Without sexual reproduction, natural selection acts upon the entire genetic makeup of an organism, but with sexual reproduction, natural selection acts upon individual genetic traits.
With sexual reproduction, less than one in four offspring will receive the best genetic traits from both parents. However, natural selection seems to be very effective at eliminating the least successful variations, especially in the harsh conditions of the wild, where only a fraction of all newly conceived offspring survive until breeding age.
Increasing the rate of evolution
Evolution generally favors organisms that evolve faster, because they can adapt more rapidly to changing conditions and compete more successfully against other variations.
The speed at which organisms evolve depends on the rate at which they mutate. However, almost all mutations are harmful. Only very rarely will a mutant be more successful at surviving and breeding than its parent.
With asexual reproduction, each ancestry is likely to accumulate many more harmful mutations than beneficial ones. And so for asexual organisms, natural selection will generally favor those with the lowest mutation rate.
However, with sexual reproduction, beneficial mutations can be separated from harmful ones, and so the reproductive systems of sexually reproducing organisms can evolve to maintain a consistently high rate of mutation.
A larger population has a higher chance of producing a beneficial mutation, but for asexually reproducing organisms, a considerable amount of time may need to pass before the mutant population grows large enough to have any chance of producing a second beneficial mutation.
Sexual reproduction removes this speed limit by allowing beneficial mutations to spread back into an existing population to be combined with other beneficial mutations, and this greatly magnifies the rate of evolution.
It is likely that without sexual reproduction, it may have taken so long for all of the right mutations to have accumulated in a single ancestry, and the rate of evolution might have been so slow, that life on earth might not have evolved past the bacterial stage.
Every complex asexual organism alive today seems to have descended at some stage in its evolutionary history from an organism that reproduced sexually.
The evolution of sex
Although bacteria simply multiply by dividing into two, they also often exchange genetic material, usually by releasing small fragments called plasmids which can be absorbed by nearby bacteria. In this way, beneficial mutations are shared. This might arguably be considered to be the earliest form of sex.
After the appearance of plants and animals in the oceans around 600 million years ago, the only practical way for these new multi-celled sea-creatures to reproduce was by releasing seeds or eggs, whose cells would then begin dividing, sticking together, and changing each time they divided, until finally forming a new adult.
The only practical time to accept potentially beneficial foreign genetic material was before the seed or egg began growing. Many sea plants and animals today release pollen or sperm into the water to be absorbed by nearby seeds or eggs, which will not begin growing until they have been fertilized.
As plants moved onto the land, their pollen was either carried by the wind or delivered directly to seeds by insects. For almost all land animals and many sea animals, sperm was more effectively delivered by injecting it into an enclosed body cavity containing eggs.
Evolution generally favors any mutation that increases the drive to find sex partners. While simple creatures like insects follow programmed patterns of behavior, more complex animals like mammals have sensitive nerves in their reproductive organs that stimulate powerful pleasure centers in their brains.
One of the more popular alternative theories for the evolution of sex is the idea that sexual reproduction generates greater genetic diversity. This would be particularly important in rapidly changing environments, where some variations might be wiped out by new conditions while other variations might be better adapted to survive.
Perhaps the most common form of this theory is the 'Red Queen Hypothesis' which says that greater genetic variation gives sexually reproducing species better resistance to rapidly adapting diseases and parasites. As valid as this hypothesis may be, it gets an undeserved amount of attention, probably because it is one of the few theories for which there might be some valid experimental evidence.
The problem with this theory is that the connection between sexual reproduction and genetic diversity is tenuous, because the diversity of a population would depend on its mutation rate rather than its mode of reproduction. For a given mutation rate, sexual reproduction would actually decrease the diversity of a population by converging towards the most successful genetic traits. The only reason why the diversity of an asexual population would be lower is because evolution would select individuals with the lowest mutation rate in order to reduce the accumulation of harmful mutations.
One of the more outrageous alternative theories is the idea that males are essentially parasites. Sexual reproduction began when some parasitic organism began injecting its genetic material into an unwitting host in order to utilize its reproductive machinery like a virus. Over the course of evolutionary history, the parasites developed a co-evolutionary symbiosis with their hosts because of the shared genetic material and the benefits of sexual reproduction. Males may look similar to their reproductive hosts, but they are still essentially parasites.
There are other theories for the evolution of sex, but most other theories are usually either subtle re-wordings of the established theories or are otherwise too easy to refute.
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