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Evolution in Structured Populations

Sewall Wright’s Seven Generalizations about Populations

Posted: May 22nd, 2014 by Charles Goodnight

Once again I seem to be reorganizing my plan of attack, and this will be a big one. I think it would be entertaining to move over to a discussion of Wright’s shifting balance theory. This is not a minor topic, and indeed, I am told that the two longest papers ever published in Evolution were on this topic (Coyne Barton and Turelli 1997, Evolution 51;643-671; Wade and Goodnight 1998, Evolution 52:1537-1548; see also Coyne Barton and Turelli 2000, Evolution 54:306-317; Goodnight and Wade 2000, Evolution 54:317-324 – FYI there is a hidden message in Goodnight and Wade. Write down the first letter of each paragraph).

It is surprising I haven’t talked much about Wright up to this point. He is one of my heroes, and one of the first luminaries of evolutionary biology I ever met—ok, lying there a bit. I was at the University of Chicago when Wade, Lande, Arnold, Teeri, and later Schemske were all starting out, but they really don’t count because none of them had tenure when I first met them. . .

Sewall_Wright

One of the classic pictures of Sewall Wright while he was at the University of Chicago.

One of the fascinating things about Wright is that, unlike Fisher, he was an experimentalist, rather than a pure theoretician. The problem with doing experiments, of course, is that they are messy and rarely fit into the simple schemes that we develop for our models. Wright, as is often noted, and as is obvious from his books, bred a lot of guinea pigs and carefully examined the genetics of their coat colors and patterns.

guinea pigs

Guinea pigs come in a wide array of colors and patterns (http://emmasguineapigs.blogspot.com/p/cavy-colours.html).

Wright development fig

Wright’s view of developmental genetics.  (Wright 1968: Evolution and the Genetics of Populations, Vol. 1)

This is a situation we have seen before. True experimentalists are confronted with complexity that theoreticians are inclined to ignore. The difference is that Wright was genius, and both a good experimentalist and a brilliant theoretician, putting him the position of being both aware of this complexity, and having the mathematical skills to actually do something with it.

I actually think that there is another piece of history that may have been realized by Crow (Crow 1998 Genetics 148:923-928), but has generally been ignored. Wright received his Ph.D. in 1915 under the direction of William Castle, first at the University of Illinois, and ending at Harvard. Parallel to this, Shull began his work on corn genetics at the University of Illinois in 1905 (and coined the term heterosis in 1914), and East was similarly working on corn genetics at the Connecticut State Agricultural College. What is important here is that Connecticut State is where the Harvard scientists went during the summer both because it was cooler, and there was more land for field experiments. This was the time when the concepts of inbreeding and hybridization and the “magic” of hybrid corn was first being discovered. Importantly, this was also the time and place where Wright was a graduate student, and my guess is that he would have been interacting with the corn breeders on a more of less daily basis. Thus, it is interesting to speculate that the development of hybrid corn gave us not only hybrid corn, but also the shifting balance theory.

In any case, whether guinea pigs or corn or both Wright came up with a set of seven generalizations about populations. In his own words (Wright 1968: Evolution and the Genetics of Populations, Vol. 1)

“There are a number of broad generalizations that follow from this netlike relationship between genome and complex characters. These are all fairly obvious but it may be well to state them explicitly.

(1)   The variations of most characters are affected by a great many loci (the multifactor hypothesis).

(2)  In general, each gene replacement has effects on many characters (the principle of universal pleiotropy)

(3)  each of the innumerable possible alleles at any locus has a unique array of differential effects on taking account of pleiotropy (uniqueness of alleles)

(4)  The dominance relation of two alleles is not an attribute of them but of the whole genome and the environment. Dominance may differ for each pleiotropic effect and is in general easily modifiable (relativity of dominance).

(5)  The effects of multiple loci on a character in general involve much nonadditive interaction (universality of interaction effects)

(6)  Both ontogenetic and phylogenetic homology depend on calling into play similar chains of gene-controlled reactions under similar developmental conditions (homology)

(7)  The contributions of measurable characters to overall selective value usually involve interaction effects of the most extreme sort because of the usually intermediate position of the optimum grade, a situation that implies the existence of innumerable different selective peaks (multiple selective peaks).”

This is the genetics that Wright envisioned. It was a world in which all traits are determined by a large number of loci (multifactor hypothesis), and each of those loci affected a large number of traits (universal pleiotropy) and interacted intensively with each other (universal interaction effects).   The conclusion from this is that there are multiple ways to achieve high fitness (multiple selective peaks). What I find remarkable about this is that this theory was laid out in 1931 (Wright 1931, Genetics 16: 97-159), and yet it is clearly the outline of a complex system model, even down to using the words like “netlike relationship”, and “complex characters”. Basically, this model was developed before computers, before DNA, and really before we knew anything about genes other than that they behaved in a Mendelian fashion. In contrast, complexity theory, being generous, traces back to the 1940s at the earliest (http://www.ralph-abraham.org/articles/MS%23108.Complex/complex.pdf).

Wright was interested in how evolution could occur in this complex system (in both the formal and informal sense) he was envisioning. His world was very different from Fishers. The big difference is that the genetic complexity he was embracing, and his belief that species tended to be divided into small semi-isolated demes (more on that on another day), resulted in his seventh generalization, that there were multiple selective peaks. In contrast, Fisher thought that migration rates were generally large enough that the species could be considered approximately a single random mating population. In this situation, regardless of the amount of gene interaction, there will be only a single adaptive peak. Thus, the big difference between their world views was whether we could model evolution as a single fitness peak (Fisher), or whether we needed to model it as multiple fitness peaks (Wright).

Adaptive Landscapes

Adaptive landscapes.  Top:  Fisher’s world view  implies a simple fitness landscape with a single adaptive peak.  No matter where it starts on the landscape with only mutation and selection a population will eventually evolve to the top of the peak.  Bottom:  Wright’s world view explicitly incorporates a complex landscape multiple adaptive peaks.  In this landscape with only mutation and selection a population will always climb the nearest peak whether or not it is the “optimal” solution.  Once on a local adaptive peak the population will be stuck there. (image taken from http://www.terrorismanalysts.com/pt/index.php/pot/article/view/30/html)

In Wright’s view Fisher’s model, in which mutation and recombination generated variation and selection sorted out the good alleles from the bad ones, was simply not adequate to describe how evolution would occur on these complex landscapes. Thus, we can imagine his goal was to describe how evolution actually did occur on this complex landscape.

Wright’s model, as I have emphasized, was published in the 1930s. A lot has changed in the last 80 plus years. In subsequent posts I will describe Wright’s shifting balance process, but not from the historical perspective of what Wright envisioned, rather from my perspective in 2014 embracing my (obviously insufficient) knowledge of modern biology. I hope people will chime in when I have missed things or gotten things wrong.

