Only a small fraction of all mutations become widespread in a species because only a small fraction of the mutations are advantageous in natural selection.
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The vast majority of mutations are either neutral, or have a disadvantageous effect, and therefore are not promoted by natural selection. For example, many single point mutations in the genome can still code for the same amino acid, therefore not affecting the function of the proteins. Additionally, many proteins can still perform their intended function even if DNA mutations coded for a few different amino acids to be used in place of others. Such neutral mutations have no detrimental or positive effect on the organism, and thus occur randomly and are sometimes passed down, but are never actively promoted and therefore do not become widespread. The mutations that are disadvantageous will hinder or destroy the organism's ability to survive, reproduce, and pass its genome to the next generation, and therefore they will be selected against during natural selection and will not become widespread. The very few mutations that do provide an advantage in an organism's survival in reproduction will therefore be passed down more frequently, and by that means will become widespread.