G.C. Williams and J. Maynard Smith conceived the maintenance of sexual reproduction differently

[This blog post was a mere comment on Sandwalk first, because I thought someone there was confused about the different conceptions of the cost of sex. See this thread, particularly towards the end. The following expands the comment, there, into its own blog post, here. But it just explicates the differences between two historically important conceptions of the cost of sex and does not pretend to offer a resolution. There's also an article here, that also does not pretend to offer a resolution of the paradox of sexual reproduction. But it describes how, despite their different conceptions of the issue, Williams and Maynard Smith were able to communicate with each other fruitfully and amicably. In fact, I find that rather amazing. Often, such situations tend to escalate into a useless dispute with both parties at cross-purpose. None of that ever happened. On the contrary, Williams and Maynard SMith were able to understand each other despite their different assumptions and advance the issue in a dialectic way rather than the contrarian fashion.]


George C. Williams
George Williams conceptualized the maintenance of sexual reproduction as a problem of selection within one species or population. He began by considering organisms that include both sexual and asexual modes of reproduction within their complex life-cycles, for example, aphids & rotifers or strawberries & corals. He conceived the cost of sexual reproduction as the cost of meiosis, that is, the cost of reducing the relatedness with the own offspring from r = 1 to r = 0.5, when these organisms meet the time or conditions for switching from asexual to sexual reproduction. With this kin-selection conception he took the maintenance of sexual reproduction as a problem of selection within a population. Herein, he disagreed with Maynard Smith: 

“I think that the primary disadvantage of sexual reproduction in relation to asexual is most fruitfully formulated as a paradox of kin selection—an organism devotes resources to the production and care of a more distant (r = 0.5) rather than a close (r = 1) relative. This formulation provides a number of advantages. In its focus on genes identical by descent, kin selection is genetically explicit and relates directly to evolution. Maynard Smith’s economic argument (resources wasted on males) makes it easy to overlook the fundamental distinction between (1) the evolutionary problem of sexual and asexual reproduction as alternative character states in a population, and (2) the purely ecological question of competition between a clone and a Mendelian population.” (Williams 1978, ‘Mysteries of sex and recombination. A review of The Evolution of Sex by John Maynard Smith.’ Quarterly Review of Biology 53: 287–289. Page 298)

“I believe that understanding has been hampered by failure to distinguish the ecological from the evolutionary problem of sexuality. In important ways, insights gained from conceptual or experimental comparisons of sexual populations and competing clones (the ecological problem) may mislead in relation to sexual and clonal reproduction as alternative processes in a population (the evolutionary question with which I am concerned here).” (Williams 1980, ‘Kin selection and the paradox of sexuality.’ In Sociobiology: Beyond nature/nurture? Ed. by G.W. Barlow and J. Silverberg. Boulder, CO: Westview: 371–384. Page 372)

The fact that William and Maynard Smith cut the cake differently gets obvious from the way in which Williams treated the maintenance of recombination as not the problem he was at all concerned with:

“I assume that observed chromosome numbers and crossover rates reflect the optimum compromise between maximizing whatever benefits there are in recombination, and minimizing recombinational load. Tighter linkage must reduce recombinational load, but it does nothing to alleviate the cost of meiosis.” (Williams 1975, Sex and Evolution, Princeton Univ. Press, p. 108)

That is, reducing replication-rate by fusing gametes is not alleviated by assuming, for example, a species with a genome consisting of one homologous pair of a giant chromosome and no crossing-over between this homologous pair. [Felsenstein and Yokoyama (1976) modelled this problem.] That would exclude recombination through segregating heterologous chromosomes as well as through crossing-over between homologous chromosomes, but it would not pay the cost of reducing r from 1 to 0.5, or the cost of males, or the cost of fusing gametes, or whatever you conceive the cost of sex to be.

John Maynard Smith
John Maynard Smith conceived the maintenance of high recombination rates (not sex) as a problem of within-population selection between alleles that increase and others that decrease recombination rates. [This differs from Williams's within-population problem of a species with a complex life-cycle and both sexual and asexual modes of reproduction within it.] He accepted Williams's criticism of group-selection arguments for this issue (what he called the "balance argument" of Williams). He agreed that this problem requires an immediate individual-level explanation. But he also maintained that the competition between a sexual population and a genetically isolated asexual clone is a case of between-population selection. At this level, he did allow for long-term or group selection to play some (limited) role.

