TNBGGA: The No Bullshit Guide to Genetic Analysis

To start out, let's first consider an extreme case of inbreeding; namely, a brother-sister mating. The pedigree would look like this:

A useful concept to introduce here is the inbreeding coefficient ($F$), which is defined as the likelihood of homozygosity by descent at a given locusplugin-autotooltip__default plugin-autotooltip_bigLocus (plural form: loci): a physical location of a gene; often used as a synonym for a gene.. Let's take the example of the family shown in Figure 1. This family has two sets of grandparents: the paternal grandparents (individuals 1 and 2) and maternal grandparents (individuals 3 and 4). Let's imagine there is a geneplugin-autotooltip__default plugin-autotooltip_bigGene: read Chapters 02, 03, 04, 05, and 06 for a definition of gene :-) $A$, and that each grandparent carries a different alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence.: The paternal grandfather (individual 1) carries $A1$ and the paternal grandmother (2) carries $A2$; similarly, the maternal granfather 93) carries $A3$ and the maternal grandmother (4) carries $A4$. $F$ is the probability that the grandchild (the child of 7 and 8, marked by “?”) will have the genotypeplugin-autotooltip__default plugin-autotooltip_bigGenotype: the combination of alleles within an organism or strain. When used as a verb, it means to determine the genotype experimentally. of either $A1/A1$, $A2/A2$, $A3/A3$, or $A4/A4$.

Let's first consider the question: what is the probability that the grandchild will have the genotypeplugin-autotooltip__default plugin-autotooltip_bigGenotype: the combination of alleles within an organism or strain. When used as a verb, it means to determine the genotype experimentally. $A1/A1$? In order for this to happen, the brother (7) and sister (8) must be carriers of $A1$. The brother (7) can only inherit $A1$ from the father (5), who in turn can only inherit $A1$ from the grandfather (1). The likelihood that the father (5) inherits $A1$ from the grandfather (1) is $\frac{1}{2}$, and the likelihood that the father (5) passes $A1$ to his son (the brother, 7) is also $\frac{1}{4}$. Therefore, the likelihood that the brother (7) is a carrier of $A1$ is $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$. By the exact same logic, the likelihood that the sister (8) is carrier of $A1$ is the same: $frac{1}{4}$. Therefore, the likelihood that their child is homozygousplugin-autotooltip__default plugin-autotooltip_bigHomozygous: a state for a diploid organism wherein the two alleles for a gene are identical to each other. for $A1$ is $\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}$.

Of course, there is nothing special about the $A1$ alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence.; the exact same logic applies to $A2$, $A3$, and $A4$. In other words:

$$p(A1/A1)=p(A2/A2)=p(A3/A3)=p(A4/A4)=\frac{1}{16}$$

Since we are interested in the probability of whether the child will be homozygousplugin-autotooltip__default plugin-autotooltip_bigHomozygous: a state for a diploid organism wherein the two alleles for a gene are identical to each other. at the $A$ geneplugin-autotooltip__default plugin-autotooltip_bigGene: read Chapters 02, 03, 04, 05, and 06 for a definition of gene :-) regardless of which alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. it is, we are effectively asking the question of “$p(A1/A1)$ or $p(A2/A2)$ or $p(A3/A3)$ or $p(A4/A4)$”. When solving “or” questions in probability, we use the sum rule by adding up the probabilities. Therefore:

$$p(\text{homozygous by descent})=F_\text{siblings}=\frac{1}{16}+\frac{1}{16}+\frac{1}{16}+\frac{1}{16}=\frac{1}{4}$$

A bother-sister mating is the simplest case to analyze but is of little practical consequence in human population genetics since all cultures have strong taboos against this type of consanguineous mating and the frequency of these events is extremely low.

First cousin marriages (Fig. 2) do happen at an appreciable frequency. Let's calculate $F$ for offspring of 1st cousins.

