Saturday, December 10, 2005

What is Population Genetics?

At the end of Wednesday's post I commented that there was one more aspect of “The tautology objection” that is worth discussing.

The basic argument was that “the survival of the fittest” is an empty tautology because the term “fittest” is simple defined as “those that survive.” We have already seen two reasons why this is wrong: (1) There is no abstract principle of natural selection used by biologists. There is nothing tautological about saying, for example, that moths possessing certain sorts of coloration leave more offspring than those lacking that coloration. (2) There are, in fact, criteria of fitness independent of mere survival. Bethell has simply confused the definition of fitness with how it is measured in certain circumstances.

But what about those abstract, mathematical formulations of evolutionary principles that fall under the rubric of population genetics? Is there anything helpful to Bethell there?

In his 1976 essay “Darwin's Mistake”, written for Harper's magazine, Bethell wrote:


The bold act of redefining selection was made by the British statistician and geneticist R. A. Fisher in a widely heralded book called The Genetical Theory of Natural Selection. Moreover, by making certain assumptions about birth and death rates, and combining them with Mendelian genetics, Fisher was able to qualify the resulting rates at which population ratios changed. This was called population genetics and it brought great happiness to the hearts of many biologists, because the mathematical formulae looked so deliciously scientific and seemed to enhance the status of biology, making it more like physics. But here is what Waddington recently said about this development:

The theory of neo-Darwinism is a theory of the evolution of the population in respect to leaving offspring and not in respect to anything else...Everybody has it in the back of his mind that the animals that leave the largest number of offspring are going to be those best adapted also for eating peculiar vegetation, or something of this sort, but this is not explicit in the theory...There you do come to what is, in effect, a vacuous statement: Natural selection is that some things leave more offspring, and there is nothing more to it than that. The whole real guts of evolution - which is how do you come to have horses and tigers and things - is outside the mathematical theory [my italics].



Before getting to the meat of the issue, we should pause to mock Bethell for syaing “qualify” when he surely meant “quantify.”

Bethell is so confused here it's hard to know where to start. Take the italicized portion of Waddington's quote, which Bethell seems to think is so important. Of course the question of how we came to have “horses and tigers and things” is outside the mathematical theory! Who ever said otherwise?

Bethell does not tell us where Waddington made this remark. I'd be curious to see the full context. But for the moment, let's think about mathematical models in general, and population genetics in particular.

The objects that mathematicians study have no physical existence at all. They are abstractly defined quantities or objects that possess only those properties mathematicians choose to bestow upon them. In defining their abstractions mathematicians typically have some real-life situation in mind. There may be no perfect circles in nature, but there are things that are sufficiently circle-like to amount to the same thing.

Students often object that it is precisely this level of abstraction that makes mathematics difficult. They have a point, but in another sense abstraction is what makes mathematics doable at all. It's understanding the real world that's difficult! Abstraction is where you say you're going to ignore most of the complicating factors that make physical phenomena difficult to predict, and instead focus on a handful of variables that you hope are the most important ones.

In population genetics the goal is to understand the short-term flow of genes in a population. In particular, suppose you find that a particular population of organisms possesses a gene we shall call A. What percentage of the organisms in the next generation will possess A?

To build a mathematical model for this situation we would ask what variables are crucial in determining A's representation in the next generation. A few things might occur to you immediately. We would want to know what percentage of organisms currently possess A, for example. We would also want to know what sort of selective advantage or disadvantage A confers on its bearers. If organisms possessing A are generally more fecund than those lacking A, for example, that will certainly be relevant to determining A's representation in the next generation.

Here are some things that are not relevant: What kind of organisms we're studying (in other words, does A occur in a population of monkeys? zebras? lions? Who cares!), or the precise phenotypic effect A has on its possessors (does A cause certain peppered moths to have darker coloration than their competitors? does A lead to more bristles on the legs of a fruit fly? Again, who cares?)

There are other factors we could mention as well.

Thus, within the mathematicla theory of population genetics a “gene” does not refer to a particular strand of DNA found in an actual critter. A gene is simply an abstract, undefined quantity, rather like the notion of a point in geometry. What we know about a gene is that it appears with a certain frequency in a population and that it competes with some number of alternative genes for representation in the next generation. We have certain other assumptions about mating behavior and fitness and so on.