Anyway, his distinction begins in the preface already:

"I am under no illusion that I have solved all the problems which I raise. Indeed, on the most fundamental question - the nature of the forces responsible for the maintenance of sexual reproduction and genetic recombination - my mind is not made up. On sex, the relative importance of group and individual selection is not easy to decide. On recombination, group selection can hardly play a significant role, but it is not clear to me whether the short-term selective forces I discuss are sufficient to account for the facts, or whether models of a qualitatively different kind are needed." (Maynard Smith 1978, The Evolution of Sex, Cambridge Univ. Press, p. ix)

"It may help to classify the various theories; first, according to the time scale on which selection is supposed to act, and then according to the 'unit of selection' - population, individual, or gene." (Maynard Smith 1978, p. 1)

"I do not find it possible to give an unequivocal answer concerning the role of group selection in the maintenance of sexual reproduction. It has played some role, as evidenced by the taxonomic distribution of parthenogens; but it is not the only relevant force, as will be apparent from the review of the balance argument in Chapter 4, section E. But, whatever one may think of the role of group selection in the maintenance of sex, it cannot explain how it started, and it cannot explain the maintenance of high levels of genetic recombination within sexual populations." (Maynard Smith 1978, p. 6)

And so throughout the book. Maynard Smith consistently distinguishes the maintenance of sexual reproduction from that of recombination, the former being an issue of selection between isolated populations and clones, the latter being one of selection between alleles within one population.

Maynard Smith's support for "some role" of long-term or group selection in the maintenance of sex (not recombination) was also defending his earlier publication from 1958 (The Theory of Evolution, Penguin Books, pp. 138-139). It is often forgotten in potted histories about the paradox of sex, that Maynard Smith did already clearly state the cost of males in this early pop-science writing and also embraced the long-term group-selection explanation of the maintenance of sex.

"If the rate of increase of an animal population were limited by the number of eggs which each female could lay, which in turn depended on how much food a female could eat and transform into eggs, then a population consisting entirely of parthenogenetic females would increase twice as fast as would a population of equal numbers of males and females. From the point of view of reproduction, males are a waste of living material. (This argument does not hold for hermaphroditic organisms, or for those animals in which both parents help to feed the young.)      The compensating advantage of the sexual process is that it increases the range of potential variation in a population, and therefore, its evolutionary plasticity." (Maynard Smith 1958, p. 138)

"Thus the sexual process is a means of ensuring evolutionary plasticity at the expense of interfering with reproduction. [...] Now if the advantage of sexual reproduction is that it increases the range of potential variation in a population, then the advantage refers to the population as a whole, and not to any particular individual in it. It follows that sexual reproduction has been established as a rule, both in animals and plants, because selection has favoured some populations at the expense of others. This forms a contrast to the the examples discussed in the last chapter, in which the 'unit' selected was the individual and not the population." (Maynard Smith 1958, p. 139)

By the way, Ghiselin (1988, p. 16, in Michod & Levin (eds): The Evolution of Sex), reminisced an instance of Williams reviewing one of his papers and telling him about the twofold cost of sex and that he [Williams] had found it in a book by Maynard Smith (1966), which must have been the second edition of the above quoted Penguin book by Maynard Smith (see also Dagg 2016, On recognising the paradox of sex. Philosophy, Theory, and Practice in Biology. DOI: 10.3998/ptb.6959004.0008.003).

The fact that John Maynard Smith never changed his mind about his hedged support for some role of group (between-population) selection in the maintenance of sex is clear from an interview of Richard Dawkins with John Maynard Smith in 1997 (deposited at the Web of Stories in 2008).  

Mis-attributing the cost of meiosis to Maynard Smith

The cost of sex has been defined as the cost of inheriting only half the genome to the offspring (Williams 1975) and the cost of wasting half the resources of a female on sons that do not contribute anything but genes to reproduction in turn (Maynard Smith 1971, 1978). These are now known as the cost of genome dilution (Williams called it the cost of meiosis) and the cost of males respectively. I already detailed how Williams continued to develop his conception of the cost of meiosis from mere genome dilution towards the sociobiological cost of cooperation between unrelated (out-crossing) gametes (see here and here). That is, gamete cooperation cannot be explained by kin selection in out-crossing species, because the gametes are not related.

While this change of Williams's conception has not been received by his peers, the ambiguity between the cost of genome dilution and the cost of males may still cause confusion.

