Editing Hardy–Weinberg principle (section)

== Derivation ==
Consider a population of [[monoecious]] [[diploids]], where each organism produces male and female gametes at equal frequency, and has two alleles at each gene locus. We assume that the population is so large that it can be treated as infinite. Organisms reproduce by random union of gametes (the "gene pool" population model). A locus in this population has two alleles, A and a, that occur with initial frequencies {{math|''f''<sub>0</sub>(A) {{=}} ''p''}} and {{math|''f''<sub>0</sub>(a) {{=}} ''q''}}, respectively.{{NoteTag|The term ''frequency'' usually refers to a number or count, but in this context, it is synonymous with ''probability''.}} The allele frequencies at each generation are obtained by pooling together the alleles from each [[genotype]] of the same generation according to the expected contribution from the homozygote and heterozygote genotypes, which are 1 and 1/2, respectively:

{{NumBlk|:|<math>f_t(\text{A}) = f_t(\text{AA}) + \tfrac{1}{2} f_t(\text{Aa})</math>|{{EquationRef|1}}}}
{{NumBlk|:|<math>f_t(\text{a}) = f_t(\text{aa}) + \tfrac{1}{2} f_t(\text{Aa})</math>|{{EquationRef|2}}}}

[[File:Hardy–Weinberg law - Punnett square.svg|thumb|upright=1.2|Length of {{math|''p'', ''q''}} corresponds to allele frequencies (here {{math|''p'' {{=}} 0.6, ''q'' {{=}} 0.4}}). Then area of rectangle represents genotype frequencies (thus {{math|AA : Aa : aa {{=}} 0.36 : 0.48 : 0.16}}).]]

The different ways to form genotypes for the next generation can be shown in a [[Punnett square]], where the proportion of each genotype is equal to the product of the row and column allele frequencies from the current generation.

{| class="wikitable"
|+ Table 1: Punnett square for Hardy–Weinberg
|-
!colspan="2" rowspan="2"|
!colspan="2"|Females
|-
! A (''p'')
! a (''q'')
|-
!rowspan="2"| Males
!A (''p'')
|AA (''p''<sup>2</sup>)
|Aa (''pq'')
|-
!a (''q'')
|Aa (''qp'')
|aa (''q''<sup>2</sup>)
|}

The sum of the entries is {{math|''p''<sup>2</sup> + 2''pq'' + ''q''<sup>2</sup> {{=}} 1}}, as the genotype frequencies must sum to one.

Note again that as {{math|''p'' + ''q'' {{=}} 1}}, the binomial expansion of {{math|(''p'' + ''q'')<sup>2</sup> {{=}} ''p''<sup>2</sup> + 2''pq'' + ''q''<sup>2</sup> {{=}} 1}} gives the same relationships.

Summing the elements of the Punnett square or the binomial expansion, we obtain the expected genotype proportions among the offspring after a single generation:
{{NumBlk|:|<math>f_1(\text{AA}) = p^2 = f_0(\text{A})^2</math>|{{EquationRef|3}}}}
{{NumBlk|:|<math>f_1(\text{Aa}) = pq + qp = 2 pq = 2 f_0(\text{A}) f_0(\text{a})</math>|{{EquationRef|4}}}}
{{NumBlk|:|<math>f_1(\text{aa}) = q^2 = f_0(\text{a})^2</math>|{{EquationRef|5}}}}

These frequencies define the Hardy–Weinberg equilibrium. It should be mentioned that the genotype frequencies after the first generation need not equal the genotype frequencies from the initial generation, e.g. {{math|''f''<sub>1</sub>(AA) ≠ ''f''<sub>0</sub>(AA)}}.  However, the genotype frequencies for all ''future'' times will equal the Hardy–Weinberg frequencies, e.g.  {{math|''f<sub>t</sub>''(AA) {{=}} ''f''<sub>1</sub>(AA)}} for {{math|''t'' > 1}}. This follows since the genotype frequencies of the next generation depend only on the allele frequencies of the current generation which, as calculated by equations ({{EquationNote|1}}) and ({{EquationNote|2}}), are preserved from the initial generation:

