6 1. INTRODUCTION

for a quasispecies to emerge and we recover the finite population counterpart of

the error threshold. Moreover, in the regime of very small mutations, we obtain

a lower bound on the population size allowing the emergence of a quasispecies: if

α ln 4/ ln σ then the equilibrium population is totally random, and a quasispecies

can be formed only when α ≥ ln 4/ ln σ. Finally, in the limit of very large popu-

lations, we recover an error catastrophe reminiscent of Eigen’s model: if σe−a ≤ 1

then the equilibrium population is totally random, and a quasispecies can be formed

only when σe−a 1. These results are supported by computer simulations. The

good news is that, already for small values of , the simulations are very conclusive.

Master

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Figure 3. Simulation of the equilibrium density of the Master sequence

It is certainly well known that the population dynamics depends on the population

size (see the discussion of Wilke [41]). In a theoretical study [30], Van Nimwe-

gen, Crutchfield and Huynen developed a model for the evolution of populations

on neutral networks and they show that an important parameter is the product of

the population size and the mutation rate. The nature of the dynamics changes

radically depending on whether this product is small or large. Sumedha, Martin

and Peliti [37] analyze further the influence of this parameter. In [39], Van Nimwe-

gen and Crutchfield derived analytical expressions for the waiting times needed to

increase the fitness, starting from a local optimum. Their scaling relations involve

the population size and show the existence of two different barriers, a fitness barrier

and an entropy barrier. Although they pursue a different goal than ours, most of

the heuristic ingredients explained previously are present in their work, and much

more; they observe and discuss also the transition from the quasispecies regime for

large populations to the disordered regime for small populations. The dependence