Molecular Evolution Activities

Neutral Theory of Molecular Evolution

Interview with Warren Ewens

The following is an edited transcript of an intervew between Warren Ewens and Anya Plutinski. The interview was conducted on December 1, 2004 in Philadelphia, Pennsylvania.

Interview Table of Contents:

Personal Background

AP: When did you first develop an interest in genetics? Who were your teachers? What texts did you read as a graduate student?

WE: I majoring in mathematical statistics at the University of Melbourne as an undergraduate. After that, I moved (in 1961) to the Australian National University to do a Ph.D. There were various persons there with whom I could do this, and I decided that I would work with Pat Moran, who at that time had developed an interest in the statistical aspects of genetics. This decision was made despite the fact that I knew essentially nothing about genetics. It was not a random choice but a reasoned one, since I was certain that genetics would be an increasingly important area of science. That view has been borne out by subsequent events, and I’m very happy that I did choose to work in genetics.

Concerning teachers, Moran was my only teacher I had when I was doing my PhD. In Australia at that time one did not do coursework as part of a PhD, but started immediately on research. Thus one’s “teacher” was one’s supervisor, with no other teacher involved in any formal way. However I did receive a lot of help from Joe Gani, who was in Moran’s department, and who had a very broad interest in stochastic processes, including an interest in genetics. In fact my first genetics paper was written with him.

So far as textbooks are concerned, there were very few readily available in Australia in those days, and such books as one could get were expensive, because they all came from overseas. Thus one often did work without the help of any textbook at all. The only textbook in mathematical genetics that I read as a graduate student was C.C. Li’s “First Course in Population Genetics”, which I found to be very useful. The lack of textbooks was mitigated to a large extent by first-class library facilities, with all relevant journals represented and up-to-date. Although I would not call it a textbook in the normal sense, Fisher’s great book “The Genetical Theory of Natural Selection” was of course extremely influential to me in my PhD. work.

AP: I was wondering if I could follow up your comment that you thought genetics was an increasingly interesting and important area. Why did you think so? Were there any social or historical factors influencing that choice?

WE: No, the factors influencing this view were largely scientific. As I said, I was not very knowledgeable about genetics generally, but I was certainly aware, as any educated person should have been, about Watson and Crick and the discovery of the nature of DNA in 1953. It did not take too much imagination to realize that, with this new evidence about the nature of the genetic material, there would need for a rewriting of the Darwinian evolutionary paradigm in terms of the actual molecular genetic material. This was also emphasized to Moran and me by David Catcheside, who was the Professor of Genetics at the Australian National University at the time.

AP: Who were your influences after your PhD?

WE: In 1964, when I completed my PhD, I did a six months post-doc at Stanford, working with Sam Karlin, who influenced me enormously. At that time he was already a famous mathematician, and fortunately for me he had recently become interested in genetics. In fact I was his first post-doc in the genetics area. His colleague Jim McGregor, with whom Karlin had written a series of important papers in stochastic processes, also influenced me greatly. At that time, Karlin and McGregor ran a regular Monday evening seminar series in mathematical genetics, and apart from them and myself, this was attended by Walter Bodmer and Oscar Kempthorne. They were both senior, well-known people in genetics, and in Kempthorne’s case in statistics as well. So this was tough company. I was influenced very much by Kempthorne, who taught me to emphasize the genetics rather than the mathematics, to make such mathematics as one did relevant to a genetic reality, rather than doing the mathematics entirely for its own sake. He was quite scathing about people who did the latter.

While I was at Stanford, Jim Crow came by to give a seminar. I got to know him very quickly, possibly because I said in the question period after his talk that I thought that what he had said was wrong! Despite this, (or perhaps because of it, because he was a great and generous man), he invited me to give some talks at Madison, which I did in June of that year (1964). He influenced me very much as well, then and later. Crow was an extremely senior figure in population genetics, and one could say that he and his close associate Kimura formed the major force in population genetics theory at that time. I have kept in close touch with both Crow and Karlin since those times, with Crow influencing me in the same way that Kempthorne did. That is to say, as a geneticist, he put the genetics first and the mathematics second, an attitude that I found very appealing. From then on, that was the way in which I tried to address mathematical genetics questions.

Genetic Load

AP: The mention of Crow brings up a further question I had. Were many biologists concerned about the problems of genetic load and mutation load at the time you began your graduate career?

WE: Yes. There was an interest in two load concepts. The first was the mutational load, and interest in that concept came from the concern about genetic damage caused by atomic bombs. This was discussed in detail by the great geneticist Muller in 1950. Jim Neal and Jack Schull had gone to Japan shortly after the 1939-45 war to conduct an examination of the mutational effects of the atomic bomb. Their work on this matter was very well known, and so the question of how much genetic damage had been produced by the bomb was uppermost in many people’s minds. That damage became analyzed mathematically as a mutational load.

