The two-fold cost of sex explains why it might not be so advantageous to be a sexually reproducing organism.

John Maynard Smith (1978) asked us to imagine a population of sexually reproducing individuals, in which a mutation occurs, allowing a female to reproduce asexually. He also makes two assumptions:

1. The number of offspring a female makes is independent of her mode of reproduction.
2. The probability that an offspring will survive to reproduction is independent of whether that individual was produced sexually or asexually.
Finally, keep in mind that the offspring of a sexual mother are half male and half female., while the offspring of an asexual female are all female.

There will be no average differences in the number of offspring between the mothers, but in the second generation, the two-fold cost of sex becomes apparent. Each daughter produced by the asexual female produces her own daughters. The sexual female, however, has only half as many daughters as the asexual female (because her other offspring were sons.) So while these daughters produce as many offspring as the daughters of the asexual, there are only half as many around to do it. This difference will become magnified each generation. This idea can be illustrated as:

```
Generation           Sexuals            Asexuals            Fraction of
population that are
asexual

1                F x M                  F                   1/3

|                    |
v                    v

2           F x M    F x M      F    F    F    F            1/2

|        |        |    |    |    |
v        v        v    v    v    v

3           F x M    F x M      F    F    F    F            2/3
F x M    F x M      F    F    F    F
F    F    F    F
F    F    F    F

```
So given this logic, asexual reproduction seems to be twice as good as sexual reproduction, and ought to quickly dominate in a sexual population. Yet it clearly has not totally taken over, so something is amiss. If the conclusion of our logic is violated, than it must be that one of our assumptions is incorrect. The first assumption, that the number of offspring which females can produce is independent of mode of reproduction might be violated in species in which males care for offspring. While there are a few cases where female fecundity seems to be limited by male parental care, in most species males provide nothing more than their genes. So the second assumption is probably the place to look.

Before I go on, I should note that in the long run, an asexual population might be doomed to extinction because it is unable to purge its unfit genotypes. (This is Muller's ratchet.) But despite this potential fate, the short term advantage of asexual reproduction should lead it to take over the population. The problem is to discover if there are short-term advantages to sexual reproduction.

If the second assumption has been violated, it means that asexual offspring should be less likely to survive and reproduce. (In other words, they should have lower fitness.) Sexual recombination helps to purge a population of deleterious mutations, so this may play a role. This can only work in the short run if the rate of deleterious mutations is high, and synergistic epistasis effects these mutations. Many others have proposed that sex is advantageous is rapidly changing environments, because it creates novel genotypes which may be more fit in the new environment.

Is there any evidence for these proposed mechanisms? The role of sex in purging deleterious mutations is consistent with evidence on mutation rates, but has hardly been tested. There is considerably more evidence for the advantage of sex in heterogeneous habitats. For example, many plants have genotypes which perform best in different microhabitats. A novel genotype may be able to exploit resources which are unused by other individuals. Experiments with certain grass species have shown that individual plants have higher fitness if grown in competition with different genotypes than with their own genotypes. This suggests that by producing novel genotypes, sex is beneficial in habitats which are heterogeneous.

Another widely cited example of the advantage of sex in heterogeneous habitats is from the geographic distribution of sexual and asexual organisms. In both plants and animals, asexual organisms are more common at high latitude and high altitude. In general, these areas are very harsh physically, but have less complex biota. It may be that sex is adaptive in complex environments, which are constantly changing. In these cases, novel genotypes are required to deal with the continuously shifting pressures of predators, parasites and competitors.

Given how wide spread sexual reproduction is, there must be some benefit to overcome the two-fold cost of sex. The ability to produce novel genotypes and the ability to purge deleterious mutations probably play a role as well. We don't have all of the answers yet, and further theoretical work, as well as experimental testing of these ideas, will be required to determine what factors overcome the two-fold cost of sex.

For more on this topic, see a good textbook on evolution, such as the following two, from where much of this discussion was drawn:

Freeman, S. and Herron, J.C. (1998) Evolutionary Analysis. Prentace-Hall Pp 191-203

Futuyma, D.J. (1998) Evolutionary Biology. 3rd Ed. Sinauer Associates. Pp 606-612.

Or see:

Maynard Smith, J. (1978) The Evolution of Sex Cambridge University Press.

I think that this is a difficult topic, and I fear that my write up is somewhat myopic. Please /msg me with questions so that I can improve it.
Statistics are mathematically simple, but applications often suffer from 'the Monty Hall problem problem': the exact assumptions from which the mathematics derive their validity often remain unstated; in such a case the whole statistical number juggling becomes a dubious affair.

Even when the assumptions are clear it's very tempting to draw false inferences - in which the reader unknowingly misreads or changes the assumptions on which the statistics are based.

This is why I love the writeup above: not only does it spell out the mathematics in great detail, it is also very careful in pointing out the exact assumptions, the exact situation in which these numbers have practical meaning.

