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Evolution in Structured Populations

What is an individual (Part 1)

Posted: September 10th, 2014 by Charles Goodnight

It seems to me that the bad-boy question of evolution that sneaks around the edges, and nobody seems to notice is what is an individual? Think about it: My definition of evolution is change in the distribution of a population due to the gain or loss of individuals. I use “individuals” as if that were a known and obvious entity. I am not alone in that. Futuyma defines evolution in terms of “individual organisms”. Other writers are less explicit, but the implication is that evolution is defined in terms of individuals, which are assumed to be organisms.

Normally, our concept of what an individual is makes sense. That is, we see a human: that is an individual. We see an oak tree: that is an individual. As usual in biology, most of the time our intuition isn’t bad. Where it falls apart is when we get to the fuzzy edges. When we see two identical twins, are they one or two individuals? What about a stand of aspen trees (which clonally reproduce), or even worse, a huge fungus that covers 2,384 acres (http://www.scientificamerican.com/article/strange-but-true-largest-organism-is-fungus/)?

Mushrooms

The worlds largest organism is a fungus, but is it an individual? (From http://www.scientificamerican.com/article/strange-but-true-largest-organism-is-fungus/ )

So, at the macro level, we have the question of whether genetic identity is enough to make two things (twins, a cutting off of a plant) the same individual, or do they need to be physically connected. Either answer raises a bunch of issues. For example, if we only require genetic identity, then I can take a cutting of your Christmas cactus, and take it with me to the other side of the world, and they are still the same individual, even though they have absolutely no contact with each other. Similarly two identical twins would probably be at least slightly offended to learn that they were the same individual.

Lindsey Lohan twins

Are they one individual or two. In this case, we can definitively answer the question, it is one individual, Lindsey Lohan, playing identical twins, but in cases of real identical twins the answer may not be as obvious. (from http://une10.blogspot.com/2011/08/visboo_29.html)

The other possibility is also difficult. With aspen clones some ramets (individual trees) are connected, others may not be. Thus, we really cannot know how many individuals are present without excavating them.

Aspen Clones[6]

How many individual aspens are in this picture? Is it three or is it hundreds, or somewhere in between? (http://watchingtheworldwakeup.blogspot.com/2008_09_01_archive.html)

It becomes even more difficult if we consider the other side of the equation. That is, it is tempting to define an individual as perhaps a set of genetically identical cells of the same species, perhaps requiring them to be physically in contact, and physically separated from other sets of cells. This is a made up on the spot definition probably won’t satisfy anybody, but it is close to the intuitive notion. However, this definition has a host of problems. Starting at an issue that nobody will lose sleep over: are mitochondria part of the individual? Mitochondria are a symbiosis between two cells, presumably an Archea like host cell and a Bacteria like symbiont, that has coevolved to the point that we no longer consider them separate entities. Instead the coevolution is so complete that we consider them to be their own (synthetic?) kingdom Eukarya.

eucaryote origins

The well established endosymbiotic theory of the origin Eukarya. (from http://www.nature.com/scitable/topicpage/eukaryotic-cells-14023963).

That is something of a straw man argument, but consider that in our bodies well over half of the cells are not human cells. Included in that are some symbionts that are passed from parent to offspring at or before birth, some that are acquired through social interactions with family and other important people in our lives, and finally some are acquired from the environment (I mean “symbiont” in the classic sense that includes both mutualists and parasites). Importantly, some are life-time residents, such as much of our gut fauna, and intra-cellular parasites, and others, such as pathogens, are temporary visitors using our bodies as environments in which to breed.

Wolbachia

“Wolbachia bacteria, green, infect the ovaries of the malaria-transmitting mosquito Anopheles stephensi.” Wolbachia are maternally transmitted, and thus effectively evolutionarily similar to mitochondria. Should they be considered part of the individual? (http://www.npr.org/blogs/health/2013/05/08/182339563/using-bacteria-to-swat-malaria-inside-mosquitoes)

In short, it seems that the boundaries of what is an individual, or even an organism start to become fuzzy when we start poking at the edges. We should hardly be surprised by this. After all, one of the hot topics in biology has been the issue of “major transitions”, which, for example, Maynard-Smith felt was sufficiently important that it deserved to be the subject of one of his last books, and (shameless self plug here) there has been a recent a volume on evolution and emerging individuality edited by Bouchard and Huneman. If individual cells evolved for group living, then eventually became an “individual” then we should expect our concept of individuality and organismality to be difficult, this is simply the nature of biology.

A final issue that we have to consider is that if that transition from groups to individuals is actually took place in the past, we should be seeing that occurring in the present as well. This goes both directions. Going downward, when should we consider a cell in a metazoan to be an individual? And going upward, when should we think of colonies of insects as individuals? Should we ever consider herds of mammals, or communities of humans as individuals?

lazy or sick

We absolve the man on the left for staying in bed calling him “sick”, whereas we blame the man on the right for being “lazy”. In both cases the result is the same: they are lying in bed, only in the second we assign causality to the person. ( left: http://donlonpharmacy.com/flu-season-is-not-over/ right: http://www.cc-chapman.com/2011/social-media-has-made-you-lazy/ )

There are a lot of questions surrounding what we mean by an “individual”, however, interestingly, we have a good intuitive notion about what an individual is. Consider the example I use in my chapter in the Bouchard and Huneman book in which I imagine two people, one who is in bed because they have the flu, and one who is in bed because they don’t feel like getting up. The first person we might excuse because they are sick with the flu, and we do not assign blame to them, since they are “sick”. We might label the second as being “lazy”, and blame them for their failure to get up, or in other words, we assign the choice not to get out of bed to them as an individual. In thefirst case we assign blame to the disease, and absolve the individual, in the second we assign blame directly to the individual. In both cases the outcome is the same: The person stayed in bed. The question becomes why is laziness a property of the individual in the second case, but not the first? I think that the phenotypic perspective, and multilevel selection have a lot to say about this question, and it is something I will be exploring in the next week or two.

Speciation in continuous populations

Posted: August 28th, 2014 by Charles Goodnight

I am in Brazil this week to give a talk about speciation in continuous populations, so I figure I will save a little effort by summarizing some of the stuff that is in that talk. By the way, Sao Carlos is a wonderful town, and if you can come up with an excuse to come to Brazil I strongly recommend it. Of course, if you are a vegetarian, while it SHOULD be a good place for you, man do these folks like their meat!

