In this month’s Insights puzzle, we invited you to rediscover a foundational principle of population genetics and to examine the validity of Richard Dawkins’s famous metaphor of the “selfish gene.” Kudos for successfully tackling both challenges!
As one reader, Lee Altenberg, pointed out, our first two problems illustrated the Hardy-Weinberg principle (also called the Hardy-Weinberg equilibrium). The principle was discovered in 1908 by the German physician Wilhelm Weinberg and, independently, by the English mathematician G.H. Hardy, who in another context mentored the legendary number theorist Srinivasa Ramanujan. The Hardy-Weinberg principle shows how, under idealized conditions of random mating and no selection pressure, two genes competing for the same spot in the genome (called “alleles”) can maintain their initial frequencies indefinitely, despite the dominance of one allele over the other. Here’s how it works mathematically.
Problems 1 and 2
1. Imagine a population of individuals with genotypes AA, Aa and aa in the proportions 0.5, 0.2 and 0.3, respectively. Imagine that there are very many individuals who mate with each other at random, so that every possible mating between genotypes takes place with the appropriate frequencies. Thus two AAs will mate with each other with a frequency 0.5 times 0.5 to produce 0.25 of the next generation — all of whom will also be AAs. What will the proportion of the three genotypes in the first generation be? In the second generation?
Figure out the proportion of individual genotypes in the first and second generations, as before assuming random mating in a large population, when the initial proportions of AA, Aa and aa are as follows:
- 0.3, 0.6, 0.1
- 0.2, 0.8, 0
- 0.7, 0.2, 0.1
- Any arbitrary proportion you can think of (the three numbers must add up to exactly 1)
2. In the above examples, notice what happens to the ratios in the first and second generations. Can you figure out why this happens? How do the final ratios depend on the original ratios?
The following table shows the frequencies of the nine possible types of matings that can take place in the first example given, with color-coded frequencies of the offspring produced.
AA 0.05, Aa 0.05
AA 0.05, Aa 0.05
AA 0.01, Aa 0.02, aa 0.01
Aa 0.03, aa 0.03
Aa 0.03, aa 0.03
Adding the frequencies of the genotypes in the first generation gives:
AA = 0.25 + 0.05 + 0.05 + 0.01 = 0.36
Aa = 0.05 + 0.15 + 0.05 + 0.02 + 0.03 + 0.15 + 0.03 = 0.48
aa = 0.01 + 0.03 + 0.03 + 0.09 = 0.16
Repeating this process with the new genotype frequencies reveals a surprise:
AA = 0.36, Aa = 0.48, aa = 0.16. The genotype frequencies remain unchanged between the first and second generations. That is, they stabilize in a single generation, as Douglas Felix pointed out. The same thing happens in the other examples:
Generation 0: (0.3, 0.6, 0.1)
Generation 1: (0.36, 0.48, 0.16)
Generation 2: (0.36, 0.48, 0.16)
Generation 0: (0.2, 0.8, 0)
Generation 1: (0.36, 0.48, 0.16)
Generation 2: (0.36, 0.48, 0.16)
Generation 0: (0.7, 0.2, 0.1)
Generation 1: (0.64, 0.32, 0.04)
Generation 2: (0.64, 0.32, 0.04)
As Felix further noted, the dominance of the gene A does not affect the new ratios of the genotypes. Whether he knew it or not, Felix effectively rediscovered the Hardy-Weinberg equilibrium!
This biologically significant result can be shown by applying elementary algebra. However, following nightrider, here’s an intuitive way to understand the result, which simplifies the algebra even further. What matters is not the initial genotype ratios, but the initial allele frequencies for the genes A and a, which we can call p and q where p + q = 1. Since mating is assumed to be random, the chance of an A encountering another A to produce the phenotype AA in the next generation will be p2; that of Aa will be 2pq, and that of aa will be q2. What does this do to the allele frequencies? AA genotypes will always provide A alleles while Aa genotypes will give A half of the time, which means that the new frequency of the allele A will be p2 + ½(2pq) = p2 + pq = p(p+q) = p; similarly the aa frequency will be ½(2pq) + q2 = q(p+q) = q. So the new allelic frequencies will remain p and q in the next generation, and will keep the phenotype ratios stable indefinitely, as long as one allele is not favored by evolution above its initial level.
