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An allele is a variation of a gene.
Alleles in the genome
Most genes only have one allele, however some have multiple variants. For example, the gene for eye color in humans (and some other animals) has several alleles: blue, green, brown, etc. In paired chromosomes, it is possible to have two alleles present, one in each copy of the chromosome. Where both alleles are the same in the pair, the organism is referred to as homozygous with respect to that gene. Where the alleles are different, it is referred to as heterozygous.
Dominant and recessive genes
Alleles can be described as "dominant" or "recessive" and a mixture of the two can appear in allele pairs in the genome. In some cases, one allele will be dominant while the other is recessive, and the dominant gene will be the only allele expressed. Only individuals with two recessive alleles of a gene (homozygous recessive) will express the recessive form. Genes may also be co-dominant (both or all alleles expressed) or have incomplete dominance (alleles "blend" to form an intermediate type in the phenotype).