In collaboration
with sam512, BaronWR, StrawberryFrog and
Andrew Aguecheek
Huser's write-up above is completely wrong,
for the reason that sam512 explains: the genes are
discrete, and cannot be mixed. However, the analysis
in the original write-up is not entirely correct either,
since it uses an insufficiently accurate model for
genetic inheritance.
Genetic model
The human DNA blueprint consists of a large number
of genes. Each gene can occur in one or more versions
called alleles. One can think of each gene as encoding
some characteristic, e.g. eye colour, and of the alleles
as the possible values of this characteristic, e.g.
"blue eyes" or "brown eyes". (Actually, eye colour is
not determined by a single gene, but that's beside
the point.)
Each gene is located on a chromosome. Humans have 23
pairs of chromosomes, and both chromosomes in a pair
contain alleles for the same genes. Any individual
therefore has two alleles of each gene (except for
genes on the sex chromosomes, as explained below).
When a male and a female reproduce,
their offspring inherits exactly one allele from each
parent for each gene. The inherited allele is selected
randomly from each parent's pair. The
details of the process are irrelevant to the
model.
The above holds for 22 of the 23 pairs, the so-called
autosomes. The last pair are the sex chromosomes.
Females have a pair of X chromosomes, while males have
one X and one Y chromosome. Essentially (outside a small
pseudoautosomal region) the X and Y chromosome are completely
different, so there are X-linked and Y-linked genes. Offspring will essentially
inherit either the X or Y chromosome of the father in its
entirety, while the alleles in the X chromosome inherited from the mother
are selected randomly from the mothers pair. A male will therefore
have only a single allele of the X-linked genes, and
this allele is always inherited from the mother. On the other hand,
the Y-linked genes of a male are inherited directly from the
great...grandfathers, modulo
mutations. Most of the
Y-linked genes are insignificant junk. Females have pairs of alleles also for the
X-linked genes; one of them is the father's unique allele,
while the other is selected randomly from the mother's
pair.
Fry's inheritance
Now let us consider a some specific gene in Fry's
genetic make-up. X-linked genes are uninteresting as they
inherited from his mother. Y-linked genes are inherited from
his grandfather, i.e. himself, giving rise to a
causal loop. Most of these genes are insignificant though.
The most fun case is therefore the autosomal genes.
Call Philip J. Fry's pair of alleles for the gene
PF and PM, where PF is
the allele inherited from Yancy and PM the one
from Fry's mother. Similarly, let YF be Yancy's
allele inherited from Fry, and YM the one he
inherits from Mildred. There are two equally likely
probably possibilities:
- PF := YM (the notation ":=" indicates a causal relationship: PF depends causally on YM). Fry inherits an allele from his paternal grandmother, just like any human would.
- PF := YF. Fry inherits a gene from his paternal grandfather, i.e. himself. There are two possibilities, each with probability 25%:
- YF := PM. Then PF = PM, i.e. Fry has inherited the same allele from his mother in two different ways. As will be explained below, this represents a degree of inbreeding.
- YF := PF. Then PF := PF, so there is a causal loop.
Inbreeding
As is commonly known, it is a bad thing for closely
related individuals to produce offspring. The reason is
that many genetic illnesses are caused by
recessive alleles. This means that
individuals with one "bad" allele B and one "good"
allele G will be healthy, but B-homozygotes, i.e.
individuals with two B alleles, are affected by the
disease. Inbred offspring are more likely to be
homozygotes for rare alleles, and thus more likely to be
diseased.
To see why, consider for example the case of two
siblings reproducing. Then each allele of the offspring
is seleceted uniformly among the 4 alleles (total) of its
two grandparents. 75% of the time the two alleles are
inherited from different places, and may or may not be
equal. But there is a 25% chance that the two alleles
are inherited from the same place, and are therefore
tautologically equal.
Hence, if a a certain allele B occurs in the general
population with
frequency 1 in 1,000, then a generic
individual has a 1 in 1,000,000 probability of being a
B-homozygote, while the offspring of a pair of siblings
has a 1 in 4,000 probability of being a B-homozygote.
Conclusion
Fry's X chromosome is normal, while (essentially)
all of his Y chromosome comes from a causal loop.
The Y-linked part of Fry's DNA could be absolutely anything,
but since there are few significant genes in the Y-linked
part it is unclear how important this is.
Of the remaining genes, 50% are normal. 25% have a
tautologically equal pair of alleles, so Fry is as
inbred as the offspring of a pair of siblings. The
remaining 25% of the genes include one allele that
exists in a causal loop.