Probability of fixation
From Genetics Wiki
This was derived in Kimura 1962.
[math]u(p)=\frac{1-e^{4N_esp}}{1-e^{4N_es}}[/math]
If we are considering the initial frequency of a single new mutation in the population p=1/(2Ne),
[math]u(p)_1=\frac{1-e^{4N_es\frac{1}{2N_e}}}{1-e^{4N_es}}=\frac{1-e^{2s}}{1-e^{4N_es}}[/math].
And if 4Nes is large
[math]u(p)_2\approx\frac{1-e^{2s}}{1}=1-e^{2s}[/math].
[math]e^{2s}\approx 1+2s[/math]
[math]u(p)_2 \approx 1-e^{2s} \approx 1-1+2s = 2s[/math].
This agrees with the results of Fisher 1930 and Wright 1931.
Notes
This is derived from
[math]u(p) = \frac{\int_0^p G(x)dx}{\int_0^1 G(x)dx}[/math],
equation 3 of Kimura 1962.
[math]u(p, t+\delta t) = \int f(p, p+\delta p; \delta t) u(p+ \delta p, t) d(\delta p)[/math]
