Probability and Genetics - Huntington's Disease

Date: 12/30/2001 at 19:35:31
From: Shirley
Subject: Probability and Genetics

I was just wondering what probability has to do with genetics - if it 
does. I am doing a math paper on Huntington's Disease, which is a 
genetically received disease. Since my paper is for math, I was 
wondering how to include math.

Thank you.

Date: 12/31/2001 at 17:42:56
From: Doctor Achilles
Subject: Re: Probability and Genetics

Hi Shirley,

Thanks for writing to Dr. Math.

This site on Mendel's Genetics shows the basics of genetic 
inheritence:

  http://anthro.palomar.edu/mendel/mendel_1.htm   

Here's the basic idea:

You have two copies of every gene. One of them came from your mom and 
one came from your dad. (They each also had two copies of each gene, 
but randomly gave one of each to you.)

The two copies you have are not necessarily identical. Brown eye color 
is a good example. Everyone has a gene that says one of two things:

  1) "Make brown eyes."
  2) "Don't make brown eyes."

So if both copies of your brown eye gene say "Make brown eyes," then 
your eyes will be brown. On the other hand, if both copies say "Don't 
make brown eyes," then your eyes will be some other color (which other 
color depends on other genes).

What happens if one copy says "Make brown eyes" and the other copy 
says "Don't make brown eyes"? You might think that you'd end up with 
something in between. But that's actually not what happens. What you 
end up with is just simply brown eyes.

So if you have one copy of your brown eyes gene (say, the one you got 
from mom) saying "Make brown eyes" and another copy (the one from dad) 
saying "Don't make brown eyes" then your eyes will be brown. In fact, 
they will be just as brown as someone who has BOTH copies saying "Make 
brown eyes."

Biologists say that brown eyes are "dominant." You can think of it 
like this. The copy that says "Make brown eyes" is a really big, loud 
bully of a gene. Whenever it gets into a cell, it pushes aside 
anything else that's there and makes sure it gets its way. On the 
other hand, the copy that says "Don't make brown eyes" is a little, 
quiet gene that lets the "Make brown eyes" gene walk all over it. The 
only time it gets heard at all is if there are two copies of it and no 
one else around to bully it.

As a convention, the two copies of a gene are written using letters.  
Capital letters stand for dominant genes, so the "Make brown eyes" 
copy would be written B, and lower case letters stand for "recessive" 
(not dominant) genes, so the "Don't make brown eyes" copy would be 
written b.

Since you have two copies, you get two letters. So an individual who 
got a B from mom and a B from dad would be written BB, someone who got 
a b from both parents would be written bb, and someone who got a B 
from one parent and a b from the other would be written Bb. Notice 
that BB and Bb individuals BOTH look exactly the same (they both have 
brown eyes), but their genes are different so their children might 
look different. Also, keep in mind that the only individuals who do 
not have brown eyes are bb individuals.

What does all of this have to do with math? You were right that the 
big link is probability. If we know what genes parents have, we can 
figure out the probabilities that their children will have different 
genes.

Let's say that there's a woman who has brown eyes and her genes are Bb 
(that is, she has one "Make brown eyes" copy and one "Don't make brown 
eyes" copy).  And she marries a man with blue eyes (since he doesn't 
have brown eyes, the only possibility is that he is bb: that is, he 
has two copies of the "Don't make brown eyes" gene). What color eyes 
will their kids have?

Remember, each kid will get one copy from mom and one copy from dad.  
So what COULD a child possibly get from dad? Well, dad is bb and he 
has to give one or the other of his copies to his kid. But his two 
copies are identical, so all he really has that he can possibly give 
is b.  So the kid has a 100% chance of getting a b from dad. 

What about mom? Well, she is Bb, so she has two "choices" - she can 
give her kid either a B or a b. Since the sorting is random, there is 
an equal chance of either event happening, so the kid has a 50% chance 
of getting a B from mom and a 50% of getting a b from mom.

So what are the possible outcomes? Well, we have to multiply the 
probabilities, so there is a 50% chance that the kid will get a B from 
mom, times a 100% chance of getting a b from dad, equals a 50% chance 
of the kid ending up Bb (with brown eyes).

