A phenotypic view of evolution Evolution in Structured Populations

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Epistasis and the Evolution of Corn

This week I want to finish up talking about epistasis so that we can move on to multilevel selection.  As you might imagine I can talk about gene interaction all day.  After all we have not yet talked about long-term selection – epistasis figures big in that, nor have we talked about cyto-nuclear epistasis.  But, hey, I am getting bored with this line of talk, and I suspect you are too. . .

So, back to today’s topic.  One type of data that a lot of people gather is QTL data on any number of traits.  One thing that can result from this is that you might end up with a pair of interacting loci.  An old example I picked up a few years ago are from a teosinte by maize cross done by Doebeli and colleagues (Doebley, J., A. Stec and C. Gustus 1995. Genetics 141: 333-346).  Aside:  As a starting graduate student my office was next door to George Beadle’s office.  He was quite old by that time, but still a charming man.  The main thing I learned from him was just how interesting corn was, and that despite what Paul Mangelsdorf might say, teosinte WAS the progenitor of corn.  As a result the Doebeli work made me very happy!.

 

corn-and-teosinte_h1

http://nrm101-summer2010.community.uaf.edu/2010/07/12/a-history-of-corn/

Doebeli identified a pair of interacting loci (BV302 and UMC 107 if you care).  He put these QTL into a teosinte background and measured the “percent of cupules lacking a spikelet”.  Don’t fool yourself, it was percent corn like kernels.   Because it was crossed into a teosinte background even under the best of circumstances the kernels mostly looked like teosinte.  In any case the values of the nine genotypes are:

teosinte maize genotypic values

Now what we would like to do is calculate the additive genetic variance, the dominance variance, and all the epistatic components of variance.  These will, of course, change as the gene frequencies change.   As a result we need to use a statistical method to calculate the variances.  It actually turns out to be surprisingly easy to do this.  My favorite program for this is JMP, although I have to say SAS has gotten more corporate over the years, and I suspect someday I will quit getting free licenses from my university and switch over to R, which is way more powerful.  I will show you the method using JMP, but it really shouldn’t take too much to translate it into any language you might be interested in.

To calculate the variance components you first need to make a table that lists the genotypes, and for each genotype its frequency and its genotypic value.  Finally, you need to list the eight independent genetic variance components.  The independent variance components should be weighted so that the variance of each contrast is equal to one (use maximum likelihood weighting N, not the BLUE N-1 weighting or the trick won’t work).  You will then want to do a linear regression weighted by the genotype frequencies of the actual genotypic values on the theoretical values

Note that the order is important since for this to work correctly you need to use type one, or sequential sums of squares.  SAS and JMP insist on type 3, and they WILL give you the wrong answer.  However, with a little digging you can ask for sequential sums of squares.   So, in JMP your data table should look like this:

JMP data sheet

Note that there are a lot of hidden columns here.  If you have JMP I am happy to send you a working file of this.  Just shoot me an email.

Your model statement should look like this:

JMP model statement

Now, here is the beauty of this little trick.  When you have done the multiple regression as I outlined it, being careful to enter the independent contrasts in the correct order, making sure they have a variance of one, and using sequential type one sums of squares, you can go to the ANOVA summary and simply read off the variance components.  That is the variance due to regression due to the Additive_A contrast is the additive genetic variance due to the A locus etc.  Thus, in our teosinte maize cross at a gene frequency of 0.5 for both loci the Anova table looks like this:

var regressions p=0.5

and the variance components can simply be read off of the table. For example the additive genetic variance due to the A locus is 0.55125.  Importantly, if the gene frequencies change so do the variance components.

var components Maize = 0.25            var components maize =0.75

Corn alleles = 0.25                                                                 Corn alleles = 0.75

The interesting point is that there is a shifting of the variance components as gene frequencies change.  Importantly in a teosinte type genotype there is almost no additive genetic variance.  As corn like characteristics are selected the additive genetic variance blossoms.  This leaves one with the interesting speculation that as corn was selected the process accelerated as the shifts in variance components enhanced the response to selection.  It is no wonder that it was so hard to find the progenitor of corn.  Teosinte does not respond to selection very well, and yet corn apparently evolved very rapidly.  Perhaps now we know why.

Calculating the local average effects is done by calculating the weighted average of replacing a randomly chosen allele in a randomly chosen individual with the allele of interest.  For example, if we wanted the local average effect of the A allele, some of the changes we would get include:

Genotype before substitution genotype after substitution
AABB AABB
AaBB ½ AaBB, ½ AABB
aaBB AaBB

The local average effect is then simply taken as the average of the difference in phenotypes before and after the substitution.  If we do this for the BV302 allele in the teosinte corn cross we get the following figure:

teosinte maize lae graph

Note that local average effects correct for the population mean, and are always a weighted deviation from zero; however be aware that the population mean is becoming much more “corn” like, so that it is actually the teosinte allele that is relatively flat and the Maize allele that is sailing up into the heights of corn-ness.  Despite the perspective issue, it can still be seen why there is no additive genetic variance in the teosinte background.  The alleles that make corn corn are nearly neutral in the teosinte background, and do not have a major effect on the phenotype until the corn suite of genes (or at least UMC107) become more common.

I am rather over-interpreting the data since it is only a single pair of loci being examined rather than the whole genome, nevertheless, this provides some interesting speculation that many of the loci that originally gave rise to corn was originally nearly neutral variation in the ancestral teosinte, and it was the actual process of domestication that released this variation that was locked up in epistatic combinations.  I am left wondering just how much genetic variation is hidden in loci that are nearly neutral in one genetic background, but become decidedly non-neutral as a population responds to selection

As with the JMP program, if anybody is interested in am excel spread sheet for calculating local average effects and local breeding values for two locus systems shoot me an email and I will be happy to send you one that I have.

Physiological and Statistical Epistasis

Last week I gave an informal seminar for our Friday lunch group, and realized that not everybody knows what I am talking about when I talk about epistasis.  I didn’t do a “what is epistasis” post originally because this was supposed to be a blog about the phenotype, and well, epistasis is genetics.  That said, much of what I want to talk about is about epistasis, so here is a bit of remediation for my blog.  If you know all about this stuff you can skip it. . .

Our original models of population genetics were based on the “additive dominance” model.  That is, all loci act independently.  We have to allow dominance, since its existence has been obvious since Mendel’s time.  This assumption of only additive effects was originally quite reasonable.  Fisher fully understood that there were interactions among loci, and he is in fact the person who first termed these interactions “epistasis”.  However, in developing his model he assumed an infinite number of loci, random interactions, and an infinitely large population size.  This is a reasonable first set of assumptions for developing the (then) brand new fields of population genetics and quantitative genetics.  Physicists and complex systems people will know this as the “mean field approximation”, and they will both laud the successes based on the mean field approximation, and lament the limitations if it.

