Ordered Genotypes: An Extended ITO Method and a General Formula for Genetic Covariance

Abstract

Traditionally, the stochastic ITO transition matrices provide a simple general method for obtaining the joint genotype distribution and genotypic correlations between any specified pair of noninbred relatives. The ITO method has been widely used in modern genetic analysis; however, since it was originally derived for unordered genotypes, it is not very useful in some new applications—for example, when one is modeling genomic imprinting and must keep track of the parental origin of alleles. To address these new, emerging problems, here we extend the ITO method to handle ordered genotypes. Our extended method is applied to calculate the covariance in unilineal and bilineal relatives under genomic imprinting, and some generalized linear functions of the transition matrices are given. Since the ITO method is limited to biallelic loci and to unilineal and bilineal relatives, we derive a general formula for calculating the genetic covariance using ordered genotypes for any type of relative pair.

The “ITO” paper by Li and Sacks¹ provides an elegant algorithm for deriving joint genotype probabilities between pairs of relatives. With the ITO method, given the genotype of an individual, it is possible to derive the conditional probability of the genotypes of any noninbred relative of that individual: P(G₂|G₁), where G_i denotes the genotype of the ith person. The ITO method was extended to handle multiple alleles and was generalized for inbred populations.² The ITO method was generalized for multiple loci and was also extended to handle consanguinity.³

Although the ITO method has been widely used to solve various problems in human genetics, it uses only unordered genotypes; that is, the genotypes are unordered in the sense that maternal and parental contributions are not distinguished. However, some new applications require the use of ordered genotypes; for example, when one is modeling genomic imprinting, one must keep track of the parental (ordered) origin of alleles. Genomic imprinting occurs when the functional activity of a person’s allele depends on whether it was inherited maternally or paternally. Strong genomic imprinting renders an imprinted locus effectively haploid and thereby causes certain genetic diseases, including disorders affecting cell growth, development, and behavior.⁴ From the point of view of quantitative genetics, the effect of genomic imprinting is to make the phenotypical values of reciprocal heterozygotes different, which means various basic values of genetic quantities, as well as correlations, are not the same as the standard values. This difference may be crucial, especially in human quantitative genetics.

Although Campbell and Elston³ did attempt to extend the ITO method to handle ordered genotypes, their extension is flawed, because of an incorrect assumption that does not generalize (as we explain in detail below). Li⁵ revised the 4×4 Li-Sacks matrices¹ to 2×2 matrices by focusing on allele identity by descent (IBD) instead of genotype IBD. However, Li⁵ still did not consider ordered genotypes.

More-accurate modeling of underlying biological processes should lead to more-accurate and more-powerful inferences. In the present study, we extended the ITO method to handle ordered genotypes. We derived some generalized linear functions of the transition matrices for deriving the probabilities of an individual’s genotype, conditional on a relative’s genotype. In the application part of this work, our extended method is applied to calculate the covariance between both unilineal and bilineal relatives under imprinting. Although the ITO approach is pleasing in terms of its clarity and understandability, it is difficult to extend it to handle loci with multiple alleles, as well as to handle very complex inbred relative pairs. Therefore, we also derive a completely general formula for the genetic covariance, using ordered genotypes for any type of relative pair; this uses the approach of Gillois,⁶ as more recently elucidated by Lange.⁷ The resulting covariance equations can be easily applied in a variance component-based linkage analysis that takes genomic imprinting into account (not shown here).

Here, we notationally distinguish ordered and unordered genotypes. Accordingly, we let A/a, with a slash, represent an unordered genotype and let A|a, with a vertical bar, denote an ordered genotype. Here, without loss of generality, the maternal allele is listed to the left of the vertical bar and the paternal allele is listed to the right.

