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The p53 gene has a mutation which results in the cell having a cancerous phenotype. This effect is dominant as one allele being mutated is enough to result in this effect. However when both alleles are mutated then the phenotype changes / become more pronounced so it's not a simple dominant mutation. What is this kind of mutation called? Is it hemi-dominant or something?

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Allelic effects

What is the name of a gene whose effect [..]

The concept of dominance make sense for alleles. The gene has no name to refer to the allelic interaction.

Patterns of dominance

Patterns of dominance and recessivity are typically used for quantitative traits. Can you somehow put a a number into the phenotypes of all three genotypes? Making some kind of association table would help even if the phenotypes are not quantitative. See below.

Attempt to answer the question

It looks like you are considering a case where the phenotypes

aa -> healthy
Aa -> sick
AA -> very sick

Under this model, it is hard to tell whether you would consider sick half way through healthy and very sick or not. It feels like you want to consider sick as being closer to very sick than to healthy so if we put numbers we could get something like

aa -> 1 (healthy)
Aa -> 2 (sick)
AA -> 2.5 (very sick)

As such, A shows partial dominance (aka incomplete dominance or semi-dominance).

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