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In biological evolution, mutation refers to a change in the sequence of bases or base pairs on a nucleic acid strand. Such changes are usually random, caused by copying errors or some external cause such as radiation or chemical action. They are "random" in the sense that the changes are not preferentially distributed with respect to the result — they are not "goal-oriented".
The majority of these changes have no effect on fitness (that is, they are "neutral mutations"). Examples of neutral mutations would be those which change between synonyms in the genetic code (sequences which produce the same protein), which affect non-coding regions of the chromosome (see gene expression), or which result in inconsequential changes (such as blood type or eye color in humans). Of the minority which are not neutral, most are harmful and a few are beneficial (see below). Whether the mutation is harmful or beneficial (and to what degree) may depend upon what circumstances the resulting organism happens to live in (a standard example is the loss of sight being beneficial for a cave-dweller living where there is no light).
In the biological context, a lethal mutation refers to any and all mutations that prevent an organism from reproducing in any way. Lethal mutations do not have to kill the organism to classify as lethal.
An example of a lethal mutation in humans is Turner Syndrome, a disorder whereby the 23rd chromosome pair only contains an X chromosome. The vast majority of these result in a spontaneous abortion, and if the person is born she will live but, with only very rare exceptions, will be infertile.
A potentiating mutation is one that, while not affecting the organism in itself, can later be utilized by a further mutation to produce a beneficial (see below) effect. An example of this can be found in duplicate genes; initially this mutation may have no effect upon the organism, but if the duplicate gene is passed along and undergoes change itself, it may induce a feature or ability that was previously not present.
That is to say; there are two potential mutations, Mutation A and Mutation B. Individually, neither of these mutations change the organism's life in any significant way, but if the organism has undergone both A and B, it becomes markedly more successful, and evolution happens.
- Richard Lenski's famous bacteria developed a mutation which, while being initially irrelevant to the organism, allowed them to metabolise citrate following a subsequent mutation. As evidence of a potentiating mutation, citrate-metabolising bacteria would re-evolve only in samples taken from after the original experiment's 20,000th generation. Based on the speed of the mutation, Lenski hypothesized that two potentiating mutations may have been required before the final beneficial mutation can achieve its benefit.
- Apples had all their chromosomes duplicated during about the same period the dinosaurs were dying off. Approximately fifty million years ago, the predecessor plant of modern apples doubled from nine to eighteen chromosomes and lost one to result in a seventeen chromosomes species. While not strictly a mutation, the duplication allowed for much more space (nearly twice as much), for mutations to form and take hold.
A beneficial mutation is exactly what it says it is: a mutation in an organism's genome that produces a beneficial effect. Specifically, it affects the organism in some way as to increase its chances of reproductive success, and therefore the chance of the mutation in question being passed along. It is safe to say that the vast majority of mutations in an organism are not beneficial. Often a mutation is neutral, producing neither a beneficial effect nor a negative effect, although it could be a potentiating mutation (see above). Many are negative mutations - producing a disorder in the organism that significantly lowers its survival and reproductive capabilities. In the case of species depending on sexual reproduction, a negative mutation can prevent the embryo being correctly conceived or even the gametes being functional.
Relevance to natural selection
Despite the odds against an individual mutation being beneficial, the driving force of natural selection means that when one does appear, it can quickly dominate a population. This is the entire point of natural selection, which invalidates any argument that "beneficial mutations" are too rare to ever have any effect.
In 2008 microbiologist Richard Lenski produced a beneficial mutation in the lab, when his E. coli developed a trait that enabled them to utilize citrate as a carbon source. While E. coli has internal cellular mechanisms that can process the citrate, one hallmark of the species is that it cannot transport citrate across the cell wall. This mutation enabled them to do so.
Vivid examples of beneficial mutations
- Lactase persistence - why humans with significant European or Kenyan ancestry can digest milk as adults.
- Antibiotic resistant bacteria — at least beneficial from the point of view of the bacteria.
- Radiation-resistant fungi (and perhaps other lifeforms) inside Chernobyl
- "German Superboy", an individual example of a human mutation that not only doesn't cause any visible disfigurement or impairment, but if anything will probably make it easier to maintain a muscular physique and/or low weight. These are characteristics that could be considered desirable in the modern day, when food is abundant.
- The ccr5-Δ32 mutation confers HIV-1 resistance to those with a double copy of the allele (homozygous). The mutation also confers resistance to plague and smallpox while increasing susceptibility to West Nile virus.
Directed mutation is a theory that species are able to consciously mutate their genome in one direction when under environmental stress. The theory is widely dismissed (and possibly considered pseudoscience; it has been picked up by intelligent design advocates and new agers.
Creationists often claim that mutations can only be harmful, or at best, neutral. Evidence of useful mutations is either studiously ignored or treated as "decay", somehow. They take it for granted that truly useful mutations are negligible. However, a 2000 study on Arabidopsis thaliana found that as many as half of non-neutral mutations can be beneficial.  A 2008 Mutation-Accumulation study on yeast found 25% of mutations with detectable fitness effect to be beneficial. 
Peppered moths anyone? Nylon-eating bacteria, anyone? Antibiotic-resistant bacteria, anyone? LENSKI'S GODDAMN BACTERIA, anyone?
Consider that all mutations have their costs, but some mutations are beneficial as well. For example, the cost of flight is the energy required to get a flying animal off the ground, so flighted birds have to be small. All the large birds become flightless. Some flying animals (like the hummingbird and dragonfly) are so adept at flying that they can't walk anymore. The benefits of having leg muscles no longer outweigh the costs (since they can fly backwards), so the legs become vestigial landing gear.
Humans' large brains also come with a cost. They consume much of our body's energy, and they have cravings (like music) that have no biological significance. However, they have allowed intelligent people to increase the human lifespan and to make the modern world (relatively) peaceful. All mutations have costs, but some have benefits. It could be speed, a new organ, or even simply being more attractive to mates. When those benefits outweigh the costs, the trait survives.
Another excuse often used by creationists is that genetic mutation only loses information. Evolution of new information is, on the contrary, quite possible.
- Blount, Borland, and Lenski 2008. Historical contingency and the evolution of a key innovation in an experimental population of Escherichia coli
- Giovanni 2010. Harvesting the Apple Genome
- Glass WG, McDermott DH, Lim JK, Lekhong S, Yu SF, Frank WA, Pape J, Cheshier RC, Murphy PM. CCR5 deficiency increases risk of symptomatic West Nile virus infection J. Exp. Med. volume 203, issue 1, pages 35–40, January 2006. doi: 10.1084/jem.20051970.