For example, Ullica Segerstrale mis-attributes the genome dilution conception of the cost of sex to John Maynard Smith.

In 1971 John Maynard Smith had usefully characterized the problem when it came to sexual versus asexual reproduction as 'the twofold cost of sex'. (this refers to the fact that in meiosis half of the genetic material is lost, whereas in asexual reproduction it is kept intact.)⁹

9 Narrow Roads, vol 1, p 362. This was also Williams' view. For Hamilton, see n 15. Segerstrale 2013, p. 225 and p. 396 n 9

Additionally, she explains the twofold cost of sexual reproduction in the glossary as follows:

"twofold cost" of sexual reproduction

For Hamilton this was the fact that a sexually reproducing species loses half of its biomass through the production of males, compared with an asexual species that produces entirely female offspring. (This is a consequence of the sex ratio in species in which only the female rear offspring). However, George Williams regarded the twofold cost as the cost of meiosis, that is, loss of the genetic material, following Maynard Smith (1971). Segerstrale (2013, p. 422)

Descending down the cascade of footnotes yields statements of Hamilton corroborating that he regarded the cost of sex as that of investing into lazy males in 1975, 1988 and 1996, but that does not give him priority over Maynard Smith (1958, 1971). The following statement of Hamilton (1975), however, may explain how Segerstrale has been mislead to believing Maynard Smith held a genome dilution conception of the cost of sex:

The crucial snag for more facile theories of sex is that which John Maynard Smith first brought fully to our attention: what Williams refers to as the twofold cost of meiosis. Hamilton 1975 reprinted in Hamilton 1996, p. 362)

Okay, this is nit-picking, but the above quotes of Segerstrale are simply wrong  in claiming that John Maynard Smith held a genome dilution conception of the cost of sex -- and Hamilton's quote apparently mislead her in this respect.

While Williams (1975, p. 8)  did cite Maynard Smith (1971) as a "more exact" source for those readers who doubt his explanation of the cost of sex, he clearly did not follow Maynard Smith in his conception of the cost of sex as wasting resources on lazy males and instead conceived it in terms of genome dilution. Maynard Smith, however, never wavered in his conception of the cost of sex as that of wasting resources on males. This is not only true for 1971 but even for 1958, when the paradox of sex had not yet been noticed.

If the rate of increase of an animal population were limited by the number of eggs which each female could lay, which in turn depended on how much food a female could eat and transform into eggs, then a population consisting entirely of parthenogenetic females would increase twice as fast as would a population of equal numbers of males and females. From the point of view of reproduction, males are a waste of of living material. (This argument does not hold for hermaphroditic organisms, or for those animals in which both parents help to feed the young). Maynard Smith (1958, p. 138)

Later, Maynard Smith (1971, p. 170) reformulated the cost of males as accruing to individuals, propped the argument up with some math and reconsidered his claim concerning hermaphrodites. Still later, Maynard Smith (1978, p. 3) repeated his argument from 1971 verbatim, but added a criticism of the idea that the cost of sex is due to genome dilution by showing that genome dilution costs nothing in isogamy (gametes of equal size). This in turn prompted Williams to re-consider and change his concept of the cost of meiosis from genome dilution to gamete cooperation, which cannot be explained by kin selection in out-crossing species.

References

• Hamilton W.D. 1996. Narrow Roads of Gene Land, vol. 1. W.H. Freeman.
• Maynard Smith J. 1958. The Theory of Evolution. A Pelican Book, Penguin Books.
• Maynard Smith J. 1971. "The origin and maintenance of sex." In: Williams G.C. (ed) Group Selection. Aldine-Atherton.
• Maynard Smith J. 1978. The Evolution of Sex. Cambridge Univ. Press.
• Segerstrale U. 2013. Nature's Oracle. Oxford Univ. Press.
• Williams G.C. 1975. Sex and Evolution. Princeton Univ. Press.

Maynard Smith (1958) on the advantage of sexual reproduction

[Update 15.07.2017: See also this article on the history of the making of the paradox of sex by George C. Williams and John Maynard Smith including evidence from their correspondence: Joachim L. Dagg (2016) On recognising the paradox of sex. Philosophy and Theory in Biology 8: e703.]

Sometime in the late 1950s it dawned on evolutionary biologists that their explanation for the evolutionary advantage of sex implied group selection, but that was not yet seen as an anomaly. One record of this awakening to the implication stems from R. A. Fisher (1958[1999]). Another particularly lucid passage comes from The Theory of Evolution by John Maynard Smith. He first describes the cost of sex as halving the rate of increase of a population and then describes the advantage of sex as more than doubling a population's range of potential variation (Maynard Smith 1958, p. 138f).

"If the rate of increase of an animal population were limited by the number of eggs which each female could lay, which in turn depended on how much food a female could eat and transform into eggs, then a population consisting entirely of parthenogenetic females would increase twice as fast as would a population of equal numbers of males and females. From the point of view of reproduction, males are a waste of living material. (This argument does not hold for hermaphroditic organisms, or for those animals in which both parents help to feed the young.)     The compensating advantage of the sexual process is that it increases the range of potential variation in a population, and therefore its evolutionary plasticity." Maynard Smith (1958, p. 138)

He adds a numerical example showing that 10 mutations would yield only ten variant genotypes in an asexual population but 3^10 in a sexual one, because each mutation will yield three genotypes AA, Aa and aa. Maynard Smith is aware that seeing the advantage of sex in increasing population plasticity implies group selection,

How a Kuhnian missed a crisis (on Ghiselin)

According to George C. Williams (1975, Sex and Evolution. Princeton University Press, p. v) there has been a crisis simmering in evolutionary biology ever since the recognition of the paradox (anomaly) that sexual reproduction is the prevalent mode of reproduction in higher animals and plants although, all else equal, asexual mutants should gain an  immediate,  twofold reproductive advantage.
   From my student days,  perusing the primary literature in order to grasp the scientific issues, I only remembered that Michael T. Ghiselin (1974, The Economy of Nature and the Evolution of Sex. Univ. California Press) introduced his book with chapters on the history and philosophy of science showing that he was a dyed-in-the-wool Kuhnian. Therefore, on rereading his book with a historical interest recently, my expectation was that he would also have seen the paradox of sex and perceived a crisis.
   Alas, Ghiselin (1974) marched through the eras with such a wide gait, that he stepped over this major problem of his time.

A very short history of evolutionary maintenance problems

The following serves to show that even an abridged history does not need to gloss over differences, here, between what was thought about the relation between sexual reproduction and heritable variation at different times.

1. Darwin's problem
How could heritable variation in fitness relevant traits (which is a prerequisite for natural selection) be maintained, if sexual reproduction halved it every generation because of blending inheritance (Jenkin 1867, 158)? The remedy, here, was the inheritance of acquired variation via Pangenesis, that is, a Lamarckian source of variation in fitness relevant traits.

2. Weismann's solution
[slightly extended because Weismann's is often regarded as identical with later views]
Weismann rejected the inheritance of acquired variation. He saluted Nägeli's idioplasm, a concept of the minute molecular structure of the basis of life, but rejected the internal self-changing force that Nägeli thought this idioplasm must have.

"Nägeli has very ingeniously worked out his conception of idioplasm, and this conception is certainly an important acquisition and one that will last, although without the special meaning given to it by its author. [...] The only proof that idioplasm must necessarily change, in the course of time, as the result of its own structure, is to be found in the fact that Nägeli has so constructed it; and no one will doubt that the structure of idioplasm might have been so conceived as to render any transformation from within itself entirely impossible."  (Weismann 1886; translated in Weismann 1891, p. 264; see also Appendix I, pp. 306-308)

Indeed, Nägeli's idioplasm minus the internal force of change is nothing but Weismann's germ-plasm.

"The germ-plasm or idioplasm of the germ-cell (if this latter term be preferred) certainly possesses an exceedingly complex minute structure, but it is nevertheless a substance of extreme stability, for it absorbs nourishment and grows enormously without the least change in its complex molecular structure." (Weismann 1891, p. 278)

These rejections posed two problems for Weismann. Firstly, heritable variation could not enter the germ-plasm via the inheritance of acquired differences. Secondly, it could neither emerge from an internal self-changing force of the germ-plasm. Therefore the problem of the maintenance of heritable variation in fitness came back to him with a vengeance. Where do heritable variations come from?

"We are clearly compelled to find some other source of hereditary individual differences, or the theory of natural selection would collapse, as it would if hereditary individual variations did not exist." (Weismann 1891, p. 274)

Fortunately, Weismann's non-blending conception of inheritance also pointed a way out. Without blending inheritance he could regarded sexual reproduction as the source of heritable variation in fitness.

