Measures of genetic variability

Nucleotide diversity (\(\pi\)) - One common measure of genetic diversity is the average number of single nucleotide differences between haplotypes chosen at random from a sample. This is called nucleotide diversity and is often denoted by \(\pi\). For example, we can calculate \(\pi\) for our ADH locus from Table \ref{Table:ADH} above: we have 6 sequences from D. simulans (a-f), there’s a total of 15 ways of pairing these sequences, and

\[\pi=\frac{1}{15} \big( (2 + 1 + 1 + 1 + 0 ) + (3 + 3 + 3 + 2 ) +(0 + 0 + 1) + (0 + 1) + (1) \big)=1.2\overline{6}\]

where the first bracketed term gives the pairwise differences between a and b-f, the second bracketed term the differences between b and c-f and so on.

Our \(\pi\) measure will depend on the length of sequence it is calculated for. Therefore, \(\pi\) is usually normalized by the length of sequence, to be a per site (or per base) measure. For example, our ADH sequence covers \(397\)bp of DNA and so \(\pi = 1.2\overline{6}/397=0.0032\) per site in D. simulans for this region. Note that we could also calculate \(\pi\) per synonymous site (or non-synonymous). For synonymous site \(\pi\), we would count up number of synonymous differences between our pairs of sequences, and then divide by the total number of sites where a synonymous change could have occurred.Technically we would need to divide by the total number of possible point mutations that would result in a synonymous change; this is because some mutational changes at a particular nucleotide will result in a non-synonymous or synonymous change depending on the base-pair change.

Levels of genetic variability across species

Two observations have puzzled population geneticists since the inception of molecular population genetics. The first is the relatively high level of genetic variation observed in most obligately sexual species. This first observation, in part, drove the development of the Neutral theory of molecular evolution, the idea that much of this molecular polymorphism may simply reflect a balance between genetic drift and mutation. The second observation is the relatively narrow range of polymorphism across species with vastly different census sizes. This observation represented a puzzle as the Neutral theory predicts that levels of genetic diversity should scale with population size. Much effort in theoretical and empirical population genetics has been devoted to trying to reconcile models with these various observations. We’ll return to discuss these ideas throughout our course.

The first observations of molecular genetic diversity within natural populations were made from surveys of allozyme data, but we can revisit these general patterns with modern data. For example, Leffler et al. (2012) compiled data on levels of within-population, autosomal nucleotide diversity (π) for 167 species across 14 phyla from non-coding and synonymous sites (Figure 2.3). The species with the lowest levels of π in their survey was Lynx, with π = 0.01%, i.e. only 1/10000 bases differed between two sequences. In contrast, some of the highest levels of diversity were found in Ciona savignyi, Sea Squirts, where a remarkable 1/12 bases differ between pairs of sequences. This 800-fold range of diversity seems impressive, but census population sizes have a much larger range.

Genotypic sharing between pairs of individuals

Our \(r\) coefficients are going to have various uses. For example, they allow us to calculate the probability of the genotypes of a pair of relatives. Consider a biallelic locus where allele \(A_1\) is at frequency \(p\), and two individuals who have IBD allele sharing probabilities \(r_0\), \(r_1\), \(r_2\). What is the overall probability that these two individuals are both homozygous for allele 1? Well that’s

\[\begin{aligned} \mathbb{P}(\textrm{both } A_1 A_1) = & \mathbb{P}(\textrm{both } A_1 A_1 | 0 \text{ alleles } \mathrm{IBD}) \mathbb{P}(0 \text{ alleles } \mathrm{IBD})  + \mathbb{P}(\textrm{both } A_1 A_1 | 1 \text{ allele }IBD) \mathbb{P}(1\text{ allele }IBD) + \mathbb{P}(\textrm{both } A_1 A_1 | 2 \text{ alleles }IBD) \mathbb{P}(2\text{ alleles }IBD)\end{aligned}\]

Or, in our \(r_0\), \(r_1\), \(r_2\) notation:

\[\begin{aligned} \mathbb{P}(\textrm{both } A_1 A_1) = & \mathbb{P}(\textrm{both } A_1 A_1 | \textrm{0 alleles IBD}) r_0 + \mathbb{P}(\textrm{both } A_1 A_1 | \textrm{1 alleles IBD}) r_1 + \mathbb{P}(\textrm{both } A_1 A_1 | \textrm{2 alleles IBD}) r_2 \label{eqn:initial_relly_IBD_calc}\end{aligned}\]

