Basic equine genetics for nonscientists...

First, some basic concepts:
1) Mendelian theory defined the 'dominant' and 'recessive' labels for genes based on inheritance patterns long before modern molecular biology existed.
2) Every horse has 2 copies of every gene.
3) Every foal inherits one copy of each gene from their sire and the complementary copy from their dam.
4) A gene is dominant if one copy from either parent is enough to cause a specific trait to be expressed, for example brown eyes.
5) A gene is recessive if two identical copies, one from each parent, are needed to cause the expression of a specific trait, for example blue eyes.

The existence of a 'dominant curly gene' has been well established by a dominant inheritance pattern in hundreds of pedigrees. In other words, you can breed a curly-coated horse carrying only one copy of the curly gene to a straight-coated horse and have a 50% chance of getting a curly foal...Mendelian genetics. If one parent has 2 copies of the dominant curly gene and the other is straight-coated, then all offspring will be curly, having inherited one copy of the dominant curly gene and one copy of the corresponding straight-coated gene.

If one parent carries the recessive curly gene (1 or 2 copies) and the other carries the dominant curly gene (1 or 2 copies), then the offspring could carry both mutations at the same gene locus (location). Likewise, traits encoded by the recessive curly gene will only manifest themselves if the foal carries 2 copies of the recessive gene, one from each parent.

Part of the confusion in this field comes from the fact that neither gene has been DNA sequenced. It is very possible the dominant curly gene is a mutation in one gene that affects coat traits and that the recessive gene is a mutation in an entirely different gene that also affects coat traits. Then, just to make life more interesting, the expression of every gene (the traits you can see in an offspring, such as brown eyes) can be modified (changed) by other genes in the DNA make-up of each individual. This can lead to false assumptions that the underlying Mendelian inheritance isn't happening. There are hundreds of genes that affect the look and feel of a horse's coat, not just one or two.
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Gene Linked to Gaitedness in Horses

In August 2012 the cover story for the science journal Nature was the discovery of the first gene conclusively linked to gaiting in horses [1]. As described in the publication, all horses can walk, trot, and canter/gallop, but some horses have the ability to perform extra gaits, such as the pace (moving the two legs on the same side of the body in unison). The genetic basis of 'gaitedness' in horses was explored in one of the oldest and most inbred horse breeds in the world, the Icelandic. Because Icelandics have many genes in common, it is easier to find genes that cause differences between horses in this breed than it would be in more genetically diverse horse breeds.

An international group of equine genomics experts spearheaded by Swedish scientists identified a link between a premature stop codon in the
DMRT3 gene (in other words a shorter-than-normal version of the DMRT3 protein is produced) and the ability to perform alternative gaits. This mutant gene was found to be permissive for 'gaitedness' in horses [1,2]. This means, there's no guarantee a horse will be able to gait (pace, fox trot, running walk, etc.) when they carry this mutant gene, but it does appear to be a necessary condition for 'gaitedness' in the breeds studied so far.

In Icelandics [1], the horses with 2 copies of the mutant
DMRT3 gene (homozygous, A/A) were 5-gaited, while most 4-gaited Icelandics were heterozygous (1 copy of the mutant DMRT3 gene and 1 copy of the wild-type DMRT3 gene, C/A). The authors concluded that A/A is required for pacing in this breed. Almost all horse DNA samples from the other gaited breeds studied had at least one copy of the mutant DMRT3 gene (Kentucky mountain saddle horse, Missouri fox trotter, Paso fino, Peruvian paso, Rocky mountain horse, Tennessee walking horse). In contrast, none of the 8 non-gaited breeds carried A.

Mouse studies showed that the wild-type (full length)
DMRT3 gene controls development of the nervous system in the fetus in such a way as to produce coordinated limb movements [1]. In contrast, mice born entirely without the DMRT3 gene had normal muscle coordination and balance, but had alterations in how they used their legs to move.

In 2013, ICHO arranged to have a variety of DNA samples from Curlies tested by the Swedish research group. Early results showed that all 6 of the curly Missouri fox trotters on my ranch at the time that were tested and one straight-coated offspring of a curly Missouri-fox trotter sire were homozygous for the mutant
DMRT3 gene (A/A). In contrast, non-gaited Sporthorse DNA samples from another ICHO member were either heterozygous (C/A) or homozygous wild-type (C/C). The Swedish scientists have indicated that the majority of horses with C/A that they tested are non-gaited or somewhat gaited. Commercial testing for this gene has recently become available.

One thing to remember is that this gene discovery is probably only the tip of the iceberg. Certainly, Icelandic horses gait differently than Missouri fox trotters or Tennessee Walkers do. Just look at the differences in conformation among the different gaited breeds and it's clear there must be other genes controlling gaits besides this one. We don't know how important the
DMRT3 gene mutation is in the overall scheme of things, but it's still an exciting discovery and sheds light on an area we knew nothing about (genetically-speaking) before this discovery.

References
1. Andersson LS,
et al. Mutations in DMRT3 affect locomotion in horses and
spinal circuit function in mice. Nature Vol. 488, August 2012, pp. 642-646. Free for downloading at http://www.nature.com/nature/journal/v488/n7413/full/nature11399.html
2. Petersen JL,
et al. Genome-wide analysis reveals selection for important traits in domestic horse breeds. PLOS Genetics Jan 2013 vol 9, issue 1. Free for downloading
at
http://www.plosgenetics.org
For readers with an ongoing interest in this topic, I recommend periodic searches of the scientific literature at PubMed (http://www.ncbi.nlm.nih.gov/pubmed/). PubMed is a public portal that allows anyone with an internet connection to search the medical/scientific literature in a (U.S.) national database called MedLine that's maintained by the National Library of Congress and the National Institutes of Health. It concentrates on human medical articles, but also captures many animal studies pertaining to human health subjects, like gene mapping projects.