To start with a simple model, consider the genome of an organism to be
a simple bit string:
The thing to notice is that as mutations accumulate, descendants
become increasingly different from their ancestors; and if an
individual dies without offspring, all the mutations unique to it
disappear from the population.
If we assume that mutations are random, each link in the tree is a
chance for a mutation, so we only need to look at the genealogical tree
itself, not the states, to determine the properties of diversity. From
here on, we do not consider the states, only the links.
So to get the properties of uniqueness, we model the genealogical tree
of populations. The model is as follows:
- There are a fixed number of sites, each with an individual
(spatial or well-mixed)
- At each time step, the current population replaced by new
generation
- A new individual is offspring of a random individual
- Inheritance can be from one or multiple parents - one is used for
illustration
Thus, if we model the populaton on a lattice, the ancestry of each
individual, traced back in time, is a random walk. But when the
lineages of two individuals collide, they coalesce, so it is a
coalescing random walk.
This is the model in 1D. Time goes down the page, with the line of
descent of individuals in the present shown in bold; those that have no
descendants are dashed:
Here is a larger 2D simulation (population size 300).The latest
generations are magnified:
