!
+1/2 (2pq)
= p!
+pq
= p (p+q)
= p
Similarly, the frequency of a gametes produced by the population will be
= all the gametes from aa individuals + ½ gametes from Aa individuals
= q!
+1/2 (2pq)
= q!
+pq
= q (p+q)
= q

3. Fundamentals of Plant Breeding (PB-212) 3 | P a g e
Dr. Asit Prasad Dash (Associate Professor), Department of PB & G, IAS, SOA (DU)
(dashpraasit@gmail.com)
Random Mating among Genotypes
Let us suppose that a Mendelian population has the genotypes AA, Aa and aa in the frequency p2
,
2pq and q2
, respectively. Since the mating among genotypes is assumed to be random, the frequency of
mating between males and females of any given genotypes will be the product of the frequencies of the
concerned genotypes in the population. Therefore, the frequencies of mating in which both males and
females have the same genotype, e.g. AA×AA, Aa×Aa and aa×aa will be equal to the square of the
frequencies of the concerned genotypes (e.g. p2
×p2
=p4
, 2pq×2pq=4p2
q2
and q2
×q2
=q4
respectively). But
the frequencies of those mating in which males and females have different genotypes, e.g. AA×Aa,
AA×aa, and Aa×aa, will be twice the product of the frequencies of two genotypes involved in the
mating (e.g. 2×p2
×2pq=4p3q, 2× p2
×q2
=2p2q2 and 2×2pq ×q2= 4pq3, respectively). This is because
such mating can occur in two different ways: (I) females of one genotype and males of other and (2)
vice-versa. For example, the two ways for mating AA and Aa are; (I) AA×Aa and (2) Aa×AA.
Therefore, the frequency of this mating will be the sum of the frequencies of these two mating
combinations.
Table: Result of random mating among the three genotypes (AA, Aa and aa)
Mating Probability
Frequency of progeny
AA Aa aa
AA×AA
(p2
×p2
)
p4
p4
AA×Aa
2(p2
×2pq)
4p3
q 2p3
q 2p3
q
AA×aa
2(p2
×q2
)
2p2
q2
2p2
q2
Aa×Aa
(2pq×2pq)
4p2
q2
p2
q2
2p2
q2
p2
q2
Aa×aa
2(2pq×q2
)
4pq3
2pq3
2pq3
aa×aa
(q2
×q2
)
q4
q4
The frequencies of different genotypes in the progeny obtained by random mating among the
genotypes can now be determined.
The frequency of AA progeny would be,
= P4
+2p3
q+p2
q2
= p2
(p2
+ 2pq + q2
) (p2
is taken common)
= p2
(since, P2
+ 2pq + q2
= 1)
Similarly, the frequency of aa progeny will be,
= p2
q2
+ 2pq3
+ q4

4. Fundamentals of Plant Breeding (PB-212) 4 | P a g e
Dr. Asit Prasad Dash (Associate Professor), Department of PB & G, IAS, SOA (DU)
(dashpraasit@gmail.com)
= q2
(p2
+ 2pq + q2
) (q2
is taken common)
= q2
(since, p2
+ 2pq+q2
= 1)
and the frequency of Aa progeny will be,
= 2p3
q + 2p2
q2
+ 2p2
q2
+ 2pq3
= 2p3
q + 4p2
q2
+ 2pq3
= 2pq (p2
+ 2pq + q2
) (2pq is taken common)
= 2pq (since, p2
+ 2pq+q2
= 1)
Obviously, random mating among the genotypes AA, Aa and aa having the frequencies of p2
, 2pq and
q2
respectively, yields the same three genotypes in the same frequencies. (p2
AA, 2pq Aa and q2
aa).
Factors affecting equilibrium frequencies
The equilibrium in random mating populations is disturbed by (1) migrations, (2) mutation,
(3) selection and (4) random drift. These factors are also referred to as evolutionary forces since they
bring about changes in gene frequencies, which is essential for evolution to proceed. Obviously, a
population in which gene and genotype frequencies remain constant over generations cannot
evolve any further, unless its gene and genotype frequencies are disturbed.
Migration
Migration is the movement of individuals into a population from a different population.
Migration may introduce new alleles into the population or may change the frequencies of existing
alleles.
Mutation
Mutation is the ultimate source of all the variation present in biological materials. Mutation
may produce a new allele not present in the population or may change the frequencies of existing
alleles.
Selection
Differential reproduction rate of various genotypes is known as selection. In crop
improvement, selection is very important because it allows the selected genotypes to reproduce, while
the undesirable genotypes are eliminated. In a random mating population, if plants with AA or aa
genotypes are selected, the frequency of A allele in the selected population would be 1 or 0,
respectively.
Random drift
Random drift or genetic drift is a random change in gene frequency due to sampling error.
Random drift occurs in small populations because sampling error is greater in a smaller population
than in a larger one. Ultimately, the frequency of one of the alleles becomes zero and that of the
other allele becomes one.