Gene Expression

Ruchira Paul sent me an email asking me to clarify this exposition of how gene selectionism can explain 50:50 sex ratios. First, I would like to second the author of the original post’s injunction to read Richard Dawkins’ The Selfish Gene, it is a masterpiece of scientific exposition. For many people an encounter with the The Selfish Gene is a K-T scale event, the world is changed after the encounter. That being said, it surely isn’t the last word. A friend of mine mentioned how she noted that Dawkins heaps scorn on Amotz Zahavi’s Handicap Principle, but a generation later this model has gained a modest level of acceptance. The Selfish Gene was written in 1975, and though it is a classic, it is not timeless in all its particulars.

With that caution and exhortation out of the way let us move on to sex ratios. We know mechanistically that meiosis produces equal numbers of precursor gametes of the two kinds in the heterogametic sex. To use a concrete example in humans it is the father who determines the sex of the offspring through the production of “Y” and “X” sperm, whose ratio is approximately 50:50. From the character of this process we can infer that the proximate result will naturally be an equitable sex ratio because there are equal numbers of male and female sex determinative sperm produced. Nevertheless, one can imagine a scenario where males could bias their production of sperm so as to favor generation of one, or females could vary the rates of fertilization or spontaneous abortion contingent upon sex. So the ultimate question is why don’t these mechanisms evolve to “optimize” sex ratio?

By optimize, consider the fact that many species are highly polygynous so that there is a large effective population difference between males and females. In other words, in any given generation a much higher proportion of females pass on their genes to the subsequent generation than males. Since the production, raising and feeding of offspring is costly it would seem plausible that a group of organisms could maximize their efficiency by simply manifesting a sex ratio favored toward females since that sex is the rate-determining step in natural increase. But generally this doesn’t occur. Why?

To answer that question we need to go back to considering the possibility of sex biasing mechanisms and the incentives toward production of males and females for a given pair of organism. I will repeat R.A. Fisher’s verbal exposition on the prevalence of 50:50 sex ratios presented in The Genetical Theory of Natural Selection. In short, let us assume a priori that the cost of production of males and females is equitable. Now, let us assume that we run experiments where in the initial generation we select an arbitrary ratio of males to females and allow them to breed in a closed system. For example, we could have three populations where the ratio of males to females are 20:80, 50:50 and 80:20. Let us name them “More Females” (MF), “Equal Ratio” (ER) and “More Males” (MM). Within the MF population with generation 1 there will be a variance in the number of males and females an organism is the parent of. The extent of this deviation from expectation is naturally inversely proportional to the number of offspring that an organism typically has. Let us assume that reproduction is on the “human scale,” so the variance will be rather high. In each scenario by the luck of the draw some parents will have mostly female offspring while others will have mostly male offspring. Mechanistically let us assume that the expectation is 50:50, so one assumes that the population would naturally equilibrate to that ratio. So let us assume that in our experiment we add more of one sex in the extreme cases to keep the ratios out of balance. Now the different populations will exhibit different internal dynamics. In the MF population the parents who tended to give rise to male broods will have more copies of their genes passed on simply because polygyny will be the natural state of things in a population where females are in excess. In contrast, the inverse will be true in the MM scenario as many males compete for a few females. In the ER situation there is no difference.

The first and last dynamic, where there are initial biased sex ratios which we can artificially maintain (though one could posit natural selective processes which result in greater mortality for one sex over the other), offer the opportunity of an evolutionarily opportunistic biasing mutant to arise. Consider a female who carries a mutation which results in discrimination against male spermatoza attempting to fuse with the ovum. Her offspring will invariably be female. In the MM scenario, where there is an excess of males, her exclusive production of females results in no “waste” due to the generation of males with low reproductive values (the odds being stacked against them). Because the female who carries this mutant has more descendants that those who do not carry this mutant (recall that her daughters will invariably reproduce, while her sisters who produced a great many sons will see most of them fail to reproduce) the mutant itself will spread through the population. The increased frequency of the mutant will begin to bias the sex ratio expectation for broods away from 50:50 and the imbalance will begin to close rather soon even if we continue to engage in addition in the biased scenarios so long as we do not increase our rate of addition (that is, we add enough females in the 20:80 scenario to generate 20:80 assuming that the previous generation with the population would have a 50:50 production of offspring). Now, what happens when the number of males and number of females is equal? Now the expectation of reproduction of male and female offspring are equal, and the mutant exhibits no advantage (in practice the variance of male reproduction is generally higher in most species). If the mutant is extant at a high frequency within the population naturally the number of females will then go into excess, at which point the mutant will be at disadvantage as males will now have a higher reproductive value as they replicate more copies of their genes to the next generation. So what you see here is a form of negative frequency dependent selection where sex biasing mutants are at advantage in situations where they are biasing the organism toward production of the rare sex. As they increase in frequency they naturally diminish their own rareness and decrease their fitness to the point where they no longer increase in proportion. Any deviation from 50:50 will tend to result in these incentives, and evolutionary pressures will tend to converge upon this value from both directions. The 50:50 ratio is like the valley bounded by two hills.

