The simplest explanation usually for how alleles are either recessive or dominant is based on whether the protein they express has a higher than average function (gain of function) or lower than average function (loss of function).

Think about it through the lens of genetic disease. Oftentimes dominant genes are gain of function mutations. In Huntington’s disease, patients express a version of the protein huntingtin that is toxic to neurons and causes the condition. If you have two copies of alleles for huntingtin, but only one of them makes a toxic protein, you’d still have the toxic effect even though it represents only half of the expressed protein. That’s because this toxic effect is a “gain of function.” That’s what makes this allele dominant.

On the other hand, cystic fibrosis is a recessive disease. The gene for cystic fibrosis encodes a protein that transports sodium across cell membranes. Defects in this gene, called CFTR, can cause non-functional protein to be expressed and decrease the transport of sodium leading to a whole bunch of systemic effects. This is a loss of function. However, if you only have one mutant copy of the allele, then the other allele expresses a functional protein that can still do what the gene is supposed to do. This makes the disease recessive, because you need two “broken” alleles for a complete loss of this protein function.

Edit: punctuation

It's dependent on protein expression relative to the recessive allele. So will it be determined by factors controlling the rate of expression of that gene, It's transcription factors, the % of guanisine-Cytosine within the gene, it's structural availibility for transcription, It's epigentic modifications (DNA methylation) and many many more factors which control Gene expression.

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If one gene has more benenficial factors expression or if the other has as negative expression factors, the dominant gene's trait will be produced.

Dominant alleles are alleles that overtake the other one in their function. The body doesn't know which is which, its an inherent trait to the gene.

Nearly all genes come as pairs, and if one of the pair can outperform the other, then its considered dominant. There are many different ways this can happen both normally and in disease. Here are some examples:

Blood type - In the ABO blood type group there are three alleles, A, B and O. A and B are dominant over O in that if you are AO or BO then your blood type is A or B. On a cellular level, the ABO locus encodes a glycoprotein. A and B alleles will express this glycoprotein (although slightly differently) while O will express no glycoprotein. Thus if you have an A or B allele, it will dominant over the O allele.

Eye color - The OCA2 gene has a large role in eye color determination. The brown eye allele will cause the cells in your iris to produce melanin and turn your irises brown while the blue eye allele will produce much less melanin. Thus if you have one brown eye allele, it will dominant over the blue eye allele by producing a lot of melanin.

Alzheimer's disease - The protein APP is involved in forming amyloid plaques seen in patients with Alzheimer's. In some families, this protein is mutated such that it forms these plaques much faster than normal. Faster plaque formation means faster disease onset and progression. Since only one allele is needed to be mutated to cause disease (it doesn't matter there's a normal one), this mutation is considered to be dominant.

Like others have mentioned, there is a tendency for dominant alleles to be "gain-of-function" which means these mutations (or normal variations) cause a new function. In ABO alleles, its production of a glycoprotein. In eye color, its production of melanin in the iris. In APP, it's the increased tendency to aggregate into plaques. This tendency is because of the redundancy of having two copies of your genes. If one loses its function, you already have another copy to take its place. Thus loss-of-function mutations are less likely to cause disease. But this isn't always the case.

Hemophilia A - This clotting disorder is caused by a mutation in the factor VIII gene. This gene is located on the X chromosome the mutation is considered recessive. If you have at least one copy of the normal factor VIII gene, you won't develop disease. However, if you have only one copy of the X chromosome (IE males) and you have the factor VIII mutation, then you will develop the disease as you only have one copy of those genes.

Cancer - The protein p53 is an important cancer preventing gene in your body. It's involved in making sure your cells are dividing at an appropriate time and rate. It functions by oligomerizing (four of the proteins get together). If there is a mutation in p53, then the mutant form will poison the complex and prevent its function. IE if one of the four proteins that get together to form the complex is mutant, the whole complex won't function. Thus having a normal allele will not prevent disease.

ABO blood group - https://www.ncbi.nlm.nih.gov/books/NBK2267/

Eye color - https://www.sciencedirect.com/science/article/pii/S0002929707626822

Alzheimer's genetics - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5453386/

Hemophilia A - https://medlineplus.gov/ency/article/000538.htm

p53 mutations - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3135636/

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