Mutations lead to introduction of new genes leading to genetic differences. These new genes introduced due to mutations may or may not persist depending upon their utility. The gene frequencies will alpo depend upon this factor. A change in gene frequencies due to mutations will depend upon mutation rate. This can be illustrated with the help of a hypothetical example. If a dominant gene 'A'
mutates to 'a'
with no reverse mutation, then frequency of 'a'
will eventually replace 'A',
if constant mutation rate persists for a long time in a population of constant size. Quantitatively, let 'p0'
be the initial frequency of 'A'
be the mutation rate with which 'A'
changes to 'a'.
In such a case, 'a' will appear with a frequency of u
in the first generation. The frequency of 'A'
will, therefore, be reduced by a factor p0u
and become p0-p0u
). In the next generation there will be further change due to the change of 'A'
thus further reducing the frequency of 'A'
by a factor p0
) x u,
so that the earlier frequency p0
)will now become p0
) x u
) = p0
In this manner, in 'n'
generations, the frequency of 'A'
will be reduced to p0
Eventually, even if the value of u
is small, the term (l-u
will approach zero so that 'A'
will disappear after several generations, if no reverse mutation takes place and the mutant allele experiences no selection pressure against it.
However, if the reverse mutation also takes place with a frequency "v"
and the initial frequency of 'A'
respectively, in one generation the frequency of 'A'
will become p0+ vq0 - up0
and that of 'a'
will become q0+ up0 - vq0.
It is obvious that 'a'
gains a fraction up0
and loses a fraction vq0
at the same time. Similarly, 'A'
gains a fraction vq0
and loses a fraction up0.
Let us now consider the fate of the frequency of 'a'
in the following generations. Let the change in the frequency of 'a'
be represented as Δq = up0
is relatively larger than q0,
is relatively larger than v
would be fairly high and the frequency of 'a'
would increase rapidly. This will lead to a situation, where 'q'
becomes larger than 'p',
so that the value of 'vq'
will increase and that of 'up'
will decrease. As a consequence, 'q'
would diminish gradually and at a certain point 'mutational equilibrium'
will be reached where Δq'
would become zero. At mutational equilibrium, the expected frequency of 'a'
^ can be expressed as