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and reproduce, the others tend to disappear. By doing so, the average tness in
general increases, so that a genome that is good at a certain time will become more
common and the relative tness more similar to the average one. The eects of
this generic tendency depend on the form of the tness. For instance, for ospring
production, it is more convenient to invest in females than in males. But as soon
as males become rare, those that carry a gene that increases their frequency in the
population will be more successful, in that the females fertilized with this gene will
produce more males, that in average will fertilize more females and so on.
The presence of sexual reproduction itself is dicult to be interpreted as an
optimization process. As reported in the Introduction, the ospring production
of sexual species is about a half of that of asexual ones. The main advantage
of sexual reproduction (with diploidicity) is that of maintaining a genetic (and
therefore genotypic) diversity by shuing paternal and maternal genes, without the
risk associated with a high mutation rate: the generation of unviable ospring, that
lead to the error threshold or to the mutation meltdown, Section 15.3.3. This genetic
diversity is useful in colonizing new environments or for variable ones [29], but is
essential to escape the exploitation from parasites, that reproduce (and therefore
evolve) faster than large animals and plants [30].
Therefore, sex can be considered an optimization strategy only for variable envi-
ronments. The fact that an optimization technique inspired by sexual reproduction,
genetic algorithms [31], is so widely used is somewhat surprising.
15.3.2. Evolution and statistical mechanics
When one takes into consideration only point mutations (MM
s
), Eq. (15.7)
can be read as the transfer matrix of a two-dimensional Ising model [14, 15, 32],
for which the genotypic element x
(t)
i
corresponds to the spin in row t and column i,
and z(x; t) is the restricted partition function of row t. The eective Hamiltonian
(up to constant terms) of a possible lineage x = (x
(t)
)
t
= 1
T
from time 1tT
is
!
T1
X
X
L
x
(t)
i
x
(t+1)
i
+ H(x
(t)
)
H=
;
(15.9)
t=1
i=1
where =ln(
s
=(1
s
)).
This peculiar two-dimensional Ising model has a long-range coupling along the
row (depending on the choice of the tness function) and a ferromagnetic coupling
along the time direction (for small short range mutation probability). In order
to obtain the statistical properties of the system one has to sum over all possible
congurations (stories), eventually selecting the right boundary conditions at time
t = 1.
The bulk properties of Eq. (15.9) cannot be reduced in general to the equilibrium
distribution of an one-dimensional system, since the transition probabilities among
rows do not obey detailed balance. Moreover, the temperature-dependent Hamilto-
