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 'p₀' 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 p₀in the first generation. The frequency of 'A' will, therefore, be reduced by a factor p₀u and become p₀-p₀u = p₀(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 p₀(l-u) x u, so that the earlier frequency p₀(l-u)will now become p₀(l-u) -p₀(l-u) x u = p₀(l-u)(l-u) = p₀(l-u)². In this manner, in 'n' generations, the frequency of 'A' will be reduced to p₀(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 'p₀' and 'q₀' respectively, in one generation the frequency of 'A' will become p₀+ vq₀ - up₀and that of 'a' will become q₀+ up₀ - vq₀. It is obvious that 'a' gains a fraction up₀and loses a fraction vq₀at the same time. Similarly, 'A' gains a fraction vq₀and 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- vq₀. If p₀ is relatively larger than q₀, and u is relatively larger than v, Δq 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 Δq' would become zero. At mutational equilibrium, the expected frequency of 'a' i.e., q^ can be expressed as

, and with u = v the equilibrium frequencies p and q will be equal (i.e. p = q = 0.5, because p + q = 1). But the rate of achievement of equilibrium frequencies is quite slow, and can be obtained from the formula, (u+v)n = log_e {(q₀-q^)/(q_n-q^)}, where n is the number of generations required to reach a frequency q_n with an initial frequency of q₀, and e = 2.1718 (log_e = 2.303 log₁₀).