The effect of linkage on the additive by additive covariance between relatives1

- The additive x additive relationship coefficient needs to be calculated in order to compute genetic covariance between relatives. For linked loci, the computation of this coefficient is not as simple as for unlinked loci. Recursive formulae are given to compute the additive x additive relationship coefficient for an arbitrary pedigree. Based on the recursive formulae, numerical values of the desired coefficient for selfed or outbred individuals are examined. The method presented provides the means to compute the additive x additive relationship coefficient for any situation assuming linkage. The effect of linkage on the covariance was examined for several pairs of relatives. In the absence of inbreeding, linkage has no effect on the parent-offspring covariance. All of the other relationships examined were affected by linkage. As recombination rate increased from 0.1 to 0.5, in descending order of percentage change in the covariance, the relationships ranked as follows: first cousins, double first cousins, grandparent-grandoffspring, half sibs, aunt-nephew, full sibs, parent-offspring. With inbreeding, the parent-offspring covariance is also affected by linkage.


INTRODUCTION
Genetic covariance between relatives can be expressed as a linear combination of genetic variance components. In order to compute the covariance between relatives, coefficients associated with the variance components need to be calculated from pedigree relationships. Additive and dominance relationship coefficients can be computed through several methods for arbitrary pedigrees [4-6, 8, 10!. The additive x additive relationship coefficient between unlinked loci can be obtained as the square of the additive relationship coefficient (7!. When loci are linked, the additive x additive relationship coefficient cannot be computed simply as the square of the additive relationship coefficient. Now this coefficient may depend on the recombination rate and it has been derived for several common relationships [2,3,12). A general approach for computing the additive x additive relationship coefficient for collateral relatives has been developed by Schnell [9]. For general pedigrees, this approach becomes very complicated. More recently, Thompson !11! has described a recursive approach for computing two-locus identity probabilities that can be applied to any pedigree.
In this paper we present independently derived recursive formulae that are different from those of Thompson for computing the additive x additive relationship coefficient for an arbitrary pedigree. These formulae can be used to examine the effect of linkage on the additive x additive relationship coefficient for any pair of relatives. Based on the results obtained in this paper, the situations when the effect of linkage on the additive x additive covariance between relatives can be ignored are examined. Some examples are given here and a C + + implementation of the recursive method with some numerical examples is available on the Web at http://www.public.iastate.edu/ N rohan by following the link Software.
2. THEORY Additive x additive coefficients are generated by epistatic effects in the covariance. Consider a two-locus model with an arbitrary number of alleles at each locus. The additive x additive genotypic value of an individual I with alleles k and k' at the first locus and alleles 1 and I' at the second locus can be written as where 6 is the additive x additive effect. Similarly, the additive x additive genotypic value for an individual J with alleles n, n', p and p' is The additive x additive contribution to the covariance between I and J can be written as a sum of 16 covariances.
Each of the 16 covariances can be written as the product between one-fourth of the additive x additive variance component (V AA ) and a probability that pairs of alleles are identical by descent (IBD). For example C OV (b kl , 6,,,) in equation (3) is where Pr(kn, Ip) is the probability that the allele k of individual I is IBD with allele n of individual J and allele of I is IBD with allele p of J. The additive x additive relationship coefficient (!I,!) is one-fourth of the sum of the 16 IBD probabilities corresponding to the 16 covariances in equation (3).
Each of these probabilities can be obtained recursively as explained below.