 

 

Soft Selection: Why it is Multilevel Selection

Posted: May 14th, 2014 by Charles Goodnight

It has come to my attention that it makes sense to spend a blog entry talking strictly about contextual analysis and soft selection. The problem which, as Okasha (2006 Evolution and the levels of selection) puts it, some “theorists find deeply counter-intuitive” is that in soft selection every group puts out exactly the same number of offspring individuals each generation. As a result there can be no variance in fitness among groups, and yet, using contextual analysis one would conclude that despite (and actually because of) this lack of variation in fitness among groups that there must be group selection acting. As evidence I can stand up a few of my favorite straw men, West, Griffin and Gardner (2007, J. Evol Biol 21:374-385, p. 380):

However, now consider that, because of localized resource competition, all groups have a fixed productivity (soft selection; Wallace, 1968) and all competition for reproductive success occurs within the group. . . . Contextual analysis therefore identifies both an impact of individual-genotype and also an impact of group-genotype on the individual’s fitness, and hence diagnoses the operation of both individual and group selection. Again, this is undesirable, as group selection should not be in operation when all groups have the same fitness.

I have to mention they earn full credit for cluelessness, which is immediately obvious in this quote with the discussion of “individual” and “group” genotype. (Um, we are phenotype view evolutionary biologists. Get with the program guys.), but also because the quote is from a section titled “There is no formal theory of group selection”. (Dang, and all this time I thought quantitative genetics WAS a formal theory.)

Clueless 2

Clueless: A classic movie that, of course, is not relevant to this weeks post.  (Hey, diss group selection, prepare to be dissed back)

Soft selection is one of a selection scheme put forward by Wallace (1968 Polymorphism, population size, and genetic load. In Population biology and evolution. Lewontin RC (ed.), pp. 87-108) in which there is a set of populations. At the end of each reproductive cycle each population produces the same number of migrants, which are the winners of individual selection acting within the population.

soft selection schmatic

Soft selection: There is individual selection within each population for a fixed number of migrants that will be produced at the end of the reproductive cycle. This is contrasted with hard selection in which individual fitness is unconstrained by group membership. (http://pedrovale.wordpress.com/2013/07/08/killing-them-softly-managing-pathogen-polymorphism-and-virulence-in-spatially-variable-environments/)

The point is, because there is no variation in output of the different groups, there is by definition no variation in mean group fitness, and as a result it is reasoned that there is no group selection. Indeed this is the conclusion reached by Wade (1985 Am Natur. 125, 61-73), in which he states “For soft selection this covariance [between group mean relative fitness and group mean phenotype] is zero by assumption. Even if the within-group genotypic fitnesses, Wij, were frequency dependent, the assumption [of constant mean group fitnesses] would prohibit the operation of group selection.” I doctored the quote to remove math notation that is specific to that paper. However, this is an interesting problem. Intuitively, in soft selection the fitness of an individual is indeed a function of group membership. After all it is the phenotype of the individual relative to the population mean that determines its fitness, with an intermediate phenotype individual having a high fitness in a group of low phenotype individuals, and a low fitness in a group of high phenotype individuals.

So, what is going on? I think the easiest thing to do is to do some math. Further, to remove the politics, lets not think about contextual analysis. Instead, think of selection on two correlated traits, say body length, and body weight. These we can imagine to be correlated because generally longer animals are also bigger. In this population we can imagine that the population has a mean length of 60 inches and a mean weight of 100 pounds. Further, there is a correlation between length and weight of 0.5, giving a phenotypic covariance matrix of (Phenotypic NOT genotypic covariance matrix. Also, keeping it as simple as possible):

Soft Selection eq 1

So, now the problem. Because these traits are correlated, there is not only direct selection, that is if you select on length, length will change, but also indirect selection: Selecting on weight will also change body length. It turns out we can separate the effects of selection acting directly on a trait (direct selection) from selection acting on another correlated trait (indirect selection). The vector that does this is called the selection gradient, β. It is just like the selection vector, except that it mathematically removes the effects of indirect selection. Thus, if we have selection only acting on body length then the selection gradient might look like this (again keeping it simple)

Soft Selection eq 2

Indicating a selection strength of 1 on Length, and a selection strength of 0 on weight. Of course, we are asserting this direct selection, but what we actually observe is S, the selection vector, which includes both the effects of direct AND indirect selection. So what would S look like? To do determine this we need to recognize that the gradient is actually equal to β = P-1S, where P-1 is the inverse of the phenotypic covariance matrix:

Soft Selection eq 3

Then doing a bunch of algebra nobody needs to know we can solve for S, and discover that:

Soft Selection eq 4

In other words, if we apply selection only on length because the traits are correlated we will also see a change in weight.

OK, Now contextual analysis and soft selection. In the case of soft selection by assumption we have:

Soft Selection eq 5

In other words, we are assuming that there is no covariance between the group mean trait and group mean fitness. I am running out of space, but basically if we use only simple covariances:

Soft Selection eq 6

But, of course, what is important is not S, but β. Thus, if we want to calculate the selection gradient we need to calculate β:

Soft Selection eq 7

The point is that in order to have no variation in fitness among groups you need to have selection at the group level to remove that variation in fitness.

I could have made the same story with the original example of length and weight. If you want to select for longer animals without changing their weight you will need to select against weight. That is, you will need to select for long skinny animals. In other words, lack of covariance between fitness and a trait correlated with one under selection does not come for free. You need to actively apply selection to remove that covariation.

Thus, the point of this whole story is that the lack of covariance between group mean fitness and the group trait is NOT evidence for lack of group selection. Far from it, it is in fact evidence that group selection is acting. To believe otherwise is to simply not understand how selection on correlated characters works.

Why I Like the Multilevel Selection Approach

Posted: May 9th, 2014 by Charles Goodnight

For the past two weeks I have been rather destructionist (is that a word), with my diatribe against kin selection. It seems to me that if you are going to tear down a structure and declare it not useful then you had better be willing to provide an alternative and explain why your alternative is a better choice. With that in mind, this week I will be talking about why I think the multilevel selection approach is the best, and possibly only legitimate, approach for studying social evolution.