So, on to speciation. This is a project I did with Maggie Eppstein, currently chair of the University of Vermont Computer Science department, and Josh Payne, then a computer science graduate student and now a postdoc at the University of Zurich. In other words, it was big on computer science, and maybe a bit less big on biology, but the implications for biology are important. (Payne, J.L., Eppstein, M.J., and Goodnight, C.J. “Sensitivity of Self-Organized Speciation to Long Distance Dispersal”, Proceedings of the 2007 IEEE Symposium on Artificial Life (Alife’07), pp. 1-7, 2007.  Eppstein, M.J., Payne, J. L., and Goodnight, C.J., “Underdominance, Multiscale Interactions, and Self-Organizing Barriers to Gene Flow”, J. Artificial Evolution and Applications (special issue on Biological Applications), Volume 2009, Article ID 725049, 13 pages, 2009

This project stemmed out of a discussion I had with several people at NECSI (http://necsi.edu/). So, in talking to them I came to realize that when you had two types that were incompatible, they were distributed randomly on a plain, interactions were local, and the types spread contagiously that there would be this interesting process of coarsening. That is if you started out with the types randomly distributed the locally common type would have an advantage and increase, whereas the locally rare type would have a disadvantage and decrease. The net result would be your random distribution would devolve into regions that were primarily one type, and other regions that were the opposite type. I also found out that the boundary between these regions would wander about randomly across the plain, but it would get trapped by any sort of boundary

Sayama coarsening

Coarsening. In this example the fitness of an individual is proportional to the number of neighbors of the same color. This results in a process of coarsening in which the once uniform distribution becomes clumped with low fitness boundaries between the different color regions.

In any case the question becomes whether speciation this process of coarsening in populations with local interaction lead to speciation. We decided to look at the very simplest case, that of simple underdominance. That is, can we take a population with within locus underdominance, and have it self organize into two species.

By the way, as with any good theoretician, I will use Mayr’s biological species concept as my definition of a species. I know that this is really not a great definition, but heck, I got to meet Professor Mayr on several occasions and he said nice things about my research, so I stand by my decision. . .

First off, the trivial case is where the underdominance is so extreme that the heterozygote is lethal:

Genotype       A1A1             A1A2           A2A2

Fitness               1                   0                   1

Obviously, such a population will immediately split into two reproductively isolated “species”. Yes, this works, but it is biologically totally unconvincing.

I won’t go into the details, however, what we can show is that it is quite easy to introduce mild underdominance into a viscous population. Thus, we would be hardly surprised to see underdominance such as:

Genotype       A1A1             A1A2           A2A2

Fitness               1                   0.9                   1

The problem is that this is not an effective barrier to gene flow. Indeed we were able to show that it takes a complete reproductive isolation to prevent the movement of a neutral gene across the boundary.

gene penetration

Gene flow of a neutral gene across a underdominant boundary. Note that there is a linear relationship between the degree of underdominance and neutral gene penetration, and importantly, a discontinuity when gene flow is zero.

Thus, we are only going to allow mild underdominance, yet we want the population to divide into two completely reproductively isolated populations. It turns out that it is not that hard if we allow multiple loci with mild underdominance. To illustrate this consider two loci with underdominance of 50%, so that the double heterozygote is lethal.

two loci no epistasis

Two underdominant loci. Double homozygotes have the highest fitness, single heterozygotes (yellow) have lower fitness, double heterozygote (white) is lethal.

If we set up this system on a 100 by 100 field with nearest neighbor mating and dispersal we see “coarsening”, but because the coarsening is not focused there is no speciation: coarsening

Local mating leads to “coarsening”, patches of one double homozygous genotype (A1A1B1B1 etc) separated by hybrid zones.

In other words, simply by having localized mating and underdominance we are half way there. We get the coarsening, and regions of the two species, but in all cases there is a pathway through a viable hybrid zone between any two regions.

This is where we had to introduce some epistasis. In our next iteration we added a bit of epistasis so that the two opposite corner genotypes were favored:

Two loci, epistasis = 0.1

Two underdominant loci. Double homozygotes of the same numerical value have the highest fitness, single heterozygotes (yellow) have lower fitness, double heterozygote (white) is lethal.

This now results in coarsening and eventually speciation:

Specation

Local mating leads to coarsening, and domination of the two most fit genotypes (A1A1B1B1 and A2A2B2B2). In this case all of the hybrids are lethal, and the two populations are reproductively isolated.

This is actually very interesting, because you will note that in the early stages all four homozygous forms are formed, but the single locus heterozygous boundaries wander around randomly. When two such come into contact they coalesce into a single stronger boundary, eventually leading to speciation. Importantly, this is not limited to only two loci. It turns out that this coarsening and coalescence of leaky boundaries will continue regardless of how many loci are involved. Thus, even extremely mild underdominance at a large number of loci will eventually lead to speciation in this model:

speciation with multiple loci

The effect of number of loci on speciation. In a panmictic population (black line) the population always fixes on one of the two best genotypes, although the time to fixation changes as the number of loci increases. In spatically structured populations speciation always occurs, with the time to speciation being a function of the number of loci required and the amount of epistasis.

This is a very simple simulation, yet it makes the important point that when there are genetic incompatibilities speciation can easily occur. Indeed, this implies that the speciation may be the expected outcome for a widespread species with limited gene flow.

Wright’s Shifting Balance Process revisited one last time

Posted: August 13th, 2014 by Charles Goodnight

It is shaping up to a busy summer, mostly because I am packing up my house so I can go to Brazil for a year. And that is my rather lame excuse for neglecting my blog.

In any case, what I did so many weeks ago was go through the three phases of Wright’s shifting balance process one at a time. Perhaps it is time to do a summary post.

To review, the three phases are (1) the phase of random drift, (2) the phase of mass selection, and (3) the phase of interdeme selection. As I pointed out before Wright’s initial model is undoubtedly naïve, nevertheless, it forms a good basis for thinking about evolution. Importantly, Wright thought all three of the phases were acting simultaneously, so in a sense the phases might be thought of as a human construct based on our need to categorize things. The question becomes how can all of these phases be acting simultaneously when they apparently have very different requirements, what with drift working best with small population size, and selection working best with large population size, and needing isolated demes for phase one, and differential migration for phase three, etc. Any way, if other people can tell just so stories so can I. The scenario I am going to paint may work, then again, it may not. I am not going to defend it either way.

I am inclined to suspect that the answer may lie in how additive genetic variances change over the adaptive topography. In particular consider the adaptive topography. If we choose an adaptive topography in which the axes are phenotypic then at any point we can take the tangent to the multidimensional surface, and I would argue that the slope of that tangent is the phenotypic variance for fitness. If we then projected that on to axes of the heritable components of the phenotype and again made an adaptive topography then the tangent would be the (effective) additive genetic variance for fitness.

12_EVOW_CH17

The reason that this is important for the shifting balance process is that this is the element that has been missing from discussions of the shifting balance process. The point being that what is important is not population size or strength of selection per se, but rather the ratio of selection to drift. When NeS <<1 a gene is effectively neutral. In words if the effective population size (Ne) times the strength of selection (S) is much less than one the gene can be considered effectively neutral. There is an interesting subtext here, in that Ohta was thinking about selection acting directly on genes (yea, we phenotypic selectionists don’t believe in such nonsense), so in effect the heritability in this case is one. In the case of traits should neutrality be related to Ne(h2S)? I am at my in-laws, so I don’t have time or facilities to work that out.

