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### Probability and Genetics - Huntington's Disease

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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.
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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
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.

back.

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