Razib Khan One-stop-shopping for all of my content

June 14, 2011

Band of brothers at war

The fruits of human cooperation

ResearchBlogging.orgThe Pith: Human societies can solve the free rider problem, and generate social structure and complexity at a higher level than that of the band. That implies that much of human prehistory may have been characterized by supra-brand structures.

Why cooperation? Why social complexity? Why the ‘problem’ of altruism? These are issues which bubble up at the intersection of ethology and evolution. They also preoccupy thinkers in the social sciences who address fundamental questions. There are perhaps two major dimensions of the parameter space which are useful to consider here: the nature of the relationship between the cooperators, and the scale of the cooperation. An inclusive fitness framework tracks the relation between altruism and genetic relatedness. Reciprocal altruism and tit-for-tat don’t necessarily focus on the genetic relationship between the agents who exchange in mutually beneficial actions. But, in classical models they do tend to focus on dyadic relationships at a small scale.* That is, they’re methodologically individualistic at heart. So all complexity can be reduced to lower orders of organization. In economics a rational ...

May 4, 2011

We, Robot & Hamilton’s Rule

The original robots

We are haunted by Hamilton. William D. Hamilton specifically, an evolutionary biologist who died before his time in 2000. We are haunted because debates about his ideas are still roiling the intellectual world over a decade after his passing. Last summer there was an enormous controversy over a paper which purported to refute the relevance of standard kin selection theory. You can find out more about the debate in this Boston Globe article, Where does good come from? If you peruse the blogosphere you’ll get a more one-sided treatment. So fair warning (I probably agree more with the loud side which dominates the blogosphere for what it’s worth on the science).

What was Hamilton’s big idea? In short he proposed to tackle the problem of altruism in social organisms. The biographical back story here is very rich. You can hear that story from the “horse’s mouth” in the autobiographical sketches which Hamilton wrote up for his series of books of collected papers, Narrow Roads of Gene Land: Evolution of Social Behaviour and Narrow Roads of Gene Land: Evolution of Sex. ...

September 23, 2010

Simple rules for inclusive fitness

ResearchBlogging.orgWith the recent huge furor over the utility of kin selection I’ve been keeping a closer eye on the literature on inclusive fitness. The reason W. D. Hamilton’s original papers in The Journal of Theoretical Biology are highly cited is not some conspiracy, rather, they’re a powerful framework in which one can understand the evolution of social behavior. They are a logic whose basis is firmly rooted in the world of how inheritance and behavior play out concretely. But because of their formality and spareness inclusiveness fitness has also given rise to a large literature derived from simulations “in silico,” that is, evolutionary experiments in the digital domain.

375px-Green_Beard_GeneOne can elucidate inclusive fitness through Hamilton’s Rule, but it is also rather easy to exposit verbally via a “gene’s eye view.” Imagine for example a dominant mutation in a diploid organism which produces the behavior of altruism toward near kin. Initially the altruist will have offspring whose probability of carrying the dominant mutation is 50%, because there is also the probability that they will carry the ancestral non-altruistic variant. Imagine an altruistic behavior which incurs a small, but not trivial, cost to the individual performing the behavior, and a large gain to the individual who is on the receiving end of the altruism. The logic of favoring near kin is such that in the initial generation the parent which behaves altruistically toward near kin is increasing their own “inclusive fitness” because their offspring share 50% of their genes identical-by-descent (in the case of a diploid sexually reproducing organism). But from a gene’s eye perspective what is really occurring is that there is a 50% chance that the gene which fosters altruism is promoting the fitness of a copy of itself. So inclusive fitness operates by modulating the parameters of costs and gains to focal individuals as a function of their relatedness, but it is the genes, the “replicators,” which persist immortally across the generations. We “vehicles” are just the ocean through which genes sail.

