4 1. INTRODUCTION
that is of order
σm.
For the discovery time, there is no miracle: before discovering
the master sequence, the process is likely to explore a significant portion of the
genotype space, hence the discovery time should be of order
card { A, T, G, C } = 4 .
These simple heuristics indicate that the persistence time depends on the selection
drift, while the discovery time depends on the spatial entropy. Suppose that we send
m, to simultaneously. If the discovery time is much larger than the persistence
time, then the population will be neutral most of the time and the fraction of
the master sequence at equilibrium will be null. If the persistence time is much
larger than the discovery time, then the population will be invaded by the master
sequence most of the time and the fraction of the master sequence at equilibrium
will be positive. Thus the master sequence vanishes in the regime
m, +∞ ,
m
0 ,
while a quasispecies might be formed in the regime
m, +∞ ,
m
+∞ .
This leads to an interesting feature, namely the existence of a critical population size
for the emergence of a quasispecies. For chromosomes of length , a quasispecies
can be formed only if the population size m is such that the ratio m/ is large
enough. In order to go further, we must put the heuristics on a firmer ground and
we should take the mutations into account when estimating the persistence time.
The main problem is to obtain finer estimates on the persistence and discovery
times. We cannot compute explicitly the laws of these random times, so we will
compare the Moran model with simpler processes.
In the non neutral populations, we shall compare the process with a birth and death
process (Zn)n≥0 on { 0,...,m }, which is precisely the one introduced by Nowak
and Schuster [32]. The value Zn approximates the number of copies of the master
sequence present in the population. For birth and death processes, explicit formulas
are available and we obtain that, if , m +∞, q 0, q a ∈]0, +∞[, then
persistence time exp
(
m φ(a)
)
,
where
φ(a) =
σ(1
e−a)
ln
σ(1
e−a)
σ 1
+
ln(σe−a)
(1 σ(1
e−a))
.
In the neutral populations, we shall replace the process by a random walk on
{ A, T, G, C } = 4 . The lumped version of this random walk behaves like an
Ehrenfest process (Yn)n≥0 on { 0,..., } (see [5] for a nice review). The value Yn
represents the distance of the walker to the master sequence. A celebrated theorem
of Kac from 1947 [21], which helped to resolve a famous paradox of statistical
mechanics, yields that, when ∞,
discovery time 4 .
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