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Fig. 15.2. Scaling of
2
with the sparseness factor s, for three values of genome length L.
`
= 0:1
and
s
= 0:1.
nian (15.9) does not allow an easy identication between energy and selection, and
temperature and mutation, what is naively expected by the biological analogy with
an adaptive walk.
A Ising conguration of Eq. (15.9) corresponds to a possible genealogical story,
i.e., as a directed polymer in the genotypic space [16], where mutations play the
role of elasticity. It is natural to try to rewrite the model in terms of the sum over
all possible paths in genotypic space.
As shown in Ref. [33], in the case of long-range mutations, this scenario simpli-
es, and the asymptotic probability distribution p is proportional to the diagonal
of A
1=
`
:
H(x)
`
p(x) = C exp
;
(15.10)
i.e., a Boltzmann distribution with Hamiltonian H(x) and temperature
`
. This
corresponds to the naive analogy between evolution and equilibrium statistical me-
chanics. In other words, the genotypic distribution is equally populated if the
phenotype is the same, regardless of the genetic distance since we used long-range
mutations. The convergence to equilibrium is more rapid for rough landscapes.
For pure short-range mutations, this correspondence holds only approximately
for very weak selection or very smooth landscapes. The reason is that in this case the
coupling in the time direction are strong, and mutations couple genetically-related
strains, that may dier phenotypically.
Since short and long-range mutations have an opposite eect, it is interesting
to study the more realistic case in which one has more probable short-range mu-
tations, with some long-range ones that occur sporadically, always between the
same genomes (quenched disorder). This scenario is very reminiscent of the small-
world phenomenon [34], in which a small percentage s (sparseness) of long-range
links added to a locally connected lattice is able to change its diusional properties.
Even in the limit of s!0 (after long times), one observes a mean-eld (well-stirred)
