The moulding of senescence by natural selection is not one of William D. Hamilton's favorite papers. In the biographical introduction he notes that both Peter Medawar & George C. Williams covered the same ground in the 1950s; a fact that he was not aware of by the time he had already invested a great deal of thought on the topic at hand. The general mathematical treatment within this paper extends the arguments of Williams in particular; but Hamilton admits that his value-add is on the margins and likely not worth the mathematical formalism which he spun out to converge upon insights analytically. Speaking of which, I'm going to skip it and go to the descriptive conclusions. If the algebra in my first post was opaque, the actuarial functions in this paper are banal and tedious. The basic logic behind the ideas are pretty clear, and Hamilton admits that his formalistic treatment really didn't push the ball that much further in any case, so I don't see any point in splashing out his grab-bag of variables and integrals onto to the web (the constant switching between discrete and continuous functions is also a bit jarring).
One of the most interesting things about this paper is that Hamilton was prompted to explore this question further than Medawar and Williams because of a rather peculiar reason, and I will quote Hamilton's exposition here:
...This latter seems to have been taken over uncritically [by Medawar and Williams] from Fisher who had written that he thought it 'probably not without significance ... that he death rate in man takes a course generally inverse to the curve of reproduction value' ....
...I hope to make it clear that the correspondence to which Fisher draws attention in the above statement is really largely trivial and that in the context to which they were restricting themselves the idea which tacitly and Medawar explicitly assumed is without foundation....
I think R. A. Fisher's The Genetical Theory of Natural Selection is probably the second most fruitful work after The Origin of Species in the history of evolutionary science. But Fisher's ~300 pages are very dense; and I've talked to friends about how sometimes one has to wonder what exactly Fisher meant here & there, and the game of exegesis beings. Remember, Fisher is the man who laid down the first brick for the Modern Synthesis with his work which fused the Mendelian tradition with that of the Biometricians. He was also the most original thinker in early 20th century statistics; ever heard of ANOVA? So when someone like Fisher puts pen to paper you need to engage in some close reading, particularly because his work is packed so many novel, ground-breaking and abstract ideas. But great scientists do make mistakes; there were even minor mathematical errors in the first edition of The Genetical Theory of Natural Selection (e.g., Sewall Wright pointing out to him that the denominator should have been 2N and not N and so forth, which was acknowledged in subsequent editions with a correction). It is perhaps a cautionary tale which reminds us that science is a fundamentally human activity and so error is part and parcel of progress.
In any case, the central question which drives this paper is the extent to which positive selection leads to the emergence of senescence. That is, is death inevitable because it is adaptively necessary? One encounters these sorts of musings in popular folklore; the old must die so that the young may flourish, and so on. But George C. Williams argued that antagonistic pleiotropy was a driving force behind the emergence of diseases late in life and hastened the breakdown of organisms as time progressed. The logic is that the reproductive value of an organism peaks early on, near the origination point of sexual maturity, and any trait which increases fitness during this period will naturally be selected. But there is no such thing as a free lunch, and many genes which may confer reproductive fitness in youth may entail physiological debility later in life. Consider an allele which increases metabolic activity and so allows an organism to engage in greater nutritional intake and heightened reproductive output. The increased metabolic activity might also correlate with earlier organ failure and cancer as the organism ages. The trade off is usually acceptable because reproductive value is so much higher early on during the life of an organism (consider a mouse, even if a mouse was physiologically immortal and could reproduce unto eternity the likelihood is that a predator would kill it before long, so it pays to live hard and breed early).
To this Hamilton generally says yes, mostly true, and the maths don't add much. But, he does suggest that one should be careful about focusing on directly pleiotropic genes. That is, those which have multiple physiological implications which have first order fitness effects. Here the fact that William D. Hamilton was a deep Malthusian comes through; he notes that any allele which increases population will eventually come up against Malthusian checks. If density is increased one may assume that this will have an effect where lifespan decreases as intraspecific competition increases because of conflict over finite resources. In contrast, if the population is somehow kept below the Malthusian limit then one can envisage a scenario of plentitude, and lifespan may reach a physiological limit far more often. An example from human history seems to be the period after the Black Death in Europe, where a drastically reduced population increased average standard of living for those who survived! There was more land to go around per person, and more value to any unit of labor. Hamilton's point is a gene-centric one; replication will push against a Malthusian limit inevitably in most situations because artificial attempts to keep the population in check will be swamped by fecund cheaters who can evade the controls or refuse self-imposed limits. As the population approaches the limits of the resources to support any given individual, to maintain a stationary state death rates will increase on the margins; the old, the young, the weak, and so on.
One feature of organisms which Hamilton touches upon briefly is that of post-reproductive life. In humans this is mainly a concern with post-menopausal females. Hamilton notes that 15 potential active post-reproductive years is somewhat anomalous; but, he seems skeptical that the reproductive impact on fitness would be great enough to select for this particular physiological trait. Rather, he suggests that perhaps antagonistic pleiotropy is the prime factor here, as a physiological switch in youth somehow exhibited age dependent effects so that the menopausal cascade was likely later on in life. He does point out that other organisms, such as aphids, exhibit post-reproductive lives, so though he does not find the maths compelling in predicting these tendencies, Hamilton seems to suspect that some sort of fitness increase is likely to other aphids which are still in their reproductive years if the older ones remain.
When he considers infant mortality his Malthusian and eugenic biases come to the fore. Though he asserts that Fisher's implication about the inversion between mortality and reproductive curves was trivial, Hamilton does not dismiss the possibility that some genes might induce early death. The logic is that a sickly infant which is bound to die at some point is more likely to drain resources from other offspring; so from an inclusive fitness perspective it is more optimal for the infant (vehicle for the allele) to remove itself from the population earlier through mortality. It doesn't seem that Hamilton is convinced by these sorts of arguments and does not work out how it would be implemented physiologically or justify that the fitness effects would in fact be positive (after all, susceptibility to disease and greater mortality might kill a sickly infant earlier, but it is hard to imagine how this could not be detrimental toward its fitter siblings). Rather, here Hamilton notes that "bio-economic" considerations operant on the level of social traditions are the primary voluntary mediators of infant mortality. That is, infanticide, a practice which Hamilton is somewhat positively predisposed towards (see much more on this in Narrow Roads of Gene Land II).
Finally, Hamilton ends with a few brief comments on higher moments of the distribution around the central tendency in regards to reproductive value as a function of age. Though his formal treatment has some implications, he seems (rightly I think) skeptical as to the efficacy of selection in shaping the nature of kurtosis of the distribution and what not. He does note that the formal treatment implies that there will be a skew to the right in a decreasing population and an inverse in a stationary or increasing population; but empirically and theoretically there are reasons to not take this inference too far. At the end Hamilton also admits that much of the treatment here does not cover the nature populations which fluctuate in size cyclically and/or erratically.
HAMILTON, W. (1966). The moulding of senescence by natural selection. Journal of Theoretical Biology, 12(1), 12-45. DOI: 10.1016/0022-5193(66)90184-6
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