Introduction
G. H. Hardy was a very successful mathematician of the early 20th century. Even today he is highly esteemed among professional mathematicians. He made seminal contributions to a variety of subdisciplines, including number theory and analysis, and he was among the most important British mathematicians of his day. His immortalization in the title of the founding theorem of population genetics therefore seems quite appropriate. But as is well-known among experts, it’s actually quite ironic.
To start with, Hardy’s contribution to the Hardy-Weinberg theorem was nothing more than a trivial mathematical observation expressed in a letter to the editor in Science. Two short paragraphs in the 1-page letter sufficed to convey the substance of the argument. The remainder is devoted to a contrived example and an expression of astonishment that “biologists” were not aware of something so straightforward. In fact, some biologists already knew of the result. In Germany, Wilhelm Weinberg [3] made the same observation at about the same time, but even he wasn’t the first. Karl Pearson was. But Hardy was led to the belief that biologists were in the dark by an uncharacteristically misbegotten notion published by G. Udny Yule [1], who expressed a mistaken intuition, common among neophytes, that any dominant trait would eventually come to represent 3/4 of the population if the trait was not under selection. Hardy’s motivation was to show that Yule was wrong, which required nothing more sophisticated than grade-school algebra. So the brilliant mathematician forever became linked by name to a mathematically trivial result.
And mathematically trivial it is. But one should not confuse mathematical simplicity with biological triviality. The Hardy-Weinberg theorem quite rightly serves as the foundation for population genetics, and among other things provides us a tool to identify traits that are evolving in real populations. So it is, I think, important to understand precisely what Hardy contributed to the theory.
Hardy’s Result
In 1908, one approached problems in population genetics–the seeds of which had just been sown–often from different perspectives than the modern one. In this case, Hardy focused on absolute numbers of individuals of each genotype and neither on their frequencies (despite subsequent misinterpretations) nor the allele frequencies, as we tend to do now. So, first we’ll look at his result using his notation and argument, and then demonstrate how it can be obtained more directly with modern ideas.
Hardy defined the numbers of individuals with genotypes AA, Aa and aa as p, 2q and r, respectively. Keep in mind that p and q do not mean what we typically teach in modern textbooks. These are numbers of individuals, not proportions. So, the total number of individuals in the population is p + 2q +r. If mating is random, the population is really large (or we ignore statistical fluctuations), sex ratios for each genotype are all unity, and there’s no selection, then the number of matings among each genotype can be obtained by expanding
(1) (p + 2q + r)2,
and the number of individuals of each genotype in the next generation can be obtained by segregating their expected proportions. If we work it all out (which Hardy left the reader to do), we obtain the results in the following Table:
Now we just use Table 1 to partition up the mating frequencies to determine the number of each genotype in the next generation. For example, let pn be the number of AA individuals in generation n, and we want pn+1. We know that all offspring from AAxAA matings will be AA, while only half of the offspring from AAxAa matings and 1/4 of those from AaxAa will be AA. So,
pn+1 = pn2 + (1/2) (4pn qn) + (1/4) (4qn2) = (pn + qn)2.
This is the brute force approach. A more elegant way is to rewrite expression (1) as
(2) ((p+q) + (q+r))2.
If we square the first term, then we get the number of AA offpspring. Squaring the second term gives the number of aa offspring. The number of heterozygotes is the sum of the two cross-terms. Either way you look at it, you get the following:
Hardy did none of this, nor did he quite use this notation. He assumed the reader grasped it without his help.
Our goal is to find a fixed point of system (3). Let that fixed point be (p, q, r). If Yule were right, then p + 2 q = 3r (there would be three times more dominant phenotypes than recessives at equilibrium). But, notice that a fixed point, by definition and equations (3a) and (3c), satisfies
Substituting these results into equation (3b) tells us that at a fixed point we must also have
This evidently is not Yule’s result. In addition, Hardy also stated without demonstration (because it was easy enough for the reader to figure out, or so he thought) that the numbers of each genotype would not change in successive generations following the first. Then he contrived an example to show in a concrete way how it worked.
The modern way of dealing with this is similar to Hardy’s, although instead of numbers of each genotype, we focus on allele frequencies. So, beware of a change in notation here. Let p now represent the frequency of allele A and q be that of allele a. We assume these are the only alleles for the trait in the population, so p + q = 1. If mating is random, alleles are partitioned identically between the sexes–i.e., p is the same for both males and females–the population is large with no migration or selection, and mutation can be neglected, then the frequencies of all genotypes in the next generation are generated by the binomial square:
This is already a fixed point since, for any p,
and therefore qn+1 = qn. The genotype frequencies may shift in the first round of breeding, as Hardy noted, but the allele frequencies are fixed from the beginning. In the modern theory of dynamical systems, every point in the phase space, p in [0,1], is a neutrally stable fixed point. Where you start is where you stay forever.
One can also easily recover Hardy’s condition, but before showing that let’s disambiguate the notation. Let NAA, NAa and Naa be the number of individuals of each genotype. Hardy’s observation was that “the condition for” system (3) to have a fixed point is
(Remember, in Hardy’s notation, q is half the number of Aa individuals.) In the modern approach, random mating and all the other assumptions are expressed mathematically as the expansion of (p + q)2, so that the genotype frequencies after a single mating must be
In other words, random mating and all the other assumptions are true if and only if these are the genotype frequencies in all rounds of mating after the first. From here it is trivially easy to see that these frequencies hold if and only if
which is precisely Hardy’s condition up to a constant equal to the population size. In summary then, if we are at an equilibrium for the genotypes, then condition (8) holds, and if condition (8) holds, we are at an equilibrium. Therefore, Hardy’s somewhat cryptic, “it is easy to see that the condition for [the distribution being at an equilibrium] is q2 = pr” means that his condition is necessary and sufficient for an equilibrium.