"I believe that such a source is to be looked for in the form of reproduction by which the great majority of existing organisms are propagated: viz. in sexual, or, as Häckel calls it, amphigonic reproduction." (Weismann 1891, p. 279)

Sex was no longer the problem but the solution.

3. Fisher's & Muller's theory
Fisher (1930) and Muller (1932) knew that the ultimate source of variation was mutation, and they explicitly rejected Weismann's idea of recombination as the ultimate source. In their theories, sexual recombination accelerates adaptation in an arms race between asexual and sexual populations.

4. Maynard Smith's & Williams's paradox: Fisher's and Muller's theory implies a benefit of sex to species not individuals. Their theory could not be accepted and group selection be rejected at the same time.

• Fisher, Ronald A. [1930] 1999. The genetical theory of natural selection, a complete variorum edition. Edited by Henry Bennett. Oxford: The Claredon Press.
• Jenkin, Fleeming. 1867. “Art. I.—The origin of species.” The North British Review, June, 46:149-171. [http://archive.org/details/northbritishrev24unkngoog]
• Maynard Smith, John. 1978. The evolution of sex. Cambridge: Cambridge University Press.
• Muller, Hermann J. 1932. “Some genetic aspects of sex.” The American Naturalist 66:118-138.
• Weismann, August. 1886. Die Bedeutung der sexuellen Fortfplanzung für die Selektions-Theorie. Jena: G. Fischer. In translation:  Weismann A (1891) The significance of sexual reproduction in the theory of natural selection. Pp. 257-342 in: Poulton EB, Schönland S, and Shipley AE (eds)) Essays upon Heredity and kindered Biological Problems. Authorised translation, second edition. Claredon Press, Oxford.
• Williams, George C. 1975. Sex and evolution. Princeton: Princeton University Press.

sex + outbreeding = gamete altruism - kin selection

The above equation aptly summarizes George C. Williams' socio-biological conception of the cost of meiosis, after the genome-dilution conception had been rejected by John Maynard Smith.

Consider male gametes to be intra-specific parasite, then female gametes act altruistically in accepting fertilization. The paradox of sex is then the occurrence of this cellular altruism in many out-breeding sexual species, that is, in situations where kin selection cannot explain altruism.

John Maynard Smith's initial criticism that the cost of meiosis only arise in species with large eggs and small sperm (anisogamy) no longer applies to this socio-biological conception of the cost of meiosis, because fertilization between gametes of equal size (isogamy) would still be a case of co-operation among not related gametes. Why does not one of the gametes cheat instead and accept the resources but not the haploid genome of the gamete with which it fuses?

This is how Williams rebutted Maynard Smith's criticism. Unfortunately, the controversy did not continue but instead a potted story is being transmitted that was already obsolete at its inception, telling that Williams's cost of meiosis is the false and Maynard Smith's cost of males the correct concept of the cost of sex.

For example, see Lehtonen et al. (2012) or this Haiku associated to it at Hanna Kokko's web-site:

1.

Forget dilution.

Males are the cost of sex.

Wasting resources

2.

Males, a waste of time

Meiosis takes time too, but

it's not always bad

3.

World so full of sex

Genes not diluted at all

Males a waste of space 

P.S.: For quotes and references of the original arguments by Williams and Maynard Smith see here.

The cost of meiosis by G.C. Williams

Re-blogged from Philosophy & Theory in Biology.

In 1971, George C. Williams conceived the evolutionary cost of sex as the cost of reducing the genome during meiosis:

“[...] in meiosis, the number of chromosomes and constituent genes is reduced by half. Each resulting gamete, and zygote that is formed by fertilization, will have a sampling of half the genes of the individual that provides the gametes. In the usual mitotic divisions, each resulting cell preserves the entire genome intact. [...] These parthenogenetic eggs would each contain twice as much of the mother's genotype as is present in a reduced and fertilized egg. Other things being equal, the parthenogenetic female would be twice as well represented in the next generation as the normal one. […] Sexual reproduction is analogous to a roulette game in which the player throws away half his chips at each spin. The game is fair as long as everyone behaves in this way, but if some do and some don't, the ones who keep their chips have an overwhelming advantage and will almost certainly win.” (Williams 1971, p. 13)

Here, Williams starts with a 50% vs. 100% genome transmission argument, also known as genome dilution, but at the end the sociobiological notions of cheating slips in.