Moran Process - Neutral Drift

Neutral Drift

Neutral drift is the idea that a neutral mutation can spread throughout a population, so that eventually the original allele is lost. A neutral mutation does not bring any fitness advantage or disadvantage to its bearer. The simple case of the Moran process can describe this phenomenon.

If the number of A individuals is given by i then the Moran process is defined on the state space i = 0, ..., N. Since the number of A individuals can change at most by one at each time step, a transition exists only between state i and state i − 1, i and i + 1. Thus the transition matrix of the stochastic process is tri-diagonal in shape and the transition probabilities are

$\begin{align} P_{0,0}&=1\\ P_{i,i-1} &= \frac{N-i}{N} \frac{i}{N}\\ P_{i,i} &= 1- P_{i,i-1} - P_{i,i+1}\\ P_{i,i+1} &= \frac{i}{N} \frac{N-i}{N}\\ P_{N,N}&=1. \end{align}$

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 at the definition of the process which states that always one individual will be chosen for reproduction and one is chosen for death. Once the A individuals have died out, they will never be reintroduced into the population since the process does not model mutations (A cannot be reintroduced into the population once it has died out and vice versa) and thus . For the same reason the population of A individuals will always stay N once they have reached that number and taken over the population and thus . The states 0 and N are called absorbing while the states 1, ..., N − 1 are called transient. The intermediate transition probabilities can be explained by considering the first term to be the probability to choose the individual whose abundance will increase by one and the second term the probability to choose the other type for death. Obviously, if the same type is chosen for reproduction and for death, then the abundance of one type does not change.

Eventually the population will reach one of the absorbing states and then stay there forever. In the transient states, random fluctuations will occur but eventually the population of A will either go extinct or reach fixation. This is one of the most important differences to deterministic processes which cannot model random events. The expected value and the variance of the number of A individuals X(t) at timepoint t can be computed when an initial state X(0) = i is given:

$\begin{align} E &= i \\ Var(X(t)|X(0) = i) &= 2i/N(1-i/N) \frac{1-(1-2/N^2)^t}{2/N^2} \end{align}$

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

For the expected value the calculation runs as follows. Writing

$\begin{align} E &= (i-1)P_{i,i-1} + iP_{i,i} + (i+1)P_{i,i+1}\\ &= 2ip(1-p) + i(p^2 + (1-p)^2) \\ &= i. \end{align}$

Writing and, and applying the law of total expectation, Applying the argument repeatedly gives or

For the variance the calculation runs as follows. Writing we have

$\begin{align} V_1 &= E - E^2 \\ &= (i-1)^2p(1-p) + i^2(p^2+(1-p)^2) + (i+1)^2p(1-p) - i^2 \\ &= 2p(1-p). \end{align}$

For all t, and are identically distributed, so their variances are equal. Writing as before and, and applying the law of total variance,

$\begin{align} Var(Y) &= E + Var(E) \\ &= E + Var(Z)\\ &= (2E/N)(1-E/N) + (1-2N^2)Var(Z). \end{align}$

If, we obtain . Rewriting this equation as

yields

as desired.

Read more about this topic: Moran Process

Famous quotes containing the words neutral and/or drift:

“I feel the carousel starting slowly
And going faster and faster: desk, papers, books,
Photographs of friends, the window and the trees
Merging in one neutral band that surrounds
Me on all sides, everywhere I look.”
—John Ashbery (b. 1927)

“But now they drift on the still water,
Mysterious, beautiful;”
—William Butler Yeats (1865–1939)