A second form of the load concept was introduced by the British biologist-mathematician Haldane who claimed, in 1957, that substitutions in a Darwinian evolutionary process could not proceed at more than a certain comparatively slow rate, because if they were to proceed at a faster rate, there would be an excessive “substitutional load.” Since Haldane was so famous, that concept attracted a lot of attention. In particular, Crow and Kimura made various substitutional load calculations around 1960, that is at about that time that I was becoming interested in genetics.
Perhaps the only disagreement I ever had with Crow concerned the substitutional load, because I never thought that the calculations concerning this load, which he and others carried out, were appropriate. From the very start, my own calculations suggested to me that Haldane’s arguments were misguided and indeed erroneous, and that there is no practical upper limit to the rate at which substitutions can occur under Darwinian natural selection.

AP: Can I follow that up? Can you, in layman’s terms, explain why you think that there is no upper limit in the way that Haldane suggested?

WE: I can, but it becomes rather mathematical. Let me approach it this way. Suppose that you consider one gene locus only, at which a superior allele is replacing an inferior allele through natural selection. In broad terms, what this requires is that individuals carrying the superior allele have on average somewhat more offspring than the mean number of offspring per parent, otherwise the frequency of the superior allele would not increase. This introduces a concept of a “one-locus substitutional load,” and a formal numerical value for this load is fairly easily calculated. However, the crux of the problem arises when one considers the many, perhaps hundreds or even thousands, substitution processes that are being carried out at any one time. In his mathematical treatment of this “multi-locus” situation, Kimura, for example, in effect simply multiplied the loads at the various individual substituting loci to arrive at an overall total load. The load so calculated was enormous. This uses a reductionist approach to the load question, and to me, this reductionist approach is not the right way of doing things. Further, the multiplicative assumption is, to me, unjustified. It is the selectively favored individuals, carrying a variety of different genes at different loci, who are reproducing and being required to contribute more offspring than the average. If you consider load arguments from that individual-based, non-reductionist basis, the mathematical edifice which Kimura built up just evaporates, and in my view the very severe load calculations which he obtained by his approach became irrelevant and misleading. The individual-based calculations that I made indicated to me that there is no unbearable substitutional load.

AP: Did you or your teachers have much concern about the effects of the bomb?

WE: Not in the sense that any such concern motivated my own theoretical work. On the other hand, as I said earlier, there was of course a general concern: any rational person would obviously be concerned about possible genetic damage caused by atomic bombs. But in the 1960’s I was too young to understand the relevant mathematical calculations, and I think that I also felt that the data were not sufficiently firm at that stage for definitive conclusions. So, although I had a general “citizen’s” interest about mutational damage, I did not have a professional interest in it in the sense of carrying out calculations concerning it.

Diffusion Processes and Molecular Evolution

AP: What were your main research interests as a graduate student?

WE: As I have said, my background was in statistics, so my research interests were largely in the statistical or mathematical side of genetics, and in particular evolutionary genetics. One of the things I did in my Ph.D. thesis was the following. The stochastic genetical model which was analyzed in some detail at that time is the so-called “Wright-Fisher” model, which is a simple Markov chain model. One of the problems with that model is that many quantities which you would like to calculate cannot be calculated exactly. The mathematics is just too difficult, and to this day nobody has been able to calculate even quite straightforward properties of this model. However, it has been known since the 1920’s that you could approximate the Wright-Fisher model by a so-called diffusion process. A diffusion process in this context is one for which time is taken to be continuous, and the frequency of any allele is also taken to be continuous, that is capable of taking any value in a continuous range of values. Although in reality time is of course continuous, the frequency of any allele must be discrete, that is it can take only a discrete set of values. So one of the questions that was in the air at that time was how accurate calculations made for the diffusion might be for the corresponding unknown values for the Wright-Fisher model. This is a purely mathematical question. This was one of the problems that I took up in my thesis, and I was able to get quite specific answers for it. Broadly speaking, what one found was that these approximations were extremely accurate, even in very small populations, and that a bound could often be placed on the error incurred by making them. Since the diffusion calculations are very simple, and the formulae which you get from them are very transparent and make very clear to you the effects of various parameters, such as mutation rates and selection differentials, one was very happy to have very simple, albeit approximate calculations, rather than exact Wright-Fisher model values that in any event are impossibly difficult to calculate.

AP: Can I follow that up? You mentioned two advantages of the diffusion approximations of the Wright-Fisher model – ease of calculation and results that made clear the effects of various parameters on those results. Were there different assumptions that one needed to be sensitive to in using diffusion models in some applications?

WE: No, I don’t think so. The main difference was in replacing a model that is discrete in time and space by one that is continuous in time and space. This sort of thing is done often in applied work when some calculation using one approach is easy and the corresponding calculation using the other approach is difficult or impossible. Further, it has been argued that diffusion formulae would probably give results closer to reality than the Wright-Fisher formulae in that diffusion processes assume continuous time, which is more appropriate than the discrete time Wright-Fisher assumption.
Another reason that one would be happy to use diffusion results is that the Wright-Fisher model is only a model. Nobody would claim that it represents true reality, because it makes many simplifying assumptions. Therefore, any calculation made by using that model need not necessarily represent a true “real-world” value at all closely.

AP: When did you first hear about techniques for observing genetic variation? What did you expect that work would find? Were you surprised?

WE: The first time I heard about results concerning the amount of genetic variation in natural populations was when I read the well-known papers of Lewontin and Hubby in 1966. In these papers they showed, by using electrophoretic techniques, that there was a lot more genetic variation in natural populations than perhaps many people had previously thought would be the case. I had no a priori particular view on the amount of variation existing in natural populations, because as a mathematician, I had no specialized knowledge about that. Many biologists had claimed that there would be very little variation, that we would all be very similar genetically, and I was quite prepared to believe that. However, this view was taken not only for purely observational reasons, but also for theoretical reasons: calculations made in the 1960’s using the concept of segregation load – a third concept of genetic load – led to the view that there could not much genetic variation from one person to the next genetically. I very quickly came to disagree with this load calculation, as I did also for other forms of load calculations, as I have mentioned earlier. I therefore saw no load-based limit to the amount of genetic variation that could exist in natural populations.

All the same, I was quite surprised at the extent of variation revealed by the Lewontin and Hubby results. At that time, in 1966, two years before Kimura put forward the neutral theory, I started thinking about whether one could use the patterns of genetic variation which they observed to assess whether that variation was neutral or selectively induced. However, my work on that at that stage was very naïve. I did not have the mathematical equipment to approach that question in depth, so I did not publish anything on that in the 1960’s. Insofar as I did do any (unpublished) work on it, my calculations suggested that the variation which had been revealed by Lewontin and Hubby in Drosophila - and also by Harris’s work on humans at essentially same time – looked to me as though it could, on the whole, reasonably be regarded as selectively neutral. Although, as I said, I published nothing on it at the time, I was fascinated by the fact that there was all this variation, and I was clear in my mind that the pattern of variation which one observed should tell you something about the nature of the evolutionary that had led to these patterns.

AP: Were you familiar with work in the 50’s and 60’s by molecular biologists such as Margoliash and Fitch on the molecular clock?

WE: I knew Walter Fitch quite well in Madison in 1971, when I visited there to work with Crow. Although I was fascinated by his work, I did not work in the same area that he did, having other interests at that time. It was however clear to me that, following the work of Watson and Crick, one would have to recast evolutionary theory, and specifically the mathematical aspects of it, in a molecular framework, and Fitch was a leader in that effort at that time.

My own interests in molecular genetics were more technical and mathematical, and focused on Kimura’s so-called infinite sites model, now called the infinitely many sites model. This model formed the start of a recasting of the type of evolutionary theory in which I was interested on a molecular basis. The word “site” of course referred to a nucleotide site, and “infinitely many” referred to the approximation that in any genome there might be billions of nucleotide sites. As an approximation you could take this to be infinitely many sites, an assumption that made the mathematics a bit easier. It was a great credit to Kimura that he in effect said: “we have to rethink through classical evolutionary population genetics theory on a molecular basis.” His first major paper on this model was more or less contemporary with the work that Fitch and Margoliash were doing. Kimura’s work could be described as intra-populational and Fitch and Margoliash’s as inter-populational.

AP: Kimura built upon your own work. What was the relationship between your original work on diffusion modeling and Kimura’s?

WE: I would definitely not say that Kimura built on my work. People built on Kimura’s work. Kimura was extremely brilliant and did not need to rely on other people. The relationship between Kimura and myself at that time was that he was the top dog and I was just a junior person involved in the same field. And, I might add, he would make that sort of thing very clear to you.

So the influence went entirely the other way. I subsequently had many disagreements with Kimura, since his work, although path-breaking, was often incorrect, and the relationship between us was pretty tense for decades. But that did not stop me from admiring his work very much and in effect saying this in my published work. For example, in my 1979 book “Mathematical Population Genetics”, I made as many positive references to his work as I could. As just one example, I said that his work heralded a rebirth of population genetics theory, which I believed then, and still believe now, to be a true statement. In cases where I disagreed with his work, I said so quietly and blandly, but also firmly, in that book

AP: It is true, though, isn’t it, that the work you did on diffusion models, in particular, the testing of the diffusion approximation the Wright-Fisher model, was something that Kimura re-derived – in particular, the results of the infinite alleles model with the diffusion approach.

WE: No, he did not borrow diffusion work from me. He had been doing diffusion work himself well before I even got into the field - he was a master of diffusion work. There was no way in which he used me or anybody else in using diffusion theory. Frequently I did independently find some of the results which he had previously found and of which I was unaware. On the other hand, after a while I did get some results that he had not obtained, in particular the sampling formula named after me.

AP: When did you first encounter the neutral theory? Were you surprised when it was suggested?

WE: I first encountered it when I read Kimura’s 1968 paper proposing the theory, which in effect claimed that many allelic substitutions that occurred in evolutionary history were the result of purely random processes and had no selective significance. I was surprised when he put this view forward, because most people in the field were in effect selectionists at that time, and thought that selection was the major vehicle for allelic frequency change. So the view that the great majority of allelic substitutions were not driven selectively but were just random chance events, was, to me, bold and innovative, perhaps even reckless. He was criticized by many people, including Dobzhansky, the famous geneticist, who was a confirmed Darwinian. Dobzhansky said something like: “We hear these irritating comments from people from time to time. They are just derived from theory, which is irrelevant. It will be found later that the great majority of replacements are selective.”
I personally took no position on this matter, ever. The reason for this was my interests were focused on the purely mathematical side of the question. I was more interested in addressing the question of what patterns of gene variation would you expect to see if indeed many of the substitutions were neutral. In particular, I was curious as to whether the Lewontin and Hubby patterns look selective or neutral.

On the other hand, one problem that I had with the neutral theory was that the main argument that Kimura put forward for it was a theoretical and mathematical one, and derived from the Haldane substitutional load argument and calculations that we discussed earlier. Kimura argued that the very large number of substitutions which one knew were going on at any one time in many populations could not be selectively induced, but had to be purely random, because of the large substitutional load that he claimed would arise if they were selective. To the extent that I thought the whole substitutional load argument was incorrect, I thought that that was an extraordinarily weak basis for the neutral theory arguments. I said so then, and have said so frequently since.

Tests of Neutral Molecular Evolution

AP: So, was part of your motivation for developing a statistical test for neutrality because you thought load arguments were ineffective?

WE: No, I was just interested in the purely mathematical question of the patterns of variation to be expected under neutrality. This interest derived from two things, as I stated earlier. The first was the data becoming available in large volume on genetic variation, and that one wanted to explain the patterns exhibited by this variation, and the second was the mathematics of the neutral theory. And so it became to me an obvious problem to think about.

In doing this I approached this question as a statistician. In statistical terms the null hypothesis is that the variation one observed was caused neutrally, or randomly. Another aspect of randomness that has to be allowed for in the calculations is that the variation that is observed was not from an entire population, but from a sample of a population. The sampling process brings about a second level of randomness. It thus became necessary to work out what patterns of variation you would expect to see from a random sample of the genetic material in some population in which allelic frequencies were changing purely randomly. Having worked that out, it then because a statistical question of developing a test to see whether the patterns actually observed are consistent with those that the neutral theory predicted were likely.

It was, for me at least, quite difficult work to do, and I think that I was very lucky to be able to do it. (Now, of course, the analysis can be done very quickly and efficiently, using Kingman’s concept of the coalescent.) Once it was done, however, I did not pursue that work very strongly after I published my paper describing my conclusions, because it was immediately clear that the test of neutrality that I devised was not, in statistical terms, very powerful. That is to say, even if there is quite a reasonable amount of selection involved, it’s very hard to observe that by the statistical procedures that I developed. The reason for this is rather hard to discuss without showing the mathematical formulae, but I thought, if you can’t really pick up very much evidence of selection from these tests, if the tests are inherently not very powerful, then why pursue this topic? So, I moved to other things. Other people subsequently developed further tests, which can describe, which are slightly more powerful than the test I produced, and which therefore have been useful in testing the neutral theory.

AP: Perhaps I should ask you then about those other tests, and their relationship to yours?

WE: My test assumed several things. First, it is based on the so-called infinitely many alleles model, and not the infinitely many sites model that I described earlier. I did this because this was more appropriate for testing the Lewontin-Hubby form of data for neutrality. It differed from tests now available that are based on the infinitely many sites model, and which use nucleotide sequence data. Since these are the ultimate form of genetic data, these tests can be expected to be more efficient than my test. Secondly, I considered the genetic variation at just one gene locus only. The lack of power in my test arises in part because of this, and a test that uses data from many loci can be expected to be more powerful than mine. Some current tests do this. There is however a problem with doing so, since the hypothesis tested now becomes whether all of the loci are selectively neutral. To me, on the other hand, the important question was: is this one particular gene locus selectively neutral? So you have to give up something in terms of the question which you ask if you use multi-locus tests.

One reason why essentially all tests of neutrality are not very powerful centers around the fact of co-ancestry. The allelic types found in a sample of, say, 100 genes are not analogous to the results, or types (heads or tails) found in 100 tosses of a coin. The different tosses of a coin give independent results, but the corresponding result is not true of genes. The genes in any sample of genes have an ancestry, and this ancestry implies a dependence from one gene to another in the nature of the genetic material. This means in effect that even though you might have one hundred genes in your sample, you only have perhaps what might be called five or ten independent observations. Think of the simple example of identical twins, who have identical co-ancestry. The genetic data in both twins tells you no more than the genetic data in one of the twins. Although we are not all identical twins to each other we are all related, even if distantly. Returning to the coin case, if you toss a coin only five or ten times you have only almost no power to assess whether the coin is fair or not. Correspondingly, in the genetic case, if in effect you only have a small number of independent genes, then you don’t have very much power in your test.

AP: You said your test was developed as an infinitely many alleles test, and not an infinite sites model. Can you explain why that shift occurred and what the distinction is between these two?

WE: My test was designed for the electrophoretic data that Lewontin and Hubby, as well as many others, were obtaining in the late 1960’s, and used the infinitely many alleles model. Strictly speaking, this model and thus this rest were not appropriate for these data, but I felt that a revised version of my test might be constructed for them.