But like Darwin’s theory of evolution through natural selection the fruit of these logics are in the details. A new paper in The Proceedings of the Royal Society puts the focus on different means by which inclusive fitness may be maximized. In particular, the paper offers up a reason for why what Richard Dawkins termed the “green-beard effect” is not more common. Selective pressures for accurate altruism targeting: evidence from digital evolution for difficult-to-test aspects of inclusive fitness theory:

Inclusive fitness theory predicts that natural selection will favour altruist genes that are more accurate in targeting altruism only to copies of themselves. In this paper, we provide evidence from digital evolution in support of this prediction by competing multiple altruist-targeting mechanisms that vary in their accuracy in determining whether a potential target for altruism carries a copy of the altruist gene. We compete altruism-targeting mechanisms based on (i) kinship (kin targeting), (ii) genetic similarity at a level greater than that expected of kin (similarity targeting), and (iii) perfect knowledge of the presence of an altruist gene (green beard targeting). Natural selection always favoured the most accurate targeting mechanism available. Our investigations also revealed that evolution did not increase the altruism level when all green beard altruists used the same phenotypic marker. The green beard altruism levels stably increased only when mutations that changed the altruism level also changed the marker (e.g. beard colour), such that beard colour reliably indicated the altruism level. For kin- and similarity-targeting mechanisms, we found that evolution was able to stably adjust altruism levels. Our results confirm that natural selection favours altruist genes that are increasingly accurate in targeting altruism to only their copies. Our work also emphasizes that the concept of targeting accuracy must include both the presence of an altruist gene and the level of altruism it produces.

Using the Avida software platform the researchers ran trials of the evolution of populations of artificial life which varied in fitness, coefficient of relatedness, as well as their phenotypes. In one set of trials the organisms operated through conventional means of kin selection, whereby the heuristic was to favor those to whom an individual was closely related. This will result in a fair amount of “false positives,” as everyone knows that near kin can be selfish and “cheat.” Remember that in the toy example above 50% of the offspring who will gain from altruism will themselves lack the altruism gene. A second set of organisms look to total genetic similarity. This is the sort of thing which humans could engage in if they had immediate knowledge of the genomic sequences of those around them. Even among near relatives genetic similarity is only correlated with, not perfectly correspondent with, coefficients of relatedness. Some full siblings may share more identity-by-descent than others. This is trivially obvious in the initial illustration, as there will be a great deal of intra-familial variance on the gene which produces altruism. To focus on the dynamics of the specific gene, the authors also looked at a green-beard effect, whereby a there is a correlation between altruism, a gene, and a visible phenotype. In other words, you know altruists by a correlated physical trait. If the correlation between a phenotype and a genotype is close enough you don’t need do a typing of their genome because you know the state of their genotype, and so have expectations as to whether they’re truly altruists or not. Presumably using the green-beard effect one could side-step the usage of kinship or relatedness as a proxy. In many cases those more distantly related could be more phenotypically similar on the traits of interest than those who are genetically closer.

What did they find? Figure 1 shows the outcomes of various sets of trials:


Their expectations were that in regards to the evolution of altruism kin selection should be inferior to genetic similarity which should be inferior to the green-beard effect. The reasoning is straightforward, as you progress across these sequence of dynamics the false positive rate of aiding those without the altruism conferring gene should decrease. That is not what they found, at least not initially.

What was happening is that they were focusing on the wrong parameters in framing their expectations. That’s why you run the model: human intuition often fails. Green-bearding is very precise as a dichotomous indicator of whether an individual carries a particular gene identical-by-descent, but mutation could produce variation in levels of altruism. What they found was that when green-bearding was dichotomous the levels of altruism tended to converge upon a lower equilibrium as individuals were focused on being just altruistic enough to count as real altruists and so gain advantages from those who were more generous. A concrete example of this would be an “affinity con”. An individual is a member of a group, and they leverage the trust which comes from being a member of the group to exploit the group. Baked into the cake of the original model is that altruists who also had a green-beard had to have donated at least once, and that is the target which green-beards converged upon. In contrast the strategy of genetic similarity resulted in greater donations, and because the model had non-zero sum dynamics (altruism increased everyone’s fitness greatly, though cheaters could exploit this to “free-ride”) the strategy which maximized donations was more successful. The researchers made green-bearding more competitive by simply increasing the donation threshold to match the equilibrium which emerged with the other strategies. So making all things equal the intuition about green-bearding was then vindicated.

