Assortative Mating And Gametic Disequilibrium

Why Mating Patterns Can Connect Different Loci

Alleles at two loci can be statistically associated even when the loci are on different chromosomes. One route is nonrandom mating: if individuals choose partners with similar phenotypes, alleles that contribute to the same phenotype can become correlated across the two members of a mating pair. Offspring inherit one gamete from each parent, so these cross-parent correlations can produce covariance between their genotype dosages at different loci.

This is distinct from physical linkage. Linkage concerns the locations of loci on chromosomes and the probability that recombination separates them. Assortative mating concerns how partners are paired. It can create cross-locus associations even for unlinked loci, although the strength and persistence of those associations depend on inheritance and the mating pattern.

Definition: For two biallelic loci, let \(U_1\) and \(U_2\) indicate whether a gamete carries a specified allele at locus 1 and locus 2, respectively. The gametic disequilibrium is \(D_{12}=\operatorname{Cov}(U_1,U_2)=P(U_1=1,U_2=1)-P(U_1=1)P(U_2=1)\). A positive value means the specified alleles occur together in gametes more often than expected from their marginal frequencies; a negative value means they occur together less often.

Gametic disequilibrium describes an association within gametes. A different, related quantity is covariance between genotype dosages in offspring. Let \(U_l\) and \(V_l\) be the transmitted allele indicators at locus \(l\) from the two parents. The offspring dosage is \(X_l=U_l+V_l\), taking values 0, 1, or 2. The covariance \(\operatorname{Cov}(X_1,X_2)\) is a genotype-level quantity; it is not, by definition, the disequilibrium within either parental gamete.

Decomposing Offspring Genotype Covariance

Consider a population of mating pairs and their offspring. Assume each offspring receives one gamete from each parent, and use the same allele coding for both parents. Write \(D_M=\operatorname{Cov}(U_1,U_2)\) and \(D_F=\operatorname{Cov}(V_1,V_2)\) for within-gamete covariances for the two parental transmissions. Define the cross-parent, cross-locus covariances \(K_{12}=\operatorname{Cov}(U_1,V_2)\) and \(K_{21}=\operatorname{Cov}(V_1,U_2)\). These quantities are measured across mating pairs and their transmissions.

No special genetic mechanism is required for the following covariance identity; it follows from the offspring dosage definition. The identity makes clear which kinds of association contribute to offspring genotype covariance.

\begin{aligned} \operatorname{Cov}(X_1,X_2) &=\operatorname{Cov}(U_1+V_1,\ U_2+V_2)\\ &=D_M+D_F+K_{12}+K_{21}. \end{aligned}

The first two terms are associations between loci within each transmitted gamete. The last two terms connect an allele transmitted at one locus to the allele transmitted by the other parent at the other locus. When both parental gametes are in linkage equilibrium, \(D_M=D_F=0\), but assortative pairing can still make \(K_{12}\) and \(K_{21}\) nonzero. In that case, the offspring dosage covariance is \(K_{12}+K_{21}\).

Worked Example: Cross-Parent Assortment and Offspring Covariance

Suppose two unlinked loci each have allele frequency 0.5. In a hypothetical set of mating pairs, assume the two transmitted alleles from each parent are independent, so \(D_M=D_F=0\). Suppose the cross-parent covariances are \(K_{12}=0.03\) and \(K_{21}=0.03\), in allele-indicator units squared. These are modest associations: for example, with allele indicators having variance 0.25, each covariance is smaller than the marginal variance.

The offspring covariance is \(\operatorname{Cov}(X_1,X_2)=D_M+D_F+K_{12}+K_{21}\). Substituting the values gives \(0+0+0.03+0.03=0.06\) allele-counts squared. Thus offspring carrying more copies of the specified allele at locus 1 tend, on average, also to carry more copies at locus 2. The within-gamete disequilibrium in each parental transmission is nevertheless zero in this example: the covariance arises from the mating-pair terms.

Assortment on phenotype can create these cross-parent terms because each partner's genotype contributes to the phenotype used, directly or indirectly, in mate choice. For an additive trait with effects \(a_1\) and \(a_2\) per copy of the specified alleles, alleles with effects in the same direction can contribute to positive cross-locus covariance under positive phenotypic assortment. If the effects have opposite signs, the corresponding covariance can instead be negative. The sign is not guaranteed solely by the phrase “positive assortative mating”; it also depends on which alleles are coded as the specified alleles and how their effects relate to the trait on which mating occurs.

How Dosage Covariance Enters Genetic Variance

For two loci, consider the additive genetic value \(A=a_1X_1+a_2X_2\), where each \(a_l\) is the change in genetic value per additional copy of the specified allele at locus \(l\). Expanding the variance gives

\operatorname{Var}(A) =a_1^2\operatorname{Var}(X_1) +a_2^2\operatorname{Var}(X_2) +2a_1a_2\operatorname{Cov}(X_1,X_2).

The last term is the cross-locus contribution. It is positive when \(a_1a_2\) and the dosage covariance have the same sign, and negative when they have opposite signs. This calculation concerns the variance of the additive genetic value under the specified allele coding and effects; it does not imply that the loci are physically linked.

