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Sometimes the magnitude of a heritable problem is not immediately obvious. It can be brought more sharply into focus with something called the Hardy–Weinberg law. This law helps to predict genotype frequencies with a simple algebraic formula, where p ² is the frequency of the homozygous dominant gene pairing, 2pq is for heterozygotes, and q ² is the frequency of the homozygous recessive gene pairing, with the individual gene frequencies, p + q, equaling 1. The Hardy–Weinberg law applies as long as no mutation (allelic changes), migration (“new blood”), or active selection for a trait occurs. Although the Hardy–Weinberg law is very useful in canine genetics, it must be remembered that dog breeding invalidates some of the basic tenets of the law because matings do not occur randomly or without selection, the way they might in nature. Also, the population is rarely in equilibrium because breeders often bring in breeding animals from outside the local population. As such, gene frequencies predicted by the Hardy–Weinberg law may not be completely accurate. Still, much useful information is provided.

Let's examine this with the American car‐chasing terrier (ACCT) and the prevalence of an autosomal recessive trait, von Willebrand disease, in this breed. Although the genetic mutation associated with von Willebrand disease has been determined in many breeds, it has not yet been characterized in the ACCT, but pedigree analysis suggests that it is autosomal recessive in nature. Because von Willebrand disease in the ACCT is autosomal recessive, it is difficult to distinguish between the homozygous dominant and the heterozygote; von Willebrand factor testing does not convincingly differentiate the two in this breed, and DNA mutation testing is not yet available. The only phenotype that can conclusively be demonstrated is the homozygous recessive, those affected with von Willebrand disease.

We did a survey with the local ACCT club and found that of 1000 dogs, 53 had von Willebrand disease. The ACCT club seemed happy that the prevalence of the disorder in the breed was of the order of only about 5%. Using the figures from the survey, the actual genotype frequency for homozygous recessive (q ²) is 0.053 and the gene frequency (q) is the square root of 0.053, or 0.23. We know the sum of p and q equals 1, and q equals 0.23, so p must equal 0.77, and p ², the proportion with the homozygous dominant genotype, equals 0.59. On the basis of these numbers, 2pq, or 0.35 (35%), would be predicted to be the likely proportion of heterozygous carriers in the population.

What does this exercise in algebra tell us? Well, it tells us that although only 5.3% of ACCTs actually have von Willebrand disease, an incredible 35% of the ACCT population are carriers of the trait, which is not detectable by conventional means. The breed club has a potentially serious problem and would be best advised to invest money in a genetic test to detect heterozygotes. In the interim, however, the goal is to identify heterozygotes by other means so that animals that are homozygous normal can be discerned. We'll have to do the best we can with von Willebrand disease testing and progeny testing to differentiate normal homozygotes from normal‐appearing heterozygotes until that DNA test becomes available.

Identifying dogs that are homozygous recessive is not a problem. They are the ones with von Willebrand disease. We also know that both normal‐appearing parents of these affected animals are heterozygotes, carriers of the trait. Nevertheless, most of the other heterozygotes, which we would like to avoid breeding together, are clinically indistinguishable from the homozygous normal animals that we would like to use in our breeding program. For example, because we know that the parents of affected animals are carriers, it follows that at least one of each of their parents (grandparents of the affected animal) must also be a carrier. Inferring genotype from a family tree is known as pedigree analysis.