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The error catastrophe of aging, originally proposed by Leslie Orgel in 1963, argues that copying errors in DNA and the incorrect placement of amino acids in protein synthesis could aggregate over the lifetime of an organism and eventually cause a catastrophic breakdown in the form of obvious aging. Experimental tests which have attempted to determine differences in the nucleotide sequences of specific proteins that correlate with age have always failed, so the theory has largely been dismissed.
Copying errors have been selected against extremely strongly over the course of evolution because the genetic material is the most evolutionarily important part of the entire organism, and the organism can in fact be viewed as a "survival machine" for the genetic material. In vertebrates, evolution has had to wrestle with the most imaginable negative consequence of copying errors imaginable - cancer - and thus has DNA copying mechanisms that work at extremely high fidelities. Because these mechanisms are tuned so well, the error catastrophe of aging, while an interesting theory, isn't a real phenomenon.
Sometimes the phrase "error catastrophe" is used in reference to smaller organisms, such as viruses. Error catastrophe in viral populations is similar to the error catastrophe of aging, but with respect to virions rather than cells. But similar to the error catastrophe of aging, error catastrophe in viral populations cannot be convincingly shown to occur. A given virus has a given genome, and if the mutation rate among viruses were so high that the whole species breaks down, then such viruses wouldn't exist in the first place. It also contradicts the idea that a specific genome is associated with each species of virus, which has been shown experimentally to be true.
Through mathematical calculations, we can determine mutation rates which would cause error catastrophes if they were the real values, but they are not. All viruses, and all human cells, can be shown to have mutation rates significantly less than those which would predict error catastrophe to eventually occur. Error catastrophe of aging is therefore a discredited theory, but the means of discrediting it is an important educational story for biologists and geneticists.