In MLS theory the distinction between selection and the response to selection explicit. MLS theory is an outgrowth of quantitative genetics. The classic breeders equation, R=GP-1S, divides the response to selection, R, into the ecological process of selection, P-1S, and the mechanism of inheritance G. This is important because it also provides a guide to research. The reason that we did experimental studies of group selection in the laboratory is that it provided a means of studying G. That is, if we experimentally apply group selection did it cause a response to selection. The answer, of course is yes. I could go on with how this response was explored in some detail, but the point is these lab studies were explicitly designed to study G, the inheritance part of the breeders equation. On the other hand, the contextual analysis studies I have been talking about are primarily useful as phenotypic analyses, that can be applied to natural populations. Thus, we have a growing number of studies demonstrating that multilevel selection is quite common in nature. These studies tell us nothing about the inheritance, for the simple reason that the research is specifically designed to inform us about P-1S. The point is that experimental studies of inheritance are, both from a conceptual and practical perspective, very different from studies of selection. It is thus this distinction is not a minor triviality of the mathematics, it is a central and useful feature of the theory.

In MLS theory fitness is seen as a function of phenotype. In kin selection theory the modeled relationship is between fitness and gene. Efforts have been made to relax this, but ultimately the method is about the effect of single genes or at least very simple genetics on fitness. In MLS theory the modeled relationship is between fitness and phenotype. This is much more realistic. Phenotypes are what we measure in real populations. The relationship between phenotype and genotype is potentially very complex, and certainly not knowable in real field studies. The MLS approach acknowledges this reality, and as a result it is a method that can realistically be used to study natural selection in the wild. This is an area where kin selection simply fails.

In MLS theory the relationship between genotype and phenotype is acknowledged to be complex. In kin selection theory a single gene (or aspect of the genotype) is considered to affect both the individual trait (the cost) and the group trait (benefit). In MLS theory the group and individual trait are considered to be separate but correlated traits, and the genetics are expressed in the form of the G matrix. This allows for the simple system of kin selection (in which the correlation between expression at the two levels is 1), but allows for the possibility that they are not perfectly correlated. As an example, in a typical kin selection model you are either selfish or an altruist. However, if the correlation is not perfect, then you could get different degrees of “efficiency” for altruists. Then selection might favor the phenotype (note not genotype!) that provides the maximum altruism for the minimum cost. In the putative Haldane case, rather than sacrificing his life for his two brothers maybe he just has to cut off his arm. As importantly, as we discover added complexity in nature of inheritance the “inheritance” part of the equation can be modified as needed, either by modifying the G matrix, or when a clever enough theoretician comes along, replacing it with something new (Anyone want to take on coskewness tensors?).

MLS theory focuses on similarity regardless of cause. Hamilton’s rule in kin selection format:

Why MLS HR KS version

can be reconstructed in MLS format in which case the equation comes down to:

Why MLS HR CA version

this makes the point that any cause of variance among groups, or equivalently (because of the bizarre nature of statistics) any cause of similarity within groups can be on the right side of this equation. Kin selection, with its focus on shared genes as the cause of similarity falls short here. What else can cause similarity within groups? How about shared cultural heritage, or traditions? How about policing enforcing uniformity?   This hugely broadens the range under which altruism can evolve.

The MLS approach is not obsessed with the evolution of altruism. The case where group and individual selection are acting in opposition is certainly interesting, but in kin selection it is the ONLY thing that is interesting. This is because it is an optimality approach and when they are acting in the same direction the equilibrium (fixation of the over all good gene) is trivial and uninteresting. Because the MLS approach measures selection as it is acting there is no need to focus solely on altruism. In general the word “altruism” is relatively rare in the MLS literature. It is an interesting sidelight, not the main focus of the research. This has lead to some interesting findings that are generally not appreciated outside of the MLS community. For example, individual selection interferes with group selection and is itself often ineffective due to indirect genetic effects. As a result, the overall response to simultaneous group selection and individual selection acting in concert is often less than simply the response to group selection acting along, and both are generally greater than the response to individual selection acting alone.

The MLS approach treats selection as a competing rates problem. Optimality approaches such as kin selection can at best tell us where a population “ought” to go, all things being equal. The problem is that all things are not equal, populations are not optimal, and there are a thousand contingencies keeping them away from simplistic optima. Because the MLS approach deals with the here and now, and how things are changing under the present conditions many of these problems go away. MLS approaches can be used to study stabilizing selection – either classic stabilizing selection at either the group or individual level, or stabilizing selection due to group and individual selection acting in opposition. However, note that I am not calling the stable equilibrium the “optimum”. In the example of group and individual selection acting in opposition the equilibrium point will be determined by a combination of the strength of selection at the two levels, the heritability of the group and individual level traits, and the genetic correlation between them.

I sum I think that the MLS is simply a better conceptual framework for thinking about social evolution. MLS theory fits firmly into the phenotypic approach, whereas kin selection theory, because of its focus on genes is basically incompatible with the phenotypic approach. MLS is by no means a mature theory, and there is much still to be done. But, heck that makes it exciting. The important point is that unlike kin selection theory, which is sadly stuck back in the 1960’s, MLS theory is an open ended field that is ripe to grow along with our increasing understanding of the subtleties of evolution.

 

Dynamical models of multilevel selection: Another problem with Kin selection

Posted: May 1st, 2014 by Charles Goodnight

First off, if you haven’t seen it check out the American Museum’s on line collection of photographs. I haven’t had a chance to really explore the hundreds of thousands of photos they have, but I am certain there are some real gems in there.

G. G. Simpson

One of the photos from the American Museum of Natural History. This is G. G. Simpson at his desk, at the museum. AMNH has over a million photographs on line.

It turns out I am not quite done “dissing” kin selection, although my discussion this time is nothing I would have thought of as a problem. What I want to talk about is a pair of papers that appeared in Evolution in a special section on multilevel selection that I edited.

The first of these is my own paper on direct fitness and contextual analysis (Goodnight. 2013 Evolution 67, 1539-48). In this paper I work through the relationship between direct fitness and contextual analysis. It turns out that both of these approaches are using multiple regression to analyze selection. In direct fitness the equation is:

dyamical models eq 1

Where W is absolute fitness, (ind) is the individual trait, (grp) is the trait in interacting partners, and x is a measure of genotype. Without loss of generality I converted absolute fitness to relative fitness (come on guys, working with absolute fitness is for chumps!), and I recognize that because these models are so naïve there must be a function relating genotype to phenotype. Thus, there is a value dynamical models eq 2that relates a change in phenotype to a change in genotype. So multiplying through by dynamical models eq 2we get the equation for contextual analysis:

dynamical models eq 3

which is really the same equation, but to me much more aesthetically pleasing for two reasons. First, as I said, working with absolute fitness is for chumps (AND it makes a difference for contextual analysis), and second, get real, we cannot measure “genotype”, hell, I don’t even know what that means, whereas I have a very clear idea of what I mean when I measure the phenotype.