Then we ask, given the assumptions we have made about these genes, what does logic and mathematics tell us about the representation of the gene in subsequent generations? Of course, any assumptions you make about selective advantages and the like are going to be highly time-sensitive. Environments are constantly changing, after all. This means that models of the sort I am describing are only likely to be applicable over relatively short time periods.

Once you begin to understand the abstract nature of mathematical models, you begin to see the absurdity of Bethell italiczing the last part of Waddington's statement. Horses and tigers are real world entities made from large numbers of genes. They evolved over millions of years in changing environments. How could a simple mathematical model that focusses on a handful of genes over short periods of time possibly tell us anything about why we have horses and tigers, as opposed to centaurs and unicorns? The mathematical models were never intended to answer such questions.

So what, then, did Fisher accomplish in The Genetical Theory of Natural Selection? It's almost a sure thing that Bethell never read the book, would not have understood it if he had, and has no sense of its historical significance. Hence his silliness about biologists requiring a handful of mathematical formulae to make them feel good about their subject.

At the time Fisher was writing there was no widespread agreement that natural selection was an important mechanism of evolution. Lamarckism in particular was still a popular contender. The principles of Mendelian genetics were still a new thing. And the origins of genetic variation were not well understood.

As a result evolutionary biology was foundering a bit. It was clear what the questions were, but answers seemed as distant as ever. So part of what Fisher accomplished was to show that the abstract machinery of mathematics could shed some light on what had previously been seen as purely empirical questions. The book actually begins with a discussion of blending versus particulate inheritance. When Darwin wrote blending inheritance was still a viable option. Darwin recognized that it was diffuclt to reconcile his theory of natural selection with blending inheritance, since it implied that new variations would quickly be diluted. But particulate inheritance, which is at the core of Mendelian genetics, is quite different. Thus, natural selection is a viable theory under a Mendelian scheme, but not viable (barring some mechanism for producing vast quantities of new variation) under a blending scheme.

Nowadays that's a commonplace observation, but at the time it was much overlooked. And Fisher put these casual observations onto a firmer footing than had previously been accomplished.

Fisher subsequently devised notions of natural selection and fitness that were amenable to inclusion in mathematical equations. Contrary to the assertions of the tautology mongers like Bethell, he did not define the fittest organisms as being those which survived. Actually, he never talked about the fitness of an organism at all, but rather the fitness of a particular gene. In his equations the greater fitness of one gene relative to another was represented as an increased probability of being represented in future generations.

The result of all this mathematizing was to show that natural selection was unique among proposed evolutionary mechanisms in being able to explain the gradual accumulation of small variations. He was also able to give a precise meaning to the idea of a population increasing its fitness over time. The result of all this was a previoulsy unknown level of rigor in drawing conclusions about evolution, coupled with a demonstration that the then proposed non-Darwinian mechanisms of evolution were theoretically inadequate.

More than this, he inaugurated a new way of thinking about biological questions. Fisher established the field of population genetics, which remains a major area of study today. It provided powerful analytical tools for pracitcing field biologists, and you can find numerous applications of its techinques in any textbook on the subject.

And now we can come full circle and see precisely why Bethell is so full of it. First, Fisher did not redefine selection. Selection meant exactly the same thing after Fisher that it meant beofre Fisher. He merely provided a formulation of it that was amenable to mathematical analysis. He was not trying to explain the origins of specific organisms or even specific adaptations. Rather, his intent was to explore the theoretical implications of Mendelian genetics for various theories of evolution. And even within the mathematical theory, he did not define anything in tautologous ways.

One final point. In applying the theories of population genetics, it is true that the fitness of a gene is measured by its representation in future generations. But this is far different from saying that fitness is defined that way. Fitness refers to a statistical tendency for certain organisms to bear more young than others. There is nothing tautological in that definition. Furthermore, it is a standard result in probability theory that if your sample size is sufficiently large, the measured freqeuncy of the particular gene in the next generation will be a good approximation to its theoretical value. (That's known as the law of large numbers, but that's a separate post). So there's nothing logiclally suspect either in the theory or the practice of population genetics.

Bethell closes his 1976 essay by writing:


Darwin, I suggest, is in the process of being discarded, but perhaps in deference to the venerable old gentleman, resting comfortably in Westminster Abbey next to Sir Isaac Newton, it is being done as discreetly and gently as possible, with a minimum of publicity.