The problem comes when we start relaxing these assumptions, and allow finite population sizes, non-random interactions, and a finite number of loci.  In this case gene interaction starts to become much more important.  When it comes to gene interaction we know that enzyme chains can be quite long, and that there are lots of other subtle interactions, thus, from one perspective there must be epistatic interactions involving tens or even hundreds of loci.  Although these interactions are undoubtedly real, there are a number of reasons to think that they can be reasonably ignored, at least over short evolutionary time scales.  The main one, is that as a variance component the contribution of additive type epistasis (AXA, AXAXA etc) to the variance among demes for an N way interaction is approximately proportional to 2 times the inbreeding coefficient raised to the Nth power (≈(2f)NVAx. . . ).  In other words, if there were large variances do to high order epistasis populations would suddenly blow up and additive genetic variance would blossom out of nowhere when populations were inbred.  This doesn’t happen.  As a result, I work with two locus models, figuring that going above two loci is a lot of work for very little return.  In any case these low order models are sufficient to move beyond the rarified additivistic world we grew up in (that’s supposed to be a pun on atavistic).

In any case, if we have a system with two alleles at each of two loci there are 9 possible genotypes.  The genotypic values are numbers, and if you know your statistics you know that a set of nine numbers has a mean and 8 degrees of freedom.  In other words these 9 numbers can always be divided into a set of 8 vectors that are independent of each other.  Any set of 8 vectors that are orthogonal will do, but I like to use the following 8 (sigh.  If I was any good with HTML this would be formatted better. . .):

Additive A locus                                                     Additive B locus

A1A1 A1A2 A2A2 A1A1 A1A2 A2A2
B1B1 1 0 –1 B1B1 1 1 1
B1B2 1 0 –1 B1B2 0 0 0
B2B2 1 0 –1 B2B2 –1 –1 –1

 

Dominance A locus                                                Dominance B locus

A1A1 A1A2 A2A2 A1A1 A1A2 A2A2
B1B1 –1 1 –1 B1B1 –1 –1 –1
B1B2 –1 1 –1 B1B2 1 1 1
B2B2 –1 1 –1 B2B2 –1 –1 –1

 

Additive by Additive Epistasis                        Additive by Dominance Epistasis

A1A1 A1A2 A2A2 A1A1 A1A2 A2A2
B1B1 1 0 –1 B1B1 1 0 –1
B1B2 0 0 0 B1B2 –1 0 1
B2B2 –1 0 1 B2B2 1 0 –1

 

Dominance by Additive Epistasis               Dominance by Dominance Epistasis

A1A1 A1A2 A2A2 A1A1 A1A2 A2A2
B1B1 1 –1 1 B1B1 –1 1 –1
B1B2 0 0 0 B1B2 1 –1 1
B2B2 –1 1 –1 B2B2 –1 1 –1

 

Statisticians in the crowd will hate these, since these are not traditional orthogonal contrasts (they don’t all add up to zero), but they are independent of each other, and they do work.  (Making them orthogonal contrasts that obey the rules changes the intercepts, but otherwise has no effect other than making them less aesthetic).

These values are what Jim Cheverud calls “physiological epistasis”.  That is they are fixed genotypic values that are constants regardless of the characteristics of the population in which they are measured.  This is not particularly interesting for studying evolution.  Instead we need to think about “statistical epistasis”.  Statistical epistasis is genetic variance that can be attributed to gene interaction.  Unlike physiological epistasis, statistical epistasis is a property of the individual and the population in which they are measured.

A quick analogy is appropriate here.  I have a fixed height (five feet 5 inches if you must know).  I have that height whether I am measured in Holland or Peru.  However, if I want to watch a parade, well, in Holland, where people tend to be tall, I am short and since I am an adult at the back of the pack I probably won’t see much.  In contrast, in Peru, where people tend to be much shorter, I will be relatively tall, and very likely I will be able to see the parade.  My height of 5’ 5” is a fixed value similar to the genotypic values of physiological epistasis, whereas whether I am tall or short is similar to statistical epistasis.

So to convert physiological epistasis into statistical epistasis you need to partition the variance in genotypic values into statistical variance components.  This is done by the old platitude of “doing a regression of phenotype on genotype”.  Following Falconer and MacKay, a good graph of the additive genetic variance for one locus with dominance:

epistasis tutorial fig 1

In this figure the additive genetic variance is the variance due to regression, and the dominance variance is the residual variance.  For additive by additive epistasis we need to do a multiple regression, and resulting in a 3 dimensional graph:

Epistasis tutorial fig 2

This graph shows two slopes.  The red plane is the regression of phenotype on genotype for the frequency of the A1 allele = 0.5, and the brown plane is the regression for the A1 allele = 0.25.  For both regressions the frequency of the B1 allele = 0.5.

The important point is that the statistical components of variance change as gene frequencies change.  In the above example, when the gene frequency of the A1 allele changes from 0.5 to 0.25 the additive genetic variance at the B locus changes from zero to being non-zero with the B2 allele favored.  If the gene frequency of the A1 changed to 0.75 the additive genetic variance for the B locus would also have increased, but instead it would be B1 allele that was favored (and the graph would not have been as pretty).  In the real world we generally do not have access to the actual interacting loci, or their gene frequencies.  Thus, we simply have to recognize that the additive genetic variance changes in a complex way as inbreeding, selection and drift act to change the underlying gene frequencies in complex manners.

Metapopulation Quantitative Genetics

Last week I talked about how Fisher’s original definition of additive genetic variance failed when average effects don’t equal average excesses. Basically this happens when ever there is some form of linkage disequilibrium. This gives us the easiest framework in which to think about this situation, that is in a “metapopulation”. A metapopulation is a population of populations that are partially connected by some level of migration, although being derived from a single ancestral population also works.

So, in theory Fisher’s additive genetic variance would technically be measured across the entire metapopulation. This would, of course be useless at best, and more likely down right misleading. The obvious solution is to measure the additive genetic variance separately for each population, but that leaves us with the problem of what do we do with migrants? In particular, what is the fate of genes as they move from one population to other. In an additive world we know what will happen. The gene will have the same effect on the phenotype as it did before.

Because the average effect is a function of the population mean, it will change, but that shift will be consistent for all alleles. Thus, if we considered two alleles. In a particular sub-population of interest allele A1 on average subtracts 1.5 inches to the height of an individual relative to the mean, and allele A2 on average adds 2 inches to the height of an individual relative to the mean. In this case the difference between the two alleles is 3.5 inches. If we move both of those alleles into another deme and measure their average effects will change because the population mean will change, but the difference between them will remain a constant 3.5 inches.

met quant gen fig 1

What this means is that if we are selecting for taller stature the A2 allele is always better than the A1 allele, regardless of which population we measure it in.

Fisher’s average effects would technically be measured as a deviation from the metapopulation mean averaged across all of the subpopulations. This would of course be a useless measure. Instead, we want to measure a separate average effect for each subpopulation, but continue to measure it as a deviation from the metapopulation mean. To be consistent with Fisher I have re-named these as “local average effects”.