Instead of setting a specific threshold there was another way that green-bearding could beat the other strategies to maximize inclusive fitness: vary the green-bearding trait and altruism continuously in a correlated fashion. In other words, the greener the beard, the more altruistic. This is a classic way that one could beat the cheaters: develop detection and discernment mechanisms. Why doesn’t this matter for the two other more “primitive” techniques? Kin selection and genetic similarity are more robust because they’re not fine-tuned, organisms with similar genome content are likely to have similar altruism levels. The genetic relatedness of altruists in green-bearding populations is going to be lower because they’re looking for a very specific genotype and its correlation with a phenotype. Green-bearding is more precise, but it’s also somewhat more complicated, and as a more precisely engineered solution it may not always be as robust.

And that necessity of fine-tuned intelligence in design may be why green-bearding is not more common. The authors note that in theory one could imagine mutations leading to concomitant variations of the magnitude of green-bearding and altruism in the same direction, but in a real evolutionary genetic context with normal parameters of mutation and effective population sizes this may not be plausible. Many people would argue that evolution is littered with kludges because natural selection makes recourse to “quick and dirty” solutions which are simple but effective, and kin selection and genetic similarity are closer to that than green-bearding. In theory selection may lead to a world of green-beards with infinite population sizes and generations, and persistent and consistent selection, but the world may be too protean for this optimal equilibrium to ever arise. So until then, we’ll make do with social evolution’s duct-tape: “I against my brother; my brother and I against my cousin; I, my brother, and my cousin against the stranger.”

Citation: Clune J, Goldsby HJ, Ofria C, & Pennock RT (2010). Selective pressures for accurate altruism targeting: evidence from digital evolution for difficult-to-test aspects of inclusive fitness theory. Proceedings. Biological sciences / The Royal Society PMID: 20843843

Image Credit: Burningrey

August 25, 2010

E. O. Wilson against Hamiltonian inclusive fitness

There is a new paper in Nature which is a full frontal attack on the utility of William D. Hamilton’s inclusive fitness framework in explaining eusociality. Martin A. Nowak, Corina E. Tarnita, & Edward O. Wilson are the authors. Wilson is famous in large part for his authorship of Sociobiology: The New Synthesis, and is arguably the doyen of American organismic biology. He is both an active scientist, and, a premier public intellectual. So with that in mind, I notice that Dienekes Pontikos alludes to “E.O. Wilson’s change of mind about group selection.” This is conventional wisdom, but it is I think wrong (though from what I can tell Wilson has not done much to disabuse the press of the notion). In Defenders of the Truth Ullica Segerstrale notes that Wilson did not expunge group selection thinking even in Sociobiology. In Evolution for Everyone David Sloan Wilson recounts that it was in fact E. O. Wilson who pointed out a group selective interpretation of data he was presenting at a conference, helping to push him early on in a rather unfashionable direction. From what I have heard Wilson always believed that the empirical data was not adequately explained by a pure inclusive fitness model, and simply waited until things shook out before pushing back with more theoretically trained colleagues who had the same skepticism.

From page 30 of Sociobiology:

…….Nevertheless, Williams’ distaste for group-selection hypotheses wrongly lead him to urge the loading of the dice in favor of individual selection. As we shall see in chapter 5, group selection and higher levels of organization, however intuitively improbable they may seem, are at least theoretically possible under a wide range of conditions. The goal of the investigation should not be to advocate the simplest explanation, but rather to enumerate all of the possible explanations, improbable as well as likely, and then to devise tests to eliminate some of them.

And page 129, the last paragraph in the chapter on group selection (quoted in full so there’ll be no confusions as to whether I’m pulling it out of context):