Two diploid outbred related individuals may share (1) both genes IBD, (2) one gene IBD, or (3) no genes IBD. If we denote transition matrices as matrices of conditional probabilities, then the three basic transition matrices corresponding to the number of identical genes shared in common by the two relatives are as follows.¹

The matrix I:

	G2=
G1	A/A	A/a	a/a
A/A	1	0	0
A/a	0	1	0
a/a	0	0	1

The matrix T:

	G2=
G1	A/A	A/a	a/a
A/A	p	q	0
A/a
a/a	0	p	q

The matrix O:

	G2=
G1	A/A	A/a	a/a
A/A	p2	2pq	q2
A/a	p2	2pq	q2
a/a	p2	2pq	q2

Following convention, in these three matrices, p and q represent the allele frequencies of A and a in the population, respectively, with p+q=1. The first matrix, I, P(G₂|G₁,share 2 IBD), gives the genotype transition probabilities for two relatives when they share two alleles IBD with person 1’s genotype given. In such a case, their genotypes are necessarily identical. When one is given to be A/A, the other must be the A/A, etc. The second matrix, T, P(G₂|G₁,share 1 IBD), gives the transition probabilities from one relative to the other when they share one gene IBD. Suppose the given relative is of genotype A/a (second row of T), then the other relative must share allele A or allele a in common with probability 0.5 plus a random allele from the population. The third matrix, O, P(G₂|G₁,share 0 IBD), gives the conditional probabilities when the two individuals do not have any alleles IBD in common. Hence, they are genetically unrelated individuals. Regardless of the genotype of one individual, the probabilities of the other individual having the genotypes (A/A, A/a, a/a) remain simply p², 2pq, and q², respectively, under Hardy-Weinberg equilibrium (HWE).

With these three basic matrices, it is then straightforward to find the joint distribution and correlation between any pair of unilineal relatives (unordered genotypes) in a random-mating population.¹ For autosomal genes, the general expression of the transition matrix for a specific pair of relatives is given as

where c_I, c_T, and c_o are the probabilities that the two specified relatives share both, one, and no genes IBD, respectively, with c_I+c_T+c_o=1.

Since the original ITO matrices were derived for unordered genotypes, they are not useful when one is modeling imprinting, since one must then keep track of parental origin. Thus, to take genomic imprinting into consideration, we must track where the IBD gene comes from by using ordered genotypes (where we list the maternal allele first). Since the heterozygote A|a may have a different genotypic value from a|A, Campbell and Elston³ introduced four basic transition matrices with dimension 4×4 (instead of three 3×3 matrices), which are listed below. Each matrix element represents the probability of sibling 2 having the specific ordered genotype conditional on the genotype of sibling 1. The subscripts m and f represent male and female. We use the same notation I and O as defined in the 3×3 matrices discussed above. The matrix S_m, P(G₂|G₁,share 1 alleleIBDthroughfather), specifies the probabilities of sibling 2’s genotypes conditional on sibling 1 sharing one allele IBD through the father. Similarly, the matrix S_f, P(G₂|G₁,share 1 alleleIBDthroughmother), specifies the probabilities of sibling 2’s genotypes conditional on sibling 1 sharing one allele IBD through the mother.

The matrix I:

	G2=
G1	A|A	A|a	a|A	a|a
A|A	1	0	0	0
A|a	0	1	0	0
a|A	0	0	1	0
a|a	0	0	0	1

The matrix S_m:

	G2=
G1	A|A	A|a	a|A	a|a
A|A	p	0	q	0
A|a	0	p	0	q
a|A	p	0	q	0
a|a	0	p	0	q

The matrix S_f:

	G2=
G1	A|A	A|a	a|A	a|a
A|A	p	q	0	0
A|a	p	q	0	0
a|A	0	0	p	q
a|a	0	0	p	q

The matrix O:

	G2=
G1	A|A	A|a	a|A	a|a
A|A	p2	pq	pq	q2
A|a	p2	pq	pq	q2
a|A	p2	pq	pq	q2
a|a	p2	pq	pq	q2

Campbell and Elston³ proposed that the transition matrix R for any specified pair of relatives could be derived by the formula

However, their formula for R is incorrect for some pairs of relatives, as we explain in detail below. Furthermore, a more complete derivation of the ITO method for ordered genotypes requires the specification of two additional matrices, T_f and T_m, which concern parent-offspring transitions. These matrices are as follows.

