Numb3rs Season 3 Episode 12: Nine Wives

Many mathematical topics are mentioned in this episode, including random walks, but we will focus on how the statistics of genetics and inbreeding coefficients.

Recessive Genetic Disorders

It is well known that when a couple have ancestors in common, there is a heightened possibility for their children to have two copies of a recessive gene, greatly increasing the possibility for disease. The reason for this is that if a particular gene which has a deleterious effect is recessive, then there can be little to no evolutionary pressure to wipe the gene out of a population. The result is that certain families can pass on genes which by themselves are harmless because of the presence of a normal allele, but can be dangerous when combined with a copy of itself. This is precisely what happens with inbreeding.

Each child gets half of his DNA from his father and half from his mother. Every person has two copies of each gene. For every gene, both the mother and father give one of their two copies to the child to make up the two copies that the child possesses. If a certain deleterious gene is both recessive and rare in the population, then it can be passed along from generation to generation for a remarkably long time without causing trouble.

Activity 1: Expression of Recessive Genes

Suppose that there is a gene with a certain recessive allele whose relative frequency in the population is p. That is to say that every one of these genes has a probability of p of being this particular allele.

Activity 2: Allele Frequency

Sickle-Cell Anemia in the U.S.

While sickle-cell anemia is still common in the United States among those with African ancestry, its prevalence is 8% less than in Africa and is slowly dropping. In Africa, being heterozygous for sickle-cell anemia is evolutionarily advantageous for the protection it gives against malaria, giving evolutionary pressure to increase the relative frequency, and being homozygous results in expression of sickle-cell anemia and most likely, an early death, which provides evolutionary pressure to decrease the relative frequency of the allele. Over many millenia, these two pressures have reached a balance in Africa (along with the Middle East and India) that depends sensitively on the rate of occurance of malaria. In the United States, malaria is exceedingly uncommon, so the genes that give rise to sickle-cell anemia are purely disadvantageous, and as such, they are very slowly being selected out of the population.

Is it possible that an eradication of malaria in Africa will result in a slow decline in the number of cases of sickle-cell anemia there, too? Perhaps, though on the time scales in question here, one would also expect that medical technology might advance to the point that sickle-cell anemia is an entirely treatable condition and that eventually there would be no evolutionary pressure exerted either way.

There are many reasons why a recessive genetic disorder may persist in a population against evolutionary pressure to exterminate it. Sometimes, being heterozygous for a particular disorder actually gives an evolutionary advantage in some other area. A classic example of this is with sickle-cell anemia.

Sickle-cell anemia is a rare genetic disorder that causes an affected person's red blood cells to change into a sickle shape upon deoxygenation. Their red blood cells' surfaces become rigid, and then are much more likely to get stuck in small blood vessels and veins, obstructing the flow of blood. It's a debilitating disease, and while advances in medicine have prolonged the lives of sufferers, in earlier times complications from the disease would often kill before the age of forty.

The relevant gene that causes sickle-cell anemia is incompletely recessive. In this case, this means that a small portion of the red blood cells in a heterozygous person will have the sickle-cell shape. These people typically have very few complications from their small number of misshapen blood cells. However, these few cells do give a rather substantial resistance to malaria. It is interesting to note that the map of populations that contain more than a modicum of the gene responsible for sickle-cell anemia overlaps identically with the regions that have historically had regular epidemics of malaria (Africa, the Middle East, and India). Additionally, populations from higher altitudes within these regions have been free of sickle-cell anemia. This is thought to be because malaria does not typically occur in these colder regions because of malaria's dependence on mosquitoes for half of its life cycle.

Besides the heterozygous form actually being advantageous in certain circumstances, another possibility allowing for the persistence of deleterious recessive genetic disorders is the simple fact that evolutionary pressure is ineffective at reducing their relative frequency once the relative frequency is too low. Recall from the first activity that if the relative frequency of a given allele is p, then the chance that a particular person will be homozygous with this allele is p2. If p is small, then p2 will be very small. For example, if a certain recessive allele has a relative frequency of 1 in 1,000, then an average of one in one million children in the population will express the trait. As such, an extraordinarily small amount of evolutionary pressure will be brought to bear against the gene. In an extreme case, if there is only one copy of the gene, then all children of the individual are guaranteed to either be heterozygous or not have the gene at all, and there is precisely no evolutionary pressure.

