Posts Tagged ‘history’

Diffusion in population genetics: who was first?

July 25, 2013

We know Motoo Kimura beause of the neutral theory. (Or should I say that we know the neutral theory because of Kimura?) But before his classic paper in 1968, Kimura was already a prominent figure in evolutionary genetics, mainly because of his productive use of the diffusion method to study the change of gene frequencies in populations. Why is diffusion so important? And who was actually the first to apply it to population genetics?

Diffusion and probability

Suppose that we have a large population, large enough that you can assume that it’s infinite in size. By using the so-called deterministic models one can easily compute the effects of evolutionary forces in the gene composition of such a population. Now suppose that the population is rather small. In that case, small fluctuations due to sampling will obviously influence the evolution of genes. This is known as genetic drift, and its study requires extensive computations. If you want to know how genetic drift affects to a population with mutation, selection, epistasis and/or linkage of multiple alleles, the computations become impractical or even impossible. But if you assume that the change in a gene frequency is very small during a short fraction of time, you can treat your gene frequencies as if they were particles diffusing in a continuum of probability states. This is (in a very simplistic way) the principle underlying the diffusion method.

Fisher’s diffusion approach

Ronald Fisher first thought that the probability of a given gene frequency could be modelled in a continuous space (as I indicated above). He borrowed the heat diffusion equation from thermodynamics and adapted it to genetics. This is, as far as we know, the first use of diffusion in population genetics. However, a diffusion process is an approximation, and Fisher’s approximation wasn’t accurate enough. Indeed, Sewall Wright noticed some discrepancies with Fisher diffusion approach. After this, Fisher found an error and was just in time to correct the equation right before the publication of his 1930 book. Fisher thanked Wright and admitted:

“I have now fully convinced myself that your [Wright’s] solution is the right one” Letter from Fisher to Wright, quoted in Provine, 1986

However, Fisher’s ego was already quite damaged, and he soon  stopped communicating with Wright. In later years, Fisher developed a bit further his diffusion method, but mostly for his own amusement, and never brought it to the fore of population genetics. (One may speculate that he was still affected by his early mistake.)

(Non-) Artistic depiction of how Fisher interpreted gene frequency distributions as continuous diffusion processes.

(Un-) Artistic depiction of how Fisher interpreted gene frequency distributions as continuous diffusion processes.

Proper diffusion: Kolmogorov and Feller

In 1931, the prolific mathematician Andrey Kolmogorov published his celebrated probability diffusion equations (although under a different name). A few years later, William Feller fully explored the potentials of Kolmogorov’s diffusion and coined the terms forward and backwards equations to refer to the two most popular forms of these equations. As Feller noticed, Kolmogorov added an additional term to the standard heat diffusion equation. That term, precisely, is the part that Fisher missed in his first approximation and that he added later on.

Kolmogorov quickly realized of the potential of his own equations to described evolutionary dynamics and published a paper about it (see comment in Feller 1951). Kolmogorov sent a reprint of his paper to Sewall Wright, who rapidly published a paper in PNAS using Kolmogorov forward equation to calculate the stationary distribution of gene frequencies. However, Wright himself preferred his integration method and his paper received little attention. Feller and Malécot showed later that Fisher’s diffusion, Wright’s integrals and the classical branching models all converge to the Kolmogorov forward equation. That is, they are mathematically equivalent.  The path was ready for someone to fully exploit the potential of Kolmogorov’s equations in genetics.

Kimura enters the game

Motoo Kimura first read about diffusion processes in Wright’s 1945 paper and quickly started to develop these equations for his own purposes. Kimura’s first diffusion paper was indeed communicated to the National Academy of Sciences by Sewall Wright himself. But if Kimura surprised the other theoreticians was because of his use of Kolmogorov’s backward equation to calculate the probability and time of fixation of new genes in a population. Kimura provided a new horizon to explore the evolution of finite-size populations in a time in which computers were not powerful enough. However, he was aware of his limits as a mathematician, and invited others to join his particular crusade:

“I cannot escape from this limitation […] but I hope it will stimulate mathematicians to work in this fascinating field”. Kimura 1964

The years that followed were dominated by the diffusion method, and ‘proper’ mathematicians joined the ‘diffusion crew’ (Karlin, Ewens and Watterson, to name but a few). The diffusion method gained in rigour and precision. Today the diffusion method has lost some interest in favour of computational simulation. However, they are still at the core (and the heart) of theoretical population genetics.

