What is the rough probability for a megafauna specimen to exhibit a gene not present in either parent?
I'm looking, ideally, at the chance for one specific individual to be the first in a breeding population to have a given gene.
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You need to define “gene” for your question to make sense. In humans, each individual has roughly 100 novel mutations (that neither parent had) (Human mutation rate revealed). However, those are not necessarily “new genes”, since many occur in non-functional DNA, even those that occur in functional regions may not alter any function, and even those that alter function may not be part of a gene, depending on your personal definition.