Extraordinary sex ratios

Extraordinary Sex Ratios is the paper that William D. Hamilton seems most proud of if the effusive self-praise in the biographical preface can be trusted. In it Hamilton claims his theoretical insight peaked, and it was within this paper that his ideas exhibited the most pluralism of purpose as he began to perceive the shape of his future research. Extraordinary Sex Ratios also marks the beginning of Hamilton's long utilization of computer simulations to push through the impasses of analytic intractability and empirical unverifiability. On occasion he even claims that in this particular area, sex ratio theory, evolutionary biology comes close to a level of precise certitude and projective power on par with more exact sciences. Here are the main points which are purportedly touched upon within Extraordinary Sex Ratios:

1. The levels-of-selection debate.

2;. The idea of conflict within the genome.

3. The 'evolutionarily stable strategy' or ESS (referred to in this paper as the 'unbeatable strategy').

4. The initiation of game theoretic ideas in evolutionary biology.

5. Finally and more indirectly, by emphasizing the costliness of male production for females and for population growth, as well as the every-ready 'option' (among small insects, for example) of parthenogensis, it helped to initiate debate over the adaptive function of sex.

I would say that 1 & 5 are implicit within the narrative structure, but 2, 3 and 4 receive explicit treatment. On occasion one stumbles upon primitively expressed concepts and ideas that receive book length treatments in later works by other thinkers, such as Genes in Conflict and Evolution and the Theory of Games. Though I would not class Hamilton with R. A. Fisher in breadth and depth of intellect, in the specific area of evolutionary biology he was clearly an individual of uncommon creative fertility. In describing the self-taught genius George R. Price he says:

...Price had not like the rest of us looked up the work of pioneers when he first became interested in selection; instead he had worked out everything for himself. In doing so he had found himself on a new road and amid startling landscapes.

Though Hamilton was certainly not self-taught, it is clear that his peripatetic intellectual travels were in many ways rather similar. It seems that like a gardener pruning weeds and selecting new varietals his intellectual progress combined both an element of chance and underlying purpose. Reading the two volumes which possess biographical introductions it is also clear that the phylogeny of Hamilton's ideas can often be easily traced; each journey catapulting him over the horizon and into an alien domain.

Like much of William D. Hamilton's early work this paper is an extension, exploration and to some extent a refutation of R. A. Fisher's conclusions as presented in The Genetical Theory of Natural Selection. I've already covered the most general theory of why sex ratios are as they are; the verbal argument is simple and derives from Fisher. As the ratio of males to females deviates from 1:1 any individual that possesses an allele that may skew the ratio toward the less numerous sex may increase their fitness. That is, in a population where the male to female ratio is 1:1000, any female which produces only males will have a far higher than mean fitness. Her genes will multiply, to the point where the sex ratio will eventually equalize so that likely there will be an overshoot and those with male bias will be less fit negative frequency dependent selection). Eventually a stable equilibrium will emerge around 1:1 so that the reproductive value of males and females will be in balance, in expectation if not outcome (males generally manifest greater reproductive skew).

So what's the problem here? Assumptions. Hamilton shows that Fisher's model does not work for sex-linked genes in the heterogametic sex. Males are heterogametic, we have an X & Y on the 23rd chromosome. In other taxa females can be the heterogametic sex. The important point here is that Y chromosomes (the example Hamilton employs) are passed only through the male; the female descendants of the male are irrelevant to the Y chromosome because it will never be passed through them (an X chromosome will spend 1/3 of the time in a male, and an autosomal chromosome 1/2 of the time, assuming equal sex ratios). Hamilton's logic is simple, if a Y chromosomal gene can distort the sex ratio so only males are produced from the point of fertilization then the Y can increase its fitness. Of course, there's an obvious problem here: once all the females disappear the males can not replicate. The Y chromosome is a selfish gene in a classic and somewhat malevolent sense here, as its interest may result in the extinction of a lineage (see meoitic drive). But remember, evolution only sees a few steps ahead. Similar principles operate for distorters on the X and autosomal chromosomes, though because of the fact that they are passed through females the selection process is far weaker. Hamilton's simulations show that catastrophic crashes occur much faster with Y sex ratio distorters.

In his survey of the literature Hamilton notes that the relative lack of Y sex ratio distorters. He also observes that generally the Y chromosome is genetically inactive (ergo, sex linkage of traits). He posits that intragenome dynamics are at work here; modifiers and inactivation has been selected for over time so that the Y can no longer make mischief. It's been cut off at the knees, so to speak. X and autsomal distorters can be found at some frequency within the population. The whole topic of selfish genes and below individual level operation of selection (i.e., within genome conflicts, cooperation and other dynamics) is a fertile field which Hamilton only hinted at in this paper (see David Haig's work).