So, who was first?

So far we could concluded that Fisher was the first using diffusion to approximate stochastic processes, not only in genetics but in probability theory. However, as also noticed by Feller, the original Fisher’s diffusion equation was first used in a probabilistic context by Albert Einstein in his classic paper about Brownian motion of particles, almost 20 years before Fisher’s account! It may be that probability as a diffusion process was a popular topic among mathematicians in the early 20th Century, and that Fisher was smart enough to adapt it to genetics before anyone else. Whether Fisher knew or not about Einstein’s approach, I have no idea.

For an historical discussion on the use of diffusion equations in genetics I recommend Kimura’s review on the topic and Felsenstein free textbook on Theoretical Population Genetics. For a technical account it is often recommended Warren Ewens classic text, but I’ve found more useful the recent manual by Otto and Day on mathematical models. The later is, in my opinion, the best textbook in mathematical biology I’ve read so far.

So, what is the take home message? Who we have to thank for the diffusion method in population genetics? It’s hard to summarize the contributions of the different people involved. But if I have to write a single sentence I would conclude: Fisher was the first, Kimura did the best!


Chromosomes and reduplication: how genetic linkage was discovered

May 15, 2013

We just accept that genetic linkage exists. I mean, it looks obvious that genes are linked to other genes because of the chromosomes. Until very recently I assumed that the concept of genetic linkage naturally emerged from Mendel’s experiments, and that perhaps Mendel himself already suggested that linkage may happen. Alas, how wrong I was!

I recently came across a concept I’ve never heard before: reduplication. By looking for some more information about it I found that ‘reduplication’ and ‘linkage’ were competing interpretations to explain the coupling and repulsion of alleles. These two ‘schools’ were championed by two of the most important geneticists at the time: William Bateson and Thomas Morgan. Here I briefly discuss the origin of the controversy and its development, and how the modern concept of ‘genetic linkage’ emerged.

Bateson and Punnett discover the coupling and repulsion of alleles

When Mendel’s laws were rediscovered at the beginning of the 20th century, many scientists begun to test their favourite organisms to see whether they also followed a Mendelian pattern of heredity. Among them, the embryologist William Bateson initiated a very successful research program, convincing a full generation of scientist that Mendelism was true. As a matter of fact, it was Bateson who first translated into English Mendel’s original paper. Together with Reginald Punnett (another early defender of Mendelism) they discovered a strange phenomenon that escaped Mendel’s attention: the coupling and repulsion of characters. What they found is that, when they crossed sweet pea plants with different characters, some of the characters always appeared together. For instance, in crossing plants with red flowers and round pollen with plants with blue flowers and long pollen grains, they found that, contrary to Mendelian expectations, ‘blue’ and ‘long’ were always present together. They called this ‘coupling’. When a pair of characters (alleles) never appeared together, they said these alleles were on ‘repulsion’.

William Bateson

Reginald Punnett

Thomas Morgan

The idea that genes may be encoded in the chromosomes was already suggested by Walter Sutton and Theodor Boveri. Based on this theory, some scientists (including Hugo De Vries and Walter Sutton himself) already predicted the existence of genetic linkage. The findings of Bateson and Punnett may have confirmed genetic linkage, and apparently they were very close:

“…there must be an order of precedence among factors composing such system, and the suggestion is plausible that this order will follow the grade of coupling in which the factors are accustomed to be linked.” Bateson and Punnett (1911)

They came out, however, with a completely different interpretation.

Reduplication versus linkage: the Bateson-Punnett-Morgan debate

When Thomas Morgan started to work with Drosophila, he didn’t believe that chromosomes were carrying the genetic information. However, after he discovered the first Drosophila mutant: white eyes (flies generally have red eyes) he changed his mind. The white mutant allele was linked to female flies, in particular to the X chromosomes. Inspired by Theodor Boveri hypothesis, Morgan started to believe that chromosomes were, indeed, the carriers of the genetic information. (Ironically, as Mayr pointed out, his last paper criticising the chromosome theory was published after his paper describing the white mutant, due to a delay in the processing of the former.) Morgan (and his people) soon realized that a chromosomal inheritance implied linkage between genes.

