Moran Process - Selection

Selection

If one allele has a fitness advantage over the other allele, it will be more likely to be chosen for reproduction. This can be incorporated into the model if individuals with allele A have fitness and individuals with allele B have fitness where i is the number of individuals of type A; thus describing a general birth-death process. The transition matrix of the stochastic process is tri-diagonal in shape and the transition probabilities are

$\begin{align} P_{0,0}&=1\\ P_{i,i-1} &= \frac{g_i (N-i) }{f_i \cdot i + g_i (N-i)} \cdot \frac{i}{N}\\ P_{i,i} &= 1- P_{i,i-1} - P_{i,i+1}\\ P_{i,i+1} &= \frac{f_i \cdot i}{f_i \cdot i + g_i (N-i)} \cdot \frac{N-i}{N}\\ P_{N,N}&=1. \end{align}$

The entry denotes the probability to go from state i to state j. To understand the formulas for the transition probabilities one has to look again at the definition of the process and see that the fitness enters only the first term in the equations which is concerned with reproduction. Thus the probability that individual A is chosen for reproduction is not i / N any more but dependent on the fitness of A and thus . Also in this case, fixation probabilities when starting in state i is defined by recurrence

$\begin{align} x_0 &= 0\\ x_i &= \beta_i x_{i-1}+(1-\alpha_i-\beta_i)x_i+\alpha_ix_{i+1}\quad i=1,\dots,N-1\\ x_N &= 1 \end{align}$

And the closed form is given by

$\begin{align} x_i = \frac{ {\displaystyle 1 + \sum\limits_{j=1}^{i-1}\prod\limits_{k=1}^{j}\gamma_k } } { {\displaystyle 1 + \sum\limits_{j=1}^{N-1}\prod\limits_{k=1}^{j}\gamma_k } } \qquad \text{(1)} \end{align}$

where per definition and will just be for the general case.

For a mathematical derivation of the equation above, click on "show" to reveal

Also in this case, fixation probabilities can be computed, but the transition probabilities are not symmetric. The notation, and is used. The fixation probability can be defined recursively and a new variable is introduced.

$\begin{align} x_i &= \beta_i x_{i-1} + (1-\alpha_i - \beta_i)x_i + \alpha_i x_{i+1} \\ \beta_i (x_i - x_{i-1} ) &= \alpha_i (x_{i+1} - x_i ) \\ \gamma_i \cdot y_i &= y_{i+1} \end{align}$

Now two properties from the definition of the variable can be used to find a closed form solution for the fixation probabilities:

$\begin{align} \sum\limits_{i=1}^{m} y_i &= x_m &1\\ y_k &= x_1 \cdot \prod\limits_{l=1}^{k-1}\gamma_l &2\\ \Rightarrow \sum\limits_{m=1}^{i}y_m &= x_1 + x_1 \sum\limits_{j=1}^{i-1}\prod\limits_{k=1}^{j}\gamma_k = x_i &3 \end{align}$

From (3) and the knowledge follows

$\begin{align} x_N = x_1 \left( 1 + \sum\limits_{j=1}^{N-1}\prod\limits_{k=1}^{j}\gamma_k \right) &= 1 \quad \Rightarrow \quad x_1 = \frac{1}{ 1 + \sum\limits_{j=1}^{N-1}\prod\limits_{k=1}^{j}\gamma_k } \\ x_i &= \frac{ {\displaystyle 1 + \sum\limits_{j=1}^{i-1}\prod\limits_{k=1}^{j}\gamma_k } } { {\displaystyle 1 + \sum\limits_{j=1}^{N-1}\prod\limits_{k=1}^{j}\gamma_k } } \end{align}$

Read more about this topic: Moran Process

Famous quotes containing the word selection:

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—Anthony Trollope (1815–1882)

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