Let us now turn to the quandary that gave this column its title: Can genes be reasonably described, metaphorically, as selfish, as Richard Dawkins suggested? Some readers essentially repeated my words about genes’ being insentient, others castigated me for pushing a metaphor too far, and still others even accused me of misunderstanding Dawkins’s idea. I would like to gently remind you that the metaphor that has now become a universally popular meme was Dawkins’s, not mine. Of course, there is nothing wrong in describing an inanimate object with an adjective that is usually applied only to humans if it is used to clarify or emphasize a characteristic, for example calling a sculpture “graceful” or a sound “insistent.” For a word to be applied this way, the underlying metaphor must have two characteristics: It must be accurate and must not lend itself to being misunderstood. Dawkins’s metaphor fails on both counts. First, it does not emphasize the object’s most distinctive humanlike features. As I’ve described, the human beings that genes resemble most in their activity are players on engineering and sports teams, fighting for places based on their skill or abilities, and working together to reach unimaginably complex goals. Human beings who do these things are not called selfish: They are in fact praised as paragons of team play. Hence Dawkins’s metaphor is clearly wrong. Dawkins applies the epithet “selfish” when he means “competitive,” which is how we describe players who vie for the same spot. Even this idea of rivalry applies, for any given gene, to only a very, very small fraction of other genes — just those that are its alleles. And this brings us to Problem 3.
Can a gene act in a metaphorically selfish way and unfairly undercut one of its competitive alleles? Do readers think the behavior of any genes can be characterized as selfish?
I enjoyed reading Lee Altenberg’s answer to this question. First Altenberg emphasizes that there is no intentionality here by saying, “I think we have to say ‘metaphorically unfairly’ as well.” I agree completely. Next he defines “unfair” as the use of any factor other than intrinsic differences in viability. Again, this is absolutely correct — it can be the only possible way to characterize selfishness rather than mere rivalry. Then Altenberg homes in on meiotic drive or segregation disruptors, which allow an allele to appear in more than 50 percent of the germ cells, thus depriving its competitor allele of its normal or “rightful” share. This is truly the kind of behavior that we would characterize in humans as “selfish.” These kinds of phenomena are, however, rare. Even rarer and more speculative are some of the other interesting phenomena that Altenberg proposed.
But apart from their rarity, there is another factor to be considered. As I said before, evolution is ruthlessly, incorruptibly meritocratic. If such phenomena exist, they do so because they are either evolutionarily neutral, or perhaps possess other advantages for the short-term or long-term survival or procreation for the species that we do not know. This is the only thing that evolution is “interested” in. If that were not the case, the selfish-appearing behavior would have been ruthlessly weeded out. So perhaps the right thing to say about genes is this: They are mostly amazingly cooperative in producing animals. And even when they have a tendency to stray and be “selfish,” natural selection, the ultra-efficient police officer, quickly gets them to toe the line. Natural selection just does not let genes be selfish to the detriment of the organism as a whole or to other useful genes.
Douglas Felix suggested that genes are not selfish toward other genes, but toward their hosts. We would like to live longer, for example, but the complaint is that our genes won’t let us. But this is in no way the genes’ doing: It is again natural selection that has fixed our life spans, for complex reasons based on its aeons-long calculus, again based on the narrow things it is interested in. You can be sure that if a mutant gene arises tomorrow that increased our life spans and thereby tangibly increased our evolutionary success, it would be an instant hit. And remember that it is natural selection itself that has imbued us with the desire to “live long and prosper” — tendencies that are, for the most part, really very important to it and, therefore, to us.
The worst thing about the metaphor of the selfish gene is that it has misled and continues to mislead millions of lay people and even some scientists, leading them to conflate the genes’ putative selfishness with the selfishness of the animals they create. Dawkins himself was guilty of this conflation when he wrote in his book, “to build a society in which individuals cooperate generously and unselfishly towards a common good, you can expect little help from biological nature. Let us try to teach generosity and altruism, because we are born selfish.” This is nonsense, and Dawkins has since retreated from it. Genes are very capable of producing highly altruistic species, as they have done in ants and bees. And the reserves of altruism we find in ourselves, despite our selfishness, can be deep.
With several insightful contributions from readers like Nightrider and Lee Altenberg, it was hard to pick the Quanta T-shirt winner this month. I’ve decided to award the T-shirt to Douglas Felix, who worked diligently through the problems and exhibited genuine joy and amazement in rediscovering the insights of the Hardy-Weinberg equilibrium.
Thank you to all who contributed. See you soon for new insights.