The shift occurred because sequence data became available. As I mentioned earlier, these are ultimate form of genetic data, so that any test based on them uses more information than a test using the infinitely many alleles model. Such a test would then be based on the infinitely many sites evolutionary model, not (as mine was), on the infinitely many alleles model. I was of course aware of this at the time that I introduced my test, but I did not work out the mathematics of a test of neutrality based on the infinitely many sites model because I felt that detailed sequence data would be a long time coming.

AP: When Kimura first proposed his theory, he was thinking of neutrality of sites that have functional roles, or loci, not neutrality – in the way we think of it today – at the molecular level, where you can have synonymous substitutions. Do you think that that was part of the reason why it was so controversial?

WE: I don’t think so.

AP: Or, was there confusion about what was being proposed as neutral?

WE: I asked him this question several times, and he gave me very brusque answers. In fact he never answered that question, at least to me.

There is one case where neutrality could be regarded as trivial, or uninteresting. Because of the redundancy properties of the genetic code, you might have two different nucleotide sequences which produce exactly the same amino acid sequence. And since it is the amino acids which are the relevant entities, you can argue that two different nucleotide sequences, that is two different alleles, that code for the same amino acid sequence, might well be taken to be selectively equivalent. I therefore sometimes asked Kimura if the neutral theory was a theory concerning nucleotide or amino acid sequences, but he never gave me an answer.

On a different point, the word “neutrality” in the theory means selective neutrality, meaning, for example, no differential viability or fecundity. This is as opposed to no difference in function. Now, I could perhaps with some difficulty imagine two different alleles whose functions were different but which were selectively equivalent. In this case one would say that, in terms of the theory, that they were selectively neutral with respect to each other. On the other hand, and this was the argument that Dobzhansky made, one might find it very hard to imagine a situation where two alleles have a different function and that there be no selective implication of that difference in function. Claims that different alleles with different functions are selectively equivalent have of course been frequently made, and the issue has been hotly contested.


AP: One of the things I noticed when you discuss the neutral theory is that you mention neutral substitutions and you mention causing something to be substituted neutrally, but you haven’t use the word “drift.” I’m curious why, when you discuss Kimura’s theory, you didn’t talk about drift causing substitution of neutral alleles.

WE: There is no strong implication about that, because it is more or less implicitly assumed that neutral substitutions arise effectively only by random drift. Neutral changes are in effect just drift changes – the two expressions are used interchangeably. So if I did not use the words “drift substitution” and used an expression like “neutral substitution,” that would just mean drift substitution.

AP: Can I follow that up? It seems, in the case of classical population genetics, that drift has a very distinct meaning. That is, random binomial sampling, and thus the effects of drift increase with smaller population sizes. Whereas for the neutral theory, substitution via drift is independent of population size. So it seems like the concept of drift in those two cases are very different. What causes drift in one case is different from what causes drift in the other. Is that a confusion on my part?

WE: I would not place such a big difference on the two situations as you do. The word “drift” originated probably with Wright, and was related to his theory of evolution as taking place best in a large population divided into smaller sub-populations. One component of his theory was drift, that is to say random changes in allelic frequencies, in these small sub-populations. It is certainly true, as you state, that drift is a more important factor in small rather than in large populations. So, the word “drift,” coming from Wright, did tend to have the connotation of arising in small populations. (As a side issue, it is, however, important to note that Wright used drift arguments only in one part of his evolutionary argument, and used selection in much of his theory.)

It is also certainly true, as you state, that the size of the population played comparatively little part in Kimura’s neutral theory. He would have claimed that neutral substitutions occur in populations of any size. So to that extent you could say that the implication of the word “drift” is slightly different in Kimura than it was in Wright.

However, the important point is this. The mechanism of the drift is the same in both cases – random sampling of genes from a parental generation making up the daughter generation. To that extent there is no difference between Wright’s and Kimura’s use of the word “drift”. That is why I do not place any real difference between the use of drift by Wright and Kimura.

AP: Would you say that the cause of substitution via drift in the neutral theory is simply neutrality of the alleles, whereas the cause of substitution in classical population genetics of alleles is reduction in population size? For Wright, for instance, a gene could be substituted via drift that did have a selective effect. Whereas, for Kimura, the alleles were neutral in their effect, that that explains why they are substituted at a regular basis.

WE: Yes, you could say that. In the case of Wright, everyone would agree that drift is relatively important in a very small population, and that even if there is a modest amount of selection, the effect of drift will dominate that of selection in a sufficiently small population. On the other hand, in a very large population small selective differences have a much more important effect, and in a huge population, it is quite unlikely that a selectively unfavored allele will randomly go to fixation. This arises because such a long time would be needed for it to go to fixation that the cumulative force of the selective disadvantage would make it very unlikely that that allele would in fact go to fixation. And, to return to your question, to the extent that Kimura claimed that the neutral theory applied to large as well as small populations, he would require a much more rigid concept of neutrality, maybe complete selective equivalence not allowing for even very small selective differences.