RECURSIVE COMPUTATION OF IBD PROBABILITIES
The principle underlying the recursive method for computing IBD probabilities is first described for a single locus. Then we show how to compute recursively IBD probabilities for two loci.
where, for example, A I f -A B is the condition that A 1 is a copy of !4!. If J is not a descendent of I, equation (5) can be simplified to However, equation (6) is not true when J is a descendent of I, because now the IBD relationships between A! and AD and between A! and AD depend on whether AI is a copy of AD or of !4!,. In order to take advantage of equation (6), it is necessary to determine whether I or J is younger, and always recurse on the younger allele. Using this procedure the recursion can be performed until both alleles in an IBD relationship are from founders. In founders, the IBD probability between two different alleles is defined to be null and is unity for an allele with itself. Several authors have used recursion to compute IBD probabilities between alleles at a single locus [6, 8, 10!. probability the pair of alleles from I is of the non-recombinant type. This pair is a copy of either one of the two non-recombinant or one of the two recombinant gametes of D. Thus, using recursion, this probability can be written as where r is the recombination rate between A and B. The pairs of alleles from I in the first eight probabilities are of the non-recombinant type, and can be computed as shown in equation (8). The pairs of alleles from I in the last eight probabilities are of the recombinant type. For example, in the ninth probability the pair of alleles from I is (Am, BI ). In this pair (Am) is either the maternal or the paternal allele of D, and (BI ) is either the maternal or the paternal allele of S. Thus, using recursion, the ninth probability can be written as This probability is not a function of the recombination rate between A and B because (A1 ) and (Bf) are inherited independently from D and S.
In the two IBD probabilities computed above, the pair of alleles that were traced back were from the same individual. However, when recursion is continued it will be necessary to trace back alleles that belong to two different individuals. For example, if S and D are younger than J, computing the first probability in equation (9) will require tracing back alleles from S and D to alleles of their parent. General rules to compute IBD probabilities that accommodate all cases encountered in recursion are described below.
Consider computing Pr[(Ax, B!) == (Aw , Bz)], where alleles in the first pair are from individuals X and Y, alleles in the second pair are from individuals W and Z, and superscripts !, y, w, z = m or f . Without loss of generality, we assume that X is younger than W and Y is younger than Z. All cases encountered in recursion can be classified into two types: where (A X , BY) is of the non-recombinant type (type-1); or where (!4!-,B!) is of the recombinant type or where A X and BY are from different individuals (type-2). Rules for recursion will be described separately for type-1 and type-2 cases.

Recursion for type-1 cases
Type-1 cases are encountered when X = Y and x = y. Now, if the condition is true, then Pr[(A x , By) (Aw , BZ)! = 1; if the condition c is not true, but all four alleles are from founders then, Pr!(AX, By (Aw, B')] = 0, because different alleles in founders are assumed to be not IBD.
Suppose condition c is not true, none of the four alleles is from a founder, and alleles at one of the two loci are the same. For example, if X = W,Y ! Z, x = w = m and z = f, then Pr!(AX, BY ) _ (A!,, Bz)! can be recursively computed as where P is the mother of X. Here, AX and !4! are the same allele, and, therefore, in the desired probability we have only three different alleles. As a result, only Hi is not traced back to its parental alleles. Note that here and in all type-1 cases both alleles A X and BY are traced back to the same parent; as a result, recombination rate enters into the formula for recursion.
Suppose condition c is not true, none of the four alleles is from a founder, and alleles at neither of the two loci are the same. For example, if X # W, Y # Z x = m, w = m and z = f, then Pr!(AX, BY ) -(Am, B i )] can be recursively computed as where P is the mother of X. This is the same situation given by equation (8). where R is the mother of Y. Here, A X and !4! are the same allele, and it is not traced back to parental alleles because X = W is a founder. As a result, only By is traced back to its parental alleles. Note that here and in all type-2 cases the alleles Ax and BY are traced back to different parents; as a result, recombination rate does not enter into the formula for recursion. Now suppose condition c is not true, none of the four alleles is from founders, but alleles at one of the two loci are the same. For example if, X = W, Y ! Z, x = w = m, y = m and z = f, then alleles at locus A are the same and fr[(!4!,-B!-) = (Aw , Bz )] can be written recursively as where P is the mother of X and R is the mother of Y. Again, !4!-and !4!, are the same allele, and as a result in the desired probability we have only three different alleles. Thus, the only allele that is not traced back is Bfzl Finally, suppose condition c is not true, none of the four alleles is from a founder, and alleles at neither of the two loci are the same. For example, X:A W, Y # Z, x = m, y = m, w = m and z = f, then Pr!(AX, BY ) _ (!4!, B i )] can be recursively computed as where P is the mother of X and R is the mother of Y. Now, in the desired probability we have four different alleles, and only AX and By are traced back.

