Coefficients of Inbreeding and Consanguinity

What do you mean, inbreeding?
Inbreeding is mating between relatives. In a population of finite size, some degree of inbreeding is inevitable. Mating between close relatives, however, is often harmful, particularly in species that normally outcross. Mammals are generally outcrossing, and humans are no exception. Almost every human society has taboos or mores that prohibit mating between closely related individuals.

Why is inbreeding harmful?
We need to think about types of mutations. Mutations are random errors in DNA replication, leading to changes in genes. Whenever we see that more than one form of a gene exists (different alleles), we know that these have resulted from mutations. Mutant alleles can produce various effects in the organisms that have them. First, and most commonly, mutations may produce no observable effect at all. The next most common result of a mutation is an allele that codes for a non-functional, or less optimally functional, protein. These are called deleterious mutations. Least commonly, a mutation may produce an allele that codes for an protein that gives the organism possessing it an advantage in some environmental conditions.

Why are harmful mutations usually recessive?
Deleterious mutant alleles are most often recessive to the normal allele. This makes sense because these deleterious alleles often code for a non-functional protein. If you have two alleles, one normal and one mutant, you will product two forms of the protein -- some of it functional, some non-functional. Usually the amount of functional protein produced by a single copy of the normal allele is enough to make the organism more or less normal. There are exceptions; the allele that causes a common form of human dwarfism is dominant, for instance. Still, the vast majority of harmful mutant alleles (e.g. those causing sickle-cell anemia, cystic fibrosis, hemophilia A) are recessive.

So why shouldn't I marry my cousin?
Mutations are common enough that all of us carry some deleterious mutant alleles. If you mate with an unrelated (or distantly related) person, that person is unlikely to have harmful mutant alleles for the same genes that you do. Therefore, though both you and your spouse may pass deleterious mutant alleles to your kid, if you don't pass harmful alleles of the same genes to the kid, the kid will be heterozygous for the genes in question. This means that the kid has two different alleles for these genes -- in this case, one normal and one mutant allele. Since the harmful mutant alleles are usually recessive, your kid will be normal.

If you marry a closely related individual, though, then both you and your spouse are likely to have the same harmful mutant alleles (you got them from the same recent ancestors). You are likely to pass the same alleles to your kid, who will then be homozygous for the mutant allele. That is, the kid gets the same mutant allele for some gene from both of you, and has no normal allele for that gene. The kid will be lacking functional protein that the gene should code for, and will have some genetic abnormality.

So, how close is too close?
We talk about degrees of consanguinity in marriage -- consanguinity means "relatedness", but the root words mean literally "shared blood." This is a remnant of the old "blending inheritance" idea that held that the inherited traits were carried in some liquid form, often thought to be the blood -- the same idea survives in expressions like "full-blooded Cherokee". Of course, we know now that heritable traits are passed in the form of sections of the DNA, that is, genes. A parent passes half of his or her genes to a child (because meiosis produces gametes containing half of the parent's genes). You can use this fact to figure out the probability that two individuals will have the same allele of a gene, or will pass the same allele to their offspring. To a geneticist, when you ask "how closely related are these two people?" the question means "what's the probability that these two people have the same allele for a given gene?"

Sue and Bob are sister and brother; their parents are Harvey and Edna. Harvey's genotype for some gene is A¹A², and Edna's genotype for this gene is A³A⁴, so the two of them have four different alleles for the gene A. Sue and Bob each have two alleles of the gene A. Let's pick one of Sue's alleles at random. What's the probability that Bob has the same allele? Well, Sue got that allele from Harvey or Edna. The probability that she got it from Harvey is 1/2; the probability that she got it from Edna is 1/2. Take the case that she got it from Harvey. Bob got some allele from Harvey; the probability that he got the same one that Sue did is 1/2, since Harvey had two alleles and gave Bob only one. Thus the probability that Bob has the same allele as Sue does because both got it from Harvey is (probability that Sue got the allele from Harvey) x (probability that Bob got the same allele from Harvey), or 1/2 x 1/2 = 1/4. But this isn't the only that Bob could have the same allele as Sue; Sue might have gotten the allele we chose from Edna, and Bob could have gotten the same allele from Edna. The probability of this is 1/4 also. All together, the probability that the allele we picked to look at in Sue is the same as one of Bob's alleles is (probability that both got it from Harvey) + (probability that both got it from Edna) = 1/4 + 1/4 = 1/2. Thus, overall, full siblings share half their alleles because of common ancestry.

In population genetics, we're more interested in the probability that the offspring of a mating between relatives will be homozygous for some gene. Degrees of relationship are usually measured by geneticists in two rather similar ways. One is called the coefficient of consanguinity, usually abbreviated f. The coefficient of consanguinity for two individuals is the probability that, if the two mated and produced an offspring, the offspring would be homozygous for a particular gene because of common ancestry. We say that such an individual has alleles that are "identical by descent".

The other way of measuring relationship is the coefficient of inbreeding (F). This is calculated for a single individual, and is simply the coefficient of consanguinity of the individual's parents.

The coefficient of consanguinity for two individuals is 1/2 times the probability that they have the same allele of a given gene. Think back to Bob and Sue. Sue gives some allele to their kid, Egbert. The probability that Bob has the same allele is 1/2, as we've already seen. The probability that Bob will give that allele to Egbert (if Bob has it) is 1/2 also, because Egbert only gets half of Bob's alleles. Thus the probability that Sue and Bob will give the same allele to Egbert is (probability Bob has the same allele Sue gives Egbert) x (probability Bob gives that allele to Egbert) = 1/2 x 1/2 = 1/4. This is the coefficient of consanguinity for the two siblings. It is also the coefficient of inbreeding (F) for their child.

Isn't there some easier way to figure these out?
Rather than go through all of the probability reasoning to figure out these coefficients, we use a simpler method called path coefficients. The chart below shows the paths by which genetic material is transferred from Harvey and Edna to Bob and Sue (and potentially to their kid, Egbert).

At each step, the probability that a given allele is transferred is 1/2, since there are two alleles in each individual, and only one in a gamete. The easiest way to figure the path coefficient is to start from one of the individuals whose consanguinity you're interested in and proceed along each possible path connecting that individual to the other one. For Bob and Sue, start with Bob. Here are the paths connecting Bob to Sue:

Bob - Harvey -Sue

Bob - Edna - Sue

In each path, count the number of individuals. The coefficient of consanguinity resulting from that path is 1/2 to that power. For each of these paths, there are 3 individuals, so each contributes (1/2)³, or 1/8. Sum up the coefficients from all paths to get the coefficient of consanguinity -- in this case, 1/8 +1/8, or 1/4.

To get the coefficient of inbreeding, do the same thing, but the paths connect the parents of the individual you're interested in. Thus Egbert's coefficient of inbreeding is the same as the coefficient of consanguinity of Bob and Sue. Of course, if we want to know the proportion of genes that are the same in Bob and Sue, we double their coefficient of consanguinity, so it would be 1/4 x 2, or 1/2, as we've seen before.

Huh?
Let's try another example to see if it gets any clearer. Check out this family:

Bud and Mabel are half-sibs. Try using path coefficients to figure out the coefficient of inbreeding of their son, Orville. When finished, go to the answer page.