AP: Do you think that part of the reason why there might have been confusion or dispute about the neutral theory had to do with Kimura’s deployment of Wright’s concept of drift? Or, do you think that the neutralists might have been more sympathetic to Wright’s view of evolution, and the anti-neutralists might have been more sympathetic to a Fisherian view?

WE: That is possible. There might have been an overlap between those who adopted what you might call the Wrightian view of evolution in classical genetics and those who adopted the Kimurian view of neutral evolution. However, one should not take that line of argument too far, because I’ve heard it from people who discussed it with Wright that Wright himself actually disagreed with Kimura on the neutral theory. If he did, he did not say so very strongly. So it might be a mistake to see a too close an association between the Wrightian paradigm and the neutral theory.

AP: How exactly did the shift to nucleotide data change the debate concerning the neutral theory? Once nucleotide data became available, did the tests for neutrality change, did the debate change? Why, if so, in your view?

WE: My answer here is something of a guess, because I moved almost entirely to human genetics questions before there was a significant amount of molecular data.

The best-known test of neutrality these days is the so-called Tajima test, which is based on nucleotide sequence data. Because of that, one could certainly say that the form of the data which was used to discuss the neutral theory did indeed change towards nucleotide sequence data. This change was inevitable, because the ultimate tests of the neutral theory would presumably have to be based on that ultimate form of data.

AP: You speak in your 1979 paper about a shift to testing for “generalized” neutrality as a better null hypothesis than “strict” neutrality. I’d was wondering what you meant by that distinction, first, and second, does that bear on Ohta’s shift to the nearly neutral theory?

WE: The strict neutrality theory would claim that essentially all the substitutions which have taken place were strictly neutral. The generalized theory would allow for the fact that many mutations, when they first arose, were perhaps slightly deleterious, but that the selective differential is so slight that these mutations could, just by chance, increase in frequency and become fixed in the population. I would call that the “generalized” theory.

That is essentially the Ohta theory. Her claim could be said to be stronger than Kimura’s, in that she would have claimed, I believe, that most substitution processes derive from slightly deleterious alleles rather than strictly neutral ones. Now, you might say, why could it possibly be that a slightly deleterious allele is more likely to fix than a strictly neutral one? Of course, any one deleterious mutation is less likely to fix than a strictly neutral mutation, but her argument is based on the view that since there are so many more slightly deleterious mutations than strictly neutral mutations, there will tend to be more fixations overall of deleterious rather than neutral alleles.

What I did in the 1979 paper was to consider what the effect would be on my test for neutrality if indeed many alleles were slightly deleterious. What I found (and this is tied up with the fact that these tests are not very powerful) that you would not easily be able to pick up, by using purely statistical methods, the fact that an allele was slightly deleterious as compared with strictly neutral.

AP: Have there been tests developed now that can discriminate between the slightly deleterious theory and the strictly neutral theory?

WE: I don’t know of any such tests. I think that if there were any, they would need huge amounts of data before and such slight difference could be established.

AP: What is your present view on the neutral theory? How has it changed in the past 25 years?

WE: As I said, I left the field quite some time ago, so I don’t have an informed view on that matter, and I don’t know how much it has changed in the last 25 years. But, the theory has clearly been very influential. For example, you could argue that junk DNA is selectively neutral. (There are reasons why one might not make that argument, but it would at least be a plausible one to entertain.) If that argument were true, then the neutral theory would become important, because, as just one example, you could use junk DNA to assist you in the reconstruction of phylogenetic trees. All the mathematical approaches to that reconstruction that I know of in effect assume neutrality, and this is so whether the data refer to junk DNA or DNA coding for genes. So, to that extent, you can think of the neutral theory as being important. Similar arguments apply to much of the theory surrounding the concept of the coalescent. Coalescent theory is very simple and elegant in the selectively neutral case, but quite difficult in the selective case.

Of course there is a strong possibility that the neutral theory is assumed not because it is appropriate but because the math of that theory is so very simple compared to the math applying for any selective theory.

AP: Can I follow that up? Do you think that that has lead to models of phylogenetic change that is not very well supported by the evidence?

WE: I think that that is quite possible. However, here we enter into another question. In mathematical population genetics theory you know from the very start that you are making big simplifying assumptions. You are in a very different position from a physicist, who might believe that his mathematical models describe reality exactly. No sensible population geneticist would make any claim along those lines. He or she is forced to simplify, because reality is so complicated that you don’t know it in any detail, and even if you did know it and used math describing it faithfully, the analysis would be impossible to carry through. So simplification is unavoidable. I do not know whether the use of the neutral theory is too much of a simplification and has lead us to incorrect and distorted views about the true evolutionary tree, it’s shape and dimensions, but I suspect that there has been quite a significant distortion.

Tape 2:

AP: I had further follow-up questions for you about notions of drift operating in classical and molecular population genetics. In particular, I want to ask about the use of different kinds of models for drift. I was wondering whether you can discuss what the assumptions or implications would be for using a classical random sampling model versus a diffusion model?

