This post is a continuation of this discussion of the error threshold.
Consider a population of sequences of the length ; each sequence is composed from s and s, therefore there are different sequences. Consider the following mutation scheme:
and each sequence is characterized by its fitness . In (1) is the Hamming distance between and . Assuming the parallel mutation-selection dynamics, one has
where is the vector of frequencies of different sequences, is a diagonal matrix with entries on the main diagonal and matrix has the entries defined by (1).
If is such that it depends only on the number of s and s in the sequences and not on their particular order, then the fitness landscape is called permutation invariant. In this case instead of -dimensional system (2) it is possible to consider system
where now is the frequency of the class (i.e., the frequency of sequences with exactly s), whose fitness is . Matrix is a diagonal matrix with the entries on the main diagonal , and is a three diagonal matrix, whose main diagonal has the entries , and two other diagonals are given by and . Here is a proof of the last statement.
Mutations to class are summed from classes and according to (1). There are total equations in (2) for sequences from class . Each equation has summands, from which are from class and from class (because we need to mutate at positions in the first case and at positions at the second case). Therefore summands are for the mutations from each class to and mutations from each to class . There are total different sequences in class and in class . Therefore, finally, we obtain that the rate of mutations from class to class per one sequence is given by and from class to class is . Simplifying, we obtain that and .