If we add gene interaction things become more complicated. I will save you the math, but if you are interested in Goodnight 2000 (Heredity 84: 587-598) I used a Markov chain two-locus two-allele approach to examine these local average effects for all of the possible two-locus genetic effects. In this I found that the best thing to measure was the “variance in local average effects”, which conceptually is the variance in what a single allele does to the phenotype after correcting for the population mean. As you might expect from the figure above the variance in the local average effect in an additive system is zero. In a non-additive system it is very much non-zero. For a system with dominance or epistasis the result would look more like this:

met quant gen fig 2

That is, the relative difference between our pair of alleles shifts as they are moved among subpopulations, and in some cases may even reverse. In other words, the variance in local average effects is decidedly not zero.

The beauty of the Markov chain approach is that ANY one or two locus interaction an be examined. The first step is to examine the 8 “pure” forms of genetic effects (additive A locus, additive B locus, dominance A locus, dominance B locus, additive by additive epistasis, additive by dominance epistasis, dominance by additive epistasis and dominance by dominance epistasis). To examine these I start at a gene frequency of 0.5 for both loci, and allow them to undergo drift, which eventually leads to fixation and loss of all variation.

The first conclusion is that all of the forms of gene interaction show a conversion to additive genetic variance. Strictly additive effects decline linearly as inbreeding coefficient increases (yes, the traditional (1-f)σ2 works in the additive world). The other effects show a maximal conversion at an intermediate inbreeding coefficient.

drift and epistasis fig 2 VA

More interesting is the variance in the local average effects of alleles:

Drift and epistasis LAE graph

Importantly the variance in the local average effects for an additive system is zero, but it is non-zero for all other forms of genetic effects.

So, what this means in common terms is that the same allele is doing different things in different subpopulations. In some subpopulations it may increase the size of an individual, and in others it may decrease the size of an individual. An allele may be the “good gene” in one subpopulation, and the “bad gene” in another population.

In short, what this is saying is that there are two types of population differentiation. We are all familiar with populations differentiating for their mean. In fact, in introductory population genetics we are told that the genetic variance among demes is 2fσ2. This is simply the expected variance among demes in which only additive effects are acting. It is not surprising that we never learned about differentiation for average effects, if for no other reason than it does not occur in additive systems. With non-additive systems the differentiation for average effects is important, and often very uncoupled with the variance in population means. Compare the variance in population means below with the variance in local average effects above.

means by inbreeding

Note that there is little correlation between differentiation for population means and differentiation for local average effects. Additive effects cause substantial differentiation for population means, but none for local average effects. Conversely dominance by dominance epistasis causes virtually no differentiation for population means, but has a huge effect on local average effects.

There has always been the question about how differentiated do two populations have to be before they can be considered separate species. When that question is asked people are usually asking how different do the population means have to be before they can be considered different species. Note that this includes questions such as how great a genetic distance do you need for speciation to occur. It should be clear at this point that this is the wrong question to ask. It is rather like asking what color car can go the fastest. A better question would be how differentiated for local average effects do two populations have to be before they can be considered different species. I don’t have the answer for that, but this is a question that has a meaningful answer and ends up being a substantial change in perspective.

car colors

Which color is fastest?

Finally, returning to my blog of two weeks ago, I suggested that one migrant every four generations was optimal. At the time I based it on the additive genetic variance:

VA by migration color

but now I can add the second part of the story:

V LAE by migration color

The variance in local average effects declines as the number of migrants increases. At one migrant every 4 generations the additive variance is maximized, and the variance in local average effects is still very large. Basically one migrant every four generations is the sweet spot between maximizing both the additive genetic variance and the variance in local average effects.

Of Average Effects, Average Excesses and Additive Variance

Today’s post is a bit of a CYA (cover your a__) post.  I once got in a bit of a kerfuffle with Warren Ewens over my loose use of the concept of additive genetic variance.  It was a little like me bringing a knife to a gunfight.  I was hopelessly outclassed by his intellect, and he appropriately schooled me.  This is my attempt to make sure that people like Ewens understand that I now am aware of some of the technical issues surrounding additive genetic variance.

Sir Ronald Fisher defined the additive genetic variance to be the covariance between the average effect of allele and the average excess.  When mating is random, and everything is in linkage equilibrium this rather esoteric definition is equal to four times the similarity among half sibs, and it is predictive of the response to selection.  In other words, Fisher, very elegantly made a very precise definition of additive genetic variance that has many important implications for (his) real world.

So, it is worth briefly defining the average effect and the average excess.  The genotypic value is the average phenotype of a particular genotype averaged across all of the genetic and environmental variants it might experience.  Thus, any given individual will have a phenotype that is different than its genotypic value due to unexplained causes (usually called environment), but the members of a particular genotype will average out to the genotypic value.  Thus, for a one locus two allele system you could get A1A1 = g11, A1A2=g12, and A2A2=g22.

We can now define the average effect of a gene substitution to be the average change in phenotype of an individual if a random allele is replaced by the allele in question.  It is usually measured as a deviation from the population mean, but that doesn’t actually affect the calculations.  Thus, the average effect of the A1 allele can be calculated by taking the average of a substitution into all of the possible genotypes:

A1A1 => A1A1 = g11 – g11 = 0

A1A2 => ½ A1A2 + ½ A1A1 = ½ (g12 – g12) + ½ (g11 – g12) = ½ (g11 – g12)

A2A2 => A1A2 = g12 – g22

The average effect of the A1 allele is then just the weighted average of these changes.

On the other hand, the average excess is the average phenotype measured as a deviation from the population mean (and here it is important) of an individual that contains a particular allele.  Thus, the average excess of the A1 allele is the weighted average of two times the A1A1 genotype (because there are two A1 alleles), i.e., 2(g11 – M), and the A1A2 genotype, (g12 – M) where M is the population mean phenotype.

Oddly enough if the population is in multilocus HW equilibrium and there are no genotype environment interactions or correlations the average effect is exactly equal to the average effect.  Yea, I didn’t believe it either, but it does work.  So, in our perfect world the additive genetic variance equals the covariance between average excesses and the average effects, equals the variance in average effects, equals four times the variance among half sibs, equals the twice the covariance between parents and offspring, equals the response to selection.  It’s amazing.

The difference between the average effect of an allele and the average excess of an allele is that the average effect measures the effect of an allele averaged over all genotypes based on their expected frequency, whereas the average excess measures the effect of an allele in the associations it is actually found in.

The problem comes when mating is not random and there is linkage disequilibrium. In this case the average effect and the average excess are not equal, and the relationship among the different measures is broken.  Falconer tried to work this out and ultimately declared that when average excesses did not equal average effects efforts to measure the additive genetic variance were “doomed to failure” (Falconer 1985 Genetical Research Cambridge 46: 337-347).  In short, when the average excess and the average effect are not equal, which is probably always in the real world, the additive genetic variance is an unmeasurable statistic with no predictive value.  Good thing its central to quantitative genetic theory, huh?