The important point is that Ne(h2S) can become small either by Ne becoming small (what is typically being considered), or by h2S becoming small. Thus, in the region of selective peaks and valleys Neh2S will be small. At peaks although the population size is large, the tangent is very small, and fitness is nearly neutral. In valleys both Ne and Ne(h2S) will be small, and again the trait will be nearly neutral. It is mainly on the slopes of the peaks that the effective additive genetic variance will be large enough that selection, and thus phase 2 will be important. This makes the point that phase one and phase two can coexist, with phase one being dominant at peaks and valleys, and phase two being dominant on slopes.

This still leaves the problem of escaping from a local adaptive peak, but I suspect there may be some ways around that as well. To me the big problem is getting a metapopulation in which there are populations on more than one peak. My first thought is that this may be in some sense fractal. That is, populations differing on different tiny peaks within a major overall local adaptive peak may be common, but populations differing in major adaptive peaks may be quite rare. Thus, the probability of metapopulations with subpopulations on different peaks probably depends on how different the peaks are. That said, there are a number of ways that such metapopulations might arise. The easiest to imagine is a changing environment in which a former adaptive peak suddenly becomes much lower. Such changes need not be permanent. For example, an unusual weather year might effectively change selection enough that a peak temporarily disappears leaving the populations free to drift. Alternatively, even if an environmental change didn’t change the height of a peak it could lower population sizes enough to decrease NeS sufficiently that phase one becomes important. Finally, it is quite possible that environmental variation could increase dramatically for what ever reason. This would have the effect of increasing the phenotypic variance, and lowering the heritability. This could lower Ne(h2S), and allow genetic drift to become the dominant force.

Once we get a metapopulation with subpopulations spread over several adaptive peaks then phase three can start participating. As pointed out in my previous post, this requires differential migration, which can be a force that overwhelms selection. That is, populations that export migrants will have a powerful influence on populations that import migrants. Also as I pointed out last week, whether a migrant can enter a population will depend on the fitness of the individual (not their offspring). Thus, a net exporter of migrants can be that simply because they have a larger population size and export more migrants, or because their migrants have a higher fitness, and are always able to invade the other population. Once the migrants invade they will mate and produce offspring. If the subpopulations are truly on different adaptive peaks than their offspring will be of low fitness, and there will be a general increase in non-additive variance (due to decreased inbreeding, and the introduction of new alleles). The net result is that the migrants will pull the subpopulation down the slope, decrease the heritability, and generally set up the population to be more susceptible to genetic drift. As a result this interpopulation migration may in fact support phase one due to its effect on population size and variance components. Remember, one migrant every four generations is optimal for speciation to occur!

I have run out of space, so I will sum this up by pointing out that the argument that the three phases or Wright’s shifting balance process have difficulty coexisting may have more to do with our emphasis on additive models and our lack of understanding of the effects of non-additivity and less to do with real problems with how this process actually works in the real world.

Wright’s Shifting Balance Process: Phase 3 part 2

Posted: July 25th, 2014 by Charles Goodnight

Last week I tried to establish that group selection by differential migration can work.   On both experimental and theoretical grounds we find it does work, and in fact will frequently be stronger than individual selection. The question comes where does it fit into Wright’s shifting balance process.

The first problem we need to confront is that Wright apparently thought that the group level trait would be concordant with the individual level trait. That is, he thought that with greater absolute fitness would come the production of more individuals, and with that a greater emigration rate. There really is no reason for this to be true. For one thing, the important factor is relative fitness, not absolute fitness. The problem is that as overall fitness increases in population so does the resulting competition. Thus, a population may experience a steady increase in some measure of absolute fitness, but no overall change in population size. This is actually the basis for the Alice in Wonderland (AKA the red queen) hypothesis. That is, Fisher’s idea that the environment is always deteriorating is largely due to the fact that other individuals are always evolving. Thus, the improvement in absolute fitness (survivorship, number of offspring produced) is completely offset by the similar improvement in other individuals with no resulting increase in apparent fitness or population size.

cheshire-cat

who , Me?

So, how to resolve this? I think the answer may lie in the Wright’s words. (I am away from my books at the moment, so this is a paraphrase). When Wright was investigating the effects of migration on population differentiation he stated something to the effect that it would appear that one migrant every other generation would be sufficient to destroy population differentiation. After uttering this now famous platitude he went on to say: However, given that immigrants will have a much lower survival and mating success than residents there could be many thousands of migrants and yet there will be population differentiation. OK, I have no idea what he really said, but this is how I remember it.

What this has to do with phase three is that even if migrants are moving randomly among subpopulations, they have to survive and reproduce once they get there.   Thus, if one group has a lower absolute fitness than another group migrants from the low fitness group will have a low relative fitness in the group they move into, and as a result may not survive and reproduce. On the other hand, migrants from the high absolute fitness group will have a high relative fitness in the low fitness group, and they will have a better than average chance of surviving and reproducing. Thus, even random migration can potentially lead to differential migration once within group individual selection is added in.

The second issue is rather interesting. When talking about phase three it is convenient to say something to the effect that migrants from high fitness populations send out migrants that lead or convert the low fitness population over to the new higher peak. This sounds something like the five rusty rats leading the founding of the village of cream puffs.

rusty rats

“And so, while the wind and the snow blew and the blizzard beat its icicles in their faces, they held on to the long curved tails of the rusty rats till they came to the place where the Village of Cream Puffs now stands.” If you don’t know the rutabaga stories by Carl Sandberg you should. (http://www.josephperry.net/rootabaga/01-03rustyrats.html)

The reality is that the offspring of these migrant individuals will be hybrids between the two peaks, which ought to put them smack dab in the middle of an adaptive valley. In other words, these migrants are more like genetic terrorists than saviors. What they do is trash the adaptive gene complex by adding genes that are generally bad for the current population.

chu red

I was looking for a non-controversial revolutionary. I came up with Pika Ché (http://www.redbubble.com/people/meganegi/works/8351063-chu?p=t-shirt)

So, my thought on phase three is that at the “end” of phase 2 you have a set of populations distributed about two or more peaks (I put end in quotes, because the phases all go on simultaneously). At this point they are all sending out and receiving migrants, but the ones on the higher peaks are net senders, and immigrants tend not to survive and reproduce. Those on lower peaks are net recipients of migrants since immigrants tend to have higher fitness than the locals. That said, the offspring of these migrants is low, and has the effect of dragging the low fitness population off of its local peak, and basically allowing drift to have its effect. However, in a sense it might be called directed drift, since there will be a continued input of new migrants from the higher peaks. These migrants will “encourage” drift in the direction of the new peak, but by no means guarantee it. In other words rather than the lower peak being led to the new higher peak, I see it getting dragged kicking and screaming through the adaptive valley.

