The process is named after Patrick Moran, who first proposed the model in 1958.
[1] It can be used to model variety-increasing processes such as mutation as well as variety-reducing effects such as genetic drift and natural selection.
The process can describe the probabilistic dynamics in a finite population of constant size N in which two alleles A and B are competing for dominance.
The two alleles are considered to be true replicators (i.e. entities that make copies of themselves).
In each time step a random individual (which is of either type A or B) is chosen for reproduction and a random individual is chosen for death; thus ensuring that the population size remains constant.
To model selection, one type has to have a higher fitness and is thus more likely to be chosen for reproduction.
The same individual can be chosen for death and for reproduction in the same step.
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.
The Moran process is defined on the state space i = 0, ..., N which count the number of A individuals.
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
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.
, we obtain Rewriting this equation as yields as desired.
For the simple Moran process this probability is xi = i/N.
This recursive equation can be solved using a new variable qi so that
Now ki, the total time until fixation starting from state i, can be calculated For large N the approximation holds.
This can be incorporated into the model if individuals with allele A have fitness
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.
Also in this case, fixation probabilities when starting in state i is defined by the recurrence And the closed form is given by where
The fixation probability can be defined recursively and a new variable
Now two properties from the definition of the variable yi can be used to find a closed form solution for the fixation probabilities: Combining (3) and xN = 1: which implies: This in turn gives us: This general case where the fitness of A and B depends on the abundance of each type is studied in evolutionary game theory.
Less complex results are obtained if a constant fitness ratio
is a constant factor for each composition of the population and thus the fixation probability from equation (1) simplifies to where the fixation probability of a single mutant A in a population of otherwise all B is often of interest and is denoted by ρ.
Also in the case of selection, the expected value and the variance of the number of A individuals may be computed where p = i/N, and r = 1 + s. For the expected value the calculation runs as follows For the variance the calculation runs as follows, using the variance of a single step In a population of all B individuals, a single mutant A will take over the whole population with the probability If the mutation rate (to go from the B to the A allele) in the population is u then the rate with which one member of the population will mutate to A is given by N × u and the rate with which the whole population goes from all B to all A is the rate that a single mutant A arises times the probability that it will take over the population (fixation probability): Thus if the mutation is neutral (i.e. the fixation probability is just 1/N) then the rate with which an allele arises and takes over a population is independent of the population size and is equal to the mutation rate.
This important result is the basis of the neutral theory of evolution and suggests that the number of observed point mutations in the genomes of two different species would simply be given by the mutation rate multiplied by two times the time since divergence.
Thus the neutral theory of evolution provides a molecular clock, given that the assumptions are fulfilled which may not be the case in reality.