Detailed analysis of an Eigen quasispecies model in a periodically moving sharp-peak landscape

Abstract

The Eigen quasispecies model in a periodically moving sharp-peak landscape considered in previous seminal works [M. Nilsson and N. Snoad, Phys. Rev. Lett. 84, 191 (2000)] and [C. Ronnewinkel, in, edited by L. Kallel, B. Naudts, and A. Rogers (Springer-Verlag, Heidelberg, 2001)] is analyzed in greater detail. We show here, through a more rigorous analysis, that results in those papers are qualitatively correct. In particular, we obtain a phase diagram for the existence of a quasispecies with the same shape as in the above cited paper by C. Ronnewinkel, with upper and lower thresholds for the mutation rate between which a quasispecies may survive. A difference is that the upper value is larger and the lower value is smaller than the previously reported ones, so that the range for quasispecies existence is always larger than thought before. The quantitative information provided might also be important in understanding genetic variability in virus populations and has possible applications in antiviral therapies. The results in the quoted papers were obtained by studying the populations only at some few genomes. As we will show, this amounts to diagonalizing a 3×3 matrix. Our work is based instead in a different division of the population allowing a finer control of the populations at various relevant genetic sequences. The existence of a quasispecies will be related to Perron-Frobenius eigenvalues. Although huge matrices of sizes 2(ℓ), where ℓ is the genome length, may seem necessary at a first look, we show that such large sizes are not necessary and easily obtain numerical and analytical results for their eigenvalues.