On the other hand, there is a 50% chance that the kid will get a b 
from mom, times a 100% chance of getting a b from dad, equals a 50% 
chance of the kid ending up bb (with eyes that aren't brown).

Let's try one more example. Let's say there's another woman who has 
brown eyes and her genes are Bb, and she marries a man who also has 
brown eyes and whose genes are also Bb. What color eyes will their 
kids have?

Again, each kid gets one copy from mom and one copy from dad. So what 
could a kid possibly get from mom? Well, mom is Bb, so she has a 50% 
chance of giving her kid a B and a 50% chance of giving her kid a b.  
What could the kid get from dad? Dad is also Bb, so he has a 50% 
chance of giving his kid a B and a 50% chance of giving his kid a b.

So what will the kids have? Again, we have to multiply the 
probabilities, so there is a 50% chance that the kid will get a B from 
mom, times a 50% chance that the kid will get a B from dad, equals a 
25% chance that the kid will be BB.  

And there is a 50% chance that the kid will get a b from mom, times a 
50% chance that the kid will get a b from dad, equals a 25% chance 
that the kid will be bb.

And there is a 50% chance that the kid will get a B from mom, times a 
50% chance that the kid will get a b from dad, equals a 25% chance 
that the kid will be Bb.  

And there is a 50% chance that the kid will get a b from mom, times a 
50% chance that the kid will get a B from dad, equals a 25% chance 
that the kid will be Bb.

So there are actually two ways the kid can end up Bb: he or she can 
get a B from mom and a b from dad, or get a b from mom and a B from 
dad. So the total probability of the kid ending up Bb is 25% + 25%, 
which equals 50%.

So there is a 25% chance of being BB, a 50% chance of being Bb, and a 
25% chance of being bb. What color eyes will the kids have? Well, 
the 25% that are BB plus the 50% that are Bb will all have brown eyes, 
so 75% of the kids will have brown eyes, and 25% of the kids will be 
bb and have non-brown eyes.

Wow! So it is actually possible for two people with brown eyes to have 
a kid with non-brown eyes! That can happen if both parents have a 
recessive b hiding in the background.

Try a few more marriages on your own. For example, what will happen if 
a woman with green eyes (bb) marries a man who is BB (brown eyes)?  
(Be careful when you do this that you keep in mind that one copy in 
every child has to come from mom and one copy has to come from dad.)

So that's the link between math (probability) and genetics. You can do 
a lot more crazy stuff if you have more than one gene that you're 
watching, or if you look at some genetic traits (like color blindness) 
that are "x-linked" (which basically just means that men get them much 
more often than women), but that's the basic idea right there.

What does all this have to do with Huntington's disease? This page 
gives a good introduction to Huntington's:

NINDS Huntington's Disease Information Pag
http://www.ninds.nih.gov/health_and_medical/disorders/huntington.html   

It also goes into more detail about the disease.

Huntington's Disease is dominant. That is, it's like the brown eyes 
gene. If someone has just one copy, then that bad copy will cause the 
disease even if the other copy is good. But "dominant" ABSOLUTELY DOES 
NOT mean "common."  VERY, VERY few people have even one copy of the 
dominant Huntington's disease, and I would bet that ABSOLUTELY NO ONE 
has two copies.

If you want to do probability work with Huntington's, you can use the 
letter H to represent the dominant, bad copy; and h to represent the 
recessive, good copy.

Almost all individuals in the world have hh. Very few have Hh. And 
because the H is so rare, (fortunately) no one is unlucky enough to 
have HH. So if a woman who is Hh marries a man who is hh, then what 
are the chances that their children will have Huntington's? Is it 
possible for a person with Huntington's to have a normal child? Would 
this normal child have ANY chance at all of having a child of his or 
her own who has Huntington's (assuming this child marries someone who 
doesn't have a history of it in the family)? Try to answer these 
questions on your own. Once you've answered them, let me know what you 
get and I can check your work.

There's a lot more on genetics and on Huntington's out there. You can 
find tons of articles on the Internet using most any search engine (I 
found the two pages in this answer just by doing a quick search in 
Google) and any library will have articles (and probably several 
books) available as well. Good luck doing the research for your 
project, and if you ever stumble upon something that doesn't make 
sense, let me know and I'll try to help you understand it.

Hope this helps. If you have other questions about this, please write 
back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/