A simple analytical framework to study the molecular quasispecies evolution of finite populations is proposed, in which the population is assumed to be a random combination of the constituent molecules in each generation; i.e., linkage disequilibrium at the population level is neglected. In particular, for the single-sharp-peak replication landscape we investigate the dependence of the error threshold on the population size and find that the replication accuracy at the threshold increases linearly with the reciprocal of the population size for sufficiently large populations. Furthermore, in the deterministic limit our formulation yields the exact steady state of the quasispecies model, indicating then that the population composition is a random combination of the molecules.

Given a sequence of $N$ positive real numbers ${a_1,a_2,..., a_N }$, the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. In the case that the $a_j$'s are statistically independent random variables uniformly distributed in the unit interval, this NP-complete problem is equivalent to the problem of finding the ground state of an infinite-range, random anti-ferromagnetic Ising model. We employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large $N$ limit. Furthermore, we calculate analytically the fraction of metastable states, i.e. states that are stable against all single spin flips, and found that it vanishes like $N^{-3/2}$.