1. INTRODUCTION 3
region consisting of the populations devoid of master sequences. The process will
wander randomly throughout this region for a very long time. We call this time
the discovery time of the master sequence. Because the cardinality of the possible
genotypes is enormous, the master sequence is difficult to discover, nevertheless the
mutations will eventually succeed and the process will start again with a population
containing exactly one master sequence. If, on average, the discovery time is much
larger than the persistence time, then the equilibrium state will be totally random,
while a quasispecies will be formed if the persistence time is much larger than the
discovery time. Let us illustrate this idea in a very simple model.
0 1 2
· · ·
i 1 i i + 1
· · ·
1
θ
2
1
θ
2
1
2
1
2
1
2
1
2
Figure 1. Random walk example
We consider the random walk on { 0,..., } with the transition probabilities de-
pending on a parameter θ given by:
p(0, 1) =
θ
2
, p(0, 0) = 1
θ
2
, p(, 1) = p(, ) =
1
2
,
p(i, i 1) = p(i, i + 1) =
1
2
, 1 i 1 .
The integer is large and the parameter θ is small. Hence the walker spends its
time either wandering in { 1,..., } or being trapped in 0. The state 0 plays the
role of the quasispecies while the set { 1,..., } plays the role of the neutral region.
With this analogy in mind, the persistence time is the expected time of exit from
0, it is equal to 2/θ. The discovery time is the expected time needed to discover 0
starting for instance from 1, it is equal to 2 . The equilibrium law of the walker is
the probability measure μ given by
μ(0) =
1
1 + θ
, μ(1) = · · · = μ( ) =
θ
1 + θ
.
We send to and θ to 0 simultaneously. If θ goes to ∞, the entropy factors
win and μ becomes totally random. If θ goes to 0, the selection drift wins and μ
converges to the Dirac mass at 0.
In order to implement the previous heuristics, we have to estimate the per-
sistence time and the discovery time of the master sequence in the Moran model.
For the persistence time, we rely on a classical computation from mathematical
genetics. Suppose we start with a population containing m 1 copies of the master
sequence and another non master sequence. The non master sequence is very un-
likely to invade the whole population, yet it has a small probability to do so, called
the fixation probability. If we neglect the mutations, standard computations yield
that, in a population of size m, if the master sequence has a selective advantage of
σ, the fixation probability of the non master sequence is roughly of order
1/σm
(see
for instance [31, Section 6.3]). Now the persistence time can be viewed as the time
needed for non master sequences to invade the population. This time is approxi-
mately equal to the inverse of the fixation probability of the non master sequence,
Previous Page Next Page