**Mutations**
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

*'p*_{0}' be the initial frequency of

*'A'* and

*'u'* be the mutation rate with which

*'A'* changes to

*'a'.* In such a case, 'a' will appear with a frequency of

*u* x

*p*_{0}in the first generation. The frequency of

*'A'* will, therefore, be reduced by a factor

*p*_{0}*u* and become

*p*_{0}*-p*_{0}*u* =

*p*_{0}(l-

*u*). In the next generation there will be further change due to the change of

*'A'* to

*'a'* thus further reducing the frequency of

*'A'* by a factor

*p*_{0}(l-

*u*) x

*u**,* so that the earlier frequency

*p*_{0}(l-

*u*)will now become

*p*_{0}(l-

*u*) -

*p*_{0}(l-

*u*) x

*u* =

*p*_{0}(l-

*u*)(l-

*u*) =

*p*_{0}(l-

*u*)

^{2}*.* In this manner, in

*'n'* generations, the frequency of

*'A'* will be reduced to

*p*_{0}(l-

*u*)

^{n}*.* Eventually, even if the value of

*u* is small, the term (l-

*u*)

^{n} 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'* and

*'a'* are

*'p*_{0}*'* and

*'q*_{0}*'* respectively, in one generation the frequency of

*'A'* will become

*p*_{0}*+ vq*_{0 }*- up*_{0}and that of

*'a'* will become

*q*_{0}*+ up*_{0 }*- vq*_{0}*.* It is obvious that

*'a'* gains a fraction

*up*_{0}and loses a fraction

*vq*_{0}at the same time. Similarly,

*'A'* gains a fraction

*vq*_{0}and loses a fraction

*up*_{0}*.* 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 Δ

*q = up*_{0}-

*vq*_{0}*.* If

*p*_{0} is relatively larger than

*q*_{0}*,* and

*u* is relatively larger than

*v*, Δ

*q* would be fairly high and the frequency of

*'a'* i.e.,

*'q'* 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 Δ

*q'* would become zero. At mutational equilibrium, the expected frequency of

*'a'* i.e.,

*q*^ can be expressed as