Probability
Now for a short diversion. As we've seen, either one of the two
chromosomes of each pair can randomly be selected to move into a
newly forming sperm or egg. We're going to be interested in what
happens to these sperm and eggs when they meet. We're going to want
to calculate what types of progeny will be produced and in what
proportion they will appear in subsequent generations. The branch of
mathematics used to calculate these values is called probability.
Don't worry. We're going to consider only a tissue-thin piece of this
branch of mathematics.
At the beginning of a football game, the officials toss a coin. Most coins have two sides, "heads" and "tails", and neglecting the unlikely events that the coin ends up on its edge or gets lost, there are two possible outcomes. If the two outcomes are equally likely, the chances of getting one or the other is 1 out of 2. Or, put another way, if 1.0 is certainty and 0.0 is no chance, then the probability of one of two outcomes is 0.5.
On a roll of a die (the singular of "dice"), there are six possible outcomes. Each of these is equally likely so the chance of rolling any one particular number if 1 out of 6, or about 0.166.
Do you see a pattern here? When any of a given number of outcomes are all independent of each other and equally likely, the chance of any one of them occurring is one out of the total number of outcomes. Or, in other words, if you want to calculate the chances of some event occurring, you total up all the independent events that could occur, and the chances of the particular event is 1 out of the sum of those independent events.
Two events happening together
What if you want to calculate the chances of tossing two dice
together to throw a 12? Whenever you need to calculate an "and"
outcome as in figuring out the chances of a six on one die and
a second six coming up on another in a single roll of two dice, you
multiply probabilities. Thus the chance of throwing a six from one
die is 1 out of six, and the chance of throwing a six from the second
die is 1 out of 6. And the chance of throwing a six and a six if 1
out of 36 or 1/6 times 1/6. Similarly, the chances of throwing a
single head is 1/2, and the chances of throwing two consecutive heads
(one head and then another), is 1/2 times 1/2 or 1/4.
One of several events
What if you want to calculate the chances of throwing a 12 or a 2?
Whenever you want to calculate an "or" outcome as in figuring out the
chances of a 12 coming up with one toss of the dice or a 2 coming up
you add probabilities. Thus, the chance of throwing a 12 is 1/36 and
the chance of throwing a 2 is 1/36. Therefore the chance of throwing
a 12 or a 2 is 1/36 + 1/36 or 2/36 (1 out of 18).
Heterozygotes produce sex cells with one or the other allele
OK. We're back to biology again. With this background, lets see
what happens when two flies, each of which carries one defective
Adh gene, are allowed to mate. What will be the consequences?
As we saw above, each individual will pass on one of their two second
chromosomes in their eggs or sperm. Since the process of which second
chromosome is chosen for any particular sperm or egg is random, some
sperm of the father will carry a defective Adh gene; others
will carry a functional one. The same goes for the eggs of the
mother. In fact, about half the sperm will carry a second chromosome
with a defective Adh gene and half will not. Similarly about
half of the eggs will carry a second chromosome with a defective
Adh gene and half will not. (To picture this situation,
imagine that each chromosome set in the parents is either blue or
pink. In addition, one of the second chromosomes is defective in the
Adh gene. During meiosis, one second chromosome -- either the
defective one, or the other-- will sometimes be included in a
specific sperm or egg.)
The results of crossing two heterozygotes
It's a matter of chance also whether a sperm with a defective
Adh gene happens to fertilize an egg with a defective
Adh gene. What are the chances of this happening? On the
average about one quarter of the offspring will carry two defective
Adh genes (How many will carry two normal Adh genes?
How many will carry one normal and one defective Adh gene?) To
get this value, we use the same reasoning as we did concerning coin
tosses. Remember, the chances of getting two heads in two consecutive
coin tosses is 1/4. If you do this calculation, you'll see that the
other 3/4 of the flies produced from this cross have either one or
two copies of the normal Adh gene are therefore resistant to
ordinary concentrations of ethanol.
