Probability of fixation

[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/(2N_e),

[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].

If 4N_es 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].