If our pair of relatives share \(0\) alleles IBD, then the probability that they are both homozygous is \(\mathbb{P}(\textrm{both } A_1 A_1 | \text{0 alleles IBD}) =p^2 \times p^2\), as all four alleles represent independent draws from the population. If they share \(1\) allele IBD, then the shared allele is of type \(A_1\) with probability \(p\), and then the other non-IBD allele, in both relatives, also needs to be \(A_1\) which happens with probability \(p^2\), so \(\mathbb{P}(\textrm{both } A_1 A_1 | \text{1 alleles IBD})=p \times p^2\). Finally, our pair of relatives can share two alleles IBD, in which case \(\mathbb{P}(\textrm{both } A_1 A_1 | \text{2 alleles IBD}) = p^2\), because if one of our individuals is homozygous for the \(A_1\) allele, both individuals will be. Putting this all together our \ref{eqn:initial_relly_IBD_calc} becomes

\[\mathbb{P}(\textrm{both } A_1 A_1) = p^4 r_0 + p^3 r_1 + p^2 r_2 \label{eqn:IBD_relly_calc}\]

Note that for specific cases we could also calculate this by summing over all the possible genotypes their shared ancestor(s) had; however, that would be much more involved and not as general as the form we have derived here.

We can write out terms like \ref{eqn:IBD_relly_calc} for all of the possible configurations of genotype sharing/non-sharing between a pair of individuals. Based on this we can write down the expected number of polymorphic sites where our individuals are observed to share 0, 1, or 2 alleles.

There’s a variety of ways to estimate the relationships among individuals using genetic data. An example of using allele sharing to identify relatives is offered by the work of Nancy Chen . has collected genotyping data from thousands of Florida Scrub Jays at over ten thousand loci. These Jays live at the Archbold field site, and have been carefully monitored for many decades allowing the pedigree of many of the birds to be known. Using these data, she estimates allele frequencies at each locus. Then by equating the observed number of times that a pair of individuals share \(0\), \(1\), or \(2\) alleles to the theoretical expectation, she estimates the probability of \(r_0\), \(r_1\), and \(r_2\) for each pair of birds. A plot of these are shown in Figure \(\PageIndex{13}\), showing how well the estimates match those known from the pedigree.

Sharing of genomic blocks among relatives

We can more directly see the sharing of the genome among close relatives using high-density SNP genotyping arrays. In Figure \ref{fig:first_cousin_IBD} we show a simulation of first cousins’ genomic sharing from their shared grandmother. Colored purple are regions where they have matching genomic material, due to having inherited it IBD from their shared grandmother.

First cousins will share at least one allele of your genotype at all of the polymorphic loci in these purple regions. There’s a range of methods to detect such sharing. One way is to look for unusually long stretches of the genome where two individuals are never homozygous for different alleles. By identifying pairs of individuals who share an unusually large number of such putative IBD blocks, we can hope to identify unknown relatives in genotyping datasets. In fact, companies like 23&me and Ancestry.com use signals of IBD to help identify family ties.

As another example, consider the case of third cousins. You share one of eight sets of great-great-grandparents with each of your (likely many) third cousins. On average, you and each of your third cousins each inherit one-sixteenth of your genome from each of those two great-great-grandparents. This turns out to imply that on average, a little less than one percent of your and your third cousin’s genomes (\(2 \times (1/16)^2 =0.78\%\)) will be identical by virtue of descent from those shared ancestors. A simulated example where third cousins share blocks of their genome (on chromosome 16 and 2) due to their great-great-grandmother is shown in Figure \(\PageIndex{14}\). 

