Since Dq = qn+1 - q
n,
Dq =
[(1 - h)*spnqn + (1 - s)*qn ]
/ [ 1 - 2shpnqn - sqn2 ]
- q n
(Expanded qn+1. )
Dq =
[(1 - h)*spnqn + (1 - s)*qn - qn*(1
- 2shpnqn - sqn2 )] / [ 1 -
2shpnqn - sqn2 ] (Lowest
common factor for q n.)
Dq =
[(1 - h)*spnqn + (1 - s)*qn - qn
+ 2shqnpnqn + sqnqn2
] / [ 1 - 2shpnqn - sqn2 ]
(Expanded q n product.)
Dq =
[(1 - h + 2hqn)*spnqn + (1 - s)*qn
- qn + sqnqn2 ] / [ 1 - 2shpnqn
- sqn2 ]
(Grouped spnq n products.)
Dq =
[(1 - h + 2hqn)*spnqn + qn
- sqn - qn + sqnqn2
] / [ 1 - 2shpnqn - sqn2 ]
(Expanded (1 - s)*q n.)
Dq =
[(1 - h + 2hqn)*spnqn - sqn
+ sqnqn2 ] / [ 1 - 2shpnqn
- sqn2 ]
(Canceled q n- q n .)
Dq =
[(1 - h + 2hqn)*spnqn - sqn
+ sqn*(1 - pn)qn ] / [ 1 -
2shpnqn - sqn2 ]
(Substituted (1 - p n ) for q n.)
Dq =
[(1 - h + 2hqn)*spnqn - sqn
+ sqn*qn- sqn*pnqn
] / [ 1 - 2shpnqn - sqn2 ]
(Expanded sq n*(1 - p n)*q n.)
Dq =
[(1 - h + 2hqn -qn)*spnqn -
sqn + sqn*qn] / [ 1 - 2shpnqn
- sqn2 ]
(Grouped spnq n products.)
Dq = [(1 - h + 2hqn
-qn)*spnqn - sqn + s*(1 - pn)*qn]
/ [ 1 - 2shpnqn - sqn2 ]
(Substituted (1 - p n ) for q n.)
Dq = [(1 - h + 2hqn
-qn)*spnqn - sqn
+ sqn - s*pnqn] / [ 1
- 2shpnqn - sqn2 ]
(Expanded s*(1 - p n)*q n.)
Dq = [(1 - h + 2hqn
-qn)*spnqn - s*pnqn]
/ [ 1 - 2shpnqn - sqn2 ]
(Canceled -sq n + sq n .)
Dq = [(1
- h + 2hqn -qn - 1)*spnqn]
/ [ 1 - 2shpnqn - sqn2 ]
(Grouped spnq n products.)
Dq = [(-h + 2hqn
-qn)*spnqn] / [ 1 - 2shpnqn
- sqn2 ]
(Canceled 1 - 1 and factored s.)
Dq = [(-h + (2h - 1)qn)*spnqn]
/ [ 1 - 2shpnqn - sqn2 ]
(Grouped q n products.)
Dq = -spnqn*(h
+ (1 - 2h)qn) / [ 1 - 2shpnqn - sqn2
]
(Rearranged terms and signs.)
If we choose to ignore the denominator (as Haldane did), we have:
Dq = -spnqn*(h + (1 - 2h)qn)
Summary : The change in q due to 1 generation of selection is called
delta q (Dq) is given by the equation:
Dq = -spnqn*(h
+ (1 - 2h)qn) / [ 1 - 2shpnqn - sqn2
]
where s is the selection coefficient, h is the dominance factor, p
n is the frequency of the more fit (increasing) allele, and q
n is frequency of the less fit (decreasing) allele.