So, what to do?  Ewens insists to me that since Fisher defined the additive genetic variance that is the correct definition, and we have to give it to him.  My thought is that we should do, perhaps with circular overtones, is to drop the “genetic” and define the “effective additive variance” (eVA) to be the variance that can contribute to a response to individual selection.

If we define effective eVA as the variance that can result in a response to selection just about everything heritable can be called “eVA”, including things that are decidedly NOT genetic.  One solution is to restrict the eVA to genetic components of the patterning node.  This takes care of perhaps one of the biggest problems with Fisher’s definition, which is that mating is not random.  There actually is one study that took this into account (Tonsor and Goodnight 1997.  Evolution 51: 1773-1784).  This study is interesting in that we did a standard half-sib mating design, however, instead of choosing mates from a uniform random distribution, we chose them randomly from an exponential distribution that had the same shape as the pollen flow distribution.  This modification had virtually no effect on the eVA, but that may have been the result of the choice of using a wide-spread weed.  However, it does not solve the problem that for any phenotypically based measure it will be virtually impossible to definitively distinguish “genetic” from “epigenetic” and “non-genetic”.  Thus, at this point we should probably punt, call it “effective additive variance” rather than “effective additive genetic variance”, and recognize that anything that contributes to the resemblance between parents and offspring will potentially get pooled into the effective additive variance.

This actually presents an interesting issue, since we know that different forms of inheritance have very different fidelities, for want of a better term.  That is the fidelity of nuclear DNA is tremendous, as evidenced by genetic diseases that can run in families for generations.  Conversely a single disruption of culture (such as a newborn getting adopted) can destroy a legacy of cultural inheritance.  Using this reasoning, it becomes obvious to me that it is still important to use clever experimental designs to divide the effective additive variance into components, such as effective nuclear genetic additive variance, effective maternal additive variance, effective epigenetic additive variance, and so on.

 

Sewall Wright, Speciation and Migration

Many thanks to several members of the Zufall lab for bringing me up to date on mutation accumulation experiments .  Her lab is actively involved in MA experiments (http://www.genetics.org/content/early/2013/07/29/genetics.113.153536.abstract), and what was satisfying to me, quite interested in dissecting out dominance and epistatic effects of new mutants.  Hopefully we will get more data on this in the future.  In any case, lets move on to the last, and perhaps, from a theoretical perspective, least interesting, of the four forces of evolution, migration.

To point out just how mundane it is, consider the infinite alleles model of mutation drift balance:

Migration equation 2

All we need to change this into migration drift balance is change the font for the mu from symbol to times. . .

Migration equation 1

More seriously, it is fun to consider that old chestnut “one migrant every other generation destroys genetic variation”.  I have no idea where this form comes from, but ultimately it is derived from Wright (1931. Genetics 16: 93-159) in which he says “How little interchange would appear necessary to hold a large population together may be seen from the consideration that m = 1/2N means an interchange of only one individual every other generation, regardless of the size of the subgroup.”  Interestingly he goes on to say “However, this estimate must be much qualified by the consideration that the effective N of the formula is in general much smaller than the actual size of the population or even than the breeding stock, and by the further consideration that q, of the formula refers to the gene frequency of actual migrants and that a further factor must be included if q, is to refer to the species as a whole. Taking both of these into account, it would appear that an interchange of the order of thousands of individuals per generation between neighboring subgroups of a widely distributed species might well be insufficient to prevent a considerable random drifting apart in their genetic compositions.”  By the way, if you were hoping I would say something scandalous about Sewall Wright, other than his penchant for erasing black boards with guinea pigs, well it ain’t gunna happen.  I had the great privilege of meeting him several times, the first time when I was a starting graduate student.  He is one of my heroes.

So, what was he really saying about migration.  He is absolutely correct that migration will tie a metapopulation together genetically.  Looking at his original figure you can see that what happens in a two allele system is that when m = 1/2N is that every gene frequency is equally likely.

Wright Figure

(From Wright 1931. Genetics 16: 93-159)

Note that this shows just how miss-stated that old platitude really is.  Far from destroying genetic variance, all gene frequencies are equally likely (at a gene frequency of 0.5), or in other words, anything is possible.  More important, it is a statement about variance in gene frequency, not variance in phenotype, and of course, there is no epistasis.  This last is hardly surprising as this was published 50 some odd years before the first theoretical treatment of the effect of epistasis on phenotypic variance (Goodnight 1983. Ph.D. thesis University of Chicago – I have to establish my bragging rights at some point).

If others can establish silly rules of thumb about migration and variation, I can do the same thing.  In my chapter on metapopulation quantitative genetics in Hanski and Gaggiotti’s book (Goodnight 2004. In: Ecology, Genetics and Evolution of Metapopulations. Hanski and Goggiotti. eds.) I make the perhaps rash assumption that we can solve for the standard approximation:

Migration equation 3

and using this to calculate the equilibrium additive genetic variance due to various forms of gene interactions we can get the following graph:

VA by migrants graph

Additive genetic variance within demes as a function the number of migrants per deme at equilibrium.  Total genetic variance in the outbred population (F=0) is standardized at 1.  (from Goodnight 2004. In: Ecology, Genetics and Evolution of Metapopulations. Hanski and Goggiotti. eds.)

There are a couple of things to notice about this graph.  First, as you might expect, for simple additive systems the greater the migration rate the greater the additive genetic variance.  Thus, in a purely additive world the additive genetic variance is a simple tradeoff with population differentiation.  The greater the migration, the greater the VA, and the less the population differentiation.  With other forms of genetic effects the tradeoff is not quite so simple.  Isolation leads to fixation, and with it the conversion of the gene interaction into additive genetic variance.  Too little migration and you get simple fixation, and no genetic variance, additive or otherwise.  To much migration and the populations are effectively panmictic, and there is no conversion taking place. This tradeoff is readily seen in the figure above.  What can be seen is that for simple dominance the greatest conversion takes place around one migrant every other generation (M = Nm = ½).  For the other forms of genetic interaction it can be seen that the additive genetic variance is maximized at around one migrant every 4 generations (M = Nm = ¼).

This raises an important point, and I am realizing I am out of space to really discuss it properly.  It will probably take at least two weeks, but I will discuss how there are two types of population differentiation:  differentiation for population means and differentiation for genetic effects.  It turns out that an increase in additive genetic variance with increasing inbreeding coefficient is clear evidence for a shift in what genes are doing to the phenotype in different populations.  Thus, the increase in additive genetic variance we see as a function of migration rate means that the subpopulations in the metapopulation are differentiating.  This means that we can utter our very own migration platitude:

One migrant every four generations is optimal for speciation to occur.

Do I believe it?  Well I certainly think it is more believable than the platitude I uttered at the beginning of this essay.