Note how if you compare Figure \(\PageIndex{15}\) and Figure \(\PageIndex{14}\), individuals inherit less IBD from a shared great-great-grandmother than from a shared grandmother, as they inherit from more total ancestors further back. Also notice how the sharing occurs in shorter genomic blocks, as it has passed through more generations of recombination during meiosis. These blocks are still detectable, and so third cousins can be detected using high-density genotyping chips, allowing more distant relatives to be identified than single marker methods alone. More distant relations than third cousins, e.g. fourth cousins, start to have a significant probability of sharing none of their genome IBD. But you have many fourth cousins, so you will share some of your genome IBD with some of them; however, it gets increasingly hard to identify the degree of relatedness from genetic data the deeper in the family tree this sharing goes.

Inbreeding

We can define an inbred individual as an individual whose parents are more closely related to each other than two random individuals drawn from some reference population.

When two related individuals produce an offspring, that individual can receive two alleles that are identical by descent, i.e. they can be homozygous by by descent (sometimes termed autozygous), due to the fact that they have two copies of an allele through different paths through the pedigree. This increased likelihood of being homozygous relative to an outbred individual is the most obvious effect of inbreeding. It is also the one that will be of most interest to us, as it underlies a lot of our ideas about inbreeding depression and population structure. For example, in Figure \(\PageIndex{16}\) our offspring of first cousins is homozygous by descent having received the same IBD allele via two different routes around an inbreeding loop.

As the offspring receives a random allele from each parent (\(i\) and \(j\)), the probability that those two alleles are identical by descent is equal to the kinship coefficient \(F_{ij}\) of the two parents (\ref{eqn:coeffkinship}). This follows from the fact that the genotype of the offspring is made by sampling an allele at random from each of our parents.

The only way the offspring can be heterozygous (\(A_1 A_2\)) is if their two alleles at a locus are not IBD (otherwise they would necessarily be homozygous). Therefore, the probability that they are heterozygous is

\[\mathbb{P}(A_1 A_2) = \mathbb{P}(A_1 A_2 |\textrm{I & J not IBD} ) \P (\textrm{I & J not IBD} ) = 2p q (1-F_{ij}) , \label{eq:hetGenHW}\]

The offspring can be homozygous for the \(A_1\) allele in two different ways. They can have two non-IBD alleles that are not IBD but happen to be of the allelic type \(A_1\), or their two alleles can be IBD, such that they inherited allele \(A_1\) by two different routes from the same ancestor. Thus, the probability that an offspring is homozygous for \(A_1\) is

\[\begin{aligned} P(A_1 A_1) & = \mathbb{P}(A_1 A_1 |\textrm{I & J not IBD} ) \P (\textrm{i & j not IBD} ) + \mathbb{P}(A_1 A_1 |\textrm{i & j IBD} ) \P (\textrm{i & j IBD}) \nonumber\\ &= p^2(1-F_{ij}) + pF_{ij}.\end{aligned}\]

using the Law of Total Probability (see Appendix \ref{eqn:law_tot_prob}). Therefore, the frequencies of the three possible genotypes can be written as given in Table \(\PageIndex{5}\), which provides a generalization of the Hardy–Weinberg proportions.

Multiple inbreeding loops in a pedigree

Up to this point we have assumed that there is at most one inbreeding loop in the recent family history of our individuals, i.e. the parents of our inbred individual have at most one recent genealogical connection. However, an individual who has multiple inbreeding loops in their pedigree can be homozygous by descent thanks to receiving IBD alleles via multiple different different loops. To calculate inbreeding in pedigrees of arbitrary complexity, we can extend beyond our original relatedness coefficients \(r_0\), \(r_1\), and \(r_2\) to account for higher order sharing of alleles IBD among relatives. For example, we can ask, what is the probability that both of the alleles in the first individual are shared IBD with one allele in the second individual? There are nine possible relatedness coefficients in total to completely describe kinship between two diploid individuals, and we won’t go in to them here as it’s a lot to keep track of. However, we will show how we can calculate the inbreeding coefficient of an individual with multiple inbreeding loops more directly.