In conclusion, although the theory of group selection is still rudimentary, it has already providd insights into some of the least understood and most disturbing qualities of social behavior. Like Arjuna faltering on the Field of Righteousness, the individual is forcd to make imperfect choices based on irreconcilable loyalties-between the “rights” and “duties” of self and those of family, tribe, and other units of selection, each of which evolves its own code of honor. No wonder the human spirit is in constant turmoil. Arjuna agonized, “Restless is the mind, O Krishna, turbulent, forceful, and stubbon. I think it is no more aesily to be controlled than is the wind.” And Krishna replied, “For one who is uncontrolled, I agree the Rule is hard to attain, but by the obedient spirits who will strive for it, it may be won by following the proper way.” In the opening chapter of this book, I suggested that the science of sociobiology, if coupled with neurophysiology, might transform the insights of ancient religions into a precise account of the evolutionary origin of ethics and hence explain the reasons why we make certain moral choices instead of others at particlar times. Whether such understanding will then produce the Rule remains to be seen. For the moment, perhaps it is enough to establish that a single strong thread does indeed run from the conduct of terminte colonies and turkey brotherhoods to the social behavior of man.

July 13, 2010

The Price of Altruism

Sometimes in a narrative you have secondary characters who you want to revisit. What do to do after the story is complete? An convenient “work-around” to this problem is to find the story rewritten from the perspective of the secondary character. In broad strokes the picture is unchanged, but in the finer grained shadings different details come into sharper relief. Though the exterior action may be unaltered, it gains different context, and the interior motive may radically alter, as the nature of subjective perspective matters so greatly in the last instance. In many ways Oren Harman’s The Price of Altruism reads to me like a narrative rewritten from the perspective of a character who was a supporting protagonist in other stories. George Price, almost a novelty act elsewhere, now becomes the primary point of view character.

I could almost say that Harman, a historian of science, has given us a novel from a “shared universe” of stories. That universe is the real world. The other stories are the lives of great scientists, and the plot consists of the working out of their ideas. In the acknowledgments Harman alludes to the wide range of works where fragments of George Price’s life filters through. I have read many of the mentioned works, The Darwin Wars, Defenders of the Truth, and Narrow Roads of Gene Land. In all of these George Price cuts a quixotic figure, mercurial, brilliant and exceedingly eccentric. His plain biography already peculiar. Price began his career as a chemist, shifted to journalism and became what we today would term a professional “skeptic,” then entered into a period of productivity as an evolutionary theorist of some major impact, and finally spent his last years attempting to live the life of a serious Christian who followed God’s commands to the best of his abilities. He died tragically, committing suicide in his early 50s in 1975, homeless, destitute, and serious ill.

Much of what I already know comes through the memories of William Hamilton in his collections of papers, titled Narrow Roads of Gene Land. In Narrow Roads of Gene Land Hamilton admits that he did not perceive in totality the implications of Price’s eponymous equation when he first encountered it (in particular, he did not initially comprehend that the two elements within the Price equation allowed for the possibility of group selection as you move up the nested hierarchies of organization and reassign the elements to ascending levels). In The Price of Altruism Oren Harman reiterates this reality, but, importantly he emphasizes that Price felt that it was Hamilton alone in all the world who had perceived the equation’s nature upon first encountering it. The back story, which is told in Narrow Roads of Gene Land, is that George Price had difficulty in getting his papers in this area published because the referees simply did not see the implications. Hamilton, perceiving the importance of Price’s ideas, connived to gain publication by making his own work conditional on the acceptance of Price’s paper (which he cited). As Hamilton already had a reputation the game worked.

The necessity of these strategies makes more sense in light of Price’s unconventional background and affect.  In evolutionary biology Price was self-taught, and he entered the field in large part because he was interested in the topic, and perceived that he was going to make some difference in the world. He arrived in London in the late 1960s, impressed people at the Galton Laboratory and managed to obtain a research grant and desk, and became an important stimulator of and collaborator with both William Hamilton and John Maynard Smith, arguably Britain’s two most prominent theoretical evolutionary biologists at the time. Price’s relationship to John Maynard Smith is referenced in Hamilton’s own biography, as well as third person narratives such as The Darwin Wars and Defenders of the Truth, but The Price of Altruism fleshes out many of the details. While Price extended Hamilton’s original work on inclusive fitness, for Maynard Smith he served more as a prod and collaborator as they explored the intersection of game theory and biology which eventually led to the ideas outlined in Evolution and Theory of Games. The “hawk” and “dove” morphs made famous by Richard Dawkins in The Selfish Gene go back to Maynard Smith’s work, but the terms themselves were of Price’s invention according to Harman. If I read Harman’s chronology correctly Price was already a fervent Christian by this time, having left atheism in the same period as he launched his career as an evolutionary biologist, and there is some hint that the term “dove” may have been influenced by his particular religious leanings. This possibility seems all the more amusing in light of Dawkins’ later career as an atheist polemicist. Price’s last contribution to evolutionary biology was an explication of Fisher’s fundamental theorem of natural selection. This formalism has been the subject of so much deep analysis, such that I think Price’s interest in it prefigured his later stab at Biblical textual analysis!

