**Application to
Genetics**

** **

** **

Living things inherit
from their parents many of their physical characteristics. The genes of the
parents determine these characteristics. The study of these genes is called **Genetics**; in other words genetics is
the branch of biology that deals with heredity. In particular, **population genetics** is the branch of
genetics that studies the genetic structure of a certain population and seeks
to explain how transmission of genes changes from one generation to another.
Genes govern the inheritance of traits like sex, color of the eyes, hair (for
humans and animals), leaf shape and petal color (for plants).

There are several types of inheritance; one of particular interest for us is
the **autosomal **type in which each
heritable trait is assumed to be governed by a single gene. Typically, there
are two different forms of genes denoted by *A
*and *a*. Each individual in a
population carries a pair of genes; the pairs are called the individual’s **genotype. **This gives three possible
genotypes for each inheritable trait: *AA*,
*Aa*, and *aa* (*aA* is genetically the
same as *Aa*).

** Example **in a certain
animal population, an autosomal model of inheritance controls eye colouration.
Genotypes

Each offspring inherits one gene from each parent in a random manner. Given the genotypes of the parents, we can determine the probabilities of the genotype of the offspring. Suppose that, in this animal population, the initial distribution of genotypes is given by the vector

_{}

where the components represent the fraction of animals of genotypes *AA*, *Aa*,
and *aa* initially. Let us consider a series of experiments in which we
keep crossing offspring with dominant animals only. Thus we keep crossing *AA, Aa, *and *aa* with *AA*. We are
interested in the probabilities of the offspring being *AA, Aa*, or *aa* in each of
these cases.

·
Consider the crossing of *AA* with *AA. *Since the
offspring will have one gene from each parent, it will be of type *AA.* Thus the probabilities of *AA, Aa, *and aa resulting are 1, 0 and 0
respectively. All offspring have brown eyes.

·
Next consider the crossing of *Aa *with *AA. *Taking one
gene from each parent, we have the possibilities of *AA, AA *(taking *A* from the
first parent and each *A* in turn from
the second parent), *aA*, and *aA* (taking *a* from the first parent and each *A* in turn from the second parent). Thus the probabilities of *AA, Aa,* and *aa*, respectively, are_{}_{} and 0. All offspring again have brown eyes.

·
Remains to consider the crossing of genotype *aa* with *AA.* there is only a possibility, namely *aA*. Thus the probabilities of *AA*,
*Aa*, and *aa* are 0, 1 and 0 respectively.
No offspring has blue eyes.

We conclude that crossing with
genotype *AA* will produce offspring
with brown eyes only. Our next step is to examine how the above fractions of
initial genotypes will change from one generation to another. For this, we let *X _{n}* be the distribution vector
of genotypes in the

_{}

is called the transition matrix. In general, *X _{n}=AX_{n-1}*. Explicitly, we have:

_{}

Observe that the *aa* type disappears after the initial generation and that the *Aa *type becomes a smaller and smaller
fraction of each successive generation. It is obvious that this sequence of
vectors converges to the vector

_{}

in the long run.

Now try to create a similar model for crossing offspring with a hybrid animal. You will see that some offspring will have brown eyes and some blue eyes.

If you have already seen the application ``Markov
chains'', then you would be able to recognize the vector *X * above as an eigenvector of the matrix *A *corresponding to the eigenvalue 1.