Posts Tagged ‘motoo kimura’

Theoretical Evolutionary Genetics: Now, in three different flavours!

November 5, 2013

Theoretical Evolutionary Genetics is at the heart of evolutionary biology. Francis Galton and his followers laid the foundations of theoretical evolution soon after Darwin’s Origin. This so-called biometrics school had its continuity in the work of Ronald Fisher, who amalgamated the Biometric and Mendelian schools into a robust theoretical framework. Animal breeders such as Sewall Wright independently developed methods to study heredity and genetic improvement in animal stocks. Fisher, Wright and Haldane (the latter coming from a different tradition, his own) are acknowledged to be the founding fathers of population genetics.

The fact that population genetics is the basis of theoretical evolutionary biology may be seen as an accident of the way it was introduced by the founding trio. As a matter of fact, there are ways of exploring the same biological problems other than the canonical population genetics approach. I recently bought two textbooks on evolution and I found that they used completely different approaches. After I read them I became aware that, indeed, there are at least three ways of doing things in theoretical evolutionary biology. These are the three ‘flavours’:

Flavour 1: Standard Population Genetics

This flavour is best represented by the Crow and Kimura manual, and more recently by the Charlesworths’ textbook. Evolutionary problems are tackled from the point of view of evolving populations in which allele frequencies change as a function of their relative selective fitness and sampling effects. Typically, a ‘change in allele frequency’ is defined and the resulting equation is solved for specific cases. These cases are basically equilibria situations or, in case of fixation/losses of alleles, how long will the process take and with which probability (on average). OK, this is an extreme oversimplification of population genetics, but for what I want to say it’s enough.

Flavour 2: Price’s Equation

I must admit that I never got Price’s equation. I mean, even after someone explained it to me and I thought I understood it, I didn’t see what it was useful for. I have recently changed my mind. I recently read Sean Rice’s ‘Evolutionary Theory’ and see, to my astonishment, that he approached classic problems in population biology using Price’s equation. Price’s equation is due to George Price, a strange man and even stranger scientist. He wrote down the whole evolutionary process into a single equation, accounting for the fitness of the ‘elements’ of an evolutionary system and the relationships between these elements (species, genes…). It’s basically a covariance equation. Using the appropriate definition of parameters one can reduce complex population genetics problems into a single covariance equation, and that’s the strength of Price’s approach.

Flavour 3: Game Theory

Game theory was used in biology originally by Bill Hamilton (I think) although it was definitively John Maynard Smith who fully developed the topic. According to Maynard Smith, he started using Game Theory while visiting Chicago, as there was nothing else to do in Chicago! Anyway, I always thought that game theory was useful for phenotype evolution modelling (like behaviour). However, while reading Martin Nowak’s ‘Evolutionary Dynamics’ I discovered that most of what we know from standard population genetics could be approached from the game theory perspective. In that case different alleles represent different game strategies. Winning strategies are equivalent to fitter alleles and the winner of the game is, obviously, the fixed allele in the population.

Is there a best flavour?

I should say that my preferred method is that of classical canonical standard population genetics, but after reading Nowak’s and Rice’s books I’m aware that this is only a personal preference. What these different approaches show is that there are many ways of solving the same problem, and that should be used as a powerful tool in evolutionary biology. As an example, I included a BOX below in which I derive a classical result in population genetics using these three different approaches. Richard Levins once wrote that “[deriving] alternative proofs for the same result is not merely a mathematical exercise – it is a method of validation”. Now I wonder whether all major findings in population genetics achieved during the last century could be reproduced by using these alternative flavours!

Box 1. Deriving the change in allele frequency with three different approaches.

One of the most important quantities in population genetics is  how much an allele frequency changes after one generation, or \Delta p. Let’s assume we have a large population of bacteria, and that for a given locus we have two alternative alleles A_1 and A_2 with frequencies p and q respectively. (Remember that p+q=1.) If A_1 have a selective advantage over A_2, how much the frequency of A_1 changes after one generation? This is a classic problem in population genetics, and we are going to derive an expression using three different methods.

Standard approach

The change in an allele frequency for an haploid population is given by (Crow and Kimura 1970):

\Delta p = \dfrac{(w_1 - w_2) p q}{\overline{w}}

where w_1, w_2 and \overline{w} are the fitnesses of allele A_1, A_2 and average fitness respectively. For selective advantage of A_1 over A_2 we define the following relative fitnesses:

w_1=1+s ; w_2=1

\overline{w}=(1+s)p + (1-p)=1+sp

being s the selection coefficient. Substituting, we find and expression for \Delta p as a function of the selection coefficient:

\Delta p = \dfrac{s p q}{1+sp}

For a small s we can find a linear approximation by the Taylor expansion about s=0:

\Delta p(s) \Big|_{s=0}= s \dfrac{d\Delta p(s)}{ds}\Big|_{s=0} + O(s^2) \approx \dfrac{pq(2sp + 2) - 2sp^2 q}{(1+sp)^2} \Big|_{s=0}

obtaining the classic result for haploid populations:

\Delta p\approx spq

Price’s equation approach

A common form of Price’s equation is (Rice 2004):

\Delta \overline{\Phi} = \dfrac{1}{\overline{w}}[cov(w,\Phi) + E(w\overline{\delta})]

and it says that the change in a trait or character (\Delta \overline{\Phi}) is a function of how traits covariates with fitness (cov(w,\Phi)) and how the parental and offspring traits are related (E(w\overline{\delta})). By noticing that cov(W,\Phi)=\overline{\Phi}(w-\overline{w}) and that E(W\overline{\delta})=0 for the haploid case (no inbreeding nor mating):

\Delta \overline{\Phi} = \dfrac{1}{\overline{w}}[\overline{\Phi}(w-\overline{w})]

As we are interested in the allele frequency as a trait we rewrite this formula in a more familiar form:

\Delta p = \dfrac{1}{\overline{w}}[p(w_1-\overline{w})]

which for the selection scheme defined above is equivalent to:

\Delta p = \dfrac{1}{1+sp}(p(1+s-1-sp))

which to a linear approximation leads to the same result as the classic approach:

\Delta p \approx spq

Game theory approach

Last but not least, we can consider that the two alleles are two strategies in a game played by their bacterial hosts. In game theory we first create a table of costs of the different strategies like this one:


In our case, if two bacteria play the same strategy (they have the same allele) there is no cost or benefit for any of them. However, if one bacteria has the fitter allele while the other has the alternative allele, the benefit for one will be the selection coefficient s and the cost for the other -s. Hence, the cost matrix has the form:


The instant rate of change of the winning strategy frequency is given by (Nowak 2006):

\dfrac{d p}{d t} = p q (f_a(p) - f_b(p))

For our game we have:

f_A(p) = sq ; f_B(p) = -sp

So that:

\dfrac{d p}{d t} = s p q

moving d t to the right and integrating both sides:

\int_{p_0}^{p_1}d p = \int_{t_0}^{t_1}s p q dt

and for a time interval of one generation (\Delta t = 1) we obtain the classic equation:

p_1 - p_0 = s p q (t_1 - t_0)

\Delta p = s p q

In conclusion, the three different approaches based on different assumptions yield the same result.

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!