: <math>\begin{align}
f_1(\text{A}) &= f_1(\text{AA}) + \tfrac{1}{2} f_1(\text{Aa})   = p^2 + p q = p (p+q) = p  = f_0(\text{A}) \\
f_1(\text{a}) &= f_1(\text{aa}) + \tfrac{1}{2} f_1(\text{Aa})  = q^2 + p q = q (p + q) = q  = f_0(\text{a})
\end{align} </math>

For the more general case of [[dioecious]] [[diploids]] [organisms are either male or female] that reproduce by random mating of individuals, it is necessary to calculate the genotype frequencies from the nine possible matings between each parental genotype (''AA'', ''Aa'', and ''aa'') in either sex, weighted by the expected genotype contributions of each such mating.<ref>{{cite web|url=https://www.mun.ca/biology/scarr/2900_HW_for_dioecious.html|title=Hardy–Weinberg in dioecious organisms|first=Dr. Steven M.|last=Carr|website=www.mun.ca}}</ref> Equivalently, one considers the six unique diploid–diploid combinations:

: <math>\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]</math>

and constructs a Punnett square for each, so as to calculate its contribution to the next generation's genotypes. These contributions are weighted according to the probability of each diploid–diploid combination, which follows a [[multinomial distribution]] with {{math|''k'' {{=}} 3}}. For example, the probability of the mating combination {{math|(AA,aa)}} is {{math| 2 ''f''<sub>''t''</sub>(AA)''f''<sub>''t''</sub>(aa)}} and it can only result in the {{math|Aa}} genotype: {{math|[0,1,0]}}. Overall, the resulting genotype frequencies are calculated as:

: <math> \begin{align}
&\left[ f_{t+1}(\text{AA}), f_{t+1}(\text{Aa}), f_{t+1}(\text{aa})\right] = \\
&\qquad=
  f_t(\text{AA}) f_t(\text{AA}) \left[ 1, 0, 0 \right]
+ 2 f_t(\text{AA}) f_t(\text{Aa}) \left[ \tfrac{1}{2}, \tfrac{1}{2}, 0 \right]
+ 2 f_t(\text{AA}) f_t(\text{aa}) \left[ 0, 1, 0 \right] \\
&\qquad\qquad+
  f_t(\text{Aa}) f_t(\text{Aa}) \left[ \tfrac{1}{4}, \tfrac{1}{2}, \tfrac{1}{4} \right]
+ 2 f_t(\text{Aa}) f_t(\text{aa}) \left[ 0, \tfrac{1}{2}, \tfrac{1}{2} \right]
+ f_t(\text{aa}) f_t(\text{aa}) \left[ 0, 0, 1 \right] \\
&\qquad=
  \left[ \left(f_t(\text{AA}) + \tfrac{1}{2} f_t(\text{Aa}) \right)^2,
         2 \left(f_t(\text{AA}) + \tfrac{1}{2} f_t(\text{Aa}) \right)
           \left(f_t(\text{aa}) + \tfrac{1}{2} f_t(\text{Aa}) \right),
         \left(f_t(\text{aa}) + \tfrac{1}{2} f_t(\text{Aa}) \right)^2
 \right]\\
&\qquad=
  \left[ f_t(\text{A})^2, 2 f_t(\text{A}) f_t(\text{a}), f_t(\text{a})^2 \right]
\end{align}
</math>

As before, one can show that the allele frequencies at time {{math|''t'' + 1}} equal those at time {{math|''t''}}, and so, are constant in time.  Similarly, the genotype frequencies depend only on the allele frequencies, and so, after time {{math|''t''&nbsp;{{=}}&nbsp;1}} are also constant in time.

If in either [[monoecious]] or [[dioecious]] organisms, either the allele or genotype proportions are initially unequal in either sex, it can be shown that constant proportions are obtained after one generation of random mating. If [[dioecious]] organisms are [[heterogametic]] and the gene locus is located on the [[X chromosome]], it can be shown that if the allele frequencies are initially unequal in the two sexes [''e.g''., XX females and XY males, as in humans], {{math|<var>f</var>′(a)}} in the [[heterogametic]] sex 'chases' {{math|<var>f</var>(a)}} in the [[homogametic]] sex of the previous generation, until an equilibrium is reached at the weighted average of the two initial frequencies.