Let’s say the parents of our inbred individual (B and C) have \(K\) shared ancestors, i.e. individuals who appear in both B and C’s recent family trees. We denote these shared ancestors by \(A_1, \dots,A_K\), and we denote by \(n\) the total number of individuals in the chain from B to C via ancestor \(A_i\), including B, C, and \(A_i\). For example, if B is C’s aunt, then B and C share two ancestors, which are B’s parents and, equivalently, C’s grandparents. In this case, there are n=4 individuals from B to C through each of these two shared ancestor. In the general case, the kinship coefficient of B and C, i.e. the inbreeding coefficient of their child, is

\[F = \sum_{i=1}^K \frac{1}{2^{n_i}} \big( 1+ f_{A_i} \big) \label{eqn:inbreeding_over_ancs}\]

where \(f_{A_i}\) is the inbreeding coefficient of the ancestor \(A_i\). What’s happening here is that we sum over all the mutually-exclusive paths in the pedigree through which B and C can share an allele IBD. With probability \(\frac{1}{2^{n_i}}\), a pair of alleles picked at random from B and C is descended from the same ancestral allele in individual \(A_i\), in which case the alleles are IBD. However, even if B inherits the maternal allele and C inherits the paternal allele of shared ancestor \(A_i\), if \(A_i\) was themselves inbred, with probability \(f_{A_i}\) those two alleles are themselves IBD. Thus a shared inbred ancestor further increases the kinship of B and C.

Multiple inbreeding loops increase the probability that a child is homozygous by descent at a locus, which can be calculated simply by plugging in \(F\), the child’s inbreeding coefficient, into our generalized HW equation.

As one extreme example of the impact of multiple inbreeding loops in an individual’s pedigree, let’s consider king Charles II of Spain, the last of the Spanish Habsburgs. Charles was the son of Philip IV of Spain and Mariana of Austria, who were uncle and niece. If this were the only inbreeding loop, then Charles would have had an inbreeding coefficient of \(\frac{1}{8}\). Unfortunately for Charles, the Spanish Habsburgs had long kept wealth and power within their family by arranging marriages between close kin. The pedigree of Charles II is shown in Figure \(\PageIndex{17}\), and multiple inbreeding loops are apparent. For example, Phillip III, Charles II’s grandfather and great-grandfather, was himself a child of an uncle-niece marriage.

calculated that Charles II had an inbreeding coefficient of \(0.254\), equivalent to a full-sib mating, thanks to all of the inbreeding loops in his pedigree. Therefore, he is expected to have been homozygous by descent for a full quarter of his genome. As we’ll talk about later in these notes, this means that Charles may have been homozygous for a number of recessive disease alleles, and indeed he was a very sickly man who left no descendants due to his infertility. Thus plausibly the end of one of the great European dynasties came about through inbreeding.

Calculating inbreeding coefficients from genetic data

If the observed heterozygosity in a population is \(H_O\), and we assume that the generalized Hardy–Weinberg proportions hold, we can set \(H_O\) equal to \(f_{12}\), and solve Equation \ref{eq:hetGenHW} for \(F\) to obtain an estimate of the inbreeding coefficient as

\[\hat{F} = 1-\frac{f_{12}}{2pq} = \frac{2pq - f_{12}}{2pq}. \label{eqn:Fhat}\]

As before, \(p\) is the frequency of allele \(A_{1}\) in the population. This can be rewritten in terms of the observed heterozygosity (\(H_O\)) and the heterozygosity expected in the absence of inbreeding, \(H_E=2pq\), as

\[\hat{F} = \frac{H_E-H_O}{H_E} = 1 - \frac{H_O}{H_E}. \label{eqn:FhatHO}\]

Hence, \(\hat{F}\) quantifies the deviation due to inbreeding of the observed heterozygosity from the one expected under random mating, relative to the latter.

If we have multiple loci, we can replace \(H_O\) and \(H_E\) by their means over loci, \(\bar{H}_O\) and \(\bar{H}_E\), respectively. Note that, in principle, we could also calculate \(F\) for each individual locus first, and then take the average across loci. However, this procedure is more prone to introducing a bias if sample sizes vary across loci, which is not unlikely when we are dealing with real data.

Genetic markers are commonly used to estimate inbreeding for wild and/or captive populations of conservation concern. As an example of this, consider the case of the Mexican wolf (Canis lupus baileyi), a sub-species of gray wolf.

