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which the j–th chromosome x(j) has been replaced by u: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.