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cognate tRNAs) for various proteins. One can see that the time obtained shifting
the frame of +1 or +2 basis, that produces a meaningless protein, is substantially
higher for ribosomal protein, and similar to the correct one for proteins carried by
plasmids [47].
We can simplify the problem by grouping \abundant" tRNAs by the symbol 0,
and \rare" tRNAs by 1. Disregarding \intermediate" tRNAs, we get a sequence x
of zeros and ones, and a tness, assumed monotonic with the replication time
1
,
that has the form
!
X
A(x) = exp(
1
(x)) = exp
A + B
x
i
;
i
i.e., a monotonic function of the \magnetization" of an Ising chain. Here the codons
does not interact, and their eect is additive. If one considers that the discharging
of a tRNA may slower the translation of cognate codons, one may introduce other
couplings among codons [48, 49].
15.3.8. Finite populations and random drift
In nite population, the uctuations plays a fundamental role. Let us just con-
sider a simple system x composed by N individuals belonging either to variant (or
species) A (x = 0) or B (x = 1). At each time step an individual x
i
is chosen
at random, replicates with a probability p(x
i
) and substitutes another randomly
chosen individual (Moran process [50]). We have a birth-death process of the quan-
tity n =
P
i
x
i
with two absorbing states, n = 0 and n = N corresponding to the
extinction of B and A, respectively. It can be shown [51] that the probability S(1)
that a single mutant B will take over the whole population is
1
N
;
where r = p(1)=p(0) is the selective advantage (or disadvantage) of variant B with
respect to A. If '0 is the \mutation probability" from A to B, the probability
of appearance of a mutant in a population of N A's is N, and therefore the
probability of xation of a mutant is Q = NS(1), for neutral selection (r = 1) we
get Q = , irrespective of the population size [52]. This result by Kimura provides
the molecular clock for estimating evolution times from the measure of the distance
between two genomes (with the assumption of neutral evolution).
1r
1r
N
!
S(1) =
r!1
15.3.9. Speciation in a at landscape
The reasons for the existence of species is quite controversial [53]. The denition
itself of a species is not a trivial task. One denition (a) is based simply on the
phenotypic dierentiation among the phenotypic clusters, and can be easily applied
also to asexual organisms, such as bacteria. Another denition (b), that applies
only to sexual organism, is based on the inter-breeding possibility. Finally (c), one