Essay available as a pdf here.
Philosophy of Achievement
RES GESTA PER EXCELLENTIAM
Many paths lead to achievement. People regularly achieve greatness–as measured by wealth, influence and status–by the adroit use of self-promotion, its sibling, marketing, and the handmaiden to both, deceit. Bribery, coercion and other forms of persuasion have also led many to “greatness.” All such paths are controversial, exciting some and dismaying others. But all, when navigated by the proper type of person, are demonstrably effective. In this lab, however, we have no interest in wealth, influence or status. Instead, we focus exclusively on a goal we see as thoroughly laudable and significant—the deepest possible understanding of the universe. For us there exists only one possible route to success, the signpost to which is found on a NASA mission patch, of all places.
The National Aeronautics and Space Administration loves patches. As a NASA scientist once told me, “if you don’t have a patch, you don’t have a mission.” Even those who control missions have their own patch. The Mission Control patch was designed by the legendary flight director, Eugene (Gene) Kranz, and Robert McCall, one of the greatest American artists of the 20th Century (and who contributed much of the artwork adorning the Natural Sciences buildings at SCC). Mission Control is short for Mission Operations Control Room, or MOCR (pronounced “Mow-kur”). But it was much more than just a room. Here, flight control teams worked continuously with the astronauts to monitor the spacecraft and other mission hardware, solved problems and generally protected the lives of the flight crew while focused on success of the mission. Mission Control was, quite literally, an indispensable part of the only successful attempt in Earth’s history to set humans down on another world. And Gene Kranz was indispensable to Mission Control. He has been intimately involved in the flights of all U.S. crewed spacecraft. He was assistant flight director for Mercury, flight director (head of a MOCR team) for all Gemini and Apollo missions, promoted to Deputy Director and then Director of the Mission Operations Directorate during the Space Shuttle years, and finally retired after 34 years at NASA following the successful repair of the Hubble Space Telescope by the STS-61 crew (December 1993). Kranz knows something about achievement.
McCall credits Kranz for the design of the patch, saying he (McCall) did little more than make it easily reproducible. This is probably something of an understatement–those familiar with McCall’s work would recognize his hand even if they knew nothing of his role in making the emblem–but essentially all elements of the patch reflect Kranz’s leadership style. The patch is circular, with a “legacy ring” surrounding a central picture. The picture is anchored by the Earth at the bottom, from which rises a Saturn V rocket boosting an Apollo spacecraft on a jet of flame toward the moon and sun while a satellite orbits Earth below. The “Mission Control” identifier is overlain by the greek letter sigma (∑) representing teamwork (summing up). At the bottom of the “legacy ring” sit icons for Mercury, Gemini and Apollo, all crewed U.S. spacecraft up to that time. The top of the ring holds prominent the Latin phrase, res gesta par excellentiam [sic]–“achieve through excellence.”
The emblem has changed since then. NASA turns to Michael Okuda, a graphic designer largely responsible for look of the fictional technology in Star Trek movies and The Next Generation television shows, whenever it feels the need to update the patch. During these redesigns, the Apollo-Saturn stack has been replaced first by the shuttle and then by a stylized spacecraft that, according to one website, “represent[s] the growing variety of U.S. space vehicles in operation.” (This is, of course, disingenuous–NASA has no crewed spacecraft in operation, and they are not about to put a Russian rocket on a U.S. patch.) The stars were rearranged to represent U.S. astronauts who died in the Apollo 1, STS-51-L (Challenger) and STS-107 (Columbia) missions. “Mission Control” became “Mission Operations” and then “Flight Operations.” The sun was replaced by Mars, and then both Moon and Mars were deemphasized and shown as barely visible crescents. The orbiting satellite was replaced by the International Space Station and then by…something. The legacy ring has added Skylab, the shuttle, the Russian MIR space station and the ISS. But throughout all these alterations one thing never changed: the Latin inscription at the top (except for a slight grammatical correction).
There is a reason for that, extending beyond NASA flight operations. The NASA flight directors during the Apollo era were among the greatest leaders the world has ever produced. I am sure there are those who dismisses this statement as childish hyperbole, but they are wrong. Listen to the flight director’s audio loop during Glynn Lunney’s first shift following the Apollo 13 accident, and then consider the qualities of flight directors who helped achieve a goal so extraordinary that a significant fraction of subsequent generations refuse to believe it was really achieved. But it was. And because they were great leaders, those responsible have taught us how they did it because they realized that the lesson applies to anyone accepting the challenge of significant goals: res gesta per excellentiam.
Excellence is also our path to achievement. It is the only one available to those of us compelled to study Nature honestly. But what does it mean, “achieve through excellence?” This is explained in the “Foundations of Mission Control,” published by Kranz in his book, “Failure is not an option” (Kranz 2000). Here are the foundations, slightly edited to fit our purpose:
To instill within ourselves these qualities essential for professional excellence:
Discipline. Being able to follow as well as lead, knowing that we must master ourselves before we can master our task.
Competence. There being no substitute for total preparation and complete dedication, for [Nature] will not tolerate the careless or indifferent.
Confidence. Believing in ourselves as well as others, knowing that we must master fear and hesitation before we can succeed.
Responsibility. Realizing that it cannot be shifted to others, for it belongs to each of us; we must answer for what we do, or fail to do.
Toughness. Taking a stand when we must; to try again, and again, even if it means following a more difficult path.
Teamwork. Respecting and utilizing the ability of others, realizing that we work toward a common goal, for success depends on the efforts of all.
…To recognize that the greatest error is not to have tried and failed, but that in trying, we did not give it our best effort.
This is what is meant by res gesta per excellentiam. This is our path, and we will do our utmost to remain on it at all times.