I uploaded to arXiv our paper On the behavior of the leading eigenvalue of Eigen’s evolutionary matrices. This is our second joint work with Yura Semenov and Alexander Bratus, both from Moscow, on the Eigen model. The first one was devoted a special case of permutation invariant Crow-Kimura model, and this one deals with the original Eigen’s model.
We study the eigenvalue problem
where is the mutation matrix for the sequences of length , and is a diagonal matrix specifying fitness landscape
The matrix is taken as
and hence defined by a unique parameter , which is the probability of exact copying per sequence per site per replication event. Here is the Hamming distance between sequences and .
Among many things that we studied in this text, here is one, which is not proved, but mostly interesting from biological point of view: the critical value of the mutation rate after which the distribution of becomes (almost) uniform can be found as
where is the leading eigenvalue of the eigenvalue problem that depends on the mutation probability .
The last formula gives a surprisingly nice estimates for several examples computed numerically (the examples are in the text).