Hardy–Weinberg principle

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The Hardy–Weinberg principle (also known as the Hardy–Weinberg equilibrium, model, theorem, or law) states that allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. These influences include mate choice, mutation, selection, genetic drift, gene flow and meiotic drive. Because one or more of these influences are typically present in real populations, the Hardy–Weinberg principle describes an ideal condition against which the effects of these influences can be analyzed.

In the simplest case of a single locus with two alleles denoted A and a with frequencies f(A) = p and f(a) = q, respectively, the expected genotype frequencies are f(AA) = p2 for the AA homozygotes, f(aa) = q2 for the aa homozygotes, and f(Aa) = 2pq for the heterozygotes. The genotype proportions p2, 2pq, and q2 are called the Hardy-Weinberg proportions. [Note that the sum of all genotype frequencies of this case is the binomial expansion of the square of the sum of p and q, and such a sum, as it represents the total of all possibilities, must be equal to 1. Therefore (p + q)2 = p2 + 2pq + q2 = 1. The solution of this equation is q = 1 - p.

If union of gametes to produce the next generation is random, it can be shown that the new frequency f satisfies \textstyle f'(\text{A}) = f(\text{A}) and \textstyle f'(\text{a}) = f(\text{a}). That is, allele frequencies are constant between generations.

This principle was named after G. H. Hardy and Wilhelm Weinberg, who first demonstrated it mathematically.



f_t(\text{A}) = f_t(\text{AA}) + \frac{1}{2} f_t(\text{Aa})






f_t(\text{a}) = f_t(\text{aa}) + \frac{1}{2} f_t(\text{Aa})







Table 1: Punnett square for Hardy–Weinberg
A (p) a (q)
a (q) Aa (qp) aa (q2)




f_1(\text{AA}) = p^2 = f_0(\text{A})^2






f_1(\text{Aa}) = pq + qp = 2 pq = 2 f_0(\text{A}) f_0(\text{a})






f_1(\text{aa}) = q^2 = f_0(\text{a})^2







f_1(\text{A}) = f_1(\text{AA}) + \frac{1}{2} f_1(\text{Aa}) 
              = p^2 + p q = p \left(p + q\right) 
              = p 
              = f_0(\text{A})

f_1(\text{a}) = f_1(\text{aa}) + \frac{1}{2} f_1(\text{Aa}) 
             = q^2 + p q = q \left(p + q\right)
             = q
             = f_0(\text{a})


\left[ (\text{AA},\text{AA}), (\text{AA}, \text{Aa}), (\text{AA}, \text{aa}),
              (\text{Aa},\text{Aa}), (\text{Aa}, \text{aa}), (\text{aa}, \text{aa}) \right]


&\left[ f_{t+1}(\text{AA}), f_{t+1}(\text{Aa}), f_{t+1}(\text{aa})\right] \\
  f_t(\text{AA}) f_t(\text{AA}) \left[ 1, 0, 0 \right]
+ 2 f_t(\text{AA}) f_t(\text{Aa}) \left[ 1/2, 1/2, 0 \right]
+ 2 f_t(\text{AA}) f_t(\text{aa}) \left[ 0, 1, 0 \right] \\
  f_t(\text{Aa}) f_t(\text{Aa}) \left[ 1/4, 1/2, 1/4 \right]
+ 2 f_t(\text{Aa}) f_t(\text{aa}) \left[ 0, 1/2, 1/2 \right]
+ f_t(\text{aa}) f_t(\text{aa}) \left[ 0, 0, 1 \right] \\
  \left[ \left(f_t(\text{AA}) + \frac{f_t(\text{Aa})}{2} \right)^2,
           \left(f_t(\text{aa}) + \frac{f_t(\text{Aa})}{2} \right),
  \left[ f_t(\text{A})^2, 2 f_t(\text{A}) f_t(\text{a}), f_t(\text{a})^2 \right]



Deviations from Hardy–Weinberg equilibrium[edit]

The seven assumptions underlying Hardy–Weinberg equilibrium are as follows:[3]

  • organisms are diploid
  • only sexual reproduction occurs
  • generations are non overlapping
  • mating is random
  • population size is infinitely large
  • allele frequencies are equal in the sexes
  • there is no migration, mutation or selection





Sex linkage[edit]






Generalization for more than two alleles[edit]


(p+q+r)^2=p^2 + q^2 + r^2 + 2pq +2pr + 2qr\,


(p_1 + \cdots  + p_n)^2\,

giving for all homozygotes:

f(A_i A_i) = p_i^2\,

and for all heterozygotes:

f(A_i A_j) = 2p_ip_j\,

Generalization for polyploidy[edit]


(p + q)^2\,

and therefore the polyploid case is the polynomial expansion of:

(p + q)^c\,


Table 2: Expected genotype frequencies for tetraploidy
Genotype Frequency
AAAA p^4
AAAa 4p^3 q
AAaa 6p^2q^2
Aaaa 4pq^3
aaaa q^4


Complete generalization[edit]


  p_1^{k_1} \cdots p_n^{k_n}



Application to cases of complete dominance[edit]


q = \sqrt {f(\text{aa})}


Significance tests for deviation[edit]


Example χ2 test for deviation[edit]


