CHAPTER 1
Introduction
In his famous paper [13], Eigen introduced a model for the evolution of a pop-
ulation of macromolecules. In this model, the macromolecules replicate themselves,
yet the replication mechanism is subject to errors caused by mutations. These two
basic mechanisms are described by a family of chemical reactions. The replication
rate of a macromolecule is governed by its fitness. A fundamental discovery of Eigen
is the existence of an error threshold on the sharp peak landscape. If the mutation
rate exceeds a critical value, called the error threshold, then, at equilibrium, the
population is completely random. If the mutation rate is below the error threshold,
then, at equilibrium, the population contains a positive fraction of the master se-
quence (the most fit macromolecule) and a cloud of mutants which are quite close
to the master sequence. This specific distribution of individuals is called a quasis-
pecies. This notion has been further investigated by Eigen, McCaskill and Schuster
[15] and it had a profound impact on the understanding of molecular evolution
[11]. It has been argued that, at the population level, evolutionary processes select
quasispecies rather than single individuals. Even more importantly, this theory
is supported by experimental studies [12]. Specifically, it seems that some RNA
viruses evolve with a rather high mutation rate, which is adjusted to be close to an
error threshold. It has been suggested that this is the case for the HIV virus [38].
Some promising antiviral strategies consist of using mutagenic drugs that induce
an error catastrophe [2,8]. A similar error catastrophe could also play a role in the
development of some cancers [36].
Eigen’s model was initially designed to understand a population of macro-
molecules governed by a family of chemical reactions. In this setting, the number
of molecules is huge, and there is a finite number of types of molecules. From
the start, this model is formulated for an infinite population and the evolution is
deterministic (mathematically, it is a family of differential equations governing the
time evolution of the densities of each type of macromolecule). The error threshold
appears when the number of types goes to ∞. This creates a major obstacle if
one wishes to extend the notions of quasispecies and error threshold to genetics.
Biological populations are finite, and even if they are large so that they might be
considered infinite in some approximate scheme, it is not coherent to consider situ-
ations where the size of the population is much larger than the number of possible
genotypes. Moreover, it has long been recognized that random effects play a major
role in the genetic evolution of populations [24], yet they are ruled out from the
start in a deterministic infinite population model. Therefore, it is crucial to develop
a finite population counterpart to Eigen’s model, which incorporates stochastic ef-
fects. This problem is already discussed by Eigen, McCaskill and Schuster [15] and
more recently by Wilke [41]. Numerous works have attacked this issue: Demetrius,
Schuster and Sigmund [9], McCaskill [27], Gillespie [19], Weinberger [40]. Nowak
1
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