WE: I can answer that question best by talking about the Wright-Fisher model, which we discussed earlier. Random events, and thus drift, are unavoidable in biological evolution, and this randomness is modeled in the Wight-Fisher model by this binomial sampling process central to that model.
The Wright-Fisher model is approximated by a diffusion process in the following way. The properties of a diffusion process are determined entirely by the mean change, and the variance of the change, in some quantity in a given small amount of time. When a diffusion process is used as an approximation to the Wright-Fisher model, this quantity is the frequency of a given allele in the population. The mean and the variance of the change in this frequency for the Wright-Fisher model are found using binomial sampling formulae, and these are then used for the diffusion process. One could then say, roughly, that the difference between the Wright-Fisher process and the corresponding diffusion process is just the difference between a discrete process and a continuous process that have the same mean and variance for the change in allelic frequency in a given time period.

As I mentioned earlier, it can be shown that formulae found for various quantities from the diffusion model, for example, the probability of fixation of an allele having a given selective advantage over the alternative allele, provide very accurate approximations to the (unknown) Wright-Fisher model values. They are indeed so accurate that for they are often used for the Wright-Fisher values, and formulae in textbooks claiming to be Wright-Fisher formulae are very often the corresponding diffusion formulae.

AP: Here’s my hypothesis behind that question. You can dispute this if you like. Because Kimura was looking at enormous amounts of molecular changes over long periods of evolutionary time, it might make more sense to use a diffusion model, to think of gene frequency changes as continuous instead of discrete, whereas for smaller populations, with smaller numbers of gene changes, you might want to use a binomial sampling model. Is that a confused hypothesis?

WE: I don’t think I would agree with the basic argument that you are making. Even though Kimura might have been interested in long time frames, as you observe, he grew up within the Wright-Fisher paradigm, and he did a great deal of diffusion theory within that paradigm well before he introduced the neutral theory. He was thus well aware of the convenience and simplicity involved in using the diffusion mechanism. So I don’t think he introduced diffusion theory because of the different time scale involved with the neutral theory. It was simply the most convenient mechanism to use, whatever the time scale.


Human Genetics

AP: You said earlier on in our discussion that you’ve moved on in your research, and you’re more interested now in human genetics. I was wondering if you could say a little bit about your recent research, and in what problems you think are important in population genetics, and genetics in general, today?

WE: I moved to human genetics for several reasons around about 1980. The first one was, as I mentioned, that I felt that tests of the neutral theory did not have enough power to be worth developing further. More broadly, I felt that that the discussion of the neutral theory was becoming too arcane, and that I should try to work on something more relevant and useful. I had several colleagues at Penn working on human genetics problems, and who told me that there were many mathematical problems in human genetics, for example those associated with finding disease genes, that I should take up. I thought that these suggestions made good sense and that I should move to the human genetics/disease area. I found that shift very hard to make, because one had to think about quite different questions and with quite different mathematics. Things like diffusion theory just weren’t involved. It took me so much effort to get into this new way of thinking that I soon paid very little attention to problems in evolutionary genetics.

My main interest in this (for me) new area was linkage analysis. Linkage analysis tries to establish the chromosome on which some disease gene might lie, and then to find its approximate location on that chromosome. Once this is done, an examination of the DNA in that location can be used to try to find, more exactly, the location of the disease gene. One can think of the linkage analysis part as indicating the general area in which to find a needle in a haystack, leaving it to other methods to locate it within that area.

The disease might be caused in whole or in part by the replacement of a single “good” nucleotide by a “bad” one, which in turn might mean the replacement of one amino acid by another. Sometimes it is caused by the deletion of some genetic material, in which case things can go more seriously wrong and a more serious disease eventuates. Thus molecular genetic considerations enter in, and this forms something of a link between molecular evolution analysis and human disease investigation. More broadly, evolutionary genetics and human genetics, which in 1980 barely spoke the same language, are now increasingly interrelated.

AP: You developed a test to help determine the location of disease gene in families. Could you explain that and how you came about discovering that?

WE: This is so-called transmission disequilibrium test, often called by the acronym TDT. The historical background to this test is as follows. People want to locate where a disease gene is in the genome One method for doing this is through the use of “marker” alleles. A marker locus has two important properties. First, one knows where it is on the genome, and one also knows any person’s genotype your at that marker locus. So if there are two alleles at this marker locus, which we might call M1 and M2, (M standing for marker), one can tell whether somebody is M1M1, M1M2, or M2M2 at the marker locus. One imagines that the polymorphism at the marker locus has been in existence for a long time. At some time in the past, maybe several thousand years ago, there was an original mutation causing a disease. Let us suppose that the disease locus is very close to the marker locus. Now that disease mutation will have occurred on a chromosome that had either M1 or M2 at the marker locus in the individual in whom the mutation occurred. Let’s assume that it was M1 – then there an immediate association between having M1 at the marker locus and having the disease. As time goes on, recombination events occur between disease and marker loci, and this association tends to break down. But if the marker locus is very close to the disease locus there will not be many such recombination events, even over quite long periods. Suppose now that you take a sample of individuals, some of whom have the disease (cases) and some whom do not (controls). If disease and marker loci are indeed very closely linked, there will tend to be an association between having the disease and having the M1. Such an association can be tested for by a classical 2x3 chi-squared test, where the two “row” categorizations are case versus control and the three “column” categorizations are the genotypes at the marker locus.