The Price of Altruism is a biography of a scientist, so naturally there’s a great deal of science. The meat and heart of the work is George Price’s life trajectory, with all its travails (many) and triumphs (few, but lasting and of importance). Yet the story begins with an exploration of the lives and opinions of men who seem of a different age, Thomas Huxley and Peter Kropotkin. Huxley and Kropotkin were archetypes, who anticipated two streams of evolutionary ecology and social theory which battled it out through the 20th century. Huxley was a man who saw nature as “red in tooth and claw,” the working out of amoral competitive forces, and human virtue as having emerged out and above nature, just as he had risen up from his working class origins to eminence. Kropotkin reflected a Russian viewpoint which saw cooperation as the norm, and competition as the deviation. For him virtue emerged from our natural tendencies. Lee Alan Dugatkin covers much of the same ground in The Altruism Equation. Great men who you meet elsewhere inevitably make cameo appearances in Harman’s narrative; R. A. Fisher, the brilliant cipher, J. B. S. Haldane, the hereditarian Marxist, and Sewall Wright, the American (also see The Origins of Theoretical Population Genetics). The bright lights of Price’s generation also make prominent appearances; William Hamilton and John Maynard Smith, their characters manifesting no great surprises, but also the schizophrenic genius Robert Trivers, with whom Price perhaps shares a great deal excepting his dark ending, as well as E. O. Wilson.

All of these individuals have an interest in evolutionary biology, but biology of a behavioral sort. Though molecular evolutionists such as Richard Lewontin and Motoo Kimura are references in The Price of Altruism, they’re ancillary to the thrust of the book’s central idea (though Lewontin seems to serve as a type, the brilliant scientist who saw the import of Price’s equation too late to engage in a productive exchange with George Price himself). Evolution, like theoretical physics, spans may domains of subject, from the aggregations of millions of individual life forms, to evolution of elements within individual genomes! The Price equation’s generality is such that it does speak to the phenomena which bubble just above the level of organizations of the substrate, DNA itself. But George Price’s focus was on higher, not lower, levels of organizations, human societies. Oren Harman makes this clear, for he brings to light Price’s correspondence with Paul Samuelson, one of the greatest economists of the 20th century. Before Price left for London and began his collaboration with Hamilton and Maynard Smith on altruism, he fancied reconstructing the basis of 20th century economics. By the end of his life Price suggested that he was going to go back to this initial impulse, and attempted to renew his correspondence with Samuelson in the hopes of obtaining a research fellowship of some sort. Price also engaged with the behavioral psychologist B. F. Skinner, though as with many of his encounters it seems that the two soured on each other, in part due to Price’s impolitic tendencies.

George Price’s aim was to explain human cooperation, altruism. In short, goodness. This is the domain of angels, but his analytical bent mean that he could not let the phenomenon lay. He had to break it down, reconstruct its fundamentals, and elaborate on how and why goodness, altruism, manifested itself in the world. From the details reported in The Price of Altruism I would have to admit that Price himself was a Janus-like figure, often being in a manifestly selfish fashion, abandoning his family to follow his intellectual bliss, and yet also radically altruistic, allowing himself to be exploited by the dregs of the London underclass near the end of his life because scripture told him so (or his reading of scripture). What I had previous read did not emphasize Price’s selfishness, his need to satisfy his own wants, and place his own elective priorities ahead of the mandatory ones which decency bound him to honor (e.g., supporting his wife and daughters). Harman has a rich catalog of George Price’s selfish actions and the small vendettas which wracked his soul. No saint was he. Much of what Harmon recounts was simply not evident from other sources. Perhaps in Hamilton’s case he wished to highlight the positive aspects of a good friend who had died tragically. More plausibly I suspect that Hamilton was simply not aware of the selfish sequence of acts which led George Price to the Galton Laboratory in the late 1960s. And it was during this period that George Price became a zealous Christian and a radical altruist. Hamilton’s perceptions may simply have been colored by the slice of Price’s life to which he was privy.

