Difference between revisions of "Probability of fixation"
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This is derived from | This is derived from | ||
| − | <math>u(p) = \frac{\int_0^p G(x) | + | <math>u(p) = \frac{\int_0^p G(x)\, \mbox{d} x}{\int_0^1 G(x)\, \mbox{d} x}</math>, |
equation 3 of [[Kimura 1962]]. | equation 3 of [[Kimura 1962]]. | ||
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The change in allele frequency (<math>\delta p</math>) over short periods of time (<math>\delta t</math>) is | The change in allele frequency (<math>\delta p</math>) over short periods of time (<math>\delta t</math>) is | ||
| − | <math>u(p, t+\delta t) = \int f(p, p+\delta p; \delta t) u(p+ \delta p, t) d(\delta p)</math>, | + | <math>u(p, t+\delta t) = \int f(p, p+\delta p; \delta t) u(p+ \delta p, t) \, \mbox{d} (\delta p)</math>, |
integrating over all values of changes in allele frequency (<math>\delta p</math>). | integrating over all values of changes in allele frequency (<math>\delta p</math>). | ||
| Line 34: | Line 34: | ||
A mean and variance of the change in allele frequency (''p'') per generation are defined as | A mean and variance of the change in allele frequency (''p'') per generation are defined as | ||
| − | <math>M_{\delta p}=\lim_{\delta t \to 0} \frac{1}{\delta t} \int (\delta p) f(p, p+\delta p; \delta t) d(\delta p)</math> | + | <math>M_{\delta p}=\lim_{\delta t \to 0} \frac{1}{\delta t} \int (\delta p) f(p, p+\delta p; \delta t) \, \mbox{d} (\delta p)</math> |
| − | <math>V_{\delta p}=\lim_{\delta t \to 0} \frac{1}{\delta t} \int (\delta p)^2 f(p, p+\delta p; \delta t) d(\delta p)</math> | + | <math>V_{\delta p}=\lim_{\delta t \to 0} \frac{1}{\delta t} \int (\delta p)^2 f(p, p+\delta p; \delta t) \, \mbox{d} (\delta p)</math> |
Revision as of 07:15, 23 September 2018
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)\, \mbox{d} x}{\int_0^1 G(x)\, \mbox{d} x}[/math],
equation 3 of Kimura 1962.
[math]u(p,t)[/math] is the probability of fixation of an allele at frequency p within t generations.
The change in allele frequency ([math]\delta p[/math]) over short periods of time ([math]\delta t[/math]) is
[math]u(p, t+\delta t) = \int f(p, p+\delta p; \delta t) u(p+ \delta p, t) \, \mbox{d} (\delta p)[/math],
integrating over all values of changes in allele frequency ([math]\delta p[/math]).
A mean and variance of the change in allele frequency (p) per generation are defined as
[math]M_{\delta p}=\lim_{\delta t \to 0} \frac{1}{\delta t} \int (\delta p) f(p, p+\delta p; \delta t) \, \mbox{d} (\delta p)[/math]
[math]V_{\delta p}=\lim_{\delta t \to 0} \frac{1}{\delta t} \int (\delta p)^2 f(p, p+\delta p; \delta t) \, \mbox{d} (\delta p)[/math]
