Finite - size scaling of the quasispecies model

We use finite-size scaling to study the critical behavior of the quasispecies model of molecular evolution in the single-sharp-peak replication landscape. This model exhibits a sharp threshold phenomenon at Q = Qc = 1/a, where Q is the probability of exact replication of a molecule of length L and a is the selective advantage of the master string. We investigate the sharpness of the threshold and find that its characteristics persist across a range of Q of order L−1 about Qc. Furthermore, using the data collapsing method we show that the normalized mean Hamming distance between the master string and the entire population, as well as the properly scaled fluctuations around this mean value, follow universal forms in the critical region. 87.10.+e, 64.60.Cn Typeset using REVTEX 1 Although the so-called error threshold phenomenon, which limits the length L of competing self-reproducing molecules, is acknowledged as one of the main outcomes of Eigen’s quasispecies model [1,2], the full characterization of the error threshold transition for finite L has not been satisfactorily carried out yet. In fact, similarly to the definition of the critical temperature for finite lattices, there is no generally accepted definition of the term error threshold for finite L [3]. Nevertheless, the study of the systematic deviations from the infinite length limit behavior introduced by the finite-size effects, besides being practically independent of the definition adopted, gives valuable information on the behavior of the relevant macroscopic quantities near the critical region [4,5]. In the quasispecies model, a molecule is represented by a string of L digits ~s = (s1, s2, . . . , sL), with the variables sα allowed to take on κ different values, each representing a different type of monomer used to build the molecule. For sake of simplicity, in this work we will consider only binary strings, i.e., sα = 0, 1. The concentrations xi of molecules of type i = 1, 2, . . . , 2 evolve in time according to the following differential equations [1,2] dxi dt = ∑ j Wijxj − [Di + Φ(t)] xi , (1) where the constants Di stand for the death probability of molecules of type i, and Φ(t) is a dilution flux that keeps the total concentration constant. This flux introduces a nonlinearity in (1), and is determined by the condition ∑ i dxi/dt = 0. More pointedly, assuming Di = 0 for all i and ∑ i xi = 1 yields Φ = ∑ i,j Wijxj . (2) The elements of the replication matrix W are given by Wii = Ai q L (3) and Wij = Ai q L−d(i,j) (1− q) i 6= j, (4)