Dogs are complex animals. They are composed of 39 chromosomes, as opposed to 19 for a fox. These chromosomes are made up of clusters of genes. Each dog is made up of 80,000 - 100,000 genes. It is estimated that each dog contains at least 3 lethal genes.
"The mutation rate for dogs cannot be determined readily, but from indirect evidence and extrapolation from other species, geneticists believe that mutation rates are normally on the order of 1 in 100,000 or less. For a sexually reproducing mammal, that would mean a new mutation in a particular gene would likely not occur more often than once in every 100,000 gametes. That may not seem like a high probability, but consider that most mammals are estimated to carry 80-100,000 genes. This suggests that every individual born has a good chance of carrying one new mutation in some gene." This means each individual dog is likely to add a mutated gene to the gene pool, not counting those he/she has inheirted.
Since the same conditions exist for humans, why aren't we, and all dogs sick? Well, the good news is each gene is one of a pair of called alleles (a variant gene, or one possiblity). Dogs inhereit one allele from each parent and it only takes one healthy allele for the dog to be symptom free. The healthy allele is complete and is almost always "dominant". The unhealthy allele is in some way damaged or missing something it need to perform normally and is said to be "recessive". It takes two such recessive unhealthy alleles to cause ill health in the dog. In a naturally occuring population it is almost impoaaible for this condition to exist!
Identical mutations are unlikely to occur simultaneously in the same gene from both parents (probability: < 1 in 10 billion), so any progeny will be heterozygous. (The exception being sex-linked genes, as the X and Y chromosomes are not homologous.) Dominant mutations will be expressed and any that are deleterious will be eliminated almost immediately from the population.
So how do dogs inheirit disease? The process is complicated but invariably comes from an animal inheiriting the exact same unhealthy allele from both parents. As mentioned, in normal reproduction of random matings this is almost statistically impossible. But with so much emphasis placed on conformity there is a natural migration towards inbreeding to attain consistency in conformation (appearance) charateristics.
Suppose we have a "mutant" allele that has lost only 5-10% of its normal function. In many cases, this would not produce a noticeable effect. If you made an individual homozygous for this allele, you would not even be aware that you had done so. Now consider that the same fate may befall a number of genes during an inbreeding program. Eventually, you will have an individual that is considerably less fit than one carrying the normal alleles for all (or even most of) these genes. There is no magic formula for regaining what you have lost. You must start again.
Linebreeding is merely a term used for a particular type of inbreeding that often focusses on one ancestor who was considered exceptional. Particularly if it is a male, this exceptional ancestor may end up as grandfather and great-grandfather -- sometimes more than once -- in the same pedigree. Father-daughter, mother-son, and some other combinations also result in a disproportionate number of genes coming from a single ancestor. This type of close inbreeding is less common. [In contrast, the mating of full sibs or first cousins doubles up on two ancestors equally.]
As the result of several common practices, most pure-bred domestic animals are more inbred than they really need to be. One is that some breeders own a small number of animals and breed only within their own group. A second is that many breeders have the idea that outstanding animals can be produced by inbreeding -- by doubling up on the good alleles while somehow avoiding the bad. Even if you were to point out that this is a gamble, such breeders might respond that they are simply helping natural selection.
Beyond the conventional close-relative inbreeding, there is another practice that has much the same effect, namely the popular sire phenomenon (generally over-use of a well-promoted champion). In fact, many who breed to such a dog believe they are doing a "good thing," as they will be increasing the frequency of occurrence of the genes that made him a champion. What they may not realize is that they are increasing the frequency of all genes carried by this animal -- whether they are good, bad, or innocuous -- and that champions, like any other animal, carry a number of undesirable recessive alleles (the genetic load) that are masked by wild-type alleles. The result of the popular sire phenomenon is that almost all members of the breed will carry a little bit of Jake Hugelberg, and any undesirable trait carried by Jake will no longer be rare. Finding a safe, unrelated mate then becomes an exercise in futility.
What can be done?
We can control many of the obvious genetic diseases by supporting research aimed at locating the genes and developing direct DNA tests for the mutant alleles. Test results should be employed to make certain that carriers are only mated to clear individuals, rather than for wholesale elimination of carriers, which would further impoverish the gene pool.
There are currently two DNA tests available for Dachshunds. ( See the next two links)