As a final note:  The semester has started.  I will try to keep up my weekly post, but please be patient if I start to fall short as things heat up.

Mutation accumulation experiments

It is important to recognize that the need to move over to a phenotypic view of evolution is not just about incorporating continuous inheritance into the transition equation.  Even restricting ourselves to particulate inheritance and nuclear genetics the phenotypic approach gives insights that are simply missed using a genic approach.  In this post I talk about one of the more entertaining lines of research, mutation accumulation experiments, and how the color and the direction of the research might change using a phenotypic approach.

From a phenotypic perspective what we care about when studying mutations is how they affect the phenotype.  Thus, as pointed out last week, we honestly don’t care about the details, rather we care about how mutations change phenotypes and most importantly how they change the transition equation in general, and the additive genetic variance in particular.  The way we have addressed this thus far is with mutation accumulation experiments.

In a typical mutation accumulation experiment a single inbred and highly homozygous line is replicated.  Each of the replicated lines is maintained at a very small population size (usually brother sister mating, or in plants, selfing).  These lines are maintained for many generations.  During that time mutations accumulate, and the lines generally decline in fitness, and the variance among lines increases.

Mutation accumulation expt
“A typical mutation accumulation (MA) experiment in a selfing diploid organism. An ancestral line is split into nMA lines, which are then allowed to accumulate mutations for t generations. Mutations (represented as colored blocks within chromosomes) are heterozygous when they first appear but can become fixed or lost in subsequent generations. After t generations, lines are expected to have accumulated different sets of mutations. . . .  [T]his leads to . . . an increase in among-line variation.” (copied from Halligan and Keightly 2009. Ann. Rev. Ecol. and Syst. 40: 151-172)

These data are used in two ways. The less interesting, from our perspective, is that there is usually a decline in fitness demonstrating that most mutations are at least mildly deleterious.  The other, which I want to focus on, is that the variation among lines is used to estimate the mutation variance Vm, the variance introduced into a population each generation as a result of mutation.  This in turn can be used to estimate mutational heritability, MA equation 1 , where Ve is the environmental variance.  Here in lies the problem:  It is tempting to equate Vm with the change in additive genetic variance.  To their credit Halligan and Keightly do not suggest that Vm is a measure of the increase in additive genetic variance, so this is actually not a criticism of the research, but rather the tempting interpretation of the research.

To see what is wrong with the additive interpretation of MA experiments, it is first necessary to recognize that variance components, and indeed, the “effect” of an allele is always a property of the population in which it is measured.  In a typical MA experiment new mutations will always occur in a homozygous background, thus, and interaction effect is always expressed as a main effect of the allele in this simplified genetic background.  To see this, consider the AXA genetic system I have introduced on several occasions.  Imagine a set of MA lines is fixed for the AABB genotype, and a mutation occurs converting A to a.  In this case the mutant population becomes aaBB, and the phenotype is converted from +1 to -1.

AXA epistasis MA

Notice that in this situation we only see the A1/A2 allele difference as an additive effect.  Since the B2 allele is not present in this experiment we will never see that the A and B locus epistatically interact.  Even if we cross it back to the ancestral population the best we will get is a measure of dominance (in this example there is no dominance).  The point is that mutations occur in the context of a genetic background.  When they occur in typical natural population they are occurring in a variable and, due to sexual reproduction, changing genetic background.  This is very different from the simple homozygous genetic backgrounds of MA experiments.  As they are currently executed MA experiments tell us that mutations occur, but unfortunately, they tell us little about how these mutations affect the response to selection, or what they do to the phenotype in any particular population.

One of the nice things about the Halligan and Keightly review is that they suggest future directions for research.  This is where the genic view and the phenotypic view really part company.  The suggestions they give all involve using whole genome sequencing to get a better handle on mutation rates, the distribution of genetic effects, and their effect on gene expression.  From a phenotypic perspective I would think that a better use of research energy would be to try to estimate the extent to which mutational effects are additive versus epistatic and dominant, and more importantly find a way to estimate the rate at which mutations increase the phenotypic variance and additive genetic variance in a population.  Non-additive effects could be qualitatively measured by examining the variance in local breeding values in the MA lines either by doing a diallel cross among the MA lines, or preferably by crossing the MA lines to a several outbred strains (see De Brito, Pletscher and Cheverud 2005. Evolution 59: 2333-2342 for an example of this experimental design using RI lines).  If the variance in local breeding values was high this would suggest that a mutations effect was largely non-additive, and that mutations were causing population divergence along the lines suggested in the Dobzhansky-Muller model of speciation.  As far as the input of mutations to the additive genetic variance within populations, this could probably be best done using a carefully chosen wild population.  In particular, I would look for a closed (no immigration) natural population with a known long term effective population size that can be bred in the laboratory, something along the lines of the desert pup fish, or a cave arthropod.  I would use this population to measure the additive genetic variance (and phenotypic variance), and using the infinite alleles model of mutation (mutations 1 equation 1) I would estimate the mutation rate that would give me the observed additive genetic variance (the Va should be zero in the absence of mutations).  By solving for the additive mutation rate you are really solving for the “effective additive mutation rate”.  That is the mutations themselves may be highly dominant or epistatic; however this calculation would lump both the additive effects and the converted epistatic effects into a net contribution to the additive genetic variance for mutations, and after all, that is what we really want.

A quick overview of genetic mutations

The third force of evolution is mutation.  There are a lot of platitudes and details about mutation that we just don’t need to care about.  One of my favorites is that (genetic) mutation is both the weakest and strongest evolutionary force.  Weakest because at least genetic mutations occur at a very low rate, and have a much smaller influence on gene frequency than other forces; strongest because it is ultimately the source of all heritable variation.  Others include things like Fisher’s geometric model that says that beneficial mutations tend to be small, and that most mutations are neutral or deleterious.  The details are things like the various types of mutations we recognize – insertion, deletion, point mutations etc.  As far as the phenotypic view goes, to quote Rhett Butler “Frankly, my dear, I don’t give a damn.”

dont give a damn

(http://thoughtsofnobody.files.wordpress.com/2012/03/tumblr_m0bj2zfuq81rqlkypo1_500.jpg)

That is not quite true.  There is one rumor I have heard that, and I emphasize I don’t know if this is true, apparently most of the major Drosophila mutations that Morgan and his students identified were transposable element insertions and deletions, basically with the transposon inserted you get the mutant non-functional form, and with the transposon absent you have the functional gene, or vice versa.   I love this because in traditional population genetics there are two ways of modeling mutations.  One is the two-allele model with reversible mutations:

Mutations 1 equation 2

which seems to fit remarkably well to the transposable element model of mutations.  The other is the “infinite alleles” model, which would seem to fit well with the “point mutations” that we learned about in introductory genetics.

mutations 1 equation 1

If you are unfamiliar with these equation, μ and ν are mutation rates, and the second equation is the standard drift equation found in any population genetics text multiplied by the probability that neither gene being compared is mutant.   I love that with all our new-found knowledge, our original models of mutation, developed before the discovery of DNA were essentially solid models to which modern discoveries have added little beyond validating their basic utility.   But, you say, what about all of the deep structure of the genome, the fact that most mutations are in regulatory regions yadda yadda yadda?  This is where Rhett Butler comes in.  From the phenotypic perspective these facts are fascinating, and important to know, but honestly they are better classified as gene physiology than evolutionary biology.  As an analogy, consider a scientist studying the ecology of some animal, say a rabbit.  They need to know that the rabbit has organs, such as a heart, and that the heart functions, but they do not need to know the details of how it functions.  In contrast a physiologist might care very much how a heart functions, but very little about how that heart helps rabbits avoid predators.  It is much the same for evolutionists:  They need to know that mutations occur, and that the mechanism for generating them exists, but other than that they generally will care very little about the details.  In contrast, the gene physiologist may care very much about the mechanisms that cause mutations, but care very little about what that does to phenotypic variation in a population.