Anyway, be that as it may, the end result is that the difference between the direct fitness approach (or neighborhood modulated fitness approach) and the multilevel selection approach of contextual analysis does not lie in the equations they use. Rather it lies in HOW the equations are used. In the direct fitness approach the equation is solved for the point where dW/dx = 0. Mathematically this has to be one of three types of points, a fitness maximum, a fitness minimum, or an inflection point. Simple inspection can distinguish between these three possibilities (or second derivatives if you prefer). In contrast, in contextual analysis the slope is analyzed at the point  where the population is currently residing, and dw/dz becomes a measure of the rate of change in relative fitness as a result of multilevel selection. In any case, it is quite reasonable to argue that kin selection and multilevel selection are very similar if not the same thing.

Next, we turn to Simon, Fletcher and Doebeli (2013 Evolution 67, 1561-72.). This is a dynamical model of two level selection using a continuous-time Markov chain, and a companion deterministic partial differential equation model. One of the first things I got out of this model is that Burt Simon is a better mathematician than I am, but as far as my little mind is capable of understanding such things, this model is quite complete, and an excellent general model of multilevel selection. Without going into details they develop a pair of partial differential equations, one in which it is assumed that there is not change in the number or types of groups, basically the frequency of individual types is allowed to change, but the overall change in group types is zero:

dynamical models eq 4

where αi is the growth rate (births-deaths) if the ith type of individual, xi is the trait value of the ith individual, and t is time. They then go on to argue that there are group level processes (group extinction, recolonization, fusion, fission, differential growth) that enter in to the equation.   On the other hand, if no changes in individual fitness are allowed then:

Dynamical models eq 6

Thus, and without going into detail, they then show that the overall change in the trait is:

dynamical model eq 5a

Please remember I am not doing this model justice, so, either believe that what I say is true, or read it yourself. (Word of advice: As Reagan, citing an old Russian proverb, said: “trust, but verify”) (Comment 2, I have no idea why these equations are so ugly.  Click on them for a clearer view).

The result of this is that they argue that if a selective event changes only the αi – the growth rate of the ith type without affecting the distribution of group types then only individual selection is acting, if the selective event changes the distribution of group types with out affecting the growth rate of individual types it is a pure group selection event, and finally if both change it is a multilevel selection event.

They then go through two examples that show the logic of what they are talking about, and eventually ask whether inclusive fitness, that is there in all cases a function bi that can be found that successfully combines individual fitness effects and group fitness effects. The answer to this is no. They point out that the two level approach can be solved directly, but the one level approach necessarily requires the prior solution to the two level approach. In their words, the reductionist approach is not “dynamically sufficient”, and there is a real difference between multilevel selection and kin selection models.

This is an interesting conundrum. On the one hand, the non-dynamical models of kin selection and contextual analysis arguably suggest that the two processes are the same. A dynamical model indicates that they are not the same. Who is right?

The answer seems a bit complex. First off, Goodnight and Simon et al. actually have different definitions of group selection. The Goodnight definition is that group selection is acting when the fitness of an individual is a function of group membership. The Simon et al. definition is that group selection is acting when the outcome of selection depends on group level fitness effects. However, I don’t think this is the problem. I think the bigger problem is that direct fitness and contextual analysis are statistical models that measure the conditions at the current value of the population. Contextual analysis works here because it is measuring the regression slopes at the current population values. It is certainly possible to imagine a system that overall had multilevel selection acting, but at a particular set of gene frequencies (or what have you) group selection was not acting at that moment. Thus, at least in theory, the strength of selection at the two levels may change from generation to generation, and selection at one level might even disappear briefly. This rather minor problem for contextual analysis is a huge problem for kin selection. That is, another way of saying the complaint about these regression models is that there are non-linearities built into multilevel selection. I suspect that if you could force the models to be linear that the manipulation of equating inclusive fitness with multilevel selection in a dynamical model just might work. However, because the two levels will inevitably have non linearities, and in most cases will in some way interact, the linear approximation of kin selection models are doomed to failure.

In other words, kin selection practitioners are guilty of one of the basic errors that all undergraduate statistics students are taught. They are extrapolating from the current population conditions to some point in the far distant frequency space. In short, they are extrapolating beyond their data.

 

 

Why I Don’t like Kin Selection

Posted: April 23rd, 2014 by Charles Goodnight

Sorry this is so long.  To paraphrase Mark Twain Blaise Pascal, I would have written less, but I didn’t have time.

Up to this point I have been slogging through the details of why the phenotypic perspective is a good idea, and how it resolves a bunch of technical issues around how evolution works.   One of the truisms of teaching is that nobody gets it on the first pass, so I will occasionally be going back over the technical details of the phenotypic approach, but at this point it is time to change gears. It turns out that the true strength of the phenotypic perspective is that it suddenly resolves a suite of issues that have plagued evolutionary biologists for a long time. Some of the issues are minor things like: what is an individual (hint: you get to decide), why sex (hint: genes are slaves to phenotypes, and have no rights), the origin of life (hint: the phenotype always comes first), why is DNA the molecule of inheritance (hint: the unstable enslave the stable), and a host of other equally trivial questions in biology. Dang, when we get done with this, evolutionary biologists won’t have anything to fight about. Sadly a lot of this isn’t published, so I have to come to grips with how I feel about writing down unpublished ideas. On the other hand, until the phenotypic approach is embraced by somebody other than me these ideas will never get published. . . .

This week, however, I figure I should talk about kin selection. By this point it should be obvious that I am no fan of kin selection or inclusive fitness. And, while I do like Hamilton’s work in general, a lot of his models just aren’t very good. My postdoc adviser, David Mertz, referring to optimal foraging theory, once claimed that Robert MacArthur set ecology back 100 years. I always loved that statement, both because I understood what he meant, and because it couldn’t possibly be true. MacArthur had a tendency to ignore important issues and produce models that were aesthetically appealing but unsatisfying to the deep thinkers in the field. Second, population ecology wasn’t 100 years old, so it implied that MacArthur had reset the field back to before it started. Well, I guess I rather feel a bit the same way about Hamilton when it comes to social evolution. The real damage that Hamilton did 50 years ago when he published his model is that people somehow thought that inclusive fitness was useful, and a gigantic field grew up that has actively interfered with our ability to understand social evolution. So what is it that I don’t like about kin selection?

macarthur hamilton

Robert MacArthur (Left), William Hamilton (Right). Both MacArthur’s optimal foraging theory and Hamilton’s kin selection theory are optimality models. Both suffer from the limitations that are inherent to the optimality approach.