Worked Example: A Cross-Locus Contribution to Genetic Variance

Continue with the hypothetical offspring population above, where \(\operatorname{Cov}(X_1,X_2)=0.06\). Let \(a_1=0.4\) and \(a_2=0.2\) trait units per allele copy. The cross-locus contribution to \(\operatorname{Var}(A)\) is \(2a_1a_2\operatorname{Cov}(X_1,X_2)\).

Substitution gives \(2(0.4)(0.2)(0.06)=0.0096\) trait units squared. This term is positive because both effects are positive and the dosage covariance is positive. It is only the cross-locus term; the total additive genetic variance also includes \(a_1^2\operatorname{Var}(X_1)\) and \(a_2^2\operatorname{Var}(X_2)\). It would be incorrect to report 0.0096 as the total variance without calculating those two terms.

The covariance scale matters. Dosage covariance describes association in the diploid offspring genotypes, whereas \(D_{12}\) describes association within a haploid gamete. They may contribute to one another across generations, but they are not interchangeable. To relate a genetic covariance calculation to a specific mating model, the model must specify how mating pairs are formed and how alleles are transmitted.

Assortment Versus Linkage

Linkage and assortment can both be associated with cross-locus covariance, but they refer to different features of the system. Physical linkage is determined by chromosome location and recombination. Gametic disequilibrium is a statistical property of allele combinations in gametes. Assortative mating is a property of pairing. It can generate statistical associations, including at unlinked loci, without making those loci physically linked.

For example, when alleles at two loci are in linkage equilibrium in parental gametes, their within-gamete covariance is zero. If positive assortment creates nonzero cross-parent covariance between alleles at different loci, offspring can nevertheless show nonzero genotype dosage covariance. Conversely, physical linkage does not guarantee nonzero disequilibrium: an association can be absent even for nearby loci, depending on allele frequencies and evolutionary history.

Worked Example: Interpreting an Association Without Claiming Linkage

In a hypothetical offspring sample, suppose the estimated dosage covariance between two loci on different chromosomes is 0.04 allele-counts squared. Assume the parental gametes were in linkage equilibrium, while partners were paired partly on the basis of a polygenic trait affected by alleles at both loci.

The covariance is evidence of an association between the two offspring dosages in this sample, not evidence that the loci are physically linked. Under the stated assumptions, a plausible explanation is cross-parent covariance induced by assortment: the offspring covariance decomposition includes \(K_{12}+K_{21}\), even when \(D_M=D_F=0\). Because the loci are on different chromosomes, physical linkage cannot explain the association in this example. The observed covariance alone does not establish that assortment was its cause; that interpretation depends on the stated model and other possible sources of association.

A Short Derivation of the Covariance Identity

The decomposition can be justified directly from the bilinearity of covariance. Since \(X_1=U_1+V_1\) and \(X_2=U_2+V_2\), expand the covariance of the sums into four pairwise terms. Grouping the two within-parent terms and the two cross-parent terms gives the stated identity.

Write the inherited dosages.
At each locus, the offspring receives one allele from each parent: \(X_1=U_1+V_1\) and \(X_2=U_2+V_2\).

Expand the covariance.
\(\operatorname{Cov}(X_1,X_2)=\operatorname{Cov}(U_1,U_2)+\operatorname{Cov}(U_1,V_2)+\operatorname{Cov}(V_1,U_2)+\operatorname{Cov}(V_1,V_2)\).

Identify each source.
The first and last terms are within-parent gametic covariances; the middle terms are cross-parent, cross-locus covariances. If both within-parent terms are zero, only the cross-parent terms remain.

This is an accounting identity, not a claim that assortment always produces a particular value or sign for any term. The mating mechanism and genetic effects determine the covariances. In particular, the identity does not assume that the cross-parent terms are equal; equality requires a symmetry assumption about the mating and transmission process.

Common Mistakes and Why the Distinction Matters

Equating genotype covariance with gametic disequilibrium. They are different quantities. Genotype covariance includes within-gamete terms and cross-parent terms; identify which is being discussed.
Calling every cross-locus association physical linkage. Association is statistical. Assortative mating can produce it across unlinked loci, and physical location alone does not determine whether an association is present.
Assuming positive assortment makes every allele covariance positive. The sign depends on allele coding and on whether the alleles' trait effects point in the same direction.
Using the cross-locus variance term as the total variance. The term \(2a_1a_2\operatorname{Cov}(X_1,X_2)\) is only one component of the variance of \(A\); the two single-locus variance terms must also be included.

Key takeaway: Assortative mating can connect alleles at different loci through correlations between the two parents in a mating pair. The resulting offspring dosage covariance is the sum of within-gamete and cross-parent covariance terms. This statistical association is not the same as physical linkage, and its contribution to genetic variance depends on the allele effects as well as the covariance.

Check Your Understanding

Use the definitions and covariance decomposition above to answer these questions.

In a gamete, what does a positive value of \(D_{12}\) say about the two specified alleles?
If \(D_M=D_F=0\), what cross-parent terms determine the offspring covariance \(\operatorname{Cov}(X_1,X_2)\)?
Why does a nonzero covariance between dosages at two unlinked loci not, by itself, show that the loci are physically linked?
If \(a_1\) and \(a_2\) have opposite signs, what sign of dosage covariance makes the cross-locus term in \(\operatorname{Var}(A)\) positive?
What two single-locus terms must be added to \(2a_1a_2\operatorname{Cov}(X_1,X_2)\) to obtain the full variance of the two-locus additive genetic value?