Oren Harman wonders at the end of the book if George Price may have been rather far along the asperger’s spectrum. If so, combined with his fierce intelligence, one is not surprised that Price exhibited a fixation on why and how humans behaved, and why and how it came to be that humans did not seem to be rational psychopaths. Though I do not know if, and honestly do not believe, that George Price was a rational psychopath, in The Price of Altruism Oren Harman paints a picture of a man with immediate urges and impulses, earthy hedonic priorities, and a strong tendency to discount the costs which his choices may have for those close to him. George Price was not the first man to not be a good father, but he was one who perhaps wondered why there were good fathers and bad fathers, those who followed their bliss despite the consequences to their progeny, and those who sacrificed so that their children could enjoy the comforts and pleasures which they elected to forgo. The science is well elucidated in works such as Unto Others, The Origins of Virtue and The Evolution of Cooperation. The Price of Altruism is rather a case study not of the theory of altruism, but of the concrete embodied human experience which eventually gave fruit to an important slice of the theory of altruism. From the small details of his day-t0-day actions, to the arc of his life, George Price played out some of the implications of his own intellectual edifice, both through contradiction and confirmation.

Recommended Reading: The Darwin Wars, The Evolutionists, A Reason for Everything, Narrow Roads of Gene Land, Natural Selection and Social Theory, The Origins of Theoretical Population Genetics, Sewall Wright and Evolutionary Biology, R.A. Fisher: The Life of a Scientist, Defenders of the Truth, Unto Others and The Selfish Gene.

July 1, 2010

Hamilton’s Rule vs. Increasing returns to cooperation

If you have even a marginal interest in evolutionary biology you will probably have heard of Hamilton’s Rule, a simple formal representation of the logic whereby a gene which favors altruism may spread through a population: rB > C, where r = coefficient of relatedness on the gene in question, B = benefit to those related, and C = cost to oneself. The idea is almost trivially obvious. Consider that you are in a situation where you are faced with the possibility of aiding your full sibling at a cost to yourself. Now imagine that you carry a single allele which favors altruism toward close relations. Your sibling has a 50% probability of carrying that allele identical by descent (let’s stay haploid for simplicity). From a “gene’s eye view” it benefits the allele to predispose you to helping your kin in direct proportion to the probability that your kin carry that allele. In other words the logic underlying inclusive fitness isn’t really that abstract, it is ordered around the benefits and costs to the theoretical genes which manipulate social behavior over the long term. This explains why the evolutionary biologist J. B. S. Haldane responded “…I would to save two brothers or eight cousins,” when asked if he would save his brother from drowning. The genetically relatedness to a sibling is 1/2, to a cousin 1/8. 2 X 1/2 = 1 and 8 X 1/8 = 1, basically equivalent to yourself. Evolutionary altruism is obviously somewhat different from common sense altruism, because you’re averaging out the behavior of many individuals over a time window.

The fascinating back story behind the development of this sort of formal thinking is recounted in W. D. Hamilton’s first collection of papers, Narrow Roads of Gene Land: Evolution of Social Behaviour. An elaboration upon the core logic of Hamilton’s Rule in two seminal papers revolutionized our understanding of the evolution of sociality in the 1960s; Hamilton was proud of how widely cited his original papers were. John Maynard Smith’s evolutionary game theory and Robert Trivers reciprocal altruism emerged out of the same ferment (Trivers’ acknowledges the debt to Hamilton in Natural Selection and Social Theory). More recently E. O. Wilson and David Sloan Wilson have been arguing for a rehabilitation of more complex models of the origins of sociality through multilevel selection theory.