The 3:1 ratio
Those flies that happen to be born with two defective Adh
genes will be incapable of producing a normal, functional alcohol
dehydrogenase enzyme. In the presence of ethanol, they will get drunk
and die. Thus, from the union of two flies that were able to survive
on ethanol, it is possible to get (in about one fourth of the
offspring), flies that are unable to survive because they have two
defective Adh genes. This ratio of 3 normal flies to 1
abnormal occurs whenever two heterozygotes are crossed.
Variation
Flies live on rotting fruit in the wild and its important for them
to have a working Adh gene in order to survive the high
concentrations of ethanol that they encounter. It's easy to see that
a defective Adh gene is detrimental to the organism, and that
such mutations are harmful. In this case, the mutant form of the gene
is clearly the exceptional and rare case, while the functional form
-- sometimes called wild-type -- is the normal. Yet in other cases,
it isn't clear which is which. For instance in humans, there are
individuals with blue eyes and brown eyes. Which are the mutants and
which are the wild-types? How about blond hair and brown? Type AB
blood or O? Many genes exist in the population in multiple forms --
alleles -- in many individuals. What we're trying to say is that the
science of genetics doesn't only deal with rare mutations, but also
with the large amount of normal variation in genes that exist in
population. In those cases where the variation is due to a single
gene with one or more alleles, individuals will pass on their genes
exactly as outlined above.
Crossing flies with variation in two different genes
What happens if you cross two flies together that show variation
in not one gene, but two? The answer is that it depends whether the
two genes are located on the same chromosome or whether they're
located on different ones. Taking the second case first, let's
examine the consequences of crossing two flies that are defective in
their Adh and rosy genes. (rosy is the name of a
gene that is located on the third chromosome). It encodes the enzyme
xanthine dehydrogenase or XDH. XDH is involved, among other things,
with the synthesis of some of the eye pigments of Drosophila, and
when a fly carries two defective gene its eyes take on a different
color than normal. Notice that here, for reasons similar to the case
of Adh above, individuals with just one defective rosy
gene have normal colored eyes. Apparently, having half of the normal
number of genes, and therefore about half the normal amount of XDH
enzyme, is enough for the fly to form normal amounts of pigments.
Thus rosy mutants are recessive.
Two different genes on different chromosomes
The situation is shown in the Figure at the right.
Again, we
have a fly that has one defective Adh gene (on one of its
second chromosomes -- shown as gene 1). This time, in addition, the
same fly carries one defective gene that encodes XDH on one of its
third chromosomes (shown as gene 2). During the process of formation
of the sex cells, one of the two second chromosomes and one of the
two third chromosomes find themselves in an individual sperm or egg
cell. As previously stated, about half the sperm (or eggs) will have
a defective Adh gene and half will bear a normal one.
Similarly, about half the sperm (or eggs) will have a defective
rosy gene; half will have a normal one. One key question is,
how many will carry both a defective Adh and a defective
rosy gene?
The two homologous chromosomes move independently of each
other
Since the Adh gene and the rosy gene aren't on the
same chromosome, their chances of ending up in the same sperm or egg
are independent of each other. One exercise worth doing is to draw
all the possible combinations of second and third chromosomes that
could form in this situation. As shown in the Figure below, there are
four. And only one of the four has two defective genes in it (the
combination at the extreme right of the figure. The green and red
bars are supposed to represent defective genes). Therefore the
chances of a sex cell carrying two defective genes, one for
Adh and other for rosy, is one out of four.
Crossing flies with two traits
Now for a harder question. If you cross two flies, each of which
has one defective Adh gene and one defective Adh gene,
what are the chances of having an offspring with two defective
Adh genes and two defective rosy genes? The answer here
can also be obtained by simply listing all of the possible
combinations that are equally likely. Then find the ones that where
there are two defective Adh genes and two defective
rosy genes. As we've seen, each sperm or egg can carry four
possible combinations of Adh and rosy genes (normal
Adh, normal rosy; defective Adh, normal
rosy; normal Adh, defective rosy; and defective
Adh, defective rosy). If two such individuals mate,
they can form sixteen possible offspring. For practice, you should
list all the possibilities, and indicate whether each will get drunk
in the presence of alcohol and whether its eyes will be normal. Keep
in mind that it takes two defective Adh genes for a fly to get
drunk on ethanol and two defective rosy genes for a fly to
bear an abnormal eye color.