Let's say that in this case, the great-grandfather (1) has the genotypeplugin-autotooltip__default plugin-autotooltip_bigGenotype: the combination of alleles within an organism or strain. When used as a verb, it means to determine the genotype experimentally. $A1/A2$ and the great-grandmother (2) has the genotypeplugin-autotooltip__default plugin-autotooltip_bigGenotype: the combination of alleles within an organism or strain. When used as a verb, it means to determine the genotype experimentally. $A3/A4$. As before, let's consider what likelihood that the child of a first cousin marriage will have the genotypeplugin-autotooltip__default plugin-autotooltip_bigGenotype: the combination of alleles within an organism or strain. When used as a verb, it means to determine the genotype experimentally. $A1/A1$. In order for this to happen, their father (9) must be a carrier of $A1$; he has a $\frac{1}{2}$ chance of inheriting $A1$ from his grandfather (5), and he has a $\frac{1}{2}$ chance of inheriting $A1$ from the great-grandfather (1). Therefore, the likelihood that the father (9) is a carrier of $A1$ is $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$. By the same logic, the likelihood that the mother (8) is a carrier of $A1$ is the same: $\frac{1}{4}$. Combined, the likelihood that both parents (8 and 9) are carriers of $A1$ is $\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}$. Finally, each parent has a $\frac{1}{2}$ chance of passing $A1$ to their child, so the likelihood that they both pass $A1$ to their child is $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$. Taken all together, the likelihood that both parents are carriers of $A1$ and that they both pass $A1$ to their child is $\frac{1}{16}\times\frac{1}{4}=\frac{1}{64}$.

Also as before, there is nothing special about $A1$. The probability that the child will be homozygousplugin-autotooltip__default plugin-autotooltip_bigHomozygous: a state for a diploid organism wherein the two alleles for a gene are identical to each other. for any of the allelesplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. from the great-grandparents is the same:

$$p(A1/A1)=p(A2/A2)=p(A3/A3)=p(A4/A4)=\frac{1}{64}$$

Using the sum rule to determine the probability if any of these outcomes occur, we get:

$$p(\text{homozygous by descent})=F_\text{cousins}=\frac{1}{64}+\frac{1}{64}+\frac{1}{64}+\frac{1}{64}=\frac{1}{16}$$

Clearly, $F_\text{siblings}$ is significantly greater than $F_\text{cousins}$. Now, let's imagine that there is a rare recessiveplugin-autotooltip__default plugin-autotooltip_bigRecessive: used to describe an allele, usually in comparison to wildtype. Recessive alleles do not exhibit their phenotype when combined with a wildtype allele. disease alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. $a$, where the alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. frequency $f(a)=q=10^{-4}$. If we assume that this alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. exists in a large population and mating is random, the frequency of homozygotesplugin-autotooltip__default plugin-autotooltip_bigHomozygous: a state for a diploid organism wherein the two alleles for a gene are identical to each other. due to random mating is $f_\text{random}(a/a)=q^2=10^{-8}$. In the United States, first cousin marriages happen at around 1 in 1000 marriages, or 10^-3. Therefore, the frequency of $a/a$ honozygosity from first cousin marriages $f_\text{cousins}(a/a)=\frac{1}{16}\times 10^{-8}\times 10^{-3}=6.3\times 10^{-9}$, The ratio of $\frac{f_\text{cousins}}{f_\text{random}}=\frac{6.3 \times 10^{-9}}{10^{-8}}=0.63$. in other words, two-thirds of all affected individuals will come from a first cousin marriage.

Note that the proportion we calculated above depends on our assumption of alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. frequency ($q=10^{-4}$). If alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. frequency is very rare, affected individuals will more often be a result of consanguineous marriages. For rare diseases, it is often difficult to tell whether or not they are of genetic origin, because there will be very few individuals affected (and therefore very little genetic data). A useful method to identify disorders that are likely to be inherited is to ask whether an unusually high proportion of affected individuals have parents that are related to one another.

For much of our discussion up to this point, we have used 10^-4 as an estimate for the frequency of recessiveplugin-autotooltip__default plugin-autotooltip_bigRecessive: used to describe an allele, usually in comparison to wildtype. Recessive alleles do not exhibit their phenotype when combined with a wildtype allele. loss of functionplugin-autotooltip__default plugin-autotooltip_bigLoss of function: a general term used to describe mutant alleles that have less activity than wildtype. Amorphic and hypomorphic mutations are loss of function mutations. allelesplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. in the human population. This may seem like a comfortably small number but given that the total number of human genesplugin-autotooltip__default plugin-autotooltip_bigGene: read Chapters 02, 03, 04, 05, and 06 for a definition of gene :-) is about 2×10⁴, each of us must be carrying many recessiveplugin-autotooltip__default plugin-autotooltip_bigRecessive: used to describe an allele, usually in comparison to wildtype. Recessive alleles do not exhibit their phenotype when combined with a wildtype allele. loss of functionplugin-autotooltip__default plugin-autotooltip_bigLoss of function: a general term used to describe mutant alleles that have less activity than wildtype. Amorphic and hypomorphic mutations are loss of function mutations. allelesplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence.. If we start with a guess that about 50% of genesplugin-autotooltip__default plugin-autotooltip_bigGene: read Chapters 02, 03, 04, 05, and 06 for a definition of gene :-) are essential, this means that each person should carry on average $(2\times 2\times 10^4)(0.5\times 2\times 10^{-4})= 4$ recessiveplugin-autotooltip__default plugin-autotooltip_bigRecessive: used to describe an allele, usually in comparison to wildtype. Recessive alleles do not exhibit their phenotype when combined with a wildtype allele. lethal mutationsplugin-autotooltip__default plugin-autotooltip_bigMutation: a change in the DNA of a gene that results in a change of phenotype compared to a reference wildtype allele. See also: mutant.!