The matrix T_m:

	Goffspring=
Gfather	A/A	A/a	a/A	a/a
A/A	p	0	q	0
A/a
a/A
a/a	0	p	0	q

The matrix T_f:

	Goffspring=
Gmother	A/A	A/a	a/A	a/a
A/A	p	q	0	0
A/a
a/A
a/a	0	0	p	q

The derivations of the matrices T_m, P(G_offspring|G_father), and T_f, P(G_offspring|G_mother), are straightforward. For example, if the genotype of the mother is A|a, the conditional probabilities that the offspring’s genotypes are A|A, A|a, a|A, and a|a are  p/2,  q/2,  p/2, and  q/2, respectively. The reason is that the alleles A and a from the mother each have a 50% chance of being transmitted to her offspring.

Li⁵ showed how the 4×4 S_m and S_f matrices³ can be derived as “‘external tensors” of the 2×2 matrices. However, the matrices T_f and T_m cannot be derived via outer products. We have shown that T_f(T_m) can be computed as a weighted sum of the S_f(S_m) matrix and a permuted version of S_f(S_m), where the middle two rows are switched (details omitted).

Now, with the above six ordered genotype 4×4 transition matrices, we can derive the conditional probabilities for two specified outbred relatives, in which we track the origin of both alleles at a locus. We first consider some simple unilineal relatives. For parent and offspring pair, since they share one gene identical by descent, the transition probabilities from parent to offspring for unordered genotypes are the elements of matrix T.¹ However, in the ordered genotype method, we have to consider separately the transition of paternal and maternal alleles. The elements of matrix T_m are now the transition probabilities from father to offspring, and the elements of matrix T_f are the transition probabilities from mother to offspring. Note that with ordered genotypes, P(G_father|G_offspring) is not equal to P(G_offspring|G_father), whereas these two are equal with unordered genotypes.

Consider the transition probabilities from maternal grandmother to grandchild GC through mother M. If the genotype of the maternal grandmother is A|A, the resulting genotype of her daughter M will be A|? (where ? is some unknown allele), so the total conditional probability that the grandchild GC is A|a with allele A inherited from the mother M is

The reason is that there is a 50% chance that grandchild GC receives the grandmaternal A allele from her mother M and a 50% chance that she receives her maternal grandfather’s allele, which is A with probability p. So the conditional probability of the grandchild’s ordered genotype being A|a is then

where q is the probability of grandchild GC randomly inheriting the second allele (allele a) from his/her father.

It is much easier to understand if we do the above derivation with matrix manipulation. The probabilities of the mother are the elements of first row of matrix T_f, since we are conditioning on the grandmother’s A|A genotype. For each genotype of the mother M, the elements of second column of T_f are the probabilities of the genotype of the grandchild GC being A|a. The total conditional probability for the grandchild having genotype A|a given grandmother’s A|A genotype via the grandchild's mother is the product of the first row and second column of the transition matrix T_f

By the same algorithm, the conditional probabilities for a grandchild given a specific genotype for the maternal grandmother are given by the elements of the product matrix T_f×T_f=T²_f. In the same manner, the conditional probabilities of a grandchild’s genotypes given a specific genotype for the maternal grandfather (via his/her mother) are given by the elements of the product matrix T_m×T_f.

For the above grandmother-grandchild pair relationship, Campbell and Elston's³ formula (p. 229)

gives

as the probability of the genotype of the grandchild being A|a given that the genotype of the grandmother is A|A, which is clearly wrong. For another example, for an aunt-niece pair connected through the mother, c_T= 1/2, but c_T does not split in half as Campbell and Elston suggest; rather, all its “weight” goes on the T_f, and the correct matrix is

Li and Sacks¹ showed that

which means a grandparent-grandchild pair shares one gene IBD and no gene IBD with an equal 50% chance. T² also gives the conditional probabilities for half siblings. They also showed that, in general,

where n+1 is the total number of generations between the two relatives. When ordered genotypes are used, similar equations hold. For example,

where T²_m gives the conditional probabilities for half siblings who have same father but different mothers and T²_f gives the conditional probabilities for half siblings who have same mother but different fathers. And, in general,

where n+1 is the number of generations between two relatives. When n is infinitely large, the conditional probabilities for two relatives are given by the elements of the matrix O; that is, the two relatives could be treated as two random samples from the general human population who are all unrelated to each other.

