Nazi Eugenics would have never worked!

That was one of the last things we found out this semester. Although everyone slept peacefully, when professor finished explaining, he did manage to make a lot of us laugh with that factoid. He was really good. Hopefully we will get him back for level 4 modules or else I don’t think I will graduate. :(

Here is why Nazi Eugenics would have never removed all unhealthy individuals from a population:

Let us assume the Nazis start killing albinos in Europe which (for the sake of example) currently has a $\frac{1}{20000}$ frequency of albinos. Albinism is recessive. All albinos die/get sterilized in the hands of the Nazis, so coefficient of selection $s = 1$, because all affected individuals gets selected against. Let $p$ be the initial population frequency of dominant allele $A$ and $q$ be the initial frequency of recessive allele a. Assuming Hardy-Weinberg equilibrium, this is the current situation:

$\begin{array}{cccccccc} & & Normal & & & Albino & & Total\\ Genotype & A A & & A a & | & a a & | & \\ Frequency of Expression & p^2 & & 2 pq & | & q^2 & | & 1\\ Fitness & 1 & & 1 & | & 1 - s & | & \\ Frequency of types of gametes produced & p^2 & & 2 pq & | & q^2 (1 - s) & | & 1 - sq^2 \end{array}$

$p^2+2pq+q^2=1$ because of HW equilibrium. Fitness is $1$ for normal people. Albinos get $1-s$ fitness where $s$ is the selection pressure as coefficient of selection. The last row is the frequency distribution of gametes produced by current generation. As you can see the amount of recessive gametes $a$ produced by the current generation albinos is lessened by a factor of $1-s$. The total is also reduced accordingly.

The frequency of allele $a$ (the albino gene) for the first generation is: $q_1 = \frac{q^2 (1 - s) + pq}{1 - sq^2} = \frac{q (1 - q)}{1 - q^2} = \frac{q}{1 + q}$

The above simplification is possible because $1-s=0$ and $s=1$ because Nazis kill all albinos and $p = 1- q$ because there are only 2 possible alleles.

Following the pattern, the frequency of allele $a$ for the second generation is: $q_2 = \frac{q_1}{1 + q_1} = \frac{q}{1 + 2 q}$

The last form $\frac{q}{1 + 2 q}$ is achieved by substituting $q_1=\frac{q}{1 + q_1}$ from the above findings.

For the $n$th generation, the frequency of allele $a$ is: $q_n = \frac{q}{1 + nq}$

So if Nazis keep killing albinos from now (in Europe where $1$ in $20000$ is albino), for $100$ generations, this is what happens:

Initial recessive allele frequency $q$ is $\sqrt{\frac{1}{20000}} = \frac{1}{141}$. After $100$ generations the recessive allele frequency i.e. frequency of albino gene is $q_{100} = \frac{1}{241}$ by using the above formula. Therefore the actual frequency of albinos is $q_{100}^2 = \frac{1}{58080}$!!!. TADA!

So you see, even if Nazis keep killing something as rare as a recessive condition with $1$ in $20000$ frequency for $100$ generations, $1$ in $58080$ will still have the condition! Not even the biblical god will find killing people for 100 generations economical, let alone the Nazi party.

Now my only wish is that, this piece of truth should have been taught back during the Holocaust :( We could have saved a zillion lives :(

Gautama, the Buddha was right: Ignorance is the root of all evil!