jones dragged

Indiana Jones being dragged kicking and screaming to the lost ark (http://www.propstore.com/cms/the-prop-store-collection/indiana-jones-and-the-raiders-of-the-lost-ark/harrison-fords-whip/)

So, then we can start to see what phase three is really doing. When a metapopulation is spread across multiple adaptive peaks migration will have the effect of moving populations off of those peaks. Because the fitness of the incoming migrants will reflect their population of origin, by in large low fitness populations will have more effective migrants (migrants that enter and survive and reproduce) than high fitness populations. Thus, the low fitness populations will be more likely to be driven away from their peak, and on average, they will tend to be dragged towards the new higher peak. Will they make it to the new peak? That is hard to say in a complex stochastic world. Some will at least temporarily climb back to their old peak, others may climb the new peak, and still others, having been dragged off of their local peak may drift around and discover yet a new and even higher peak.

Phase 3: Group selection by differential migration

Posted: July 16th, 2014 by Charles Goodnight

Having taken a week off (blame it on Jason Wolf – he is the one who gave me a writing assignment!) it is time to get back to it. This week we turn to the final phase of Wright’s shifting balance process, phase three the phase of interdeme selection. In Wright’s view populations centered on high fitness peaks would tend to export more migrants than populations on lower fitness peaks, thus, high fitness populations are net exporters of migrants, whereas low fitness populations importers of migrants. Nice idea, but there are lots of reasons why it might not work.

One possible concern is whether or not group selection by differential migration will actually work. There are two lines of evidence that group selection by differential migration does indeed work, one experimental, and one theoretical.

The experimental study is a study by Wade and I (Wade and Goodnight 1991 Science 253: 1015). In this study we specifically examined whether selection by differential migration could result in a change in a response to selection. In this experiment we set up three pairs of metapopulations. Each pair consisted of two metapopulations of 50 subpopulations. In the first metapopulation we would count all of the adults, take the mean number of adults per subpopulation, and then calculate the relative fitness as:

Equation

were Ni is the population size of the ith subpopulation, N is the mean subpopulation size in the metapopulation and wi is relative fitness of the ith subpopulation. We would then perform selection by multiplying the founding population size (20 adults) by the relative fitness and taking that number of individuals to found the next generation. For example if the average population size at census time was 100 beetles, a population that produced 120 beetles would have a relative fitness of 1.2, and we would collect 24 offspring from the population. Similarly, one with a census number of 100 would have a relative fitness of 1, and produce 20 offspring, and one with a census size of 80 would have a relative fitness of 0.8 and produce 16 offspring. For each subpopulation the first 20 it produced would go back to found the next generation. If it produced more than 20 the remaining beetles went into a migrant pool. Thus, our first population would be founded with 20 from the parental population and 4 would go into the migrant pool. If the relative fitness was less than 20 then all of the beetles went back in to found the next generation, and the number was then made back up to 20 by sampling from the migrant pool. For example, in the third population, all 16 beetles from the parental population plus 4 from the migrant pool would be used to found the next generation.

In the paired population the same number of migrants was used, however, they were randomly chosen from the metapopulation without respect to fitness. The idea being to maintain the same level of migration, but to not have that migration correlated with fitness.

Wade and Goodnight fig 1Wade and Goodnight fig 2

I should also add, so it doesn’t confuse things, there were three treatments, selection every generation, selection every other generation, and selection every third generation. In any case, we observed a substantial response to selection, and as usual, group selection is vindicated. It is, of course, standard fare that group selection is very effective, but prior to this study all group selection studies had been due to differential extinctions. This was the first to confirm that group selection by differential migration does indeed work.

Wade and Goodnight Fig 3
The second piece of evidence is theoretical and provided by James Crow and his colleagues (Crow, Engels and Denniston 1990 Evolution 44: 233-247). In this paper they considered a situation in which there were two subpopulations, one fixed for a favorable combination of alleles at two or more loci, the second foxed for a less favorable combination. These combinations were separated by a fitness valley. They then asked what would happen if there was migration (either adult or zygote) between the two populations. The results of this model are nicely illustrated by their figure three.

Crow et al Fig 3

In this figure the fitter gene subpopulation has a fitness of 1.2, and the less fit subpopulation has a starting fitness of 1.0. Migration from the fitter to the less fit subpopulation at first lowers the fitness of the subpopulation receiving migrants, but eventually leads to conversion over to the new fitness peak. From this they conclude that “The importance of Wright’s shifting-balance theory remains uncertain, but we believe that whatever weaknesses it may have are not in the third phase.”

Interestingly, Nick Barton, although he put a negative spin on it, confirmed Crows model of the efficacy of group selection by differential migration (Barton. 1992 Evolution 46: 551-557). He developed a model that was similar to Crows, but had migration be independent of fitness. That is he examined what happened if migration went from the less fit to the more fit subpopulation (among other things). In his conclusion he states:

“The striking results of Crow et al. (1990) are not due to selection in favor of a novel combination of genes, but rather reflect the power of gene flow over selection: the outcome is decided while the incoming alleles are at low frequency when the new well-adapted combination of genes has yet to appear.”

Barton goes on to conclude “Thus, while populations may well diversify through a “shifting balance,” it is difficult to see that this process leads to significant adaptation.”

I think one could rather conclude that his results confirm Crow et al.’s result that group selection by differential migration is very powerful, and in most cases it will overwhelm individual selection. Barton raises some important points. For example, Wright assumed that the most fit populations sent out the most migrants, however, relative fitness is a funny thing, and this certainly need not be the case. Indeed it is a bit of an unjustified leap of faith to believe that within population fitness in the form of increased survival and reproduction of individuals translates into higher among populations fitness in the form of increased emigration rate.

The Phase of Mass Selection and Long Term Selection Experiments

Posted: July 2nd, 2014 by Charles Goodnight

On to Phase 2 of Wright’s Shifting Balance Process. But before I do I should probably start with a shameless attempt to up my standings in the next Carnival of Evolution World Cup Competition by alerting the committee responsible to the following figure that I found:

precambrian-rabbit-copy

Obvious evidence of Pre-Cambrian Bunnies (unapologetically lifted from http://clubschadenfreude.com/2013/02/19/not-so-polite-dinner-conversation-part-9-the-second-half-of-19-limestone-coelacanths-and-circular-reasoning/)

OK, on to phase 2: the phase of mass selection. In Wright’s words “. . . the set of gene frequencies drifts far enough to cross one of the . . . saddles in the surface of fitness values . . . There ensues a period of relatively rapid change in this deme, dominated by selection among individuals (or families) until the set approaches the equilibrium . . ..” (Page 455, Wright 1977). Evolution and Genetics of Populations. Vol. III. Univ. Chicago Press.).

