## More on quasispecies

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

$QWp=\lambda p,$

where $Q=(q_{ij})_{2^N\times 2^N}$ is the mutation matrix for the sequences of length $N$, and $W$ is a diagonal matrix specifying fitness landscape

$W=\text{diag}(w_0,\ldots w_{2^N-1})$.

The matrix $Q$ is taken as

$q_{ij}=q^{N-H_{ij}}(1-q)^{H_{ij}}$

and hence defined by a unique parameter $q$, which is the probability of exact copying per sequence per site per replication event. Here $H_{ij}$ is the Hamming distance between sequences $i$ and $j$.

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 $p$ becomes (almost) uniform can be found as

$\displaystyle q^*=\frac 12\sqrt[N]{\frac{\sum w_j}{\max_j\{w_j\}}}=\sqrt[N]{\frac{\lambda(0.5)}{\lambda(1)}},$

where $\lambda(q)$ is the leading eigenvalue of the eigenvalue problem that depends on the mutation probability $q$.

The last formula gives a surprisingly nice estimates for several examples computed numerically (the examples are in the text).