Hamilton’s Rule

A recent poll of ASA members showed some interesting results.

1. 73% of Christian professionals in the sciences affirmed the following: “Plants and animals developed through evolutionary processes” (with natural and/or non-natural causes from ancestral forms)
2. 60% affirmed “Plants and animals developed through evolutionary processes with natural causes from ancestral forms.”
3. 61% affirmed “Biologically, Homo Sapiens evolved through natural processes from ancestral forms in common with primates.”

There was one key area where the support dropped that I want to explore. Only 27% affirmed “Human behaviors, like kindness, care for children, competition, or desire for revenge, developed through evolutionary processes with natural causes.” Note that this is less than the 40% that affirmed “Living organisms on earth developed through evolutionary processes with natural causes from non-living material more than 3 billion years ago.”

The reason for this in my opinion is that while the theory of evolution has great explanatory power there are areas where to date it hasn’t. One such area is cooperation or altruism. Furthermore, the implications of “nature red in tooth and claw” goes against our Christian beliefs. This kind of ambivalence to evolutionary theory by Christians goes all the way back to Darwin’s “Origin of Species”.

Darwin added the following quote of Charles Kingsley to his second edition:

A celebrated author and divine has written to me that “he has gradually learnt to see that it is just as noble a conception of the Deity to believe that He created a few original forms capable of self-development into other and needful forms, as to believe that He required a fresh act of creation to supply the voids caused by the action of His laws.”

On the other hand, fellow evolutionist Alfred R. Wallace advocated to Darwin to use Spencer’s term “survival of the fittest” as a synonym for natural selection. Darwin did do this starting in his fifth edition.

From the use of this term, many Christians inferred social darwinism. (Whether social darwinism really is implied is historically dubious.) It is the social darwinism of laissez faire economics more than anything else that motivated William Jennings Bryan during the Scopes trial. He said the following on the age of the earth question: “It is better to trust in the Rock of Ages than to know the ages of rock.”

Since Darwin, evolutionary theory has struggled to explain cooperative or altruistic behavior in humans and eusocial creatures. One such attempt is known as Hamilton’s Rule.

Formally Hamilton’s Rule is genes should increase in frequency when

rB > C

r = the genetic relatedness of the recipient to the actor, often defined as the probability that a gene picked randomly from each at the same locus is identical by descent.
B = the additional reproductive benefit gained by the recipient of the altruistic act,
C = the reproductive cost to the individual of performing the act.

This is known more informally as kin selection. Since this only explains sacrifice for related creatures the self-sacrifice for the “unrelated” is still unexplained. That was until an apparently little-noticed paper two weeks ago in Science.

Hamilton’s rule states that cooperation will evolve if the fitness cost to actors is less than the benefit to recipients multiplied by their genetic relatedness. This rule makes many simplifying assumptions, however, and does not accurately describe social evolution in organisms such as microbes where selection is both strong and nonadditive. We derived a generalization of Hamilton’s rule and measured its parameters in Myxococcus xanthus bacteria. Nonadditivity made cooperative sporulation remarkably resistant to exploitation by cheater strains. Selection was driven by higher-order moments of population structure, not relatedness. These results provide an empirically testable cooperation principle applicable to both microbes and multicellular organisms and show how nonlinear interactions among cells insulate bacteria against cheaters. [Emphasis mine.]

Here’s how Smith et al generalized the rule:

To bridge the gap between theory and data, we derived a generalization of Hamilton’s rule that does not assume additivity or weak selection and whose parameters are empirically measurable (21). We found that cooperators increase in frequency if

r . b – c + m . d > 0 (1)

Distributions can be described by their moments: parameters that measure their shape and location. The relatedness vector r = {r1, r2, …} measures how the distributions of social environments encountered by cooperators and noncooperators differ in each of these moments (fig. S2). r1 is equivalent to r in Hamilton’s rule (5). The other terms are higher-order relatedness coefficients (22, 23). Any smooth function can be expanded into a Taylor polynomial series whose coefficients measure its linear, quadratic, and higher-order components. The benefit vector b describes noncooperator fitness as a function of social environment (red lines in Fig. 1) in terms of its Taylor coefficients. c is the cost of cooperation when all neighbors are noncooperators. m d is nonzero when benefits depend on recipient genotype (Fig. 1C). m is the moments vector for cooperators. d is the difference between the Taylor series of cooperators and noncooperators. Unlike Hamilton’s rule, Eq. (1) disentangles fitness effects from population structure and is valid for arbitrarily complex forms of social selection. When fitness effects are additive, Eq. (1) reduces to rb – c > 0.

Figure 1

Fig. 1. Measuring the costs and benefits of cooperation in microbes. Blue, cooperator fitness; red, noncooperator fitness. (A) In Hamilton’s rule, b is the slope of fitness against the frequency of cooperators among social neighbors; c is the fitness difference between cooperators and noncooperators for a given social environment. Fitness effects are nonadditive when benefits are (B) nonlinear or (C) depend on recipient genotype.