This is not to say that the molecular knowledge of mutations has been without value.  Consider Dollo’s law (Dollo, L. 1893. Bulletin de la Société Belge de Géologie de Paléontologie & D’Hydrologie 7:164-166) in which he states that “An organism is unable to return, even partially, to a previous stage already realized in the ranks of its ancestors.”  In more recent restatements this has been interpreted to mean that a trait, once lost, can never be re-evolved in its ancestral form.  Thus, fish have dorsal fins, but when whales re-evolved them they are a new evolution with significant differences.  That is, once the dorsal fin of fishes was lost it was lost forever, and when whales returned to the sea they had to re-evolve them.  This is a nice idea, and often, as in the case of whales, appears to be true, but there have since been several good examples of apparent reversals.  This ability to re-generate lost traits actually makes sense now that we know that most mutations are in regulatory regions.  That is, it is reasonable to presume that regulatory mutations simply “turn off” the trait.  It is then presumably possible to turn it back on by evolving a new regulatory pathway.  As a caveat:  There are almost certainly limits on this because the unexpressed gene would be insulated from selection, and presumably eventually be destroyed by mutations that are neutral as long as the gene is not expressed.

Since I seem to have about used up my allotment this week talking about genetics, I will mention one of the more delightful graphs I have seen in recent years:

Components_of_the_Human_Genome

http://commons.wikimedia.org/wiki/File:Components_of_the_Human_Genome.jpg

I see two things that interest me.  First, remember all that energy you spent learning the triplicate code in genetics class?  Well it explains 2% of the genome.  98% pays no attention to it.  I am not sure what to make of that, except perhaps that there sure is a lot we don’t know about the genome, and that the McDonald/Kreitman test (McDonald and Kreitman 1991. Nature 351: 652-654), and other tests that depend on comparing synonymous and non-synonymous mutations don’t work for 98% of the genome.  The other thing I find fascinating is that LINEs, SINEs, and various transposons account for over 40% of the genome.  As far as we know these are mostly dead viruses and active or inactive transposable elements, or in other words junk and parasitic DNA.  Its not particularly relevant to a discussion of mutations, but it still amazes me.

So, to bring this full circle, from a phenotypic perspective we really don’t care very much about mechanisms of mutations, although there are a number of examples where knowledge of mechanism helped our understanding of the effect of genetic mutations on the phenotype.  However, it is really a good thing that we don’t need to know much about the mechanism, because even at this point there is little we do know about the genome and the effects of mutations.  After all, the paradigmatic studies of DNA and the discovery of the triplicate code still leaves us with 98% of the genome that we really don’t understand.  Thus the bottom line is that there is a lot we don’t know about mutations, and that is ok because in general we don’t need to know most of it.

Drift of continuous traits.

Wikipedia claims that there are no formal models of blending inheritance, although I would be inclined to disagree with this, as Fisher 1930 discusses blending inheritance in enough detail to call it a formal model.  Further, one older web site defines blending inheritance as “A discredited model of inheritance suggesting that the characteristics of an individual result from the smooth blending of fluid like influences from its parents.” (http://groups.molbiosci.northwestern.edu/holmgren/Glossary/Definitions/Def-B/blending_inheritance.html).  Nevertheless, it should be clear that any phenotypic view of evolution will have to include some continuously inherited aspects.  As I have indicated, in the phenotypic view it is phenotypes that create new phenotypes, and that new phenotypes are defined by a transition equation that is determined by the patterning node.  The patterning node, as I have described it, is specifically those influences on a phenotype that are “heritable” in the sense that a trait value in the parent/teacher phenotype in some manner defines the trait value in the offspring/student phenotype.  By the way, it seems to me that “parent” and “offspring” are loaded enough terms that I am looking for a term that would include both parents and non-parent mentors, both of which can contribute to non-genetical inheritance, and the phenotype to phenotype transition equation.

For genetically transmitted traits the rules are quite clear:  for some aspects of the transition equation the simple rules of Mendelian inheritance will suffice, for others some modification of the quantitative genetic concept of heritability will be needed.  The beauty of genetic traits is that because they have particulate inheritance, and we are diploid, there is “hidden” variation, the “segregation variance”, not to mention variance that is tied up in epistatic and dominance relationships, but can feed into the heritability (see last weeks post).  The result is that with quantitative genetics we have the best of all possible worlds.  Indeed, you can basically think of Fisher as having made the world safe for the Biometicians by giving them an explanation for why they can ignore the associated loss of heritable variance.  That is, quantitative genetics assumes “blending inheritance” with no loss of heritable variance.

The problem comes when we have traits that truly follow blending inheritance.  Mendel is not going to save us from the fact that culture and language are continuously inherited.  Fortunately a basic model was provided for us by Fisher.  Of course, Fisher was dealing only with bi-parental inheritance, but the logic still works.  The idea is that if the offspring/student learns from exactly two individuals (i.e., their parents), and those two individuals each contribute equally then the variance among the offspring will be the variance of the average of the two parents.  If we assume random mating we know that the variance of an average is simply the variance of an individual divided by the sample size, n.  Thus the variance of the offspring is equal to V(parents)/2.  Obviously, in the absence of mutational input the variance will go down by one half per generation until it asymptotically approaches zero.

Of course learned traits are never learned perfectly, thus, it is natural to assume that there will be some level of mutation in the transmission process.  In this case we can solve for the steady state where

Cultural drift eq 1

It should be apparent that this balance occurs when the variance in the trait is equal to 2*V(mutation).  Thus, in this simple case of bi-parental inheritance we can say that the heritable variance will be equal to twice the mutational variance for the trait.

More generally, an offspring/student may have multiple individuals that contribute to their patterning node.  Thus, the transition equation for the trait may be a weighted average of a larger set of parents and teachers.  In that case:

Cultural drift equation 2

where wti is the weight given to the teachings of the ith parent/teacher.  I assume that the sum of the weights is equal to one.  Note that with equal weighting as the number of parent/teachers go up the equilibrium variance goes down rapidly, and quickly becomes nearly equal to the mutation rate.

