  A particular case of the famous quasispecies model - the Crow-Kimura model
with a permutation invariant fitness landscape - is investigated. Using the
fact that the mutation matrix in the case of a permutation invariant fitness
landscape has a special tridiagonal form, a change of the basis is suggested
such that in the new coordinates a number of analytical results can be
obtained. In particular, using the eigenvectors of the mutation matrix as the
new basis, we show that the quasispecies distribution approaches a binomial one
and give simple estimates for the speed of convergence. Another consequence of
the suggested approach is a parametric solution to the system of equations
determining the quasispecies. Using this parametric solution we show that our
approach leads to exact asymptotic results in some cases, which are not covered
by the existing methods. In particular, we are able to present not only the
limit behavior of the leading eigenvalue (mean population fitness), but also
the exact formulas for the limit quasispecies eigenvector for special cases.
For instance, this eigenvector has a geometric distribution in the case of the
classical single peaked fitness landscape. On the biological side, we propose a
mathematical definition, based on the closeness of the quasispecies to the
binomial distribution, which can be used as an operational definition of the
notorious error threshold. Using this definition, we suggest two approximate
formulas to estimate the critical mutation rate after which the quasispecies
delocalization occurs.
