2 1. INTRODUCTION
and Schuster [32] constructed a birth and death model to approximate Eigen’s
model. This birth and death model plays a key role in our analysis, as we shall
see later. Alves and Fontanari [1] study how the error threshold depends on the
population in a simplified model. More recently, Musso [29] and Dixit, Srivastava,
Vishnoi [10] considered finite population models which approximate Eigen’s model
when the population size goes to ∞. These models are variants of the classical
Wright–Fisher model of population genetics. Although this is an interesting ap-
proach, it is already a delicate matter to prove the convergence of these models
towards Eigen’s model. We adopt here a different strategy. Instead of trying to
prove that some finite population model converges in some sense to Eigen’s model,
we try to prove directly in the finite model an error threshold phenomenon. To this
end, we look for the simplest possible model, and we end up with a Moran model
[28]. The model we choose here is not particularly original, the contribution of this
work is rather to show a way to analyze this kind of finite population model.
Let us describe informally the model (see chapter 2 for the formal definition).
We consider a population of size m of chromosomes of length over the alphabet
{ A, T, G, C }. We work only with the sharp peak landscape: there is one specific
sequence, called the master sequence, whose fitness is σ 1, and all the other
sequences have fitness equal to 1. The replication rate of a chromosome is propor-
tional to its fitness. When a chromosome replicates, it produces a copy of himself,
which is subject to mutations. Mutations occur randomly and independently at
each locus with probability q. The offspring of a chromosome replaces a chromo-
some chosen at random in the population.
The mutations drive the population towards a totally random state, while the
replication favors the master sequence. These two antagonistic effects interact in a
complicated way in the dynamics and it is extremely difficult to analyze precisely
the time evolution of such a model. Let us focus on the equilibrium distribution
of the process. A fundamental problem is to determine the law of the number
of copies of the master sequence present in the population at equilibrium. If we
keep the parameters m, , q fixed, there is little hope to get useful results. In order
to simplify the picture, we consider an adequate asymptotic regime. In Eigen’s
model, the population size is infinite from the start. The error threshold appears
when goes to and q goes to 0 in a regime where q = a is kept constant. We
wish to understand the influence of the population size m, thus we use a different
approach and we consider the following regime. We send simultaneously m, to
and q to 0 and we try to understand the respective influence of each parameter on
the equilibrium law of the master sequence. By the ergodic theorem, the average
number of copies of the master sequence at equilibrium is equal to the limit, as the
time goes to ∞, of the time average of the number of copies of the master sequence
present through the whole evolution of the process. In the finite population model,
the number of copies of the master sequence fluctuates with time. Our analysis of
these fluctuations relies on the following heuristics. Suppose that the process starts
with a population of size m containing exactly one master sequence. The master
sequence is likely to invade the whole population and become dominant. Then the
master sequence will be present in the population for a very long time without
interruption. We call this time the persistence time of the master sequence.
The destruction of all the master sequences of the population is quite unlikely,
nevertheless it will happen and the process will eventually land in the neutral
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