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X
u(x) =
J fi 1 :::i K g x i 1 x i K ;
(15.8)
fi 1 :::i K g
where J fi 1 :::i K g , for each dierent set of indexesfi 1 : : : i K g, are independent, iden-
tically distributed, random variables, so that for every K we are dealing with a
random ensemble of phenotypes (and of tness). The larger K, the faster the t-
ness correlations decay in sequence space, so that the tness landscape is less and
less correlated, i.e., as it is usually said, more and more rugged. In the limit k!1,
one has a complete disordered tness landscape, where the values of the tness at
dierent positions in sequence space are independent random variables (random
energy model [25]). For K = 2 this model is similar to the one introduced by
Kauman [26] for modeling the genetic network of a cells.
On an extremely rugged landscape one observes an error threshold transi-
tion [17, 27] between a state in which the population distribution is concentrated
on local maxima (localized phase), and a wide distribution. Due to the disorder,
the distribution in the localized phase depends on the initial conditions, i.e., on the
past history of the population.
15.3.1. Evolution and optimization, replicator equation
The evolution may appear as an optimization process. Indeed, if we neglect muta-
tions, Eq. (15.5) becomes
p 0 = A
hAi p ;
called the replicator equation (discrete time).
Assume that we start from a uniform distribution over the whole phenotypic
space, and that the tness shows a single, smooth maximum. The phenotypes u
with A(u) <hAitend to decrease in frequency, while those with A(u) >hAitend
to increase their frequency. Because of this, the average tnesshAiincreases with
time (Fisher theorem [28]). Notice that this results may be compatible with the
extinction of the population, either due to chaotic oscillations for high reproductive
rates, oscillations that may substantially reduce the population to a level in which
random sampling is important, or becausehAiis below one in Eq. (15.4). An
example of this is given by genes that alter the sex ratio. These genes are able
to \kill" male embryos, and thus increase their own population (since they reside
on the female chromosome) although the nal fate of the populatiion is extinction.
This eect may be the origin of the smallness of male Y chromosome in mammals:
a smaller chromosome implies less genes that may be targeted by the proteins
produced by the killer genes.
The structure of Eq. (15.5) says that the tness does not have absolute values,
except that the average tness (the basic reproductive rate) has to be greater than
one. For a given genome, the survival of individuals depends on their relative
tness: those that happen to have a tness larger than average tend to survive
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