They were extirpated in the wild during the mid-1900s due to hunting, and the remaining five Mexican wolves in the wild were captured to start a breeding program. estimated the current-day, average expected heterozygosity to be \(0.18\), based on allele frequencies at over forty thousand SNPs. However, the average Mexican wolf individual was only observed to be heterozygous at \(12\%\) of these SNPs. Therefore, the average inbreeding coefficient for the Mexican wolf is \(\hat{F} = 1 -\frac{0.12}{0.18}\), i.e. \(\sim 33 \%\) of a lobo’s genome is homozygous due to recent inbreeding in their pedigree.

Genomic blocks of homozygosity due to inbreeding.

As we saw above, close relatives are expected to share alleles IBD in large genomic blocks. Thus, when related individuals mate and transmit alleles to an inbred offspring, they transmit these alleles in big blocks through meiosis. As an example, let’s return to the case of our hypothetical first cousins from Figure \ref{fig:IBD_cousins_chr_cartoon}. If this pair of individuals had a child, one possible pattern of genetic transmission is shown in Figure \(\PageIndex{20}\). The child has inherited the red stretch of chromosome via two different routes through their predigree from the grandparents. This is an example of an autozygous segment, where the child is homozygous by descent at all of the loci in this red region.

The inbreeding coefficient of the child sets the proportion of their genome that will be in these autozygous segments. For example, a child of first full cousins is expected to have \(1/16\) of their genome in these segments. The more distant the loop in the pedigree, the more meioses that chromosomes have been through and the shorter individual blocks will be. A child of first cousins will have longer blocks than a child of second cousins, for example.

Individuals with multiple inbreeding loops in their family tree can have a high inbreeding coefficient due to the combined effect of many small blocks of autozygosity. For example, Charles II had an inbreeding coefficient that is equivalent to that of the child of full-sibs, with a quarter of his genome expected to homozygous by descent, but this would be made up of many shorter blocks.

We can hope to detect these blocks by looking for unusually long genomic runs of homozygosity (ROH) sites in an individual’s genome. One way to estimate an individual’s inbreeding coefficient is then to total up the proportion of an individual’s genome that falls in such ROH regions. This estimate is called \(F_{ROH}\).

An example of using \(F_{ROH}\) to study inbreeding comes from the work of Sams and Boyko (2018b), who identified runs of homozygosity in 2,500 dogs, ranging from 500kb up to many megabases. Figure \(\PageIndex{21}\) shows the distribution of FROH of individuals in each dog breed for the X and autosome. In Figure \(\PageIndex{23}\) this is broken down by the length of ROH segments.

Dog breeds have been subject to intense breeding that has resulted in high levels of inbreeding. Of the population samples examined, Doberman Pinschers have the highest levels of their genome in runs of homozygosity (\(F_{ROH}\)), somewhat higher than English bulldogs. In Figure \(\PageIndex{23}\) we can see that English bulldogs have more short ROH than Doberman Pinschers, but that Doberman Pinschers have more of their genome in very large ROH (\(>16 Mb\)). This suggests that English bulldogs have had long history of inbreeding as they have many small blocks, but that Doberman Pinschers have a lot of recent inbreeding as their autozygosity is contained in long blocks relatively unbroken by recombination.

Summary

• This chapter developed the relationship between allele frequencies and genotype frequencies within a generation and among relatives.
• Under random mating, we derived expectations of the genotype frequencies (Hardy-Weinberg), and we can identify deviations away from these expectations.
• Identity by descent (IBD) refers to the sharing of alleles due to a recent shared biological relationship.
• We can predict the probability and expected level of sharing of alleles IBD among pairs of relatives using mendelian transmission probabilities (as contained in coefficients \(r_0\), \(r_1\), and \(r_2\)). One useful summary of relatedness for a pair of individuals is the kinship coefficient \(F_{i,j}\).
• We can also learn about genetic relationships from the sharing of genomic segments among relatives, with many long shared segments revealing a closer relationship.
• An inbred individual has parents who are more closely related than random draws from some reference population.
• Inbreeding results in decreased heterozygosity and a complementary increase in homozygosity. We can use the kinship coefficient of the parents to estimate the distortion away from Hardy-Weinberg and the expected level of heterozygosity.
• Inbreeding coefficients can be calculated from genetic data, either for multiple individuals at a single locus or for multiple loci for a single individual.