Now, we model bilineal relatives. Let us first consider the simple but most important type: full siblings. Since full siblings have a 25% chance of sharing two genes IBD and a 25% chance of sharing no genes IBD, they therefore have a 50% chance of sharing one gene IBD. However, we are dealing with ordered genotypes, so the two siblings have a 25% chance of sharing one maternal allele IBD and no paternal allele IBD, and vice versa. Thus, the transition matrix for full siblings is

Other relatives who can share two genes IBD are double–first cousins whose parents are members of two sibships. There are six types of sibships in the general population, because of six different mating types.¹ With use of the same algorithm mentioned in figure 1 of the work of Li and Sacks¹ but with maternal and paternal alleles labeled, we derive the conditional matrix for double–first cousins, which is

Next, we try to model the relationship for some unlineal relatives in which the conditional matrix 𝒮 for full siblings and the matrices T_m and T_f are involved. We use the broad sense of the term “avuncular,” which includes uncle-nephew, uncle-niece, aunt-nephew, and aunt-niece relationships. The conditional probabilities of a nephew’s genotypes are then given by the product of 𝒮T_m(f) conditional on the uncle’s genotypes. Conversely, the conditional probabilities of the uncle’s genotypes are given by elements of the product of T_m(f)𝒮 conditional on the genotypes of the nephew. Whether T_m or T_f is the transition matrix involved depends on whether the mother or the father of the nephew is the “connecting” relative. We verified that 𝒮T_m(f)=T_m(f)𝒮, which indicates that the uncle-nephew matrix is the same as the nephew-uncle transition matrix.¹ Through further multiplication of matrices, we can also prove “the most remarkable property”^1(p352); that is,

which indicates that uncle-nephew relationships are the same as those for grandparent-grandchild or half siblings¹; whether the transition matrix T_m or T_f is involved depends on whether the uncle is the nephew’s paternal uncle or the nephew’s maternal uncle. Extension of the above equations results in other important matrices,

whose elements give the conditional probabilities for first cousins and the probabilities for the great-grandchild conditional on one given great-grandparent.

Next, as an illustration of the utility of our extended ordered-genotype ITO method, we now derive equations for the genetic covariance between siblings and for the covariance between parent and child, with genomic imprinting considered. We begin with the standard genetic model and extend it to consider the case in which a locus is subject to imprinting. To derive the covariance formulas, it is necessary first to define the QTL model and its variance components. Here, we briefly review the results of Spencer.⁸ Assume that an unobserved major gene has two alleles, allele A and allele a, with P(A)=p and P(a)=q. In the standard genetic model, the genotypic value is a if the genotype is A/A, that of the a/a homozygote is -a, and that of heterozygote A/a is d. However, under imprinting, different genotypic values are possible for the two possible heterozygotes: d₁ for A|a and d₂ for a|A (fig. 1). It is usually assumed that a⩾d₁ and d₂⩾-a. We have d₁=a (d₂=-a) when there is complete inactivation of the maternally (paternally) derived allele. A measure of imprinting is denoted as⁹

Figure 1.

Genotypic values of the four possible genotypes

Spencer⁸ derived many useful genetic components of variance under imprinting, which are summarized in table 1. When d₁=d₂=d—that is, there is no imprinting (I=0)—the above various genetic values “revert” to their standard values. We further show that

and

(no constraints on I), where σ²_a and σ²_d are the additive genetic variance and dominance genetic variance, respectively, under the standard genetic model with no imprinting (details not presented here).

Table 1.

Values of Genetic Components of Variance under Genomic Imprinting^[Note]

Name	Expression	Definition
μ	a(p-q)+(d1+d2)pq	Mean phenotype of a population in HWE
αm	a+d2q-d1p	Average effect of a gene substitution for males
αf	a+d1q-d2p	Average effect of a gene substitution for females
σ2Am	2pqα2m	Additive genetic variance for males
σ2Af	2pqα2f	Additive genetic variance for females
σ2D	pq[pq(d1+d2)2+(d1-d2)2]	Dominance genetic variance
σ2Dm	pq[pq(d1+d2)2+(d1-d2)2]	Dominance genetic variance for males
σ2Df	pq[pq(d1+d2)2+(d1-d2)2]	Dominance genetic variance for females
σ2G	pq[2αmαf+pq(d1+d2)2+(d1-d2)2]	Overall genetic variance
σADm	pqαm(d1-d2)	Covariance between dominance deviation and breeding value for males
σADf	pqαf(d2-d1)	Covariance between dominance deviation and breeding value for females