Kin selection is an optimality approach. Optimality approaches have a way of providing interesting insights but then grinding to a halt when efforts are made to apply them to experimental systems. The case in point is optimal foraging theory (Back to MacArthur. Maybe Mertz and I were cut from the same cloth). MacArthur and Pianka (1966. Am Nat 100:603-609) developed the first optimal foraging model, which was an effort to solve the problem of what an organism should do to maximize energy intake within the constraints of the ecology of the organism. Such constraints eventually included such things as search time for food, handling time etc. This was an enlightening model in that it really focused on the idea that organisms can be thought of as solving the problem of maximizing food intake while minimizing risks and costs. This led first to a large number of models on optimal foraging theory, and second, the realization that real organisms basically never follow the optimal solution. The solution to the lack of fit to real data was a series of ever more complex essentially post hoc theoretical solutions, e.g., they are maximizing limiting resources not calories, they are avoiding secondary compounds etc. The bottom line is that today you will frequently find simple optimal foraging models used as a starting point for more nuanced theoretical and experimental studies, but you would be hard pressed to find ANY papers that are solely about an optimal foraging model. I asked a colleague about this, and her response was that people probably stopped because it was just not very useful. This is the fate of optimality models of all sorts. They provide nice qualitative insights, but they simply are not very useful. Kin selection models are no exception. Yes, Haldane’s famous statement about being willing to sacrifice his life for two brothers or eight cousins is a nice qualitative insight (among other things it demonstrates that Haldane had a time machine so that he could travel to the future to bask in Hamilton’s brilliance).   However, that is as far as ANY kin selection study has ever gotten. Read a few. They do crazy hard research on the behavior of prairie dogs or slime molds, then at the end they say something to the effect of “and this is consistent with a model of kin selection.” There are never any numbers telling us just how consistent or anything else. How close does your organism have to be to the optimum before your theory is supported? Optimality models don’t provide that insight.

Note added later:  I originally attributed the two brothers, eight cousins comment to Dobzhansky.  This was simply me writing too fast.  In fact, this story may be apocryphal.  Thanks to Trevor Pierce for pointing this error out.

Kin selection can only focus on altruism. ALL kin selection models are about the evolution of altruism. To a kin selectionist altruism is when an individual increases the fitness of another individual at the expense of their own fitness. To a multilevel selectionist altruism is when group and individual selection are acting in opposition. There are ample models showing the equivalence of these two statements (e.g., Goodnight 2013 Evolution 67, 1539). Because kin selection is an optimality model the only time it is interesting to study social behavior is when the two levels are acting in opposition. Multilevel selection is much richer than this. There are plenty of times when two or more levels of selection act in the same direction. Many of these would be very interesting, however, biologists tend to ignore them because they are outside the realm of what can be studied using kin selection models.

Kin selection is a genic model.   The way that Hamilton originally developed his model, and the way it is virtually always used is based on shared genes. Individuals are altruistic towards other individuals because they are relatives and thus might share genes. Relatedness is a proxy for the probability of shared genes, but there are other means of detecting genetic similarity, such as the “greenbeard” model (Jeeze I hate that term. I hope who ever invented that term burns in hell, that is, if atheists can burn in hell). The problem is that the world doesn’t work that way. Models be damned, there is no “altruism” gene. Models that start with the assumption of a single locus with an altruistic and selfish allele are ok as a starting point for qualitative thinking, but totally useless in the real world. The problem is that the genic nature of kin selection does not give us a way to move beyond that. Sure you find lots of times where modelers will define “x” to be some measure of the genome, but if we are going to use it in the real world we need to know WHAT aspect of the genome. Also there have been attempts to move to a phenotypic based kin selection model, but these have always failed, mainly because they always go back to the thought that shared phenotype means shared genes and somewhere buried down there is an altruism gene. On top of this there are plenty of cases of culturally based altruism. Soldiers are famous for acts of altruism among genetically unrelated members of the same unit. What makes them similar is culture not genes. Genic models cannot handle this, and kin selection is no exception.

Kin selection uses a linear additive genetic model. The one thing we know for a fact is that there are tons of evolutionarily important interactions in the biological world. These include dominance, epistasis, and indirect genetic effects of all sorts. We also know that these have profound effects on selection, and especially multilevel selection. With its focus on single altruism genes and gene sharing kin selection models are relegated to the world of one behavior, one gene. Relatedness (r) in kin selection models tends to be the proportion of genes shared. In fact, “r” equates to the fraction of variance among (kin) groups, that is a measure of similarity. If there are interactions, particularly indirect genetic effects, “r” may be much larger than the proportion of genes shared.

Kin selection assumes that the cost and benefit are the same trait. In kin selection models the trait is “altruistic” versus “selfish” which has fitness consequences on the individual (cost) and on its partners (benefit). This works fine for single locus traits that are the focus of kin selection models; however, real traits are polygenic. In a polygenic setting the cost and the benefit must be considered separate, but genetically correlated, traits. Consider two individuals, both altruists, but one is a more efficient altruist than the other. That is the efficient one can help at less cost to itself. In this case both individuals give the same benefit (B), but different costs (C). This is not possible in a single locus model, but it is an expected result for polygenic models. In kin selection models, because they are the same trait determined by a single locus the only thing that can change the equation is r, which is strictly a measure of the proportion of shared genes. For a polygenic trait when the group and individual trait are considered to be separate correlated traits, I have shown that “r” is the ratio of heritabilities for the group and individual level trait. (Goodnight 2005 Population Ecology 47, 3-12.). If we take, for example, a typical metazoan, the cells within an organism are nearly genetically identical, thus the within individual heritability for cell level traits is very nearly zero. On the other hand the heritability of organismal traits is what ever it is, and very likely non-zero. In this case “r” is the ratio of the heritabilites of the organismal trait to the cell trait, which could be a very large number. In the kin selection world “r” goes from zero to one, in my world it goes from zero to infinity.

Nobody has ever or will ever measure the strength of kin selection. Kin selection is an optimality approach, and tells you what the best solution is. However, the best solution doesn’t mean much if it is unattainable. It may be unattainable for many reasons. There may not be genetic variation for it. It may be opposed by selection acting on something else.  Selection on it may be so weak as to be meaningless.  How do we address this? Well the typical way is to measure the strength of selection, and the heritability of a trait. If we do this we have data to specifically address these questions.  For example there are numerous examples of studies showing that opposing selection on different traits or different life-stages prevent a trait from changing. Central to these studies is comparing the strength of opposing selection and seeing if the actual value of the trait corresponds to the value predicted by the estimated competing rates of evolution. There are plenty of other studies in which selection is technically found to be operating, but it is so weak that it can be disregarded, thus, knowing the strength of selection is essential to being able to interpret its importance. Because kin selection is an optimality model it is not and cannot be used to measure the strength of kin selection. Indeed, because kin selection is apparently defined by Hamilton’s rule it is not at all clear what we might mean by the strength of kin selection. Unfortunately, until we know the strength of kin selection relative to other evolutionary forces any conclusions drawn from kin selection studies will be nothing more than “just so” stories.