This chi-square test (of association) was used in the past as a surrogate as a test for linkage. However, it was subsequently observed that associations can arise for reasons quite different from linkage. One such reason is population stratification. You might have a disease which occurs more frequently in one human group, let’s say Caucasians, or whites, and less frequently, let’s say, in blacks. Further, there might be a gene which occurs more frequently in whites and less frequently in blacks. If so, you will see an association between the gene and having the disease in a sample containing both blacks and whites, but this has nothing to do necessarily with linkage. This means that the test which I described a few moments ago has a serious flaw in it – the inference that the association arises because of linkage is not necessarily correct. This could be a serious problem, so several people attempted to overcome it. The test which I helped to develop was in this direction. In broad terms, the way in which we were able to overcome the stratification problem was by using data within families. We looked at a mother and a father and an affected child, a so-called trio, and were able to develop a test for linkage using “within family” data, whose properties were free of any population stratification which there might be. As I said, the details of the procedure are rather mathematical, not easy to describe verbally, but I think I’ve given you the broad background of how it works.

More recently I have moved into the area of genomics and bioinformatics, which could be described as the analysis of questions previously considered at the single gene locus level by an analysis at the whole genome level, using of course whole genome data. The bioinformatics part this work involves the use of computer technology; because one has massive amounts of data when one looks at the entire genome, one cannot easily manipulate, absorb or analyze these without the use of computers. Such analyses are often involved with the investigation of genetic diseases, so the three areas in which I have been interested in, evolutionary genetics, human genetics, and genomics, are increasingly coming together.

This increasingly implies a change of direction in research in theoretical evolutionary population genetics. The original work of Fisher and Wright is prospective – it looked to the future, attempting to and succeeding in validating the Darwinian theory within a Mendelian hereditary paradigm, and in quantifying the Darwinian theory in Mendelian terms. Of course, there are many people who do not believe in evolution, but to me it’s as solid a fact as you could imagine, largely because of this work. The emphasis has now changed in evolutionary genetics to retrospective questions, looking backwards in time. These are data-induced questions, of the form: “These are the genetic data that we now have, how did we get here?” The reconstruction of the phylogenetic tree linking all contemporary species is perhaps the most obvious example of that sort of question. Human genetics often also looks retrospectively: “We have these data about people are affected by this disease, what can we say about when and where the original disease mutation arose?” It is this similarity in research directions that has led to the increased unification that I have just referred to, especially since the data involved in approaching these questions is increasingly genomic.

AP: Can I follow that up? What do you think that this unification will entail, in terms of rejecting all or part of classical or molecular genetics? Do you think that some of the original theory will be challenged?

WE: That is a very hard question to answer. Some will have to be thrown out. Many questions will be addressed using molecular theory, because molecular data are the data that we now have. But, on the other hand, the original classical theory still has relevance, and there will always be a place for the broad implications of the classical theory. On the other hand a lot of the classical theory will have to be changed. Biological reality is unbelievably complicated, and these complications were ignored in much of the classical theory. This was perhaps inevitable, since it is natural to start with simplifying assumptions that lead to an amenable mathematical analysis.

This leads one to ask: “When is mathematics useful in a scientific discipline?” It is no coincidence that mathematics grew up strongly in physics, and most strongly in simple areas of physics, for example in considering the motion of a single planet around a sun, where one can get a fairly complete theory. More complicated areas of physics do not yield so easily to mathematics. As a trivial example, even if we know the weather today, it is impossible to state in detail what the weather will be like in three weeks: there are too many complexities involved. I think that biology is in that situation. It’s incredibly complex, and many of the simplified mathematical models considered in the past were almost hopelessly oversimplified. To me is a fascinating and difficult question to know how useful mathematics will be for the very complex biological reality that is now being investigated.

Thank you Warren.

Genetics Draft

WE: We are talking now about John Gillespie and his drift and draft views, which I have discussed with him recently. His view, which I believe is very compelling, is that if you have a favored allele which is moving rapidly to fixation at some locus, it will drag along with it genetic material at closely linked loci. This may include loci at which the alleles are selectively neutral. Under this view the changes in the allelic frequencies at those neutral loci are caused not only by pure random drift, but are also induced by the draft of the frequency change of the favored allele. An expression often used in this connection is a “selective sweep”: the selective process sweeps along with it segments of genetic material close to the selective locus. John’s argument would be that we have to recast evolutionary genetics theory in terms of “draft,” rather than “drift.”

AP: So, is he effectively saying that there is no or very little drift?

WE: You could say that he was saying what Fisher said in a somewhat different context. Fisher would always admit that there was drift at any one locus, but he would claim that selection at that locus would ultimately overcome the effects of drift. John’s argument comes to the same conclusion in a different context. He would admit that there is drift at a neutral locus, but he would claim that that is unimportant compared with the draft caused by selection, not at that locus, but at a closely linked locus. Of course many calculations have been made in connection with this argument, a central parameter being the recombination fraction between the neutral and the selected locus.

AP: Thank you.