  The evolution of the allelic proportion $x$ of a biallelic locus subject to
the forces of mutation and drift is investigated in a diffusion model, assuming
small scaled mutation rates. The overall scaled mutation rate is parametrized
with $\theta=(\mu_1+\mu_0)N$ and the ratio of mutation rates with
$\alpha=\mu_1/(\mu_1+\mu_0)=1-\beta$. The equilibrium density of this process
is beta with parameters $\alpha\theta$ and $\beta\theta$. Away from
equilibrium, the transition density can be expanded into a series of modified
Jacobi polynomials. If the scaled mutation rates are small, i.e., $\theta \ll
1$, it may be assumed that polymorphism derives from mutations at the
boundaries. A model, where the interior dynamics conform to the pure drift
diffusion model and the mutations are entering from the boundaries is derived.
In equilibrium, the density of the proportion of polymorphic alleles, \ie\ $x$
within the polymorphic region $[1/N,1-1/N]$, is
$\alpha\beta\theta(\tfrac1x+\tfrac1{1-x})=\tfrac{\alpha\beta\theta}{x(1-x)}$,
while the mutation bias $\alpha$ influences the proportion of monomorphic
alleles at 0 and 1. Analogous to the expansion with modified Jacobi
polynomials, a series expansion of the transition density is derived, which is
connected to Kimura's well known solution of the pure drift model using
Gegenbauer polynomials. Two temporal and two spatial regions are separated. The
eigenvectors representing the spatial component within the polymorphic region
depend neither on the on the scaled mutation rate $\theta$ nor on the mutation
bias $\alpha$. Therefore parameter changes, e.g., growing or shrinking
populations or changes in the mutation bias, can be modeled relatively easily,
without the change of the eigenfunctions necessary for the series expansion
with Jacobi polynomials.