There would appear not to be much controversy about this. In particular, Wright’s claim for this phase is that the population will climb the nearest peak and approach the local optimum. I doubt that Fisher would argue much with that. However, there are actually is the potential for some discussion. In particular, in an additive world the response to selection occurs strictly by changes in gene frequency of alleles with fixed effects. However, one of the points I have made before (https://blog.uvm.edu/cgoodnig/2013/07/31/drift-and-epistasis-the-odd-effects-of-small-population-sizes/) is that drift can convert epistatic variance into additive variance, and in the process change the average effects of alleles. As I mentioned last week, this may be the important role of Wright’s phase one: Drift causing shifts in local average effects. As I also discussed last week it is unlikely that these shifts will be major, since in general epistasis tends to be small as a variance component, and thus in most cases there won’t be much material for the “conversion” process to work on. This is where Wright’s phase two comes in.

To see where this is important it is useful to look at long-term selection experiments, and note two anomalies that are consistently found in such experiments. First, they work too well. That is, you typically can get 100 or more generations with a nearly linear response to selection (and MUCH more if we acknowledge the existence of laboratory selection experiments involving bacteria — Wiser, Ribeck, and Lenski 2013. Science 342:1364-1367). Second, there are typically intermediate selection plateaus.

Corn Selection copy

The results of 100 years of selection for oil and protein in corn. Note the overall long term linear response to selection (orange highlight), which is none the less punctuated by extended selection plateaus (green highlights).

Turning first to the long-term linear response to selection. This is actually expected under Fisher’s infinitesimal model, which has the odd feature that, because there are infinitely many loci of infinitely small effect, selection changes the gene frequencies of the individual loci by an infinitesimal amount, which is to say gene frequencies do not change. Of course in the real world there is a finite number of loci, nevertheless, this long-term linear response to selection implies that there are indeed a VERY large number of loci contributing to the response to selection. It turns out that there are two other possibilities. One is the ongoing input of new mutations, which due to space constraints I will not talk about, and the second is epistasis. It turns out that, as with genetic drift, selection will drive the conversion of epistasis to additive effects. The rather surprising empirical result from an old paper (Goodnight 2004 in: Plant Breeding Reviews. J. Janick, ed.) is that in epistatic systems selection seems to convert epistasis into additive effects at a more or less steady rate. Thus, this conversion of epistasis into additive variance driven by selection is a possible explanation for the extended response to selection.

VA with selection and epistasis

The additive genetic variance in four simulated populations with two loci and 100 alleles per locus under selection. Dashed lines are populations with only additive effects. Note the approximately exponential decline in the additive genetic variance. Solid lines are populations with additive-by-additive epistasis. Note that the additive genetic variance remains elevated for 10 generations before beginning an exponential decline.

The intermediate selection plateaus are also consistent with an epistatic model. The typical, and adequate, explanation for intermediate selection plateaus is that the population has run out of selectable variation, and is waiting for either a favorable mutation, or a favorable recombination event. However, consider the simple case of additive-by-additive epistasis.

VA fitness surface

The fitness surface for additive-by-additive epistasis.

 

From this fitness surface you can see that there are two possible outcomes of selection, fixation of the A1A1B1B1 or A2A2B2B2 fitness peaks. Interestingly, in a completely deterministic system there is a boundary dividing the domains of attraction for the two peaks. Thus, we can start two populations near fixation in one of the fitness valleys, such as nearly fixed for A1A1B2B2, and with arbitrarily small changes in gene frequency they will go to different peaks.

deterministic selection

The response to selection for two populations, one with a starting gene frequency of A1 = 0.99, and B1 = 0.0101 or 0.0099. Note in this deterministic model the outcome is a function of minor differences in gene frequency at the B locus. Each arrowhead represents one generation.

This simple system gives us both the results typical of long term selection experiments, that is surprisingly long responses to selection, and intermediate selection plateaus.

response to selection

The response to selection in the example described above. Note the long response to selection with an intermediate selection plateau.

Two more fun graphs, then I will conclude and leave you alone. First, the intermediate plateau is due to a lack of additive genetic variance, but not total genetic variance. The problem is that when the population nears a gene frequency of 0.5 the genetic variance is mostly expressed as epistasis, and the population cannot respond to selection. When the population is dominated by either allele at either (or both) locus the additive genetic variance increases.

VA during selection

During the selection process total genetic variance remains relatively constant (except near fixation) but additive genetic variance dominates during the early and late stages of the selection when frequencies are far from 0.5, and the epistatic variance dominates during the middle stages when the gene frequencies are near 0.5.

And, because I cannot resist, the average effects of the A1 and A2 alleles reverse over the course of the selection experiment.

figure 12 AXA LAE

The local average effects of the A1 allele. The A1 allele starts out being the low fitness allele. Half way through the response to selection the gene frequencies approach 0.5 and the local average effects reverse with the A1 allele becoming the high fitness allele.

So, returning to Wright’s phase two, the phase of mass selection, we see, again, that Fisher and Wright were seeing two sides of the same process. Within populations selection will appear to act exactly as Fisher said, as a process that refines adaptations and leading to the (locally) optimal phenotype. Between populations it must be recognized that in an epistatic world selection is a diversifying force that acts to magnify the small differences in local average effect and potentially driving populations to different adaptive peaks.

In other words, to paraphrase Dave McCauley when he was a postdoc and spiritual leader of us graduate students at the University of Chicago, it is the interaction of stochastic and deterministic evolutionary forces that give meaning to life.

Was Fisher (W)right?

Posted: June 28th, 2014 by Charles Goodnight

Once again without internet, thundering my way north on the Silver Star. It is hard to keep focus on what I intended on the subject I am writing on. The Evolution meetings were inspiring to say the least. Lots of great talks, many of which I could easily write a blog about, but I must stay on phase one for the moment!

Like Barton and Charlesworth and Carson and Templeton , Fisher and Wright would have disagreed on the importance of genetic drift (That sentence must win some sort of prize for name dropping!). Fisher would have emphasized that small population size would decrease the number of alleles and the (molecular) genetic variation within populations. He also would have, in all likelihood, argued that most meaningful genetic variation was additive, and epistasis would not be present to any great extent. In contrast, Wright would have argued that gene interaction is important in populations, and as a result genetic drift could result in shifts in genetic architecture, and with that the potential to form what Dobzhansky called adaptive gene complexes.   What I want to talk about is that, much like the blind men examining the elephant, they could both be right.