Putting this into English, it was the population structure rather than kinship that selected for cooperators of M. xanthus over cheaters, aka Richard Dawkin’s “selfish gene”. Furthermore, this work takes Hamilton’s rule from a heuristic to an empirically useful structure to look at other social structures. So, I suspect we will see a lot of followup on this in the coming years. It also challenges our intuitive concept of what constitutes the “fittest” which survives.

One thing that those of us who are critical of intelligent design are fond of saying is that it promotes a God of the gaps. The study above shows that we should be careful of our own gaps. Whether it’s research like this or the RNA World concerning origin of life or even multi-verses we should be careful that our apologetics doesn’t lean too heavily upon a lack of a natural explanation. History has shown that such an explanation often does come along even if it takes a long time. I’ll close re-quoting Kingsley’s warning to us about gaps — this time as he wrote Darwin in 1859:

I have gradually learnt to see that it is just as noble a conception of Deity, to believe that he created primal forms capable of self development into all forms needful pro tempore & pro loco, as to believe that He required a fresh act of inter-vention to supply the lacunas wh. he himself had made. I question whether the former be not the loftier thought.

14 comments to Hamilton’s Rule

  • Jon Tandy


    It is true that evolution assumes death, as does old-earth creationism (special creation of animal life over long ages prior to the creation of mankind).  T’his is the criticism that young-earth creationism has of both alternative viewpoints — they assume from the Biblical account that there was life created in the beginning without death, and that the death of animals came in at the same time as Adam’s fall.  Many Christian scholars and scientists dispute that interpretation.

    Evolutionary theory, strictly speaking, doesn’t deal with explaining the first life.  It simply explains how species change over time through mutation, and survive to produce new generations of organisms.  The origin of life has been an active field of study for over a half century, and certainly there must be life present before biological evolution can have anything to act upon, but they’re not dealing with the same sets of data and theories.

    If theistic evolutionists acknowledge the Bible as an accurate scientific record, which many do not (including as you say the “creation of life before the introduction of death”), it really would not pose them any problem anyway.  They acknowledge that biology (study of life) requires life to first be present.  This could be seen to loosely fit the pattern in Genesis — life first, then reproduction and death.

    Origin of life research deals with how non-living matter could develop into self-replicating structures, eventually resulting in what we would call “living things.”  OOL does not have a conclusive theory yet, but even if they ever do, it would not be a problem for evolutionists, theistic or not.

    Jon Tandy

  • Scot Sutherland

    That definitely clears up that question for me.  So would it be fair to say that the popularized creation-evolution debate is really not about evolutionary theory at all, but about OOL, and evolutionary theory gets caught in the crossfire.

    I’m new here so forgive me.  Where would be a good place to propose and discuss an idea I’ve been batting around about an alternative way to define “life”?  The current biological definitions don’t work very well for learning science.  I think we need a way forward to develop a theory of mind that is more useful for cognitive science.


    Sorry missed your post at first. Still mucking about learning how this works.

    I would definitely concur with evolutionary observations (laws). Based upon my understanding they are well established. It is the theory lacks explanatory power for a learning/cognition scientist. As a result we have a significant number of entrenched perspectives from numerous fields, none of which bring parsimony.

  • Randy Isaac

    Scot, in a way it is a tautology to say that life comes before death since death is the cessation of life. It’s not very meaningful to say that death comes before life. As Jon pointed out, a major challenge is to determine how life arose from non-life. I’ve recently listened to a few lectures by OOL researchers and the definition of life is a non-trivial challenge which doesn’t have broad consensus. It would indeed be an interesting thread. If you would like to start a post on that topic, Terry Gray moderates this blog and he will review and post it for you if you’d like.

  • Randy Isaac

    Bob Greenhow asked me to post this on his behalf:

      In the paragraph which begins “There was one key area where the support dropped that I want to explore” there is the survey question “Living organisms on earth developed through evolutionary processes with natural causes from non-living material more than 3 billion years ago.”

    I would like to comment on this statement by reproducing a short piece written some years ago for my own satisfaction –

    What is life ?


    A series of questions may help in answering. What is the mass of life ? that is, how much does it weigh? What are its dimensions, length, breadth, height? What color is it? Can it be dissected or analyzed in the laboratory? Where can pure life be found, or where can we get at it to examine it more closely? (A concrete example is the difference between a dead amoeba and a living amoeba.)


    Stupid questions, some might say. I reply, are they stupid because they have no answer ? Do these questions not reveal a fact, perhaps an unwelcome fact to some, that life itself, as isolated in our thoughts from material or physical entities, is beyond physics. Aristotle would have called it metaphysics, a term I avoid because of what it already means. I am content to say that it is non-physical, or that it is beyond physics.


    What is my purpose is writing all this? It is because I repeatedly come across statements or phrases that really are meaningless, such as the evolution of life, or the emergence of life. What really can be studied is the evolution of living things, or the emergence of certain living beings out of different living beings. 


    I wish that there were more evidence among biologists, especially among Christian biologists, of the realization that life itself is beyond physics, beyond nature. I hesitate to call it supernatural; perhaps, with apologies to Aristotle, we could say ‘metanatural.’



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