Note.— The table reflects the work of Spencer.⁸

With the above definitions of different genetic variance components (table 1), we now begin to derive the covariance between siblings and between parent and offspring under genomic imprinting, using our transition matrices. Spencer⁸ derived the covariance between parent and offspring under genomic imprinting. However, he did not derive the covariance between a pair of full siblings. As an illustration of the utility of our ordered ITO method, we first derive the covariance between sib pairs and also include a short part on deriving covariance between parent and offspring, to verify that our results match Spencer’s results.

Let k₀ denote the probability of sibling 1 and sibling 2 sharing no alleles IBD, k_1m denote the probability of sibling 1 and sibling 2 sharing one paternal allele IBD, k_1f denote the probability of sibling 1 sharing one maternal allele IBD with sibling 2, and k₂ denote the probability of sibling 1 and sibling 2 sharing two alleles IBD. The probability of sibling 2 having a particular genotype (G₂) given the genotype of sibling 1 (G₁) can be calculated as

which, for full siblings, is the same as matrix 𝒮 (in eq. [1] above).

Given the above conditional matrix, the joint probability of sibling 2 having a certain genotype s₂ and sibling 1 having another certain genotype s₁, P(s₁,s₂) can be derived by multiplying the specific element of the above matrix P(G₂|G₁) by the probability of having the certain genotype for sibling 1. The genetic covariance between a pair of siblings can be derived as

For full siblings, k₀=1/4, k_1m=1/4, k_1f=1/4, and k₂=1/4; we get the following model from equation (2):

As stated earlier, σ²_Am+σ²_Af=2σ²_a and σ²_D+σ_ADm+σ_ADf=σ²_d (σ²_a and σ²_d are additive genetic variance and dominance genetic variance, respectively, under the standard genetic model). Thus, equation (3) simplifies to be exactly the standard genetic model, defined as

For half siblings, we distinguish between half siblings who share a mother (k₀=1/2, k_1f=1/2, and k₂=0) and half siblings who share a father (k₀=1/2, k_1m=1/2, and k₂=0). Equation (2) gives, respectively,

In the next section, we derive equations for the genetic covariance for a parent-offspring pair as follows. Let o and p_f be denoted as the genotypic values of offspring and mother, respectively. The joint probability of the offspring having a certain genotype and the mother having another genotype, P(o,p_f), can be derived by multiplying the specific element of the matrix T_f—P(G_offspring|G_mother)—by the probability that the mother has that genotype. Then, the covariance σ_opf is calculated as

Similarly, we can derive the covariance equation for father and offspring:

Although historically attractive, the ITO method we generalized here is limited to biallelic loci and to unilineal and bilineal relatives. To tackle these limitations, we now introduce a more general way of calculating the covariance for any relative pair under imprinting. This is a generalization of the approach used by Gillois,⁶ which was more recently summarized by Lange,⁷ whose notation we use here.

First, suppose there are two or more alleles, let the kth allele have population frequency p_k and the ordered genotype k|l have the trait value w_k|l, then we can write

where α_k is the additive impact of the maternal allele, β_l is the additive impact of the paternal allele, and δ_k|l is the residual departure from additivity. Under imprinting, the identity w_k|l=w_l|k does not necessarily hold. No generality is lost if we adjust the trait mean to be 0, so that . The allelic contributions α_k and β_l are chosen to minimize the deviations δ_k|l=w_k|l-α_k-β_l. One way of doing this is to minimize the sum of squares

which is achieved by taking and (see appendix A).

Next, suppose individuals i and j are relatives. The covariance Cov(X_i,X_j) between the trait values X_i and X_j of i and j can be computed in the following steps. Here, we use the 15 detailed identity states of Gillois,⁶ Harris,¹⁰ and Jacquard.¹¹ Figure 2 (in the style of Sobel et al.¹² and Jacquard¹³) shows the 15 detailed identity states that are possible when maternal and paternal alleles are distinguished. The states vary from sharing no alleles IBD, S₁₅, to sharing all four alleles IBD, S₁.

