Mutation

Below is a flash animation I wrote last evening to demonstrait change in allele frequency due to mutation. It is still a bit buggy so it may give some funky results. For example, if there is mutation only in one direction (no reverse mutation) it will look like allele frequencies exceed 1.0. This is the allele frequency text box reporting values in scientific notation and not showing the exponent (e,g, 8.9 x 10^{-9} will show up as 8.9 making it appear that the allele frequency is greater than 1.0 which you know is not possible). The default values for mutation rates are much much greater than what would be seen in nature, which in eukaryotes has been estimated to be 2.2 x 10^{-9} mutations per base pair per year. As you play with this animation notice that as the allele frequency approaches the theoretical equlibrium point the rate of change in allele frequency slows down. Note also that allele frequency appears to not be changing at some point. This is because change is happening slowly (at decimal places to the right of what is reported.

**Things to observe:
**

a. What effect does the mutation rate have on the rate of change in allele frequency.

b. What effect does the ratio between forward and reverse muation have on the equilbrium frequency (does a mutation rate of u= 0.1 and v= 0.01 give you a different equilibrium than u = 0.01 and v = 0.001)?

c. Choose some mutation frequencies, use the formula below for determining equilibrium frequency and test your calculations.

Below the animation is the algerbra we covered in class. Make sure you understand the derivation of equilibrium frequency.

Mutations arise, and as a net result of alleles change over time (albeit very slowly).

The simplest model assumes two alleles with forward and back mutations.

Under this model allele frequencies will change each generation, eventually approacing and **equilibrium value**.

Consider the frequency of allele **A**. A fraction **(1- u)** of these alleles do not mutate, while a fraction v of allele **a** mutate to **A**. Hence the change in allele frequency is:

p

_{(t+1)}= (1- u)p_{t}+ vq_{t}

= (1- u)p

_{t}+ v(1-p)_{t}{expressing q in terms of p}

= (1- u)p_{t}+ v - vp_{t}

= p_{t}(1 - u - v) + v

The change in the frequency of the A allele can be descrbed as p or p_{(t+1)} - p_{t}

p = p

_{t}(1 - u - v) + v - p_{t}Mutation is at equilibrium when

p = 0 (no change in allele frequency) So ...

0 = p

_{t}(1 - u - v) + v - p_{t}= p

_{t }- p_{t}u - p_{t}v + v - p_{t}= -p

_{t}u - p_{t}v + vp

_{t}u = - p_{t}v + vp

_{t}u + p_{t}v= vp

_{t}(u + v) = vp at Equilibrum ( p

_{(eq)}) (the frequency of p (and hence q) at which allele frequency no longer changes)p

_{(eq)}= v/(u + v)