Generalizations Hardy–Weinberg principle

-



1 generalizations

1.1 generalization more 2 alleles
1.2 generalization polyploidy
1.3 complete generalization





generalizations

the simple derivation above can generalized more 2 alleles , polyploidy.


generalization more 2 alleles

punnett square three-allele case (left) , four-allele case (right). white areas homozygotes. colored areas heterozygotes.


consider allele frequency, r. two-allele case binomial expansion of (p + q), , three-allele case trinomial expansion of (p + q+ r).







(
p
+
q
+
r

)

2


=

p

2


+

q

2


+

r

2


+
2
p
q
+
2
p
r
+
2
q
r



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



more generally, consider alleles a1, ..., given allele frequencies p1 pn;







(

p

1


+
⋯
+

p

n



)

2





{\displaystyle (p_{1}+\cdots +p_{n})^{2}\,}



giving homozygotes:







f
(

a

i



a

i


)
=

p

i


2





{\displaystyle f(a_{i}a_{i})=p_{i}^{2}\,}



and heterozygotes:







f
(

a

i



a

j


)
=
2

p

i



p

j





{\displaystyle f(a_{i}a_{j})=2p_{i}p_{j}\,}



generalization polyploidy

the hardy–weinberg principle may generalized polyploid systems, is, organisms have more 2 copies of each chromosome. consider again 2 alleles. diploid case binomial expansion of:







(
p
+
q

)

2





{\displaystyle (p+q)^{2}\,}



and therefore polyploid case polynomial expansion of:







(
p
+
q

)

c





{\displaystyle (p+q)^{c}\,}



where c ploidy, example tetraploid (c = 4):



depending on whether organism true tetraploid or amphidiploid determine how long take population reach hardy–weinberg equilibrium.


complete generalization

for



n


{\displaystyle n}

distinct alleles in



c


{\displaystyle c}

-ploids, genotype frequencies in hardy–weinberg equilibrium given individual terms in multinomial expansion of



(

p

1


+
⋯
+

p

n



)

c




{\displaystyle (p_{1}+\cdots +p_{n})^{c}}

:







(

p

1


+
⋯
+

p

n



)

c


=

∑


k

1


,
…
,

k

n


 
∈

n

:

k

1


+
⋯
+

k

n


=
c





(


c


k

1


,
…
,

k

n





)




p

1



k

1




⋯

p

n



k

n






{\displaystyle (p_{1}+\cdots +p_{n})^{c}=\sum _{k_{1},\ldots ,k_{n}\ \in \mathbb {n} :k_{1}+\cdots +k_{n}=c}{c \choose k_{1},\ldots ,k_{n}}p_{1}^{k_{1}}\cdots p_{n}^{k_{n}}}








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