Decomposing Multilocus Linkage Disequilibrium

Corresponding author: Root Gorelick, Arizona State University, P.O. Box 871501, Tempe, AZ 85287-1501., cycad{at}asu.edu (E-mail)

Communicating editor: M. W. FELDMAN

	ABSTRACT

TOP ABSTRACT DECOMPOSITION OF MULTILOCUS... DISCUSSION LITERATURE CITED

We present a mathematically precise formulation of total linkage disequilibrium between multiple loci as the deviation from probabilistic independence and provide explicit formulas for all higher-order terms of linkage disequilibrium, thereby combining J. Dausset et al.'s 1978 definition of linkage disequilibrium with H. Geiringer's 1944 approach. We recursively decompose higher-order linkage disequilibrium terms into lower-order ones. Our greatest simplification comes from defining linkage disequilibrium at a single locus as allele frequency at that locus. At each level, decomposition of linkage disequilibrium is mathematically equivalent to number theoretic compositions of positive integers; i.e., we have converted a genetic decomposition into a mathematical decomposition.

A precise measurement of linkage disequilibrium is required for studying virtually any phenomenon in multilocus population genetics. This is especially true for explicit multilocus models that investigate the contributions of physiological epistasis to additive genetic variance (CHEVERUD and ROUTMAN 1995 ; WAGNER et al. 1998 ; WAGNER and LAUBICHLER 2000 ). Linkage disequilibrium is usually defined as the deviation from probabilistic independence between alleles at two different loci. This deviation from independence can have different causes, such as a lack of independent segregation or recombination, or any number of other evolutionary forces. The presence of linkage disequilibrium (gametic disequilibrium) is thus an indication that either stochastic (e.g., drift) or deterministic (e.g., selection, gene flow) evolutionary forces have been acting on a population (HEDRICK 2000 ; ARDLIE et al. 2002 ).

The classical definition of linkage disequilibrium, D, follows the probability theory definition of deviation from independence. Independence of two events, B and C, means that Pr(BC) = Pr(B) · Pr(C), where Pr is probability and BC is the joint distribution of B and C, so that the deviation from independence is measured as D = Pr(BC) – Pr(B) · Pr(C). Changing notation slightly to let A_k(i) designate the kth allele at the ith locus gives the linkage disequilibrium between the alleles at two loci, D₂, as D₂ = Pr(A_k(1)A_k(2)) – Pr(A_k(1)) · Pr(A_k(2)), where Pr represents probability and A_k(1)A_k(2) represents the joint occurrence of A_k(1) and A_k(2) in a single haploid gamete. In most modern interpretations of probability theory, the primitive concept of "probability" is interpreted as a relative frequency; therefore, Pr(A_k(1)) is the same as the frequency of allele k at locus 1.

The quintessential examples of linkage disequilibrium are coadapted gene complexes, in which several loci are tightly linked because they provide a large selective advantage if they occur together. In these cases, linkage disequilibrium is maintained by selection. Although coadapted gene complexes are implicit in Wright's shifting-balance hypothesis (WRIGHT 1931 ), have been used to explain outbreeding depression (DOBZHANSKY 1948 ; LYNCH 1991 ), and are frequently cited as evolutionary hypotheses (PALOPOLI and WU 1996 ; RAWSON and BURTON 2002 ), the linkage disequilibrium of these purported coadapted gene complexes is almost never quantified. This is particularly surprising given the well-cited article by GEIRINGER 1944 , in which she provides most of the algorithm for computing higher-order linkage disequilibrium coefficients. In this article, we complete and simplify Geiringer's formulation and then show how the sums of products of those coefficients equal the definition of (total) linkage disequilibrium as the deviation from probabilistic independence given by DAUSSET et al. 1978 .

Methodologically, we follow Geiringer's lead and decompose higher-order linkage disequilibrium into lower-order linkage disequilibrium terms. In other words, we take a top-down approach to defining multilocus linkage disequilibrium, rather than the bottom-up approach followed by virtually everyone since GEIRINGER 1944 . LEWONTIN 1974 is typical of the bottom-up approach. There are very few other top-down decomposition approaches such as BULMER's (1980) decomposition of multilocus epistasis or WAGNER and LAUBICHLER's (2000) character decomposition approach in population genetics.

In this article, we first define linkage disequilibrium at a single locus as the allele frequency at this locus, which greatly simplifies notation. Second, we extend the definition of linkage disequilibrium to multiple loci by invoking compositions of positive integers. Our decomposition of multilocus linkage disequilibrium is entirely consistent with the standard definitions for two loci, as well as its previous extensions to three, four, and six loci (GEIRINGER 1944 ; BENNETT 1954 ; HASTINGS 1984 ). Third, we show how this definition is entirely consistent with the notion of linkage disequilibrium as the deviation from probabilistic independence.

	DECOMPOSITION OF MULTILOCUS LINKAGE DISEQUILIBRIUM

TOP ABSTRACT DECOMPOSITION OF MULTILOCUS... DISCUSSION LITERATURE CITED

Define the one-locus coefficient of linkage disequilibrium, D₁, as D₁(A_k(i)) = Pr(A_k(i)). This definition may appear paradoxical, but it dramatically simplifies notation for the decomposition of multilocus linkage disequilibrium. In elementary algebra we have the analogous problem of defining the algebraic expression xⁿ when n = 0 (LAKOFF and NUNEZ 2000 ). Note that our definition of a locus encompasses protein-coding loci, quantitative trait loci, and even single nucleotides.

