SME
Genetics
Population genetics (Hardy–Weinberg)
Use this for “1 in 2500 newborns has cystic fibrosis. What share of people are carriers?”
Diagram
p + q = 0.98 + 0.02 = 1
p
2
+
2pq
+
q
2
p
2
+
2pq
+
q
2
=
0.9604
+
0.0392
+
0.0004
=
1
98 of 100 alleles are A.
This is an example. Type your own values to start a new problem.
q²
Affected share (aa)
calculated
nₐₐ
Affected: 1 in
example
q
Frequency of a
calculated
p
Frequency of A
calculated
p²
Share AA
worked out (not typed)
2pq
Carrier share (Aa)
worked out (not typed)
nAa
Carriers: 1 in
worked out (not typed)
Formulas
q
2
=
q
×
q
0.0004 = 0.02 × 0.02
nₐₐ
=
1
q
2
2,500
=
1
0.0004
p
=
1
−
q
0.98 = 1 − 0.02
p
2
=
p
×
p
0.9604 = 0.98 × 0.98
2
pq
=
2
×
p
×
q
0.0392 = 2 × 0.98 × 0.02
nAa
=
1
2
pq
25.5102
=
1
0.0392
Assumptions
•
Hardy–Weinberg: a large population mating at random, with no selection, drift, migration or mutation.
•
The trait is autosomal recessive, so only aa is affected: q² is the affected share.
•
Carriers (Aa) look unaffected; p + q = 1 and p² + 2pq + q² = 1.
•
A share of 1 in n is 1 ÷ n.
Step-by-step
WE KNOW
nₐₐ = 2,500
FIND
affected share (aa) (q²), frequency of a (q), frequency of A (p), share AA (p²), carrier share (Aa) (2pq), carriers: 1 in (nAa)
Step 1 · Find affected share (aa): q²
Use nₐₐ = 1 ÷ q²
One
in
n
affected
is
a
share
of
1
n
.
q
2
=
1
nₐₐ
q
2
=
1
2,500
q
2
=
0.0004
Step 2 · Find frequency of a (q)
Use q² = q × q
Only
aa
is
affected,
so
q
is
the
square
root
of
the
affected
share.
q
=
√
q
2
q
=
√
0.0004
q
=
0.02
Step 3 · Find frequency of A (p)
Use p = 1 − q
The two alleles’ frequencies add to 1.
p
=
1
−
0.02
p
=
0.98
Step 4 · Find share AA (p²)
Use p² = p × p
Two
A
alleles
meet
with
chance
p
×
p
.
p
2
=
0.98
×
0.98
p
2
=
0.9604
Step 5 · Find carrier share (Aa): 2pq
Use 2pq = 2 × p × q
A
from
one
parent
and
a
from
the
other,
or
the
other
way
round:
twice
p
×
q
.
2
pq
=
2
×
0.98
×
0.02
2
pq
=
1.96
×
0.02
2
pq
=
0.0392
Step 6 · Find carriers: 1 in (nAa)
Use nAa = 1 ÷ (2pq)
A
share
of
1
n
is
one
person
in
n.
nAa
=
1
0.0392
nAa = 25.5102
Check
Put the numbers back in. Both sides should match.
0.0004 = 0.02 × 0.02 ✓
2,500
=
1
0.0004
✓
0.98 = 1 − 0.02 ✓
0.9604 = 0.98 × 0.98 ✓
0.0392 = 2 × 0.98 × 0.02 ✓
25.5102
=
1
0.0392
✓
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Testing for Hardy–Weinberg equilibrium
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