The above was a verbal treatment (numbers being used only for illustration). Below I’ll reproduce the formal model introduced in John Maynard Smith’s Evolutionary Genetics (Box 13.1). The verbal model is good at explaining the ubiquity of the 50:50 ratio, but the algebraic model can show us with more rigor the various parameters involved in the overall process.

Assume a pair produce m sons and f daughters, where:

m + kf = C

…where k is the ratio of a cost of the daughter to a son. That is, if a daughter is 50% as costly to produce/raise/etc., then k is 0.5. C represents the total ‘expenditure’ on offspring production. Now assuming a random mating population where:

m* = typical production of sons
f* = typical production of daughters
m = production of sons when female carries M dominant mutant
f = production of daughters when female carries M dominant mutant

M has no affect on males. Now, the frequencies of the genotypes of interest will be:

P = M/+ for females (who express and bias)
p = M/+ for males (who don’t express and bias, but rather carry the allele which will express in daughters in the event that it is passed on)

Since M is a mutant its initial frequency is trivial, so we can ignore homozygotes. M will be found almost always in one copy situations, M/+, where + is the wild type allele. Since heterozygotes will initially be rare as well can assume that matings within the population are almost always between heterozygotes of both sexes and wild type homozygotes. With that in mind, the following table illustrates the outcomes of various matings of interest:

      Male   Female  
Male Female Frequency M/+ +/+ M/+ +/+
M/+ +/+ P(1 – p) ~ P m/2 m/2 f/2 f/2
+/+ M/+ p(1 – P) ~ p m*/2 m*/2 f*/2 f*/2
+/+ +/+ (1 – P)(1 – p) ~ 1 – P – p m* f*

Among the offspring the frequencies are as follows from multiplying and collecting from the above table:

M/+ males → Pm/2 + pm*/2

M/+ females → Pf/2 + pf*/2

total males m*(1-P) + mP ~ m*

total males f*(1-P) + fP ~ f*

The approximations above are justified by the fact that M is a low frequency mutant, so for example in the last expression P is so small in comparison to 1 – P that the product of the former dwarfs the latter.

Now we take these expressions and step through them for future generations with a recurrence relation. Assume that P’ & p’ represent the frequencies in the next generation for M/+ males & females respectively. Then we can deduce:

P’ = 1/2P(f/f*) + 1/2p
p’ = 1/2P(m/m*) + 1/2p

Note that males (the second element in each expression) as passing on M to half of their offspring and not responsible for any biasing of sex ratios. In contrast, females may bias the sex ratio as defined by x/x*, where the denominator is the wild type expectation of progeny of that sex produced. The proportion of males and females carrying M is variable in the next generation to the extent that females carrying M in the previous generation tilt their production of offspring toward males or females. Now, collecting the equations above:

P’ + p’ = 1/2P(f/f* + m/m*) + p = (P + p) + RP, where R is:

1/2(f/f* + m/m*) – 1

R roughly gauges the modulating effect of the mutant, M. If M has no effect, then:

1/2(1+ 1) – 1 = 0

In this case, P’ + p’ = P + p, there is no change across the generations as R = 0 is removed from the equation. But, now we have to back up and go back to the first equation:

m + kf = C

We can rearrrange so that:

f = (C – m)/k
f* = (C – m*)/k

And substitute into the relation above for R:

R = [(C – m)/k(m – m*)]/[2m*(C – m*)/k]

The k‘s cancel to leave:

R = [(C – m)/(m – m*)]/[2m*(C – m*)]

Now, if R is greater than 0 then M increases in frequency (see the original recurrence relation). If R is less than 0, then it decreases in frequency (presumably this implies changed selective environments since it must have attained some proportion for this to be relevant). So, if:

m* is less than C/2 then the mutant will invade if m is greater than m*
m* is greater than C/2 then the mutant will invade if m* is greater than m

You can check this by plugging values numerically into the algebra above. The implication is that mutants will converge upon m * = C/2, the evolutionarily stable ratio. In which case, m* = kf*:

C/2 + kf* = C, 1/2 + kf*/C = 1, kf* = C/2

In other words, the cost of expenditure for both sexes is balanced, k = 1, from the perspective of the gene over the long term. Now, presumably if there was some policing method by the population on such genetic shenanigans then one could imagine scenarios where biased sex ratios may emerge, and in the caes of haplodiploidy the genes themselves offer up a structural incentive for a difference in sex ratios. But that’s a different model….

Comments

  1. #1 CCP
    June 14, 2007

    Nice verbal exposition–I’ll take your word on the math.
    An interesting twist to this subject is provided by turtles and crocodilians with temperature-dependent sex determination. No sex chromosomes, so no biasing of sperm (or eggs, as would be the case in birds). I think Fisher’s model still works, but the mutations one posits have to affect nest-site selection.

  2. #2 Ruchira
    June 14, 2007

    Thanks, Razib.

  3. #3 David B
    June 15, 2007

    Dawkins changed his mind about the Handicap Principle – see the 2nd edition of ‘Selfish Gene’.

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