In the meanwhile, Bateson and Punnett proposed a mechanistic explanation of the observed coupling and repulsion of characters: reduplication. By this mechanisms, cellular division giving rise to gametes would be asymmetrical. For instance, if two gametes of genotypes AB and ab form a new zygote (see Figure 1, left), the new individual will produce new gametes by cellular division. But because of the asymmetric cellular division, there will be more gametes AB and ab than Ab and aB. Hence, A and B are coupled alleles. In the very same paper Bateson and Punnett proposed to abandon the use of the terms ‘coupling’ and ‘repulsion’ and adopt reduplication instead.

Reduplication and Linkage models to explain the coupling and repulsion of gametes.

Figure 1. Reduplication and Linkage models to explain the coupling and repulsion of gametes.

Morgan soon complained, and in a paper in Science he said that Bateson’s “results are a simple mechanical result of the location of the materials in the chromosomes.” The battle for linkage begun!

At first Morgan was deliberately ignored. Punnett avoid citing Morgan’s findings by limiting his research to characters in which “sex-limited inheritance is not involved”. Later on Bailey used this as an argument to attack Morgan: “their results, however, are complicated by the phenomena of sex limitation, and by differential death rate”.

But Morgan and his students had already collected dozens of Drosophila mutants and, measuring crossing-over rates, they mapped the genes into the chromosomes. The linkage model seemed to be correct (Figure 1, right). The masterpiece “The Mechanisms of Mendelian Heredity” summarized their findings. Interestingly, the frontispiece of the book is actually a chromosomal map of genes. I think that was a (very successful) provocation.

Bateson finally accepted that Morgan may be right, but he warned:

“promising though it is, must be tried by tests on a scale far wider than experience of Drosophila provides before we are able to assess its value with confident”. Bateson (1919)

The battle for linkage was over, but a new one was about to begin.

Three-dimensional genetic linkage?

What if genes were organized in a three-dimensional manner rather than in a linear fashion? You would say that this is crazy, but that was William Castle‘s interpretation of the observed linkage between characters. Castle observed that the fraction of recombinants did not always support a linear disposition of genes, particularly for long chromosomal distances. Morgan and his people predicted that double-crossing over may be creating this artifact. Castle vowed for a 3D disposition of the genes.

Alfred Sturtevant got particularly upset, as his main project was to generate a comprehensive (and linear) linkage map of the Drosophila genes. Several papers from Morgan, Sturtevant and colleagues insisted in the importance of double crossing-over in the interpretation of recombinants. Sturtevant and Calvin Bridges maps were, however, based on the analysis of double mutants. It took the fine analysis of multiple mutants by Hermann Muller to show that double and triple crossing over was, indeed, the reason why recombination distances were not fully additive.

By 1920 Castle acknowledges that double crossing over may have an impact in distances and finally accepted the linear hypothesis proposed by Morgan. Some additional discussion about the Morgan-Castle debate can be found here.

The acceptance of genetic linkage

During the 1920s the chromosomal basis of heredity and a linear disposition of the genes was well established. Even Punnet accepted it in a paper published in 1923. But how can it be that Bateson and Punnett didn’t see that genes were linearly linked? They were pretty close indeed. Punnett had an answer for that. He wrote decades after the debate was over:

“I have sometimes been asked how it was that having got so far we managed to miss the tie-up of linkage phenomena with the chromosomes. The answer is Boveri. We were deeply impressed by his paper “On the Individuality of the Chromosomes” and felt that any tampering with them by way of breakage and recombination was forbidden. For to break the chromosome would be to break the rules. So it was left for Morgan and his colleagues to make use of Janssen’s observations and by their brilliant work to link up genetics and cytology, thereby opening up a new era in these studies.” Punnett (1950)

It is amazing that the very same scientist, Theodor Boveri, inspired both Bateson and Morgan schools of thought, yet they reached opposite conclusions. After all, the generation of hypothesis is, in a manner, a matter of interpretation. Thankfully, hard work and communication is a successful way of validating hypothesis in science, although it may take some fight in the process.