Following HASTINGS 1984 , the formulas for two- and three-locus multilocus linkage disequilibrium, in which D₁(A_k(i)) was substituted for Pr(A_k(i)), are defined as

Let D_n be the coefficient of linkage disequilibrium between n loci. Then the pattern here is that D_n = Pr(A_k(1)A_k(2) ... A_k(n)) minus all possible products of lower-order linkage disequilibrium coefficients, such that each term has all of its subscripts adding up to n. The key to writing down an explicit formula for D_n is that the phrase "all possibilities of the subscripts adding up to n" refers to partitions of the positive integer n (ANDREWS 1976 ). A partition of a positive integer n is a set of positive integers that adds up to n; i.e., = {n₁,n₂, ..., n_m} such that . The set of all partitions of n is designated p(n); e.g., p(5) = {{5}, {4, 1}, {3, 2}, {2, 2, 1}, {3, 1, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}. To define multilocus linkage disequilibrium, we have to add over all partitions, excluding the trivial partition = {n}, and permute over all alleles for a given number of loci. However, the order of elements of the partition matters, and hence we construct the number-theoretic compositions c of the positive integer n (ANDREWS 1976 ). For example, all of the compositions for the partition = {2, 2, 1} are the ordered triples (2, 2, 1), (2, 1, 2), and (1, 2, 2). Using these mathematical notions we can generalize the two- and three-locus cases to define linkage disequilibrium between n loci as

(1a)

where n_i c means that n_i c is a scalar component of the vector c. Equivalently,

(1b)

The only way to decompose n into a single positive integer is c = (n). Therefore, we can also write the highest-order coefficient of linkage disequilibrium as , where the summation has only a single term and the product has only a single factor. Therefore, Equation 1a yields

(2)

which we use below.

Equation 1aEquation 1b has never been written explicitly for general multilocus linkage disequilibrium, even though special cases have been given by GEIRINGER 1944 , BENNETT 1954 , and HASTINGS 1984 . The only explicit definition previously given for multilocus linkage disequilibrium is due to DAUSSET et al. 1978 ,

(3)

which we call total linkage disequilibrium, D_n, where we have again replaced Pr(A_k(i)) with D₁(A_k(i)). We refer to D_n as total linkage disequilibrium because, as we show below, all of the nonboldface linkage disequilibrium coefficients D₁, D₂, D₃, ..., D_n can be independent from one another and contribute to D_n. Equation 3 has a simple heuristic interpretation: D_n(A_k(1) ... A_k(n)) measures how far the haploid genotype at all n loci deviates from probabilistic independence.

We are now ready to derive the relationship between D_n and D_n. In Equation 3, substitute _{all compositions c of n} [_nicD_ni (...)] for Pr(A_k(1), ... A_k(n)) (see Equation 2), yielding The last term in this equation is simply the value of _nicD_ni (...) for the composition c = (1, 1, 1, ..., 1), i.e.,

Therefore, Equation 3 becomes

(4a)

or, equivalently,

(4b)

Equation 4aEquation 4b provides the crucial link between deviations from independence (D_n) and the linkage disequilibrium coefficients D_n computed by GEIRINGER 1944 and her intellectual successors by decomposing D_n into the terms D_ni, where n_i = n.

	DISCUSSION

TOP ABSTRACT DECOMPOSITION OF MULTILOCUS... DISCUSSION LITERATURE CITED

We have converted the genetics problem of decomposing linkage disequilibrium into the mathematical problem of decomposing positive integers into their additive parts, all while maintaining the convenient heuristic definition of total linkage disequilibrium as the deviation from independence. Unlike GEIRINGER 1944 , we can write down an explicit formula for multilocus linkage disequilibrium because we invoke partitions of integers and define D₁(A) = Pr(A), thereby merging her notion of linkage disequilibrium with those of DAUSSET et al. 1978 .

One immediate consequence of our decomposition approach is that the single highest-order coefficient of linkage disequilibrium, D_n, cannot be examined in isolation. Because , we need to examine all lower-order linkage disequilibrium coefficients, D_ni (...) with n_i < n. All of the subscripted linkage disequilibrium coefficients D₁, D₂, D₃, ..., D_n can be independent from one another and all contribute to D_n, which we therefore call total linkage disequilibrium.

Multilocus definitions of linkage disequilibrium have not been used very often in empirical studies because of the large number of inputs and linkage disequilibrium coefficients that must be analyzed (2ⁿ – 1). Currently, even third-order linkage disequilibrium is seldom measured (THOMSON and BAUR 1984 ). However, explicit terms for multilocus linkage disequilibrium are of theoretical importance.

One important theoretical application is the analysis of multilocus epistasis. CHEVERUD and ROUTMAN 1995 developed a two-locus model of physiological epistasis that has been further refined by WAGNER et al. 1998 . To analyze the evolutionary consequences of epistasis in these models, one has to first define linkage disequilibrium for a subset of the loci. Thus, to extend models of physiological epistasis to multiple loci, we must first define linkage disequilibrium for that subset of loci, which we have just done. Models of multilocus epistasis will be crucial in debates over what factors maintain coadapted gene complexes, increase additive genetic variance, and foster speciation (GOODNIGHT 1988 , GOODNIGHT 1995 ; WADE and GOODNIGHT 1998 ).

	ACKNOWLEDGMENTS

We thank Phil Hedrick, Tom Dowling, and two anonymous reviewers for their helpful comments.

Manuscript received November 26, 2003; Accepted for publication December 18, 2003.

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