Kin selection confounds three different things. From a multilevel selection perspective Hamilton’s rule consists of three different elements, cost, which is the strength of individual selection, benefit, which is the strength of group selection, and r, which, for a trait and the group mean of a trait, is the fraction of variance among groups. More generally, it is the squared correlation between the group and individual trait. If it is a phenotypic selection model that would be the phenotypic correlation, if it is a genetic model it is the additive genetic correlation. The problem is that kin selection mushes these three things, group selection, individual selection, and variance explained by the group trait into a single value. The question is how does one interpret this? It tells us nothing about whether or not kin selection is important since it tells us nothing about the strength of selection. It tells us where a trait should evolve to, but if a trait is at that predicted optimum it gives no guidance as to whether that value is due to kin selection, due to something else that just happened to be at the same optimum, or if it is just passing through as it evolves to some other value. Further there are three parameters that can be manipulated, C, B and r, so for any given optimum we can presumably make a three dimensional surface of values of these parameters that will all provide the same optimal trait value. Thus we potentially can’t even easily compare two populations at the same optimum.

There are other minor concerns I could raise, but I am up to twice my normal length for a blog post so I will stop. In closing, I will say I am not going to dismiss kin selection as useless any more than I will dismiss optimal foraging theory as useless. However, like optimal foraging theory, it appears to mainly be useful in making broad stroke qualitative predictions that can be used in the introduction, or in a laudatory paragraph about how wonderful Hamilton is at the end of a paper. If you want to make quantitative statements about selection in real world populations that will contribute to our understanding of social evolution multilevel selection might be a better choice.

The Phenotypic Approach — A recap.

Posted: April 18th, 2014 by Charles Goodnight

This week I want to finish up the discussion of indirect genetic effects and contextual traits by tying them back to the theme of this blog. Going way back to the early days (yes, next week this blog is one year old!), it is important to remember that the theme of this blog is that there is much to be gained by flipping our standard way of thinking about evolution on its head. That is, we typically think about evolution as something that happens to genes. In the shellfish jeans model of evolution it is genes that make phenotypes, to carry them forward to the next generation.

walking clam

Apparently shellfish jeans are black. (from http://www.lilikoijoy.com/2013/09/an-american-hometown-parade.html)

However, living things are complex systems, and as with any complex system there are multiple ways of looking at them. Each of the different ways of looking at a complex system is a way of simplifying it so that it is interpretable to the simple minds of humans, and as a result each will have strengths and weaknesses. I think it can be argued that the genic view has been useful in developing our understanding of how evolution works, in no small part because it simplifies the inheritance to the point of triviality. Haldane’s models of selection really helped us understand how selection works; however, it did so at the expense of anything resembling reality.

In contrast to the genic view I have been arguing that we should start thinking about phenotypes creating new phenotypes, and using genes as part of a “transition equation” that creates offspring phenotypes based on the characteristics of the parental phenotypes. At some level this is just another perspective from which to study evolution, and perhaps one that loses the simplistic view of genes as the center of evolution (Dare I follow Godfrey-Smith and call them rational agents?). What is gained from this view, however, is enough to make me, at least, think that it is more than “just another view”.

So far I have primarily focused on mechanical aspects of why the phenotypic view, as I call it, is in many respects preferable. Perhaps the greatest advantage is that evolution works on phenotypes, and in most cases it is phenotypic data we access to.  It is a phenotypic perspective aligns with this reality. It always seemed to me rather irrational that we have this view of evolution based on change in a theoretical object that has little basis in reality (or as Pigliucci quoting Godfrey-Smith put it, genetic material is “a stuff not a discrete unit.”), which we are rarely in a position to measure, and when we do have access to things correlated with the genes (SNPs etc.) we more often then not discover that the “gene” is affected by a host of unidentified modifiers. How much more rational is it to construct a theory and a world view around the phenotype which is observable (or at least traits are observable), and that is the focus of selection and adaptation?

Nearly as great an advantage, however, is that the phenotype-to-phenotype transition equation is not constrained in the way that genes constrain our view to particulate inheritance. The transition equation can contain both Mendelian elements and continuous elements. The continuous elements can be either things that are truly continuous, such as culture, or they can be continuous approximations of underlying particulate traits, such as is used in quantitative genetics. This is actually more important than it appears at first blush. A theory of evolution based solely on changes in gene frequencies is simply inadequate given what we are beginning to learn about inheritance. Because we have this gene-based view of evolution we have had wildly difficult times incorporating even simple things like cytoplasmic inheritance, let alone complications such as epigenetics. Our usual approach is to study such things in isolation. Thus, we treat “cultural inheritance” as if it was somehow distinct and isolated from genic evolution. One need only look at the correlation between lactose tolerance in adults and the cultural use of cows to know that this isolation is simplistic. We also see extravagant claims that epigenetics are somehow distinct from “Darwinian” evolution. I am still looking for where Darwin discusses epigenetics in the Origin of Species.

The third advantage to the phenotypic approach is that the transition equation naturally incorporates various aspects of population structure. In the genic view mating and interaction structure are not easily incorporated since they don’t alter the structure of a gene. Instead they alter the effect of the gene on the phenotype, and how it affects heritability. It is this last area that has been the focus of recent blog posts. Starting with “measuring the heritability of contextual traits” and proceeding from there.   Because the phenotype-to-phenotype transition equation is a means of predicting the distribution of phenotypes in the next generation it can easily be modified to include the effects of mating structure or interaction structure.

Perhaps the most dramatic distinction between the genic view and the phenotypic view comes with multilevel selection. Multilevel selection really is not particularly interesting from a genic perspective. If each gene is working for its own best interests in isolation from other genes then keeping track of selection structure is of little consequence or interest. Of course the down side to the simplistic genic view is that population structure does matter, and while using the genic perspective it is easy to make models that ignore population structure, they have precious little to do with muddy boots reality. From a phenotypic perspective, however, selection structure is important, and the level of selection will alter both the rate of adaptation and the qualitative nature of those adaptations. Rather satisfyingly, experimental results strongly support the idea that level of selection matters.

The point is that the things I have been discussing in this blog are wildly complicated from a genic view but naturally fall within the logic of the phenotypic view, thus, to reiterate a theme, while the genic view may be useful, it is perhaps time to move on and try to think about evolution from another perspective.