But what about Hamilton’s original ideas, the core elements of inclusive fitness? Their spareness rendered them analytically tractable, but like all models the original formalism made some simplifying assumptions. Relatively weak selection pressures, as well as additivity of fitness effects, were two major axioms, and ones which Hamilton defended in Narrow Roads of Gene Land. A new paper in Science argues that the assumptions rendered the model too simple to be of more than qualitative or heuristic utility in most cases. They modify the Hamiltonian framework by including nonlinear fitness distributions as well as stronger selection coefficients in the context of microbes. A Generalization of Hamilton’s Rule for the Evolution of Microbial Cooperation:

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.

The bottom line here is that the authors are indicating that a simple framework with the parameters of Hamilton’s original formalism can not explain the various forms of altruism found among microbes, even ubiquitous ones such as biofilms. One should not be surprised, as the problem of altruism was not solved by inclusive fitness in its details, though many use it in a hand-waving manner, i.e., “…everyone knows….” To correct this impasse the authors modify Hamilton’s Rule:


Some of the parameters are now bold. That means they’re vectors, not scalars. Basically lists of variables. First in the list for r is the original coefficient of relatedness, with subsequent elements representing higher orders of relatedness. b represents the benefits to noncooperating morphs as a function of social environment, the frequencies of cooperators and noncooperators. The cost to the focal individual remains the same. Finally, m are the moments for the cooperators (measuring distributions of fitness in terms of their shape) and d represents the difference between cooperators and noncooperators of the distribution. When fitness effects are totally additive, that is there are no nonlinearities and conditionalities of genotype fitness on environment, the second part of the equation falls away, and r and b reduced to their first elements, so you have a classical form of Hamilton’s Rule.

Figure 1 illustrates the aspects differentiating a classical vs. modified Hamiltonian model:


Basically the simplifying assumptions in Hamilton’s original model is illustrated by panel A. The authors claim that the assumptions allow for no quantitative prediction of real structured altruism which we see. Figure 2 has some experimental data:


Here’s the text:

Parameters of the generalized Hamilton’s rule measured in an experimental population of sporulating Myxococcus bacteria. (A) Absolute fitness of a cooperator strain (blue circles) and a cheater strain (red diamonds) as a function of their frequency within groups. Data points are independent experimental replicates; lines, regression model fit to data. (B) Fitness terms in Eq. (1), calculated from the data shown in (A). Green diamonds, benefit vector b; purple circles, genotype-dependence vector d. Points show best-fit model (±SD from bootstrapped data). (C) Initial distribution of cooperators among groups for a specific experimental population. (D) Social structure terms in Eq. (1) were calculated for the population shown in (C). Blue, cooperator moments m; red, noncooperator moments mnon; black, relatedness vector r.

As you can see in panel A there’s frequency dependence going on here. Cooperators run up against a wall, but the frequencies at which they’re fitter than the noncooperators is rather high. Panel B is important because it shows that the benefits really accrue at the higher moments, now the lower additive one. This means that higher level population structure and nonlinearities when viewed on an individual scale are very important. Figure 3 illustrates the nature of frequency dependence, and the conditions where cooperators flourish and cheaters can persist:


Since higher order structure is critical parameters such as migration between groups are important to keep track of us. More experiments obviously need to be done here, I’m not convinced that one model can explain-it-all. But, there are obvious limitations to the classical Hamiltonian framework in many situations. One of the major points in this paper which jumped out at me was the following: “…increasing-returns nonadditivity allows cooperation to evolve at levels of population structure comparable to that seen among social insect colonies.” Increasing returns is a concept which is important in economics in understanding how technological innovation has allowed for productivity gains over the past two centuries. Human social systems are complex, almost baroque to a fault, and their byzantine structure can easily be dismissed as random acts of contingency. But increasing returns to cooperation may explain the ubiquity of more complex orders than we would expect. And yet here we see it on the scale of bacteria! The logic of non-zero sum is deeply rooted in the nature of life, but the next stage is to flesh out how it produces such rich behavioral phenomena. Endless behaviors most ornate!

Smith J, Van Dyken JD, & Zee PC (2010). A generalization of Hamilton’s rule for the evolution of microbial cooperation. Science (New York, N.Y.), 328 (5986), 1700-3 PMID: 20576891

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