Figure 2.

The 15 possible detailed identity states for individuals i and j with ordered genotypes. The squares represent maternal alleles, and the circles represent paternal alleles. Within each state, the top two symbols are i’s alleles, and the lower two symbols are j’s alleles. Lines connect alleles that are IBD (in the style of Sobel et al.¹² and Jacquard¹³).

Conditioning on these detailed identity states of the two relatives and using the identities , , , and , we can deduce

where δ_i is the probability of the ith detailed identity state.¹¹ A detailed derivation of equation (7) is given in appendix A. When there is no imprinting, equation (7) reduces to the general covariance equation derived by Gillois.⁶

Since trait means E(X_i)=E(X_j)=0, the covariance Cov(X_i,X_j)=E(X_i,X_j), which is given in equation (7). If we assume that neither i nor j is inbred, we have δ₁=δ₂=…=δ₈=0. The covariance Cov(X_i,X_j) then simplifies to

When there are two alleles, we can rewrite the summations in terms of our notation used above:

Thus, from our general equation (8), we obtain the same covariances as we derived above for full siblings (eq. [3]), half siblings (eq. [4]), mother-offspring pairs (eq. [5]), and father-offspring pairs (eq. [6]). Furthermore, our equation (7) can be used to generalize the variance-components model developed by Shete et al.¹⁴ to handle all possible types of inbred relative pairs.

Once the detailed identity coefficients¹¹ are computed, any relative-to-relative covariance is expressible in terms of the theoretical variances and covariances defined above. An algorithm for computing these detailed identity coefficients (under the assumption that the entire pedigree structure connecting the two individuals is known) was derived by Nadot and Vaysseix.¹⁵

In summary, in this work, we extended the ITO method,^1,3 to handle ordered genotypes in an attempt to generalize this simple but useful method. We also showed that Campbell and Elston’s previous formula³ for the transition matrix R is incorrect for some pairs of relatives. In practice, a more complete derivation of the ITO method for ordered genotypes requires the specification of two additional matrices, T_m and T_f, which we derived in this work. By tracking the paternal or maternal origin of each allele, we now have six basic transition matrices, with the help of which it is possible to derive conditional probabilities between two specified outbred relatives when we need to distinguish the two forms of the heterozygotes. In addition to providing an algorithm for deriving conditional probabilities with use of ordered genotypes, the ITO approach can be used to derive formulas for the genetic covariance between a pair of relatives. To complement the more limited ITO approach, we also derived a completely general formula, for the genetic covariance, using ordered genotypes; this formula is applicable to multiallelic loci and to any type of inbred relative pair.

We illustrated the utility of the extended ITO approach and our general covariance formula by using them to derive the genetic covariance under imprinting between parent and offspring and between sib pairs. The derived formulas for the covariance between a parent's and an offspring's genotypic values are the same as those given in the study by Spencer.⁸ The consistency of our equations with previous work proves the applicability of our proposed method for the calculation of covariance between two relatives when we have to deal with ordered genotypes—for example, when we try to model genomic imprinting in human quantitative genetic analysis. Also, it should be noted that our work could help accurately test genetic hypotheses or predict risk (for genetic counseling), given a known genetic model.³ Our extended ordered-genotype ITO method and our general covariance formula, with their easy applicability, will be helpful in modeling the complex relationship between relatives under the important biological phenomena (genomic imprinting) that need further statistical attention.

Appendix A : Covariance between Individuals i and j under Imprinting

Let the kth allele have population frequency p_k and the ordered genotype k|l have trait value w_k|l, then we can write w_k|l=α_k+β_l+δ_k|l, where α_k is the additive impact of the maternal allele, β_l is the additive impact of the paternal allele, and δ_k|l is the residual departure from additivity. Under imprinting, the identity w_k|l=w_l|k does not necessarily hold. No generality is lost if we take the trait mean

To minimize

the partial derivative with respect to α_k was taken, which gives

which is true only if

Using the above facts and because

we can conclude that . Similarly, also holds.

If X_i and X_j are traits of individuals i and j, since E(X_i)=E(X_j)=0, then the covariance is