Adventures in Math and Pregnancy

One of my devoted fans (okay, my sister!) recently made the following request:
Hey, there hasn't been much math on this blog recently. Here's a challenge: come up with a topic that's about math AND pregnancy (just 'cause I find the pregnancy really interesting!)

Ha! I knew it! Somebody reads this for the math! (Either that, or she's trying to ruin my life.) In either case, I am too much of a math junkie not to talk about math when asked. (Maybe this entry will ruin her life instead of mine!)

It seems logical to take this opportunity to talk about probability and statistics and their relationship to genetic inheritance. As you probably know already, a sperm and an egg fuse into an embryo, which divides a lot and eventually becomes a baby (assuming all goes as normal). A human being usually has 46 chromosomes in each cell, and half the genetic material originates from each parent, meaning that the mother contributes 23 chromosomes and the father contributes 23 chromosomes. The chromosomes can be paired into 23 pairs of like chromosomes.

Chromosomes in a pair basically look alike, except for in the 23rd pair, which is the one that determines the sex of the baby. Females have two X-shaped chromosomes, while males have one X and one Y. So everyone inherits an X chromosome from their mother, and the father's contribution to this final pair is what determines their sex. There is a 50-50 chance that any given sperm carries an X or a Y chromosome.

Chromosomes are tightly coiled strands of DNA, and contain sequences called genes that "control a hereditary characteristic." As of 2001, scientists believe that humans have 30-40,000 genes.

There are many ways in which genes can control what we are like. Some traits are a result of simple inheritance, in which a single pair of genes determines the outcome. Usually there are two possible traits, and one is dominant over the other, meaning that a person having one of the dominant gene and one of the recessive gene will express the dominant trait. The actual genes a person has is known as their genotype, and the expression of those genes is known as their phenotype. For example, I was talking about the fact that I am Rh-negative. What this means is that my blood lacks a certain protein that the blood of Rh-positive people contains. Rh-positive is dominant, and Rh-negative is recessive. For simplicity let's denote Rh-negative as (-) and Rh-positive as (+). What are the possible pairings of these two genes? We can have four possible combinations: (+) (+), (+) (-), (-) (+), and (-) (-). Since (+) is dominant, the first three combinations will be expressed as Rh-positive, and the last one is the only case of Rh-negative. What are the chances that our baby will be Rh-positive?

I don't actually know Jeff's blood type. If he's (-) like me, then our baby will be Rh-negative no matter what, because no matter which of his Rh genes he contributes and no matter which of mine I contribute, the baby will have a genotype (-) (-). If Jeff is Rh-positive, then he's either (+) (-) or (+) (+). [(-) (+) is equivalent to (+) (-).] If Jeff is (+) (-) then our baby will be either Rh-positive or Rh-negative with equal probability, because Jeff's gene is what will determine the baby's phenotype, and the chances of him contributing either gene are the same. If Jeff is (+) (+), though, the baby will be Rh-positive no matter what.

Sometimes there are traits where there is not a simple dominant gene, but two co-dominant genes and a recessive gene. For example, the genes for the A and B blood types are co-dominant, while the gene for the O blood type is recessive. I have two O genes, because I am type O. Both of my parents, however, are type A. How could this be possible?

My parents' blood phenotype is A, but their genotypes must both be AO. They each contributed their O genes to me, so that is why I have type O blood. We can use a Punnett square to see how this works. The first row of the square represents one parent's possible gene contributions, while the first column represents the other parent's contributions. The four remaining squares show the possible outcomes of their offspring's genes.

	A	O
A	AA	AO
O	AO	OO

As you can see, there's a 3 in 4 chance that my parents would have an A-phenotype child, and only a 1 in 4 chance they'd have an O-phenotype child. It actually worked out this way in our family, because out of four children I'm the only one with type O blood.

A and B are co-dominant, meaning that someone who inherits the gene for A from one parent and the gene for B from another, will actually express both and have type AB blood. A person with type AB blood can never have a child with type O blood (although they could have a child with type A or type B blood).

Some traits are a result of the effects of multiple genes. For example, eye color is determined by at least three sets of genes, two of which are understood so far. One set is the brown-blue pair, where brown is dominant and blue is recessive. Another set is the green-blue pair, where green is dominant over blue but brown from the first pair is dominant over this pair. I have brown eyes. My father has brown eyes, and my mother has blue eyes. Jeff has blue eyes. So the color of our baby's eyes will be determined by what he inherits from me. The question is, what genes do I have? I know that for the first pair, I must have B-b (where B=Brown, b=blue). For the second pair, I must have inherited a gene for blue eyes from my mother, because that's all she had, whereas it is not obvious what I inherited from my father, because it's masked by the brown gene. I asked him if there was anybody in his family who had green eyes, and he couldn't think of anyone, so probably (although not definitively) I inherited a gene for blue eyes from him. This means that my eye-color genes are probably the following: B-b b-b. But since we're doing this for the math, let's pretend for the moment that my eye-color genes are B-b G-b. What are the possibilities for our child? Again we do a Punnett square, with my possible contributions along the top, and Jeff's possible contributions in the first column:

	B G	B b	b G	b b
b b	Bb Gb	Bb bb	bb Gb	bb bb
b b	Bb Gb	Bb bb	bb Gb	bb bb
b b	Bb Gb	Bb bb	bb Gb	bb bb
b b	Bb Gb	Bb bb	bb Gb	bb bb

The first two columns for our offspring are all phenotype brown, the third column is phenotype green, and the fourth column is phenotype blue. So he has a 50% chance of having brown eyes, a 25% chance of having green eyes, and a 25% chance of having blue eyes. This table is not very exciting because of Jeff's pure blue eyes. But, if I were having a baby with someone whose eye-color genes were like mine, we would get a very different result:

	B G	B b	b G	b b
B G	BB GG	BB Gb	Bb GG	Bb Gb
B b	BB Gb	BB bb	Bb Gb	Bb bb
b G	Bb GG	Bb Gb	bb GG	bb Gb
b b	Bb Gb	Bb bb	bb Gb	bb bb

In this case, our offspring would have a 3/4 chance of brown eyes, as only the genotypes in the lower right 2 x 2 of the subtable would have non-brown phenotypes. The remaining 1/4 chance consists of 3/16 chance of green eyes and 1/16 chance of blue eyes.

Of course as I said above, scientists have not figured out all the genes that influence eye color. For example, my mother has a yellow ring in her blue eyes. What causes that? Nobody knows (yet).

There are some interesting sex-linked traits that are caused by genes on the X or Y chromosomes. For example, baldness is a sex-linked trait. The genes for baldness are located on the X chromosome, but baldness is recessive, so if a woman has one set of genes for baldness and one set for not-baldness, she won't be bald. The Y chromosome, however, does not have those genes so whatever baldness genes a man has on his single X chromosome is what he expresses. This means that a man inherits his baldness from his mother's genes, not his father's, so the best way to tell if a boy might grow up to be bald is by looking at his mother's family. There are some bald men in my family, such as my maternal grandfather, and a few of my paternal uncles, but my father has a full head of hair, so I'm thinking there's probably not a high probability that our son will go bald as he ages.

I don't know what our baby will be like but I'm interested in what you all think. Brown eyes, blue eyes, green eyes? Brown hair, blond hair? Curly hair, wavy hair, straight hair? A, B, or O bloodtype? Rh-positive or Rh-negative? Any guesses?