This raises the interesting point that the effects of drift on a continuously inherited trait is a function of not only the population size, but also the details of how it is inherited.  If an new phenotype learns its trait value by sampling all or most of the community, and copies it as closely as possible there were be essentially no heritable variance.  In contrast, if the trait is acquired by copying one model, and then modifying it substantially (i.e., learning it badly, and making it up so it is similar) there will be substantial heritable variance.  This is something that we rarely think about – the mating and interaction structure of a population can radically influence the amount of heritable variance.  For genetic systems this manipulation primarily takes the form of changes in mating structure, and manipulation of the degree of inbreeding (more on this later).  With blending inheritance there are many more ways that the degree of heritable variance can be manipulated.  For some traits there may be tight formal teaching by numerous teaching, with “mutations” carefully controlled.  These traits become very stable, and can change very little over long periods of time.

Balanese danceBalinese dance 2 

(Left: http://lauradavis.net/roadmap/virtual-vacation-dancing-our-way-through-bali-day-25  Right http://www.coloradocollege.edu/events/2013-05-04-a-celebration-of-balinese-performing-arts) Balinese dance is carefully taught, and has become a very stable part of the culture of Bali.

On the other hand there are traits that are very malleable, fashion comes to mind, in which a premium is placed on innovation and being unique, but not too unique.  In this case a premium may be placed on “mutation”, and individuals may choose to heavily weight some “teachers” over others.

cnn pll-fashion-four-girls-story-top

“(Here) you have Aria’s (from left) vintage rock-and-roll … Hanna’s high-end glam with her famous pops of color, you have the softness of Spencer … and my sexy, tough and modern Emily,” costume designer Mandi Line said. “Aria is my fantasy doll, Hanna is my high school me, Spencer is who I learn from the most, and Emily comes the most natural to me.” (http://www.cnn.com/2012/09/05/showbiz/tv/pretty-little-liars-fall-fashion-tv)

Drift and Epistasis: The odd effects of small population sizes

This week I want to return to specifically genetic drift (as opposed to drift in general).  First, a bit of history.  Back in 1984 there was a pair of competing articles on founder event speciation.  In that exchange Barton and Charlesworth (1984. Annual Review of Ecology and Systematics 15: 133-164) responding to Carson and Templeton (1984 Annual Review of Ecology and Systematics 15: 97-131) make the claim that genetic drift does little more than deplete genetic variance, and reduce the number of alleles.  Based on the additive models that were available at the time they were exactly right.  Genetic drift increases Wright’s inbreeding coefficient, and with it decreases the additive genetic variance.  In this additive world drift decreases the ability of a population to adapt and respond to selection.  Shortly after this exchange took place I published the first of my papers on epistasis and drift.  It is worth looking at how these models can change our perspective on the importance of drift.

Additive genetic variance is a statistical property of a population.  Thus, as gene frequencies change so does the additive genetic variance.  See Falconer and MacKay’s Quantitative Genetics (1996) ( $250.00, are you kidding me?) for a derivation of the dominance situation.  For the epistatic situation the simplest form of epistasis is additive-by-additive (AXA) epistasis.  For this form of epistasis the additive genetic variance is simply the variance of the marginals (this doesn’t work if there is dominance).  This is AXA epistasis because both loci are additive (1, 0, -1), but the direction of effects is a function of the genotype at the interacting locus.  Note that at a gene frequency of 0.5 the marginals, and with it the additive genetic variance, is zero.  If the gene frequency for both loci changes of 0.75 the marginals change as well, and there is additive genetic variance.  In other words, when there is epistasis changing the gene frequency changes the proportion of the total genotypic variance that is expressed as additive genetic variance.

epistasis and drift fig 1 AXA

It turns out that with a two-locus two-allele system there are 9 cells, which means 8 independent genetic effects.  These are Additive A locus, Additive B locus, Dominance A locus, dominance B locus, AXA epistasis, additive-by-dominance and dominance-by-additive (AXD) epistasis, and dominance-by-dominance (DXD) epistasis (Goodnight, C. J. 2000, Modeling gene interaction in structured populations. Epistasis and the evolutionary process. Wolf et al eds.).  These effects are simply numbers filled in as “genotypic values” for each of the nine genotypes.  The genetic variance components are determined by doing a regression of phenotype on genotype that is weighted by the genotypic frequencies.  As a result any given pattern of genotypic values will partition into different variance components, and the partitioning will depend on the gene frequency.   In the system I use, at a gene frequency of 0.5 each of the types of epistasis will be expressed only as its variance component.  Thus, as shown in the example above, at a gene frequency of 0.5 AXA epistasis has only AXA epistatic variance.  At other gene frequencies it will be a mix of AXA and additive variance.

One of the best ways to look at this is using Wright’s inbreeding coefficient, f.   f is formally the correlation between the gametes that combine to form an individual, but it is often convenient to think of it as the probability that two alleles are identical by descent.   If we start at a gene frequency of 0.5, gene frequencies will change randomly, but the inbreeding coefficient will increase in a regular and predictable manner.  Shown below is the expected additive genetic variance for each of these pure forms of genetic variance.   It can be applied to “real” values of pairs of QTL easily. The general method is described in the epistasis chapter cited above.

drift and epistasis fig 2 VA

When I first identified that drift could increase additive genetic variance I found it a bit disconcerting.  Additive genetic variance is the variance in the average effects of alleles (yes, I know the formal definition is a bit different), and nothing magical.  There are only two ways that the additive genetic variance can increase.  One is to increase the number of alleles (drift emphatically does not do that), and the other is to change the average effects of alleles, and not just to change them, but to make them more different from each other.  Since increasing f, by definition, decreases the effective number of alleles, drift must be increasing the additive genetic variance by spreading the average effects of alleles.

What  I found was that when there is gene interaction genetic drift does indeed spread the average effects of alleles, but more importantly in the process it shuffles the average effects.  In the figure below think of the middle line as the ancestral population, and the colored slash marks on the line the average effects of different alleles on an arbitrary scale.  The top and bottom line are two populations derived by founder events, and periods genetic drift.  Note that in both of the derived populations the additive genetic variance (length of the line) has increased, and the rank order of the alleles has been shuffled.  More importantly, the rank order has been shuffled differently in the two populations, such that an allele that is “good” in one population may be “bad” in another population.

drift and epistasis fig 3 LAE scheme

I found that a good way to measure this effect was to look at the “variance in local average effects (LAE) (Goodnight, C. J. 1988. Evolution 42: 441-454).  It turns out that in the traditional linear additive model the variance in LAE is zero, whereas when there is gene interaction it is non-zero.  If the variance in LAE is zero it means that the relative difference between two alleles in their effect on phenotype will be the same regardless of what population they are measured in.  If it is non-zero then the relative difference in the effect of the two alleles will change as you move from population to population.