Table 3: Example Hardy–Weinberg principle calculation
Number 1469 138 5 1612

From which allele frequencies can be calculated:

p & = {2 \times \mathrm{obs}(\text{AA}) + \mathrm{obs}(\text{Aa}) \over 2 \times (\mathrm{obs}(\text{AA}) + \mathrm{obs}(\text{Aa}) + \mathrm{obs}(\text{aa}))} \\  \\
& = { 3076 \over 3224} \\  \\
& = 0.954


q & = 1 - p \\
& = 1 - 0.954 \\
& = 0.046

So the Hardy–Weinberg expectation is:

\mathrm{Exp}(\text{AA}) & = p^2n = 0.954^2 \times 1612 = 1467.4 \\
\mathrm{Exp}(\text{Aa}) & = 2pqn = 2 \times 0.954 \times 0.046 \times 1612 = 141.2 \\
\mathrm{Exp}(\text{aa}) & = q^2n = 0.046^2 \times 1612 = 3.4

Pearson's chi-squared test states:

& = 0.001 + 0.073 + 0.756 \\
& = 0.83


Fisher's exact test (probability test)[edit]


\operatorname{prob}[n_{12} | n_1] =  \frac{\binom{n}{n_{11}, n_{12}, n_{22}} } {\binom{2n }{n_1, n_2}} 2^{n_{12}},


An example Using one of the examples from Emigh (1980),[4] we can consider the case where n = 100, and p = 0.34. The possible observed heterozygotes and their exact significance level is given in Table 4.

Table 4: Example of Fisher's exact test for n = 100, p = 0.34.[4]
Number of heterozygotes Significance level
0 0.000
2 0.000
4 0.000
6 0.000
8 0.000
10 0.000
12 0.000
14 0.000
16 0.000
18 0.001
20 0.007
22 0.034
24 0.067
26 0.151
28 0.291
30 0.474
32 0.730
34 1.000



Inbreeding coefficient[edit]


 F = \frac{\operatorname{E}{(f(\text{Aa}))} - \operatorname{O}(f(\text{Aa}))} {\operatorname{E}(f(\text{Aa}))} = 1 - \frac{\operatorname{O}(f(\text{Aa}))} {\operatorname{E}(f(\text{Aa}))},

where the expected value from Hardy–Weinberg equilibrium is given by

 \operatorname{E}(f(\text{Aa})) = 2 p q

For example, for Ford's data above;







The principle was thus known as Hardy's law in the English-speaking world until 1943, when Curt Stern pointed out that it had first been formulated independently in 1908 by the German physician Wilhelm Weinberg.[5][6] William Castle in 1903 also derived the ratios for the special case of equal allele frequencies, and it is sometimes (but rarely) called the Hardy–Weinberg–Castle Law.

Derivation of Hardy's equations[edit]


p + 2q + r = 1



Hardy’s equivalence condition is


for generations after the first.

Now to prove Hardy's statement: the successor generation's proportions equal, \textstyle p:2q:r = p_1:2q_1:r_1, if and only if the equivalence condition \textstyle q^2 = pr holds. If \textstyle p:2q:r = p_1:2q_1:r_1, then

\begin{align}q^2 &= q_1^2\\
&= \left[\left(p + q\right) \left(q + r\right)\right]^2\\
&= \left(p + q\right)^2 \left(q + r\right)^2\\
&= p_1 r_1\\
&= p r\end{align},

so the equivalence condition holds. This shows equal proportions imply the equivalence condition. Now suppose the equivalence condition \textstyle q^2 = pr holds.

p &= p \left(p + 2 q + r\right)\\
&= p^2 + 2 p q + p r\\
&= p^2 + 2 p q + q^2\\
&= \left(p + q\right)^2\\
&= p_1

By symmetry, r = r_1. Finally,

q &= q \left(p + 2 q + r\right)\\
&= q p + 2 q^2 + q r\\
&= q^2 + q (p + r) + q^2\\
&= q^2 + q (p + r) + p r\\
&= \left(p + q\right) \left(q + r\right)\\
&= q_1

This shows \textstyle p = p_1, \textstyle q = q_1, and \textstyle r = r_1. Thus, proportions \textstyle p:2q:r = p_1:2q_1:r_1 equal. This shows the equivalence condition implies the proportions equal. Therefore, the proportions equal if and only if the equivalence condition holds, as was claimed.

Since the equivalence condition always holds for the second generation, all succeeding generations have the same proportions.

Numerical example[edit]


\text{sum} & = {\mathrm{obs}(\text{AA}) + 2 \times \mathrm{obs}(\text{Aa}) + \mathrm{obs}(\text{aa})}
& = 1750



r & = { 5 \over 1750} = 0.00286

As checks on the distribution, compute



E_0 = q^2 - pr = 0.00382 \,

For the next generation, Hardy's equations give,

r_1 & = (q + r)^2  = 0.00668

Again as checks on the distribution, compute



E_1 = q_1^2 - p_1 r_1 = 0.00000 \,


Graphical representation[edit]


De finetti diagram.png





  2. ^ http://www.mun.ca/biology/scarr/2900_HW_for_dioecious.html
  4. ^ a b Emigh, Ted H. (1980). "A Comparison of Tests for Hardy–Weinberg Equilibrium". Biometrics 36 (4): 627–642. doi:10.2307/2556115. JSTOR 2556115. 


See also[edit]

External links[edit]