Hamilton then moves to higher levels of organization. What about population substructure? Inbreeding? Or, one of his favorite themes, spatial viscosity? These can all effect the outcomes of sex ratios. Consider a species which is characterized by distinct demes with little between population gene flow. If the flow is low enough in rate then a Y distorter would result in a local extinction. Eventually the region would be reoccupied by migrants from another deme. One supposes in this sort of scenario gene flow would have been very low, but because of Hamilton's use of entomological examples these sorts of dynamics may occur. He proposes the model of a 'host' body which migrants colonize. Now, imagine that N females settle on a host, and that their reproductive output is equal. Within this species there are two types of Y chromosome, Y^a and Y^b. Females are inseminated once, so they either carry sperm of Y^a & Y^b variety, Type a and Type b, which can distort sex ratios so that there is an expectation of proportions of x_a and x_b. Imagine a host which is shared by r Type a females and n - r Type b females (n being the total obviously). The sex ratio, X_r is give by:

[ rx_a + ( n - r ) x_b ]/n

After one sequence of host usage a particular Type a female may then be allocated a number of emigrant females proportional too:

x_a/X_r ( 1 - X_r )

What we're interested in is the fitness of Y^a. The probability distribution of Type a females within the set is defined by F_r. The average Y^a fitness, W_a is then:

Σ (start at 1 sum across n of set) F_r x_a/X_r ( 1 - X_r )

While the average fitness for the Y chromosome is:

W = 1 - X where X is the sex ratio for the whole population.

Now we're looking for the difference between W_a and W in fitness, so after some algebra:

Σ (start at 1 sum across n of set) F_r x_a/X_r + ( 1 - p )( x_b - x_a ) - 1, where p is the frequency of Y^a

Hamilton differentiates this equation (first derivative with respect to x_a) and sets x_a = x_b = c, which is zero (an equilibrium) only when:

c = ( n - Σ (start at 1 sum across n of set) F_r )/( n ( 1 - p ) )

Then he posits that settling on hosts is a random binomial process. Using the binomial expansion the previous equation reduces to:

c = ( n - 1 )/n

This is what he calls the 'unbeatable sex ratio' (the equivalent to the ESS). The analysis stops here, and Hamilton notes that simulations where n = 2 show that the ratio is around ~0.07. He notes a species where this is close to the sex ratio, but admits that the example is unrealistic and improbable. But of course the point here was to push a theoretical boundary condition; which he has done by suggesting there are alternative evolutionarily stable strategies from the 1/2 ratio that Fisher proposed. As a minor coda to this portion of the paper Hamilton introduces variation in sex distortion by females as well and assuming that inbreeding varies as a function of n (remember that that means the number of females which land on a host and produce offspring which may then mate with each other):

( n - 1 )/2n

The only difference here is the addition of 2. That means that as n → ∞ the ratio is 1/2. Why? As the population of females which land on the host increases the situation verges upon Fisher's panmictic assumption! If n = 1 that would entail obligate sibling inbreeding, and the equilibrium ratio is 0. By this, Hamilton now infers that that simply means that the rate limiting step in reproduction are the number of daughters that a female can produce, since one brother can inseminate innumerable sisters.

The next section is a breezy survey of the literature on various organisms, their sex ratios and possible correlates with parameters such as inbreeding and mating systems (e.g., polygyny). Hamilton concludes:

Although the theoretical position is even less clear for n > 2, the ratio lies at least in some respects within known bounds. Since the great majority of recorded sex ratios of Hymenoptera are in the range 1/4 to 1/2, it is thought likely that the model does at least exemplify the forces that are operating.

These early papers are a little flaccid in the empirical details when set next to the theoretical superstructure. But then, there is a reason that Hamilton exalts these as his crowning theoretical achievements; they were truly simply guides and pilots which might instruct upon the highest probability of fruit in experimental or observational endeavors. A rough & ready check of the literature was more for the rationale of falsification than verification.

Hamilton ends his treatment with a few more baby-steps in the direction of game theory. In an inbreeding situation where the offspring of an organism on forced to breed with each other:

fitness ∝ to N of inseminations by sons + N of daughters

Since one son can inseminate many daughters the logic would tell us that a female attempting to maximize her descendants fitness would skew the ratio toward daughters (since that's the "rate limiting step" in increase). What about if you have two competing organisms, parasites, which land upon a host? Assume that one arrives before the other, and the second one knows of the prior resident's presence. Then the fitness of the second organism is proportional to:

x /( x + x₀ ) [ ( 1 - x ) + ( 1 - x₀ ) ] + ( 1 - x )

Where x is the sex ratio produced by organism #2 and x₀ is the sex ratio of organism #1. Note the extrapolation from the previous equation where one assumes inbreeding of the offspring. A little calculus to find the maximum value fitness for the second organism as a function of sex ratio results in:

x^* = ( x₀ )^1/2 - x₀

Both organisms max out at 1/4 for the sex ratio, that is, 1 male for 4 females. This is a non-zero sum situation as there is a larger population on other hosts aside from the two parasites in question, it's really not the conventional inbreeding model first alluded to above. Hamilton calls this the "unexploitable" strategy. What about a "unbeatable" strategy where you are looking at a pairwise payoff? That is, where the two organisms are competing directly against each other. Here it is:

x^t = ( 2x₀ )^1/2 - x₀

So that 1/2, the conventional sex ratio, is unbeatable in this pairwise combination. If one imagines that there are two morphs in the population which exhibit ratios of 1/4 and 1/2 respectively, what happens? It depends; if the morphs land on their hosts randomly, that is, one is not conditional upon the other, 1/4 wins. On the other hand, if the 1/2 ratio morph attempts to pair with the 1/4 morph then it can maintain its frequency over time, though not when it is very rare or very common. Hamilton notes that with insect parasites, the current context, this sort of sex ratio strategizing is unlikely. That being said, the final portion of the paper here is a clear foreshadowing of the Trivers-Willard hypothesis. From reading Robert Trivers' own memories of his ideas during this period I note that Hamilton's papers were a significant influence.

Hamilton, W.D. (1967). Extraordinary Sex Ratios. Science, 156(3774), 477-488. DOI: 10.1126/science.156.3774.477