Blind men elephant 2

In the classic fable of the blind men and the elephant each man examines only a part of the elephant, and fails to understand what the whole is.  (from http://www.newsfromthehill.com/2011/06/keep-on-sunny-side.html)

First, Fisher. If you have a population with a large amount of additive by additive epistasis and send it through a bottleneck you will find that the additive genetic variance increases (yea! That’s a big part of the reason I have tenure!). However, although the total genetic variance increases a little bit, the additive variance increases a lot more. The net result is that the epistatic variance declines precipitously. The net result is that after a few generations of small population size the epistatic genetic variance basically disappears. Selection will typically do the same thing. In general both drift and selection tend to drive genetic variance to lower levels. Thus dominance by dominance epistasis tends to be converted to additive by dominance, dominance, and additive variance. Similarly additive by additive variance tends to get converted to additive variance. It is because of this tendency that there is virtually no reason to bother with modeling or measuring three way and higher epistatic variance. It also means that most populations most of the time will also tend to have relatively little two locus epistatic variance. The bottom line: Fisher was right: Within populations under most circumstances we can ignore gene interactions, and treat populations as if they were additive. This means that in many cases genetic drift will simply have the effect of reducing the variance within populations.

epistasis and drift

After a few generations of small population size the nonadditive variance (green line minus red line) becomes very small. Fisher was right, within populations we can ignore gene interaction.

Importantly, however, this view that has Fisher as correct is focused within populations. Within populations the epistasis disappears as a variance component, but it does not go away. The interactions are still there, it is just that most of the time populations will be in a gene frequency space where most of the epistatic variance has been “converted” into additive variance. The other way that you can think of this is that the epistatic variance disappears it reappears as variance among populations. The way this happens conceptually is that when one of the loci gets fixed the epistatic interaction between a pair of loci will be converted into additive effect. Of course in reality, one allele doesn’t get fixed while the other stays at intermediate frequency, however, thinking simultaneously about both loci going to partial fixation tends to hurt my head.

So it is this between population component where Wright comes in. As I said the fixation of alleles leads to the conversion of epistasis to additive variance, however, in different populations it may be different alleles that get fixed by drift. Thus, in an interaction between the A and B locus it may be the B1 allele that moves towards fixation in one population and B2 that moves towards fixation in a second population. In both populations the additive variance will increase due to conversion of epistasis to additive variance, but the increased additive variance will be different in that different alleles at the A locus will be favored in the two populations. In other words, you don’t get something for nothing. An increase in additive genetic variance is always accompanied by a shift in what alleles do. I have mentioned this before and identified this as a shift in the local average effects of alleles. However, even without the fancy name it is an important effect. In interacting systems genetic drift has the potential to send a population down a new evolutionary trajectory. Although the mathematical tools to describe it were not available to him, I believe that this is what Wright was talking about when he was thinking about phase one of his shifting balance process.

Mixing of alleles

When there is epistasis genetic drift will not only increase the additive genetic variance, it will also change the average effects of alleles.    Thus, an allele that was “good” prior to a bottleneck may be “bad” after the bottleneck.

Thus, it appears that both Fisher and Wright were correct. Within populations genetics will typically act in an additive manner, and we will see that in many circumstances population bottlenecks will do little more than decrease the ability of a population to adapt. Thus, within populations a Fisherian view is expected to entirely adequate. However, that decrease might not be as much as you might expect (for what it is worth, the VA should increase whenever VAA > 1/3 VA – that is ignoring the typo in the relevant equation), and there may be some shifting of local average effects. This shift in the effects of alleles on the phenotype is what Wright was talking about. If we are looking at a metapopulation we need to acknowledge this effect of gene interactions on the differentiation of populations.

Just to remind you this shift of local average effects due to gene interactions is an entirely different form of population differentiation than differentiation of population means. Two populations with identical mean phenotypes can nevertheless be differentiated for average effects.

In conclusion, then we can see that indeed Fisher and Wright were looking at different parts of the same elephant. Fisher was looking at the apparently additive world that is within a population, and Wright was looking at the emphatically non-additive world that is between populations. I would argue that a modern enlightened view of phase one of Wright’s shifting balance theory would combine these two views. Within populations an additive view will typically be adequate. Genetic drift will as often as not lead to a decrease in additive genetic variance, and epistatic variance will typically not be detectable. The true effects of epistasis will primarily be seen among populations, and they will be seen in the form of shifts in local average effects. These are measures we typically do not make, so until we hve more data we will not know how important epistasis is in population differentiation.

The 1984 founder event debate: Its relation to Phase 1 of Wright’s Shifting Balance Process

Posted: June 20th, 2014 by Charles Goodnight

Today I am speeding south on the Empire State in the morning and the Silver Star in the afternoon. I should be in Raleigh Durham for the Evolution meetings late this evening. For the uninitiated Amtrak trains have names that reflect where they are going. Thus, the famous Steve Goodman/Arlo Guthrie song, “City of New Orleans” is about the train that runs from Chicago to New Orleans. Finally, long distance trains have no internet, so this post is a bit sketchy. But, enough about trains, and on to phase one of Wright’s shifting balance. As I mentioned last week Wright identified three phases of his shifting balance process of how he thought populations might evolve on a complex adaptive landscape. The first of these phases is phase 1, the phase of random drift.

new_orleans_title

From an historical perspective, a pair of papers in Annual Reviews in 1984 (Carson and Templeton 1984 Ann. Rev 15:97-131, Barton and Charlesworth 1984 Ann. Rev. 15:133-164) are worth discussing. Although ostensibly about founder event speciation, they do a great job of laying out the state of the art for genetic drift in 1984. Taking the side that founder events (and by implication genetic drift) are relatively unimportant in evolution was Barton and Charlesworth, while Carson and Templeton attempted to defend founder event speciation.

Barton and Charlesworth’s points can be summarized in basically one sentence: Genetic drift and population bottlenecks reduce genetic variation, and honestly don’t change gene frequencies that much. In short genetic drift is unlikely to be important because it reduces a populations ability to respond to selection, and will cause the random loss of variation. From this perspective there is really no reason to give genetic drift an important role in evolution. In contrast Carson and Templeton argued that indeed there is something special about founder events, especially when there was epistasis, that they can drive a population to a new adaptive peak, and they should not be ignored. Here, however, is where they fell flat. In 1984 there was no theory on the effects of epistasis and genetic drift, or even a sense of how we should model it. The bottom line on these is that, in my view, Carson and Templeton were soundly defeated, not because they were wrong, but because they lacked any mathematical framework to use as scaffolding for their argument, whereas Barton and Charlesworth had the entire corpus of Fisher’s and Dobzhansky’s work to draw upon.

This pair of papers actually had quite an impact on me since they came out right after I finished my thesis, which had my original model of epistasis and founder events. I was sitting there with an unpublished manuscript on exactly the topic that would have completely changed the debate that these two pairs of authors were having (Goodnight 1987 Evolution 41:80-91).

