The brown gene determines if eumelanin has a brown or black colour.
Some essential steps in the generation of melanine are affected by the enzym tyrosinase (TYR) and the tyrosinase related proteins 1 and 2 (TYRP1 and TYRP2). Whether the two latter ones are enzymes or not is currently subject of debate.
TYR is the rate limiting enzyme in the generation of melanine. It catalyzes the hydroxylation of tyrosine to L-3,4-dihydroxyphenylalanine (DOPA) and the oxydation of dopa to dopaquinone. Dopaquinone is then converted to dihydroxyindole carboxylic acid (DHICA) via transformation of dopachrome by TYRP2 (dopachrome tautomerase). DHICA is oxidized to indole-5, 6 quinone-carboxylic acid, which is then combined into the polymer eumelanin. TYRP2 plays a role in the oxidization of DHICA but it's unsure if it's an enzyme or not.
If TYRP1 is broken the pigment is still formed but it has a slightly different colour. Instead of black it has become dark brown.
There are several mutations that result in a broken form of TYRP1. If a dog has two copies then the eumalin will have a brown (chocolate) colour.
The alleles bs, bd and bc are the most common. bs is a change of a single nucleotide which cause the TYRP1 protein to become 182 amino acids shorter. bd is a deletion of three nucleotides. bc is also a mutation of a single nucleotide which changes an amino acid in an area where parts of the protein are "glued" together and thus the structure of the protein changes it into a non-functional variant.
There are three rarer mutations which are known in specific breeds. In a family of Australian Shepherds a mutation of a single nucleotide was found that resulted in a triplet that doesn't encode for an amino acid and thus the TYRP1 protein is short and non-functional. In the Lancashire Heeler a single mutation also makes a non-functional protein; this mutation is called be. More recently variant bh, also a single nucleotide mutation, was discoverd in Siberian Huskies and Husky-crosses.
In dogs any combination of two recessive alleles produces a brown coat colour. Variations in brown colour are not contributed to the different alleles. Also, some dogs have more than one mutation. DNA tests report these cases as more than two alleles.
Little, Clarence C., The Inheritance of Coat Color in Dogs, Ithaca, New York, Comstock Pub. Associates, 1957.
Wagatsuma, T., Suzuki, E., Shiotsu, M. et al. Pigmentation and TYRP1 expression are mediated by zinc through the early secretory pathway-resident ZNT proteins. Commun Biol 6, 403 (2023). https://doi.org/10.1038/s42003-023-04640-5
Brancalion, L.; Haase, B.; Wade, C.M. Canine coat pigmentation genetics: A review. Anim. Genet. 2022, 53, 3–34. https://onlinelibrary.wiley.com/doi/full/10.1111/age.13154