Dominant Genetic Disorders

Tay-Sachs Disease

Israel pioneered the practice of genetic screening for couples during the 1970s to combat Tay-Sachs Disease, a deadly recessive genetic disorder. The most common form of the disease attacks children within within their first half-year of life. This form is a result of not inheriting from either parent the genes necessary to break down certain components of cell membranes (called gangliosides) inside of nerve cells. As the nerve cell cannot expel these gangliosides, the result is a gradual build-up of these lipids in the cell. Eventually the build up forces the nerve cell to expand and distend until it loses all function. The result is a progressive loss of sensation, cognitive function, and motor skills and ends in death.

Prior to the 1970s, Tay Sachs Disease was relatively common among Ashkenazi Jews. An estimated 1 in 30 people with such ancestry is carrier. The Jewish organization Dor Yeshorim anonymously screens for the disease with the intention that couples should not marry if both partners are carriers for Tay-Sachs. Genetic testing has been tremendously effective in eliminating the expression of Tay-Sachs Disease in Jewish populations, even though the actual prevalence of the gene remains unchanged. In the past few years, the only cases of Tay-Sachs Disease have been in non-Jewish families.

The fact that the proportion of people expressing a given rare recessive allele is much smaller than the frequency of the allele in the population does not help with diseases associated with dominant alleles. Due to the enhanced evolutionary pressure against dominant disorders, only a small proportion of people with the gene go on to develop the disease, and they usually develop the disease much later in life, after the prime reproductive years.

An example of a dominant genetic disorder is Huntington's disease, which has a mean onset age of about fifty years.

Activity 3: Expression of Dominant Genetic Disorders

Suppose a given dominant genetic disorder has a relative frequency of p.

Because of this stronger evolutionary pressure, dominant genetic disorders are much less common that recessive ones, and when they do hit, it is usually after thirty years of age in humans, probably because average human life expectancy was less than forty years until very recently on evolutionary time scales.


In the preceding, we have only considered what happens when mating patterns are essentially random within the population. However, even if a recessive genetic disorder is extremely uncommon, if an individual who is heterozygous for this disorder inbreeds, there can be a very high chance of having offspring who are homozygous for the relevant gene.

Activity 4: Proof of Inbreeding

Are all of us inbred to some degree? Suppose that you're not inbred at all. How many parents do you have? How many grand-parents do you have? How many great-grand-parents?

Suppose historically, humans have had children at about an average age of twenty years old. Then there have been about one hundred generations since the year 1 A.D.. About how many ancestors would you have, going back up your family tree one hundred generations? How many people were alive in the world then? If these numbers don't make sense, then you probably have ancestors that occupy multiple places on your family tree, meaning that you're inbred.

Practical Uses of Inbreeding Coefficients

Dog and cattle breeders both depend on the mathematics of inbreeding coefficients to strike a balance between selecting desirable traits and not expressing potentially dangerous genetic diseases. The calculations of the inbreeding coefficients can be laborious for large family trees, but computers can do the mathematics with ease. There are many programs publicly available that are capable of computing the inbreeding coefficient once given the family tree.

Anyone who is breeding a population to try to select for a given trait usually must inbreed at some point, since the cost of raising the exponential number of original individuals is exponentially prohibitive. Mathematicians developed the concept of an inbreeding coefficient to help precisely identify the risk of a given individual being homozygous with respect to any gene.

The inbreeding coefficient of an individual is the chance that that individual is homozygous with respect to a particular gene and that both copies of that gene came from a single copy of the gene from a single ancestor. For instance, the inbreeding coefficient is blind to the possibility that two unrelated individuals may both poses a copy of the same gene and may both pass that gene along to their child, making their child homozygous. In fact the only way to detect such a possibility is on a per-gene basis, either by looking for similar phenotypic effects in each of the parents' families, or by sequencing each of the parents' genomes, looking for identical genes. The advantage of the inbreeding coefficients is that the probability of the same gene being passed from ancestor to descendant in two different ways is the exact same for every gene, and can be determined very easily from the family tree.

Activity 5: Calculations of Inbreeding Coefficients

Activity 6: Odds of Genetic Disorders

There are about 5,000 known recessive genetic disorders in humans. Some are relatively common, but most are very rare. To simplify calculations, suppose that the allele frequencies of all the genes for every disorder are identical at 1 in 1,000.

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