What changed between 1984 and 1987 (actually 1995 – Goodnight 1995 Evolution 49: 501-511) was the development of formal models of the effect of genetic drift on epistatic variance.  The important detail that Barton and Charlesworth did not have available to them was that the genetic variance components are a statistical property of a population, and they change as gene frequencies change. In the rather odd world of Fisher’s infinitesimal model even selection does not change gene frequencies, and as a result in Fisher-World© (OK, its not really copyrighted) it is completely valid to assume that additive genetic variance is a constant. However, Fisher also assumed that populations were infinitely large. If they are not infinitely large then gene frequencies DO change, and my models showed that on average these changes in gene frequency lead to an increase in the additive genetic variance. More importantly, Barton and Charlesworth also assumed that the average effects of alleles were constants, which is a valid assumption in Fisher-World. In my later models I showed that indeed there is only one way that additive genetic variance can increase following a population bottleneck. That is the only way to increase variance to change the average effects of alleles. Thus, in retrospect, the problem that Barton and Charlesworth had is that they were trying to apply Fisher-World reasoning to a situation in which it did not apply.

So, the way this applies to phase 1 of the shifting balance process is that it we need to acknowledge that drift is not just change in gene frequencies. Of course at a molecular level that is exactly what it is, but what we are interested in is the phenotype, and the quantitative genetic variance components. I would even go so far as to say it is not the increase in additive genetic variance that is particularly important. Of course, increasing the additive genetic variance increases the rate at which a population can respond to selection; however, getting to your destination a few generations early does not strike me as the stuff of the shifting balance process.

Instead, I think it is the shift in local average effects of alleles that are the interesting feature of drift with respect to the shifting balance process. What this means is that if two populations are isolated from each other then genetic drift can lead to a slight increase in additive genetic variance, but shifts in average effects of alleles that are different in the two populations. These shifts mean that the same allele is doing different things in different populations. It also means that selection acting on the same phenotype in the same manner in the two populations will favor different alleles. What is good in one population may be bad in another population.

Importantly, Wright, and Fisher, and even Barton and Charlesworth and Carson and Templeton did not have access to this. They considered average effects to be unchanging, and as a consequence missed one of the major features of drift in small populations. What these models tell us is that the population genetics of gene frequencies is very different from the quantitative genetics of phenotypes, and since evolution is about phenotypes in most cases it is the quantitative genetics that will be important.

Final note: Yes, of course Barton and Charlesworth, and all good quantitative geneticists know that average effects are a function of population characteristics and gene frequency, but this is not something that will normally enter their thinking or intuition.

Wright’s Shifting Balance Process

Posted: June 13th, 2014 by Charles Goodnight

Now that I have talked about how Wright thought evolution didn’t occur on adaptive landscapes, now it is time to talk about how he thought it did occur. The 7 assumptions and the adaptive topography were all basically background for his “shifting balance” process of evolution (Wright 1977 Evolution and Genetics of Populations. Vol. III. Experimental Results and Evolutionary Deductions. Univ. Chicago Press). To quickly summarize the previous few posts, Wright thought that the evolutionary possibilities could be visualized as an adaptive topography with fitness peaks and valleys.

wsbp adaptive topography

Wright’s adaptive topography. The horizontal axes represent aspects of the genotype or phenotype, the vertical axes represents fitness. The red dot is the current position of a population on the adaptive topography. Wright’s central question was how does a population cross an adaptive valley to move from its current local adaptive peak to a second higher peak.

Wright felt that very large or very small populations, or even a single medium sized population would not be able to navigate this topography, either because very large populations would be dominated by selection and unable to cross fitness valleys, or single very small populations would be so small and isolated that they would not be able to adequately explore the landscape. Instead, he thought that a metapopulation, or population of populations, would be the ideal population structure for evolution. (Of course, he did not use the term “metapopulation”: That term was coined by Levins, 1969 Bull. Entom. Soc. Am. 15: 237-240.) He basically thought that a medium size population would have a balance of drift and selection that it would allow the population to drift away from an adaptive peak and randomly explore the adaptive landscape. However, he also thought that a single population would be inadequate since the chance of that population actually coming under the selective domain of a new higher peak would be very small. Thus, he thought that the metapopulation structure with numerous moderate size populations was necessary since collectively they would be able to drift away from an adaptive peak and adequately explore the adaptive landscape.

wsbp metapopulation

A metapopulation is a population of populations that is connected by a low level of migration

The process he envisioned is his “shifting balance” process, which he imagined as a three phase process (I prefer calling it a process since the “theory” is whether or not the “process” is important. I also prefer SBP, because as an infant my daughter Sylvia’s nickname was Sylvia Bilvia Pilvia, or SBP for short.).   The three phases he identified were:

 

(1) the phase of random drift. During this phase the subpopulations drift at random across the adaptive landscape. Drift is random with respect to fitness, thus, during this phase the populations are not constrained by selection, and can easily cross adaptive valleys.

wsbp phase 1

Phase 1, the phase of random drift. During this phase the populations move randomly without respect to fitness due to random sampling processes in small populations.

(2) the phase of mass selection. During this phase the subpopulations come under the selective influence of local adaptive peaks and are driven by natural selection to climb the closest peak. Selection is a deterministic process that always drives the population “up hill” towards higher fitness, regardless of the height of the peak relative to other peaks.

wsbp phase 2

Phase 2, the phase of mass selection. During this phase selection drives the populations towards the nearest local adaptive peak.

(3) the phase of interdeme selection. During this phase the subpopulations that are on higher peaks are more successful, and as a result are net exporters of migrants, whereas those on lower peaks are less successful and net importers of migrants. Wright thought that this differential migration would effectively export successful adaptive gene combinations to other subpopulations, and eventually shift the balance of adaptation over to the new adaptive peak.

WSBP phase 3

Phase 3, the phase of interdeme selection. During this phase the populations move randomly without respect to fitness due to random sampling processes in small populations.

In Wright’s discussion it is clear that he considered these three processes to be occurring simultaneously within a metapopulation. Presumably any given subpopulation may at some times be dominated by random drift, with selection being a relatively weak force in that population, whereas other subpopulations may be under the influence of an adaptive peak and be more strongly influenced by selection. Finally, all subpopulations would be sending and receiving migrants. A subpopulation may be a net recipient of migrants while it is in an adaptive valley, but perhaps at a later point become a net exporter of migrants as it climbed a particularly good adaptive peak. It is this constant shifting of the balance of migration and selection from one peak to another that is the reason that Wright named this the “shifting balance” process.