There is actually one more reason that the phenotypic perspective is useful. That is that there have been a number of controversies in evolutionary biology that have resisted easy analysis from the genic perspective. Many of these issues simply go away using a phenotypic approach. I will address some of these in the next few weeks. Hang on to your hat, it promises to be a wild ride.

Bond_22_The_Car_Spin_(The_Man_With_The_Golden_Gun)

The name is Bond, James Bond. (From The Man With The Golden Gun)

 

 

 

 

Indirect effects, Individual Traits and Contextual Traits

Posted: April 11th, 2014 by Charles Goodnight

Another pure essay post.  I was surprised that last weeks post didn’t generate any controversy.  I guess that that proves that the only people who read my posts are people who agree with me.  Sigh.  As the song goes “I’d love to change the world, but I don’t know what to do, so I’ll leave it up to you”

ten_years_after_-_a_space_in_time_-_front1

This week I want talk about indirect genetic effects in comparison to contextual traits, something about which I have not been particularly clear.  In general it can be dismissed in two sentences.  Individual and contextual traits are part of the phenotypic compartment, and indirect genetic effects are part of the inheritance compartment.  As such they are independent concepts.

Whether a trait is an “individual trait” or a “contextual trait” depends entirely on what it is measured.  Thus, if it is a characteristic of an individual (height, weight, sprint speed) it is an individual trait.  If it is a characteristic of the group, neighborhood or other aspect of the context that an organism finds itself in, then it is a contextual trait.  One point here is that one of the whole points of contextual analysis is that we are treating “as if” they were traits of the individual, so perhaps from a rather odd perspective there really is no difference between individual and contextual traits.

Indirect effects on the other hand occur when genes in one individual affect the expression of a trait in another individual.  This is an idea that has been around for a long time, certainly it is an underlying theme in Griffing’s work (e.g., Griffing 1977 Selection for populations of interacting genotypes. In: Proceedings of the International Congress on Quantitative Genetics, August 16-21, 1976. E. Pollak, Kempthorne O, andBailey TB (eds.) Iowa State University Press,  Ames  Iowa., and references there-in), however the modern development of the idea, and the term “indirect genetic effects” can be traced to (Moore, Brodie, and Wolf 1997 Evolution 51, 1352-62.).  Indirect effects will almost certainly affect contextual traits, but in many circumstances they will also affect individual traits.  And that is the point of this essay:  individual traits can be influenced by both the genetics of the individual and the genetics of other individuals with whom they interact.  Similarly, contextual traits can be influenced by the genetics of the focal individual, and by the genetics of other individuals with whom they interact.

Thus just because a trait is clearly measured on the individual and correctly called an “individual” trait, does not mean that the genes reside in the individual expressing the trait.  A really good example is Griffing’s study of biomass in Arabidopsis.  If you recall, in this study Griffing grew pairs of plants together in sterile agar, and measured dry weight of the plants after they were harvested Clearly, biomass is a trait measured on an individual, and must be considered an individual trait.  Just as clearly in his study the trait biomass was determined both by the “direct effects”, that is the effects of in individuals genes on itself, and indirect genetic effects, the effects of its interacting partner on its phenotype.

At this point I am basically done with the issue I wanted to raise today, but it is worth discussing this point a bit more.  Just as with contextual traits, the realized heritability of individual traits will potentially depend both on the mating structure of the population and on the interaction structure.  Thus, even apparently pure individual traits can have there heritabilities change when the interaction structure changes.

Nobody has ever done a detailed manipulative study of the effects of interaction structure on the heritability of individual traits.  This is too bad, because it potentially has some profound implications.  I will give you one:  One of the truisms of evolutionary theory is that you can get a response selecting on just about anything.  However, In my thesis I worked with Arabidopsis selecting on leaf area (Goodnight 1985 Evol. 39, 545-58).  In this study I actually got a negative response to individual selection, a result that was predicted by Griffing (1977 In: Proceedings of the International Congress on Quantitative Genetics, August 16-21, 1976. E. Pollak, Kempthorne O, andBailey TB (eds.)).  Further, the one apparent exception to the idea that you can select on anything is competitive ability.  There have been a lot of experimental studies of the evolution of competitive ability that have failed to get a response (e.g., Futuyma 1970 American Naturalist 104, 239-52.).   Perhaps now we can put that old saw that you can select on anything into a new light.  Perhaps you can select on anything when you put the organisms in an environment where competitive interactions among individuals are minimized.  Apparently in both my study and Futuyma’s study the indirect genetic effects outweighed the direct genetic effects and prevented a response to selection from occurring.

A Reprise on Contextual Analysis.

Posted: April 4th, 2014 by Charles Goodnight

My life has gotten a bit hectic these days, so as usual a bit late, and perhaps a bit short.  At this point I have gone through most of the basics that I wanted to talk about before getting into more speculative stuff, but I think that a few weeks of review and revisiting past posts is probably warranted.  What I want to talk about for the next couple of weeks is something the difference between contextual traits and indirect genetic effects.  I think that my past discussions on the difference have perhaps not been very clear, and I hate to say it, part of the problem may have been a bit of confusion on my part.

Turning first to contextual traits.  It is important to remind ourselves that the classic breeder’s equation, R = GP-1S, divides evolution by selection into the ecological process of selection, and the heritable transmission represented by the G matrix.  Contextual analysis, in its standard form, deals only with S.  Thus if a trait is measured on the individual it is an individual trait, if it is measured on the context the individual finds itself in it is a contextual trait.  The heritable (genetic?) basis is entirely irrelevant.  The beauty of contextual analysis is that it is treats a trait that is measured on the context as if it were a trait of the individual.  Thus if an individual is in a group of 16 individuals then it has the trait of “group size = 16”, if it is in a group that is 30% altruists, then it has the trait of “altruism level = 0.30”, and so on.  Perhaps the correct way to think of it is that our individual is experiencing a group size of 16 or an altruism rate of 0.3.

In the early group selection days, people like Maynard-Smith insisted that group selection could only be invoked when groups were distinct entities that had clear borders.  That is, groups were things you could walk around.  Of course this becomes problematical when experimentalists examined the effects of migration (e.g., Wade 1982 Evolution 36, 945-61), or when you had group selection by differential migration (e.g., Wade and Goodnight 1991 Science 253, 1015).  Contextual analysis allows us to resolve this issue easily.  A contextual trait is a trait measured on the context.  Classic Maynard-Smithian group selection is but one extreme of a continuum that ranges from group selection at one end to frequency dependent selection at the other extreme.  Of course, this begs the questions: when is it group selection, and when is it frequency dependent selection.  If the two are part of a continuum then where you draw the line is at some level arbitrary, and a matter of aesthetics rather than science.  This also makes the interesting point that since the selective pressures on almost all traits are at some level dependent on the context the organism is found in, it suggests that pure individual selection, in which the fitness of an individual is solely dependent on its phenotype, and not at all influenced by the phenotype of its neighbors, is probably at least as rare as group selection acting by differential extinction and recolonization of whole groups.  My guess is that viewing the evolution by natural selection outside of a multilevel selection perspective is simplistic, and frankly, wrong.