NUMERICAL EXAMPLES
The recursive formulae are used here to examine the effect of linkage on the additive x additive relationship coefficient. Cockerham [2] stated that the covariance between two relatives, where one is an ancestor of the other, is not affected by linkage. Schnell [9] as well as Chang [1] showed that the previous statement is not always true. It can be shown that the covariance between a parent and its non-inbred offspring is not affected by linkage. However, the covariance between a parent and its inbred offspring, as well as between grandparent and grandoffspring, will be affected by linkage.
Consider first the covariance between parent (W) and a non-inbred offspring (X). The additive x additive relationship coefficient (ox,w) can be computed using two-locus computations. However, of the 16 probabilities, only four are non-zero because the parents of X are assumed to be unrelated. For example, if W is the mother of X, two-locus computation reduces to where A and B are the two loci. Note that the four probabilities in equation (16) are of type 1 and as a result we can write because the recombination rate cancels out in equation (17). As a result the recombination rate plays no role in the covariance between parent and offspring. Assume now that X is inbred, its parents being full sibs. Assume also that the parents of W are unrelated. In this case all 16 probabilities in section 3.2 will have non-zero values, and !X,w is given by Note that in this case the recombination rate will affect the covariance between parent and offspring.
Consider now computing the additive x additive relationship coefficient !G,W between grandparent (W) and grandoffspring (G). Let W be the maternal grandparent of G, X the daughter of W and the mother of G, and Y the father of G. Again, O G , W can be written using two-locus computation. As in the parent-offspring case, there are only four probabilities that are non-zero because Y is considered to be unrelated to W. Applying equation (11) to the four probabilities in equation (19) gives and As a result !G yv can be written as which is a function of the recombination rate r.
The recursive method was used to compute numerical values of the additive x additive relationship coefficient for different relatives and different recombination rates (table 1). As expected, when linkage is absent (r = 0.5) the additive x additive coefficient is equal to the square of the additive coefficient. In the absence of linkage, the genetic covariance will be identical for certain pairs of relatives. For example, the covariance between grandparent-grandoffspring, half sibs and aunt-nephew, is equal to 0.25 V A + 0.0625 V AA . However, if loci are linked, the genetic covariance for these pairs of relatives will not be the same (table 1). The numerical values of the additive x additive relationship coefficient increase as the linkage becomes tighter (r becomes smaller). As a result, when we assume that linkage is absent, the additive x additive variance component will be overestimated.
Numerical values for the additive x additive relationship coefficient for full sib and for parent-offspring relationships, after several generations of selfing, are given in tables II and III. In this design, individuals in generations i are the offspring of selfed individuals from generation i -1. The numerical values in table II are for the relationship between the offspring of a single selfed individual from generation n. The numerical values in table III are for the relationship between a parent in generation n and its offspring in generation n + 1. Note that after t generations, if linkage is absent, the additive x additive relationship coefficient for full sibs has the same value as the additive x additive relationship coefficient for parent-offspring. When linkage is present the two values are different. The additive x additive relationship coefficient of a founder with any individual obtained through selfing will be always one. The numerical value of additive x additive relationship coefficient will converge to four, because each of the 16 probabilities converges to one, after several generations of selfing. As the number of generations of selfing increases, the effect of linkage decreases.

DISCUSSION
This paper describes a recursive method to compute the additive x additive relationship coefficient for arbitrary pedigrees in the presence of linkage. The additive x additive relationship coefficient can be described as one-fourth the sum of 16 two-locus IBD probabilities that can be recursively traced back to known values. We have given five recursive equations to compute these IBD probabilities, where the origin of the younger pair of alleles is traced back to the previous generation.
Thompson [11] gave six recursive equations to address the same problem. However her approach differs from ours. Some of these differences are briefly described below using our notation. Thompson's approach is based on recursive equations for two-locus IBD probabilities involving only the alleles of parent P in its offspring X or X', where P is not Y, W or Z nor an ancestor of any of them. Further her recursive equations are linear combinations of one-and two-locus IBD probabilities while our equations are linear combination of only two-locus IBD probabilities and do not involve one-locus IBD probabilities.
While all the recursive equations given in the present paper are based on tracing alleles back to the previous generation, not all of Thompson's [11] equations are based on this principle. For example, consider equation (8) in Thompson !11!, which in our notation becomes where alleles AX and Ay, are from parent P. This equation is obtained by observing that alleles AX and AX, will be the same with probability one half; if the two alleles are the same, then the two-locus IBD probability on the left hand side of equation (25) becomes the one-locus probability Pr(BY = B z ); if the two alleles are not the same, the two-locus IBD probability is P r[(Ap, By) (A P , Bz)!.
In contrast, we trace back the alleles A X and BY to the previous generation. Suppose x = x' = m and y = m, then the two-locus IBD probability on the left hand side of equation (25) becomes where P is the mother of X and X', and R is the mother of Y. This is clearly different from equation (25). Although these two approaches use different recursive equations the final results for the IBD probabilities are identical. This demonstrates that there is more than one approach to compute IBD probabilities by recursion.
Based on the recursive method described in this paper, numerical values of the desired coefficient for selfed or outbred individuals are given. Using the computer program available at http://www.public.iastate.edu/ N rohan, the effect of linkage on the additive by additive covariance can be examined for any type of relationship. This would be useful to examine the potential bias in covariance estimates when linkage is ignored. Figure 1 gives the rate of change in the additive by additive covariance for several relationships. Relationships with flatter curves are less biased by linkage. Other applications are in linkage analysis and the identification of pairwise relationships based on data at linked loci (11!.