Evolutionary Models for Simple Biosystems - Biological Networks - page 338

In the real world, only a certain kind of mutations are possible, and in this
case M ` becomes a sparse matrix M ` . We introduce a sparseness index s which
is the average number of nonzero o-diagonal elements of M ` . The sum of these
o-diagonal elements still gives ` . In this case M ` is a quenched sparse matrix,
and M ` can be considered the average of the annealed version.
Both M s and M ` are Markov matrices. Moreover, they are circular matrices,
since the value of a given element does not depend on its absolute position but only
on the distance from the diagonal. This means that their spectrum is real, and
that the largest eigenvalue is 0 = 1. Since the matrices are irreducible, the corre-
sponding eigenvector 0 is non-degenerate, and corresponds to the at distribution
0 (x) = 1=2 L .
15.2.4. The phenotype
The phenotype of an individual is how it appears to others, or, better, how it may
aect others. Clearly, the phenotype concerns the characteristics of one's body,
but may also include other \extended" traits, like the dam for beavers, termite
nests, or the production of oxygen that nally leaded ancient reductive organisms
to extinction.
This function is in general rather complex: genes interact among themselves in
an intricate way (epistatic interactions), and while it is easy to \build" a bad gene
(for instance, a gene that produces a misfolded (incorrectly folded) protein, or a
working protein that interferes with the biochemistry of the cell), a \good" gene is
good only as long there is cooperation with other genes. This is the main reason
of the \general" failure of genetic engineering: it is dicult to design a gene that
produces the desired result (and indeed this is never attempted: one generally tries
to move a gene with some eect from an organism to another), but it is yet more
dicult to avoid this gene to interfere with the rest of machinery. Only some genetic
modications do not lead to drastic lowering of the output, and most of \successful
stories" of transgenic plants concerns genes coding for proteins that inhibit some
specic poison.
Genes that have additive eects on the phenotype are called \non-epistatic";
this in general holds only for a given phenotypic trait.
Whithin our modeling, we represent the phenotype of an individual as a set
of quantitative characters u, that depend on the genotype, the age and on past
experiences. A simplication that ease the treatment of the subject consists in
assuming that the phenotype is just a function of the genotype:
u = u(g) :
With this assumption, the tness can be considered a function of the genotype,
without introducing the phenotype [9{11].
As an example of phenotypes that depend on accumulated characteristics (age),
we shall briey discuss the Penna model in Section 15.4.