Drift and epistasis LAE graph

So, I think there are two important implications to think about.  The first is that perhaps the old masters such as Mayr and Carson were on to something.  They were great naturalists and believed that founder events were important in speciation.  Theoreticians come along and tell them they are wrong because their models say that drift cannot cause important evolutionary effects.   Later we discover that simply adding gene interaction changes everything, and perhaps drift can do something important.  The lesson in my mind is that when theory and data are in disagreement then the theory is wrong, and it is the modelers job to figure out WHY their models don’t work.

The second is that this has important things to say about the genic view of evolution.  What we see is that whether a gene is good or bad often depends on the genetic context in which it is found.  In the AXA example the A1 gene is good in a population fixed for the B1 gene, but bad in a population fixed for the B2 gene.  It also shows why assigning fitness to genes is doomed to failure (I love that phrase.  I lifted it from Falconer).  In any given population we can go through the mathematical exercise of assigning a fitness to each allele at each locus, however we will discover that this assignment is critically dependent on the frequencies at other loci, and it will change in an unpredictable way as the details of the genotypic distribution change.  That is, the fitnesses assigned to individual genes will have no predictive power, and will change from generation to generation.

By the way it was MJ Wade who got the gift of his weight in M&Ms

Of um, drift, and M&Ms

Obviously there is much more to talk about concerning selection, and also a considerable amount of excitement about it.  I got an email and a facebook comment that are worth mentioning here.   One is that Michael Morrissy, a research fellow at the University of St. Andrews Scotland sent me a manuscript on using path analysis in selection studies.  It was quite good, but not yet published.  Keep an eye out for it.  The facebook comment was a suggestion that I consider turning this into a book.  The answer to that is yes,  I am thinking about it.  We will see if any publishers are interested.  That said, I want to move on from selection to drift, and come back to specific topics of selection at a later time.

Most are familiar with genetic drift, which is change in gene frequency due to random sampling.  I find that when I teach genetic drift it is one of the hardest concepts to get across, so I will spend a few minutes on it.  For starters, lets consider M&Ms.  On one extreme, we have the case of a friend of mine, who was very fond of chocolate, and well liked by the staff where he was department chair.  When he left to move institutions the secretaries gave him his weight in M&Ms as a parting gift.  He was not a small man, and assuming he weighed about 200 lbs that is about 41000 M&Ms (http://eeunix.ee.usm.maine.edu/~white/mmissue/hardevidence.html).  We can guess that if you figured out the proportions of the colors in that gift it would have been very close to 30% brown, 20% yellow, 20% red, 10% orange, 10% green, and 10% blue.  These are the proportions that the M&Ms people aim for in their bags (http://dealnews.com/features/The-Color-Mixture-in-an-M-Ms-Bag-Is-a-Precise-Science-and-Other-Candy-Facts/626727.html).   On the other hand the “fun size” M&M bags given out at Halloween have, on average 18.36 M&Ms.  Because of their smaller size we expect a great deal more variability in the proportions of each color.  Looking only at the proportion of yellow M&Ms (supposedly 20% of the mix) we find that the median is 3, and the mean is 2.96 (N=25).  Thus, the average proportion yellow in the sample is 16.12%, not too far off from the target.  That said, there is substantial variation among samples, and there is actually a 4% chance that you will not get any yellow M&Ms at all.

Proportion yellow M&Ms

(from http://www.statcrunch.com/5.0/viewreport.php?reportid=9365)

 

The point of this little exercise is that when sample sizes are very large the actual proportions are very similar to the expected proportions.  On the other hand, when sample sizes are very small the actual proportions can deviate substantially from the expected proportions.

Genes are much the same way.  Since we are talking about random sampling with no selection, it really doesn’t matter how we group the genes, that is, the standard grouping of two genes per locus in each individual is nice but not important to us.  Instead all we need to know is the number of genes in the population.  For diploid organisms that will be 2N, or twice the size of the population.  If 2N is very large, then sampling will have very little effect on gene frequencies, on the other hand, if 2N is small then random sampling may have a huge effect on gene frequencies.  Thus, we expect gene frequencies to take a “random walk”, that is the frequency should change randomly from one generation to the next, with the average size of the change being inversely proportional to the population size:

Allele-frequency

“Simulation of genetic drift of 20 unlinked alleles in populations of 10 (top) and 100 (bottom). Drift to fixation is more rapid in the smaller population.” http://en.wikipedia.org/wiki/File:Allele-frequency.png

 

Much has been written about genetic drift, so, at least for the moment, I will leave the discussion at this.  More importantly, we need to talk about how to move from the concept of genetic drift as the effect of random sampling of discrete particles to a more general concept of random change due to small population size that covers both change in numbers of particles, and random change in continuous traits.

The first question to ask is whether or not there are continuous traits that might be subject to random change due to changes in population size.  I can think of, um, one obvious example in language, and that is the use of discourse particles.  Discourse particles are words like “um”, “er”, “uh” (or perhaps “argh” if you are a pirate, and om if you are a yogi) that we unconsciously use in conversation.  These days lecturers make a concerted effort to avoid them but they are an essential part of conversational language, and have been for a very long time (Erard, M. 2007. Um . . . Slips, stumbles and verbal blunders, and what they mean. New York, Pantheon Books).  If you speak Spanish you probably say “este” or “pues”, if Japanese then you probably say “eto”.  Basically these are filler words with effectively no meaning, and words we are often unaware of saying.  As such they are perfect examples of continuous parts of our phenotype that are subject to random change, and rarely if ever under selection.

I am not much of a probability theorist, so I will only give an outline of what I am thinking about how I would model it.  My thought is that for any continuous trait, such as words like “um” each person (at a particular point in their life) can be thought of as having a fixed value of the continuous random variable that is their value of the trait.  Other developing individuals will pick up these variables as part of their patterning node, but of course since THERE IS NO SUCH THING AS A MEME (sorry for the sudden outburst of caps lock syndrome), the value of the trait that they pick up will be some random variable centered around some weighted mean of the “parents” they copy.  Clearly in a very large population every individual would have a random value of the trait that was slightly different from that of others, but the mean of the trait would change very little between generations.  On the other hand, in small populations this imperfect transmission, and the loss of older forms as individuals die will potentially lead to substantial shifts in the distribution of the continuous trait.  This may well be the reason that we no longer “hem and haw” as we did in the past, but instead “um” our way through the day.  Meanwhile, who knows what happened to “er”.

There are two things that are obviously missing from this essay.  The first is that I really had trouble coming up with a continuously inherited trait in non-social organisms.  Does anybody know of any?  Is my ignorance simply that, ignorance, or is it that they don’t exist, or that nobody ever thinks about them?  The second it is very possible that the drift process for a continuous trait has been worked out, but I simply don’t know what it is.  Perhaps some modification of the diffusion approximation will work just as well, perhaps better, for continuously inherited traits, I would love to know if anybody has any insights.

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