It is now almost exactly thirty years later and even Bethell would have to concede that both evolution in general, and natural selection in particular, are going strong. Ignorant bombast will lose out to simple truth every time.

59 Comments:

At 4:58 PM, Blogger RPM said...

I'm not sure what terminology Fisher used, but I would clearly distinguish between alleles and genes/loci. I know it may seem like petty semantics, but there is a difference between the two.

 
At 11:20 PM, Anonymous Anonymous said...

I agree with rpm completely. There's a huge difference between alleles and genes, and you're missing it.

 
At 2:18 PM, Blogger Jason said...

rpm and anonymous-

Fisher used he term gene exactly as I used it here (at least in Chapter Two of The Genetical Theory, which is what I was looking at as I wrote my essay). The term allele does not appear in the index of the book (I'm looking at Oxford University Press's Complete Variorum Edition of the book published in 1999).

I am aware of the difference between an allele and a gene, but I don't see how that distinction is relevant to anything I said. As defined by Mark Ridley in his textbook Evolution (sec. ed.) a gene is a sequence of nucleotides coding for a protein while an allele is simply a variant of a single gene. I don't see a “huge” difference there. And in his textbook Population Genetics: A Concise Guide, John Gillespie points out that many books on this subject effectively use the term gene and allele interchangably.

Everyone knows the term gene, but many people do not know the terms locus and allele. I did not want to add a vocabulary lesson to an already lengthy post. But if you can point to a specific place in my essay where it is important to distinguish between genes and alleles, please show me where. There is a decent chance I will use this blog entry for my next CSICOP column, so I'd certainly be grateful if you can point to any genuine errors.

 
At 6:58 PM, Blogger Jason said...

Since I deleted one comment and added another, people following this thread may not realize that something has changed. So I am adding this little comment just to make the comment counter tick up by one.

 
At 8:43 PM, Blogger RPM said...

Jason, I only pointed it out because it's a pet peeve of mine. I know many respected geneticists use the terms interchangibly, and it bothers me every time. Despite that, I still think you push it a bit far because you write about both genes/loci and alleles and refer to both of them as "genes". A reader without a background in the field may be confused by the terminology.

I'm not a big fan of the Gillespie text either -- it provides a decent introduction to the theoretical aspects of popgen, but doesn't really deal with applications. I prefer Hartl & Clark or Hedrick.

In population genetics the goal is to understand the short-term flow of genes in a population.

I would say that population genetics describes changes in allele frequencies within a population. Gene flow occurs between populations.

 
At 11:28 AM, Anonymous Anonymous said...

Jason remarked "Bethell does not tell us where Waddington made this remark. I'd be curious to see the full context."

Stephen Jones attributes it to the 1967 Wistar conference, but gives no more context:
http://tinyurl.com/crzmu

RBH

 
At 11:29 AM, Anonymous Anonymous said...

Jason remarked Bethell does not tell us where Waddington made this remark. I'd be curious to see the full context.

Stephen Jones attributes it to the 1967 Wistar conference, but gives no more context:
http://tinyurl.com/crzmu

RBH

 
At 12:41 PM, Blogger Jason said...

RBH-

Thanks for the link. I have a copy of the Wistar procedings, and I found the Waddington quote. It is embedded within a discussion of Murray Eden's paper. My initial impression is that Waddington's point was similar to the one I made in my post The Tautology Objection, which in turn was based on part of Phillip Kithcer's discussion of this topic in Abusing Science. Certain abstract formulations of natural selection may indeed be tautologous, but biologists don't apply natural selection in the abstract. When you look at the specific ways selectionist reasoning is used, there is nothing logically suspect about it.

Incidentally, I notice that Jones used the same quote from Bethell's 1976 article as I did. In Jones presentation, the word “qualify” has mysteriously been transformed into the correct word “quantify”, without any indication that change had been made. Go figure.

RPM-

I take your point about my reference to the flow of genes. If I turn this into a CSICOP column, I will be more careful about my phrasing.

Thanks also for the book references. I'll try to track them down. I rather like Gillespie's book, perhaps for the same reasons you don't care for it. As a mathematician interested in biology, I'm more concerned with the theoretical side of things than with the practical applications.

 
At 1:03 PM, Anonymous Anonymous said...

There are, in fact, criteria of fitness independent of mere survival. Bethell has simply confused the definition of fitness with how it is measured in certain circumstances.

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