## What does it mean to present “an exact solution” of the quasispecies model?

This post is a continuation of many previous discussions, e.g., one, two.

In a nutshell, the whole problem is to find the leading eigenvalue and the corresponding eigenvector of the eigenvalue problem

$\displaystyle (M+\mu Q)p=\lambda p,$

where $M=\text{diag}(m_0,\ldots,m_N)$, $\mu$ is a positive parameter, $p=(p_0,\ldots,p_N)$, and $Q$ is tridiagonal, with special structure. Seems like an innocent problem.

However, a huge deal of research is devoted to finding possible (partial) solutions to this problem. In particular, there are several papers, that provide “exact” solutions to it. First one is [1]. It considers a special problem when $M=\text{diag}(A,0,\ldots,0)$, which is called a single or sharply peaked landscape. But if one looks through the paper, it is quite unclear what is exactly “the exact solution.” The paper provides an implicit relation for the leading eigenvalue $\lambda$, find its expression for the limit $N\to\infty$ and gives a complicated expression for the component of $p$, which hardly can be used for any direct calculations (in the paper only $p_0$ is actually given and compared with numerical computations). Can we call this “an exact solution”?

There are more examples, e.g., [2], [3]. It would be natural, given the titles of these papers, to see expressions for $\lambda$ and $p$ in these papers for given $M$ and $\mu$, but there are none. Only some asymptotic formulas are present, and, which is also interesting, not for the eigenvector. This actually shows that it is probably simply impossible to present a simply looking solution for a general $M$.

However, it turns out [4] that for the single peaked landscape we can find simple asymptotic expressions for the eigenvector. This is just one of many new results in our text in arXiv (this is a new, very much expanded version). In particular, we prove that the limit distribution of the leading eigenvector is simply geometric:

Theorem: Consider the eigenvalue problem $(M+\mu Q)p=\lambda p$ and assume that $M=\text{diag}(N,0,\ldots,0)$, and $Q$ is tridiagonal (see its structure in the paper). Then in the limit $N\to\infty$ the eigenvector can be found as

$p_k^\infty=\lim_{N\to\infty}p_k=(1-\mu)\mu^k,\qquad k=0,1,\ldots$

for $\mu\leq 1$. If $\mu>1$ then the distribution is degenerate.

The proof in our paper is quite involved, but it seems that we actually found a way to attack this kind of problems in a simple way. Hence, this is definitely not the last my post on the quasispecies model.

Update: As I expected, these solutions were written down, probably for the first time, in [5], Eq. 55,  where the authors use some ad hoc methods. So, we should be content then that we provided a mathematically rigorous derivation of this result with a nice estimate of the speed of convergence.

[1] Galluccio, S. (1997). Exact solution of the quasispecies model in a sharply peaked fitness landscape. Physical Review E56(4), 4526.

[2] Baake, E., & Wagner, H. (2001). Mutation–selection models solved exactly with methods of statistical mechanics. Genetical research78(01), 93-117.

[3] Saakian, D. B., & Hu, C. K. (2006). Exact solution of the Eigen model with general fitness functions and degradation rates. Proceedings of the National Academy of Sciences of the United States of America103(13), 4935-4939.

[4] Bratus, A. S., Novozhilov, A. S., & Semenov, Y. S. (2013). Linear algebra of the permutation invariant Crow–Kimura model of prebiotic evolution. arXiv preprint arXiv:1306.0111.

[5] Saakian, D. B., Hu, C. K., & Khachatryan, H. (2004). Solvable biological evolution models with general fitness functions and multiple mutations in parallel mutation-selection scheme. Physical Review E70(4), 041908.