Eigen’s theory and paramuse model

This post is a continuation of this discussion of the error threshold.

Consider a population of sequences of the length {L}; each sequence is composed from {0}s and {1}s, therefore there are {2^L} different sequences. Consider the following mutation scheme:

\displaystyle  \mu(\sigma\rightarrow\sigma')=\begin{cases} \mu &\mbox{ if } H(\sigma,\sigma')=1,\\ -L\mu, &\mbox{ if } \sigma=\sigma',\\ 0 &\mbox{ otherwise}, \end{cases} \ \ \ \ \ (1)

and each sequence is characterized by its fitness {r(\sigma)}. In (1) {H(\sigma,\sigma')} is the Hamming distance between {\sigma} and {\sigma'}. Assuming the parallel mutation-selection dynamics, one has

\displaystyle  \boldsymbol{\dot{p}}=(\boldsymbol R+\boldsymbol M)\boldsymbol p, \ \ \ \ \ (2)

where {\boldsymbol p} is the vector of frequencies of different sequences, {\boldsymbol R} is a diagonal matrix with entries on the main diagonal {r(\sigma)-\bar{r},\,\bar{r}=\sum_\sigma p(\sigma)r(\sigma)} and matrix {\boldsymbol M} has the entries defined by (1).

If {r(\sigma)} is such that it depends only on the number of {0}s and {1}s in the sequences and not on their particular order, then the fitness landscape is called permutation invariant. In this case instead of {2^L\times 2^L}-dimensional system (2) it is possible to consider {(L+1)\times (L+1)} system

\displaystyle  \boldsymbol{\dot{\tilde{p}}}=(\boldsymbol{ \tilde R}+\boldsymbol{\tilde M})\boldsymbol{\tilde p}, \ \ \ \ \ (3)

where now {\tilde p_k} is the frequency of the class {k} (i.e., the frequency of sequences with exactly {k} {1}s), whose fitness is {\tilde r(k)}. Matrix {\boldsymbol{\tilde R}} is a diagonal matrix with the entries on the main diagonal {\tilde r(k)-\bar{r},\,\bar r=\sum_k \tilde r(k)\tilde p_k,\,k=0,\ldots, L}, and {\boldsymbol{\tilde M}} is a three diagonal matrix, whose main diagonal has the entries {\tilde \mu_{k,k}=-L\mu}, and two other diagonals are given by {\tilde \mu_{k,k+1}=\mu (k+1)} and {\tilde \mu_{k,k-1}=\mu(L+1-k)}. Here is a proof of the last statement.

Mutations to class {k} are summed from classes {k-1} and {k+1} according to (1). There are total {\binom{L}{k}} equations in (2) for sequences from class {k}. Each equation has {L} summands, from which {k} are from class {k-1} and {L-k} from class {k+1} (because we need to mutate at {k} positions {0\rightarrow 1} in the first case and at {L-k} positions {1\rightarrow 0} at the second case). Therefore {k\binom{L}{k}} summands are for the mutations from each {k-1} class to {k} and {(L-k)\binom{L}{k}} mutations from each {k+1} to class {k}. There are total {\binom{L}{k-1}} different sequences in class {k-1} and {\binom{L}{k+1}} in class {k+1}. Therefore, finally, we obtain that the rate of mutations from class {k-1} to class {k} per one sequence is given by {k\binom{L}{k}/\binom{L}{k-1}} and from class {k+1} to class {k} is {(L-k)\binom{L}{k}/\binom{L}{k+1}}. Simplifying, we obtain that {k-1\rightarrow k=(L-k+1)\mu} and {k+1\rightarrow k=(k+1)\mu}.


About Artem Novozhilov

I am an applied mathematician interested in studying various evolutionary processes by means of mathematical models. More on my professional activities can be found on my page https://www.ndsu.edu/pubweb/~novozhil/
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