It is clear why this process is so attractive to me, and many others. It is a theory that combines the stochastic processes of genetic drift with the deterministic processes of selection at multiple levels to lead to not only adaptation, but also to the evolution of novel solutions to the process of adaptation. That said there are more than a few reasons to be skeptical about the process (Coyne, Barton and Turelli 1997 Evolution 51: 643-671). Perhaps the most obvious is that the different phases of this proposed process require very different population sizes. For example, the drift phase will presumably be most effective if populations are small and isolated. In contrast, in phase two, the phase of mass selection, larger population sizes will make selection more effective, and random drift less important. Finally, phase three, the phase of interdeme selection is most effective with high migration rates. Thus, we have phase one requiring isolation and small population size; phase two requiring large population size; and phase 3 requiring high migration. This suggests that the three phases would be working at odds with each other, and is one of the main conceptual reasons that the shifting balance process is often discounted. My main thought on this is that, as we have seen in past posts, this is not the first time that we have seen theory being used to discredit the intuition of very smart experimentalists, and to repeat my favorite adage, when theory and experiment are in conflict, the theory is wrong.

As I have written elsewhere, we (Wade and Goodnight 1998 Evolution 52: 1537-1553.) think that it is premature to dismiss Wright’s shifting balance process based on intuition and parsimony reasoning. That said, Wright’s model is old, and it needs to be brought into the 21st century. What I will do over the next several weeks is discuss each of the phases in turn, and suggest ways in which discoveries made over the last 70 years can be incorporated and used to modify our understanding of the process originally proposed by Wright.

Of Population Structure and the Adapative Landscapes

Posted: June 5th, 2014 by Charles Goodnight

Last week I talked about adaptive topographies, and while my discussion may have done little more than add to the confusion, at least it got across Wright’s view that there are multiple selective peaks, which in essence means that there are multiple solutions to the problem of achieving high fitness.

12_EVOW_CH17

Figure taken from http://locustofauthority.wordpress.com/

Wright was interested in how a population could move from one local peak (such as the center intermediate height peak in the figure above) to another higher peak. He considered several options, which are summarized in the figure I showed last week:

SW_1988_fig4

He speculated that there were only a few ways that a population could move from one peak to another. In his figure above A, B, and C all depict very large population sizes: in the figure N is population size, U is mutation rate, and S is selection intensity, thus 4NU and 4NS large means that the population size is very large relative to the mutation and selection rates. The point of these top figures is that he thought that the only way a very large population can move from one peak to another is by a change in the environment (C).

His point is well taken in that in large populations drift will have only a small effect, and the population’s position on the adaptive landscape will be dominated by the effects of selection. Selection can only drive a population “up hill” to higher fitness, thus there is no way for the population to move down hill in fitness and cross a valley to a higher adaptive peak.

However, there are two reasons this may not necessarily be true. The first is a model put forth by Weinreich and Chao (2005 Evolution 59: 1175), that in retrospect is rather obvious (Isn’t that true of all great models?). Consider a population of bacteria (remembering they are the prokaryote equivalent of haploid) that has two loci A and B. Further imagine that A2B2 is intermediate fitness, A1B2 and A2B1 are of low fitness and A1B1 is the high fitness genotype. Based on Wright’s reasoning if we started a population off in a chemostat fixed for A2B2 it could not evolve to become A1B1 because of the mixed genotype low fitness valley. However, what Wright was not figuring on was just how large these populations are. A typical chemostat might have 106 to 109 cells per milliliter. If we imagine a mutation rate of 10-5 per locus then in each milliliter of chemostat there will be between 20 to 20000 bacteria that have mutated from A2B2 to A1B2 or A2B1. These mutants are effectively a low fitness population that is one mutation away from moving to the higher peak. Obviously it is in constant flux, new mutants are being added continuously due to mutation, but lost due to their lower division rate, and being eluted from the chemostat. Nevertheless, there is this substantial standing population of single mutant low fitness bacteria. At the higher density (109 cells) we would expect roughly 2 double mutant cells per milliliter at any given time. Note in the figure below the chemostat has 650 ml, thus such a system should have between 1.3 and 1300 high fitness individuals at any given time, even before taking into account the effects of selection.  Thus, in the very large population sizes of bacteria, two locus peak shifts, far from being rare, become nearly a certainty. Whether or not this works for the much smaller population sizes of multicellular organisms, or adaptive peaks that require the assembly of more than a few interacting loci remains an open question.

chemostat

A typical chemostat setup.  Media is added to the chamber at a constant rate, and effluent is removed.  When properly set up such chambers will maintain a constant population size of the experimental organisms. (image taken from http://openi.nlm.nih.gov/detailedresult.php?img=2906461_1471-2180-10-149-1&req=4)

The second model is Gavrilets’ “holey landscape” model (Gavrilets 1997 TrEE 12: 307). In this model Gavrilets points out that real adaptive landscapes have very high dimensionality, and that high dimension graphs do not behave the same way as the three dimensional graphs we are familiar with. He argued that with a large number of horizontal axes there would nearly always be a ridge along some dimension that connected the two points. Thus, he argued that rather than an adaptive landscape of hills and valleys we should think of adaptive topographies as flat plains with holes in them. The holes represent fitness valleys that selection would prevent the populations from entering. In this model the flat plain means that there is no selection, and the changes are neutral. Without selection all populations will drift at random over the landscape regardless of the population size (although large populations will move more slowly). For more on this check out some of Østman’s work on the evolutionary dynamics of holey landscapes. For what it is worth, my own perspective is that Gavrilets’ model may or may not be correct. Either way it does not qualitatively change Wright’s model. In Wright’s model it is necessary to cross adaptive valleys, in Gavrilets’ model there is a high dimensionality ridge connecting them. Either way it will take a combination of selection and drift to explore the landscape and move to a higher peak. After all, are we really surprised that the details of an 80 year old model may not be exactly correct?

Gavrilets-holey

Gavrilets’ holey landscape model. Taken from http://evolvingthoughts.net/2012/12/evopsychopathy-4-adaptive-scenarios/

Returning to Wright’s figure at the top of the post, parts D, E, and F consider what happens when populations are smaller. In D he imagines that a population is very small. He suspected that these would be so small that selection would be relatively ineffective, and they would be so dominated by drift, (not to mention inbreeding depression) that they would have little chance of evolving to a new peak. In figure E he suggests that a medium size population would be the ideal balance between selection and drift, with drift allowing exploration of the landscape, but selection still being strong enough to cause it to tend to climb towards peaks. The problem with the E scenario is that a single population can only explore a small part of the landscape, and it is unlikely that it would stumble upon a higher adaptive peak.

This leaves us with F, which is a metapopulation structure, that is a set of moderate size to small subpopulations that are joined together by a low level of migration. He felt that this population structure provided a set of subpopulations that were small enough to be strongly influenced by drift, and because it was a large number of subpopulations they could explore a much larger portion of the adaptive landscape. Finally, because they were tied together by low levels of migration, when a population evolved towards a new adaptive peak it could export this fitness solution to other populations.

So, this is why Wright focused on what are now known as metapopulations. He reasoned that it was only in these structured populations that you had the conditions that would allow the kind of random exploration of the adaptive landscape that he thought was essential for a population to discover and move to a higher adaptive peak.

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