Difference between revisions of "Coalescent"
(→Coalescence of Two Lineages) |
|||
| (5 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | =Coalescence of Two Lineages= | + | =The Coalescence of Two Lineages= |
| + | |||
| + | The chance two lineages coalesce in the previous generation is the chance that they pick the exact same copy of a gene to inherit which is <math>1/2N</math> because there are 2''N'' total copies of a gene in the population. | ||
| + | |||
| + | The chance they did not coalesce is therefore the remaining probability <math>1-1/2N</math>. | ||
| + | |||
| + | Each ''g'' generation there is the same independent probability of not coalescing, a total of <math>(1-1/2N)^g</math>, until a coalescent event occurs with probability <math>1/2N</math>. | ||
| + | |||
| + | Therefore, the probability of coalescence in the ''g'' generation is | ||
| + | |||
| + | <math>P(\text{coalescence at time }g)=\frac{1}{2N}\left( 1 - \frac{1}{2N} \right)^g</math> | ||
| + | |||
| + | which is a form of geometric distribution in discrete time. | ||
| + | |||
| + | In a large population over many generations this is closely approximated by an exponential distribution | ||
| + | |||
| + | <math>P(\text{coalescence at time }g)=\frac{1}{2N}\left( 1 - \frac{1}{2N} \right)^g \approx \frac{1}{2N}e^{-g/2N}</math> | ||
| + | |||
| + | |||
The PDF of an exponential distribution of the coalescence of two lineages in a diploid population of size ''N''. | The PDF of an exponential distribution of the coalescence of two lineages in a diploid population of size ''N''. | ||
| Line 5: | Line 23: | ||
<math>P(\text{coalescence at time }g)=\frac{1}{2N}e^{-g/2N}</math> | <math>P(\text{coalescence at time }g)=\frac{1}{2N}e^{-g/2N}</math> | ||
| − | For example, the probability of two lineages coalescing in a small population of 20 individuals in exactly the ninth | + | For example, the probability of two lineages coalescing in a small population of 20 individuals in exactly the ninth generation is 2%. |
Integrate to get the CDF. | Integrate to get the CDF. | ||
| − | <math>F(\text{coalescence at time }g)=\int_0^g\frac{1}{2N}e^{-g/2N}</math> | + | <math>F(\text{coalescence at time }g)=\int_0^g\frac{1}{2N}e^{-g/2N} \text{d}g</math> |
| − | <math>F(\text{coalescence at time }g)=\frac{1}{2N}\int_0^g e^{-g\frac{1}{2N}}</math> | + | <math>F(\text{coalescence at time }g)=\frac{1}{2N}\int_0^g e^{-g\frac{1}{2N}} \text{d}g</math> |
<math>F(\text{coalescence at time }g)=\frac{1}{2N} \frac{-e^{-g\frac{1}{2N}}}{\frac{1}{2N}} + C</math> | <math>F(\text{coalescence at time }g)=\frac{1}{2N} \frac{-e^{-g\frac{1}{2N}}}{\frac{1}{2N}} + C</math> | ||
| Line 18: | Line 36: | ||
<math>F(\text{coalescence at time }g)=-e^{-g/2N} + C</math> | <math>F(\text{coalescence at time }g)=-e^{-g/2N} + C</math> | ||
| − | Because the CDF must <math>\lim_{g \to \infty}\left( -e^{-g/2N} + C \right)= 1</math> and <math>\lim_{g \to \infty} -e^{-g/2N}= 0</math> then <math>C = 1</math>. | + | Because by definition the CDF area must sum to one, <math>\lim_{g \to \infty}\left( -e^{-g/2N} + C \right)= 1</math>, and the limit of <math>\lim_{g \to \infty} -e^{-g/2N}= 0</math> then the constant of integration must be one, <math>C = 1</math>. |
<math>F(\text{coalescence at time }g)=-e^{-g/2N} + 1</math> | <math>F(\text{coalescence at time }g)=-e^{-g/2N} + 1</math> | ||
| Line 37: | Line 55: | ||
<math>6N=g</math> | <math>6N=g</math> | ||
| + | |||
| + | =The Coalescence of More than Two Lineages= | ||
| + | |||
| + | =The Coalescence of an Infinite Number of Lineages= | ||
| + | |||
| + | =Coalescence in a Population of Changing Size= | ||
Latest revision as of 11:03, 18 February 2016
Contents
The Coalescence of Two Lineages
The chance two lineages coalesce in the previous generation is the chance that they pick the exact same copy of a gene to inherit which is [math]1/2N[/math] because there are 2N total copies of a gene in the population.
The chance they did not coalesce is therefore the remaining probability [math]1-1/2N[/math].
Each g generation there is the same independent probability of not coalescing, a total of [math](1-1/2N)^g[/math], until a coalescent event occurs with probability [math]1/2N[/math].
Therefore, the probability of coalescence in the g generation is
[math]P(\text{coalescence at time }g)=\frac{1}{2N}\left( 1 - \frac{1}{2N} \right)^g[/math]
which is a form of geometric distribution in discrete time.
In a large population over many generations this is closely approximated by an exponential distribution
[math]P(\text{coalescence at time }g)=\frac{1}{2N}\left( 1 - \frac{1}{2N} \right)^g \approx \frac{1}{2N}e^{-g/2N}[/math]
The PDF of an exponential distribution of the coalescence of two lineages in a diploid population of size N.
[math]P(\text{coalescence at time }g)=\frac{1}{2N}e^{-g/2N}[/math]
For example, the probability of two lineages coalescing in a small population of 20 individuals in exactly the ninth generation is 2%.
Integrate to get the CDF.
[math]F(\text{coalescence at time }g)=\int_0^g\frac{1}{2N}e^{-g/2N} \text{d}g[/math]
[math]F(\text{coalescence at time }g)=\frac{1}{2N}\int_0^g e^{-g\frac{1}{2N}} \text{d}g[/math]
[math]F(\text{coalescence at time }g)=\frac{1}{2N} \frac{-e^{-g\frac{1}{2N}}}{\frac{1}{2N}} + C[/math]
[math]F(\text{coalescence at time }g)=-e^{-g/2N} + C[/math]
Because by definition the CDF area must sum to one, [math]\lim_{g \to \infty}\left( -e^{-g/2N} + C \right)= 1[/math], and the limit of [math]\lim_{g \to \infty} -e^{-g/2N}= 0[/math] then the constant of integration must be one, [math]C = 1[/math].
[math]F(\text{coalescence at time }g)=-e^{-g/2N} + 1[/math]
[math]F(\text{coalescence at time }g)=1-e^{-g/2N}[/math]
For example, there is a 95% probability that two lineages will coalesce within 6N generations.
[math]F(\text{coalescence at time }g)=0.95=1-e^{-g/2N}[/math]
[math]1-0.95=e^{-g/2N}[/math]
[math]\log_e 0.05=-\frac{g}{2N}[/math]
[math]-2N\log_e 0.05=g[/math]
[math]-2N\times-3=g[/math]
[math]6N=g[/math]
