10
RAPHA¨
EL CERF
where M(u, v) is the probability that the chromosome u is transformed by mutation
into the chromosome v. The analytical formula for M(u, v) is then
M(u, v) =
j=1
(1 q)1u(j)=v(j) +
q
κ 1
1u(j)=v(j) .
Replication. The replication favors the development of fit chromosomes. The
fitness of a chromosome is encoded in a fitness function
A : A [0, +∞[ .
The fitness of a chromosome can be interpreted as its reproduction rate. A chro-
mosome u gives birth at random times and the mean time interval between two
consecutive births is 1/A(u). In the context of Eigen’s model, the quantity A(u)
is the kinetic constant associated to the chemical reaction for the replication of a
macromolecule of type u.
Authorized changes. In our model, the only authorized changes in the population
consist of replacing one chromosome of the population by a new one. The new
chromosome is obtained by replicating another chromosome, possibly with errors.
We introduce a specific notation corresponding to these changes. For a population
x
(
A
)m
, j { 1,...,m }, u A , we denote by x(j u) the population x in
which the j–th chromosome x(j) has been replaced by u:
x(j u) =





⎜x(j



⎜x(j


x(1)
.
.
.

1)⎟⎟
u
+
1)⎟⎟
.
.
.
x(m)








We make this modeling choice in order to build a very simple model. This type
of model is in fact classical in population dynamics, they are called Moran models
[17].
The mutation–replication scheme. Several further choices have to be done to
define the model precisely. We have to decide how to combine the mutation and
the replication processes. There exist two main schemes in the literature. In the
first scheme, mutations occur at any time of the life cycle and they are caused by
radiations or thermal fluctuations. This leads to a decoupled Moran model. In the
second scheme, mutations occur at the same time as births and they are caused
by replication errors. This is the case of the famous Eigen model and it leads
to the Moran model we study here. This Moran model can be described loosely
as follows. Births occur at random times. The rates of birth are given by the
fitness function A. There is at most one birth at each instant. When an individual
gives birth, it produces an offspring through a replication process. Errors in the
replication process induce mutations. The offspring replaces an individual chosen
randomly in the population (with the uniform probability).
We build next a mathematical model for the evolution of a finite population
of size m on the space A , driven by mutation and replication as described above.
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