Math 326 April 9 Model

Math 326, April 9 - Gene Propogation

If a disease is transmitted genetically, are we able to predict the percentage of the population which suffers from that disease over time? After all, if the disease is fatal or otherwise impedes reproduction, fewer and fewer children carrying the guilty gene will be born.

Section 7.3 models this passing of genes using the Hardy-Weinberg principle. Each gene comes with two sides, one from the mother and one from the father. Suppose for a particular gene, each side could be either A or B. Each person is either AA, AB or BB, with the proportions of each in the total population denoted P(AA), P(AB) and P(BB).

Suppose in addition that configuration is assigned a fertility rate e between 0 and 1. For instance, if AA and AB individuals are healthy but BB represents a disease where 20% of sufferers die before reproducing, then we could set e(AA)=1, e(AB)=1 and e(BB)=0.8. Each generation, we determine the number of offspring each group has based on its proportion of the population and its fertility rate. We assume mating is random, so that A's and B's are distributed in the next generation according to the probability of each configuration.

The proportions in the population do eventually approach an equilibrium, but it takes a long time. Check the proportions after 500 generations. Make a 100% stacked area graph of the first 100 generations to illustrate how the population approaches its equilibria.

Additional Questions

Hereditary Hemochromatosis is a genetic which afflicts about 1 in 64 people in Ireland and Scotland (it's about 1 in 200 in the US). While we are now better at treating it, it has traditionally resulted in a quick death once the disease emerged, especially in men. Let's assume that the fertility rate for HH sufferers (with the BB genotype) is a mere 0.4, stopping most males and some females from having children. Find fertility rates for carriers (AB) and non-carriers (AA) which result long term in approximately 1 in 64 sufferers. What biological explanation could there be for these fertility rates? ---highlight--> (It appears carriers are imbued with some slight genetic advantage, although scientists have not yet discerned its nature; they've only predicted it based on models like this.)
Suppose a small, lucky segment is miraculously born with the "Super Hero" BB genotype. Initially, this group is very small, with P(AA)=0.999, P(AB)=0 and P(BB)=0.001. However, with their super-powered health and attractiveness, the BB people are able to reproduce at 2.5 times the rate of everyone else (e(AA)=0.4, e(AB)=0.4, e(BB)=1). How many generations will it be before everyone is super?
Create a spreadsheet model which allows for the possibility of migration. For instance, A represents brown eyes and B represents blue eyes. A country starts off with a 100% brown-eyed population. Each generation, some blue-eyed immigrants are introduced to the population. How does the proportion of blue-eyed people evolve over time? You must figure out how to incorporate migration, and choose fertility and migration rates which are appropriate.
Modify you migration spreadsheet to allow migration both ways. One country starts all brown-eyed, one all blue-eyed. There is some migration in each direction each year (maybe equal in each direction, maybe different).