The logic, or should I say calculation, of Maynard Smith's paradox is indisputable. The problem I see with it - note: I am not a biologist - is that it presents a situation that is utterly unrealistic.

Asexual reproduction happens by means of cell division - the development of this complex process is in itself a major feat of evolution. Sexual reproduction can be seen as the opposite: it requires that genetic material from two different cells somehow combines into a new cell.

But the sexual process is really just an extension of the asexual one. Slightly simplified, this is what happens at the DNA level. In cell division, the genetic material splits in two, then each of the halves double themselves to form a full copy, around which the rest of the cell regroups to form two cells. In sexual reproduction, the split halves then leave the cell to combine with other halves, and a new cell has to form around it.

So at the DNA level, the process isn't very different; in a 'primordial DNA soup' without cells, both mechanisms could develop. But when cells are formed, the processes become very different: with genetic recombination, the DNA halves must leave the cell in order to combine with another matching half. (Sexual reproduction is a particular form of genetic recombination.)

Now I can see how both possibilities, that is, DNA duplication and DNA recombination, can preexist as mechanisms within the primordial DNA soup, and these processes can be retained as the process of cell formation develops, leading to cell division (asexual reproduction) and a mechanism to send out 'DNA halves' to other cells (sexual reproduction).

But I can't see how one mode of reproduction would easily develop from the other, and it seems utterly preposterous to suggest that one of the two could be created as the result of a genetic mutation. When an organism - i.e. something with one or more body cells - has both modes of reproduction, they exist as completely different mechanisms, and are not in direct competition at all.

Therefore, assumption 1 is totally unwarranted. I won't even discuss assumption 2.

By way of illustration, consider some examples.

single-cell organisms with sexual reproduction (do they exist?)
the paradox predicts that the very large majority of reproductions occurs asexually, i.e. by cell division; does actual evidence contradict this in any way?
the strawberry
offshoots from strawberry plants can form roots and completely separate plants - but there is no direct competition with sexual reproduction, which allows new strawberry plants to form in completely different locations - assumption 1 cannot even be said to apply here
human beings
soon, we will be able to clone humans; even if we could only clone ourselves, we can't expect cloning would spread as rapidly as in The Boys from Brazil - assumption 1 is demonstrably false
human body cells
this organism actually uses both modes of reproduction very successfully - asexual reproduction to form human bodies from a single cell, and sexual reproduction to spawn off new bodies before the present ones collapse; these mechanisms don't compete at all, but complement each other - both are essential to our continued existence, and to that of all sexually reproducing multicellular organisms; there is no paradox at all.

The two-fold cost of sex should actually be refered to as being the two-fold cost of anisogamy. This is because a hermaphrodite or dioecious population –– in which every individual produces both male and female gametes –– can have a growth rate nearly as rapid as an asexual population. Anisogamy is the asymmetry in the size of gametes that is based on that gamete's sex. For example, we are anisogamous because the human egg is much larger than a sperm cell, although the two gametes are genetically equivalent. The two-fold cost of anisogamy is brought about when the following conditions exist:
1) The growth of a sexually-reproducing population is limited by the number of females.
2) Sperm are less costly to produce than eggs.
3) Fertilization occurs in an open system, such that sperm are cast into a common mixed pool, and encounter eggs randomly.

The first condition is always true for a sexually-reproducing population in which there is a sufficient number of males to fertilize all females, and for which paternal care has a negligible effect on the survival of offspring. The second is generally true because eggs are energetically costly to produce. Energy and resources needed for the growth of the embryo must be parcelled into each egg cell, regardless of whether or not it becomes fertilized. Sperm, on the other hand, only require a store of energy for travel to the egg (if the demand exists –– the flagella of human sperm, or pollen tube growth) and a structure capable of persisting in a harsh environment.

The final condition is required because individuals that cheat can obtain an evolutionary advantage. Suppose that the resources required to produce one egg can instead be converted into ten sperm. A population of hermaphrodite individuals might initially tend to produce 9 eggs and 10 sperm; it is a fair system, because there is a balanced ratio of gametes to ensure fertilization, and everyone gains an equal number of fertilizations. If a mutation arises that causes an individual to produce 8 eggs and 20 sperm, however, they obtain a disproportionately greater number of fertilizations by saturating the pool of male gametes. This sort of mutation would may be met in some species, such as in echinoderms that release sperm into the sea water, or angiosperms that release wind-borne pollen. It is circumvented in others, that have evolved a closed system of mating. For example, although hermaphrodite earthworms fertilize externally, some undergo a lengthly courtship ritual before allowing the mutual exchange of a single sperm.

Otherwise, cheaters succeed until the number of female gametes produced is halved. Thus, the hermaphrodite population incurrs a two-fold cost of anisogamy. It is important to note that although sexual reproduction is a necessary condition for this scenario, it is anisogamy that actually causes this cost.

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