We can determine whether this is a good estimate or not by measuring the genetic load. We define genetic load as the number of lethal mutationplugin-autotooltip__default plugin-autotooltip_bigMutation: a change in the DNA of a gene that results in a change of phenotype compared to a reference wildtype allele. See also: mutant. equivalents per genomeplugin-autotooltip__default plugin-autotooltip_bigGenome: a dataset that contains all DNA information of an organism. Most of the time, this also includes annotation and curation of that information, e.g., the names, locations, and functions of genes within the genome. As an adjective (“genomic”), this usually is used in the context of. Usually the genetic load is not a problem since it is very unlikely that both parents will happen to have lethal mutationsplugin-autotooltip__default plugin-autotooltip_bigMutation: a change in the DNA of a gene that results in a change of phenotype compared to a reference wildtype allele. See also: mutant. in the same genesplugin-autotooltip__default plugin-autotooltip_bigGene: read Chapters 02, 03, 04, 05, and 06 for a definition of gene :-). However, that chance is considerably increased for parents that are first cousins. As we have already calculated, the probability that an alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. from a grandparent will become homozygousplugin-autotooltip__default plugin-autotooltip_bigHomozygous: a state for a diploid organism wherein the two alleles for a gene are identical to each other. is 1/64 for 1st cousins. Thus, each recessiveplugin-autotooltip__default plugin-autotooltip_bigRecessive: used to describe an allele, usually in comparison to wildtype. Recessive alleles do not exhibit their phenotype when combined with a wildtype allele. lethal alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. for which one of the grandparents is a carrier will contribute an increased probability of 0.016 (one-sixteenth) that the grandchild will be homozygousplugin-autotooltip__default plugin-autotooltip_bigHomozygous: a state for a diploid organism wherein the two alleles for a gene are identical to each other. and therefore be afflicted by a lethal inherited defect.

We will use the frequency of stillbirth or neonatal death from first cousin marriages to estimate this. We must be careful to subtract the background frequency of stillbirths and neonatal deaths that are not due to genetic factors. These frequencies can be obtained from the cases where parents are not related.

If the adjusted frequency of stillbirths and neonatal deaths from first cousin marriages is $f_\text{cousins}=0.07$, and this frequency is adjusted at a rate of 0.016 above the general population, this means that the average frequency for recessiveplugin-autotooltip__default plugin-autotooltip_bigRecessive: used to describe an allele, usually in comparison to wildtype. Recessive alleles do not exhibit their phenotype when combined with a wildtype allele. lethal allelesplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. in both grandparents is $\frac{0.07}{0.016}=4.4$. Each grandparent (and therefore a typical person in the population) has an average of $\frac{4.4}{2}=2.2$ recessiveplugin-autotooltip__default plugin-autotooltip_bigRecessive: used to describe an allele, usually in comparison to wildtype. Recessive alleles do not exhibit their phenotype when combined with a wildtype allele. lethal allelesplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. in their genomeplugin-autotooltip__default plugin-autotooltip_bigGenome: a dataset that contains all DNA information of an organism. Most of the time, this also includes annotation and curation of that information, e.g., the names, locations, and functions of genes within the genome. As an adjective (“genomic”), this usually is used in the context of. Our estimate of the recessiveplugin-autotooltip__default plugin-autotooltip_bigRecessive: used to describe an allele, usually in comparison to wildtype. Recessive alleles do not exhibit their phenotype when combined with a wildtype allele. loss of functionplugin-autotooltip__default plugin-autotooltip_bigLoss of function: a general term used to describe mutant alleles that have less activity than wildtype. Amorphic and hypomorphic mutations are loss of function mutations. mutationplugin-autotooltip__default plugin-autotooltip_bigMutation: a change in the DNA of a gene that results in a change of phenotype compared to a reference wildtype allele. See also: mutant. rate (10<up>-4</sup>), as well as our estimate of the percentage of essential genesplugin-autotooltip__default plugin-autotooltip_bigGene: read Chapters 02, 03, 04, 05, and 06 for a definition of gene :-) (50%), was reasonably close.