I should clarify one aspect of the frequency dependent selection issue.  In mathematical modeling of selection there are frequency dependent models in which fitnesses change as gene frequencies change.  Call this mathematical frequency dependence.  In these models there is only one group, and as a result there can be no multilevel perspective.  Importantly, these models cannot be used in (short-term) studies of real populations for the simple reason that gene frequencies rarely change fast enough to see this mathematical frequency dependence.  To study frequency dependent selection in nature we need to find different populations that have different frequencies of the different phenotypes in different populations.  This is statistical frequency dependence.  I would argue that statistical, but not mathematical, frequency dependence should be studied as multilevel selection.

Another interesting point about contextual analysis:  it comes from another field. The earliest reference in my endnote is (Przeworski  1974 Contextual models of political behavior. Polit. Method. 1, 27-61), although the more definitive reference is (Boyd and Iversen 1979 Contextual analysis:  Concepts and statistical techniques. Wadsworth, Belmont, CA.).  Since that time there have been a number of developments, and independent derivations of the technique.  In 1987 Contextual analysis was introduced to the biological world (Heisler and Damuth 1987 Am. Nat. 130, 582).  In 1996 contextual analysis was reinvented and called direct fitness, later called neighborhood modulated fitness (Taylor and Frank 1996 J. Theor. Biol 180, 27-37, arguably, Queller 1992 Evolution 46, 376-80.).  In 2010 it was again rediscovered, although from a more genetic perspective, and labeled social selection (McGlothlin, Moorad , Wolf, and Brodie 2010 Evolution 64, 2558-74.).  Bottom line:  These are all the same thing. Contextual analysis has the precedence by nearly a decade over every other misbegotten term.  Can we please just call everything by one name, and can it please be the name that crosses back to other scientific disciplines, and can it please be the one that respects precedence?  All that those different things are contextual analysis.  It is the only term that fulfills all those criteria, can we please just use contextual analysis.  It is the correct term!

Finally, there is the interesting question of what is the correct trait.  In our original contextual analysis papers (e.g., Goodnight, Schwartz, and Stevens 1992 Am. Nat. 140:743-761) we used the group mean of the trait, whereas McGlothlin et al. chose to use the mean of the group excluding the focal individual.  Both of these make sense in the context that they were used.  In our theoretical studies using the raw group mean considerably simplified the math, and made our message much clearer, whereas in the McGlothlin study they were considering social interactions explicitly, and it made sense to leave out the focal individual and only include those they interact with.  Either and both of those are contextual traits, and as with any selection analysis the choice of which traits to include in the analysis depend on the situation.

OK, I will quit ranting.  Next week I will move on to the indirect genetic effects I meant to get to this week.

Measuring the heritability of contextual traits.

Posted: March 28th, 2014 by Charles Goodnight

First a plug for an upcoming conference.  If you are interested in artificial life there is a conference, Alife 14, being run by, among others, a friend of mine, Hiroki Sayama.  There is one week left submit abstract, and a good time should be had by all.  .  This meeting will take place in New York at the end of July, beginning of August.

I have been looking for a data set that could be used to illustrate calculating the heritability of contextual traits.  Happily one came along at the last minute, although I had to do some hard thinking to figure out how to interpret it as a contextual trait. . .

The paper I am talking about is a new one in Evolution,  Edward, Poissant, Wilson and Chapman 2014, Sexual conflict and interacting phenotypes:  A quantitative genetic analysis of fecundity and copula duration in Drosophila melanogaster. Evolution doi:10.1111/evo.12376  (http://onlinelibrary.wiley.com/doi/10.1111/evo.12376/abstract).  This is a well-done and analyzed article that is well worth reading.  However, as is my wont, I will misuse their data for my own purposes.  Thus, the caveat of the day is that I am in no way complaining about what they published, just trying to use their data to illustrate a point.

drosophila-mating

In the interests of having a picture of real organisms, I am including a pair of mating Drosophila.   The Edward et al (2014 Evolution) study is about the genetics of mating in Drosophila.  (Picture taken from http://www.wired.com/wiredscience/2011/06/flies-alter-their-ejaculate-to-get-the-best-bang-for-the-buck/)

What they did in this study was to use a half sib design, crossing each of 16 sires to 3 dams.  They then took the offspring from these crosses and put them in something that I once called an “intermixing ability” type design (sad story why I didn’t call it “ecological combining ability”) (Goodnight 1991 Am. Nat. 138, 342-54).  That is daughters and sons from each cross were crossed in all possible manners in a manner similar to a combining ability study, except that the progeny were not collected and raised, rather the mating behavior of the pair was studied.

This design is an important conceptual shift.  In effect they are treating the mating pair as a group, and the productivity of that group is a function of both the male and female phenotype, and the interaction between them.  My one complaint about their design, which I am sure is a not so much an oversight on their part, but a consequence of the already large size of the experiment, is that they only had one replicate for each full sib family pair, thus it isn’t possible to fully analyze the interaction between cross types.  Given that doing this would have at least doubling the size of the experiment, the decision not to have replicates within cells is hardly surprising.

They then measured three traits.  First, for each female they measured the egg laying rate of the females while they were still virgin, and before being placed with the male, the duration of mating, and the egg laying rate after mating.   Here is where I am going to do a little bit of perhaps inappropriate slight of hand.  First, I am going to call the virgin egg laying rate an “individual trait” of the female since it is done when there is no possible interaction, then second I am going to call the mating duration a “contextual trait” since it is a function of the interaction between the male and female, and third, I will call the post fertilization egg laying rate the “fitness trait”.

Doing this we can then do a regression of post-mating oviposition rate (fitness) on pre-mating oviposition rate (individual trait) and mating duration (contextual trait):

CA of Dros. mating

Click on the table for a clearer view.

It would have been great if the duration (Dur) and the interaction had been significant, but that’s what you get for using data designed for another purpose.  What this is basically telling us is that to the extent that selection is acting on egg laying rate, there is strong selection on female fertility (as measured by premating egg number) but no detectable selection on the copulation duration.

Even though there is no selection on the contextual trait of duration, we can nevertheless measure the heritability of this trait (remember selection and heritability are different things!).  What we need to do is simply do a nested ANOVA of duration, and since we are focusing on the females we will only include the sire and dam of the female.  We shouldn’t expect much since males are assigned to females in all possible combination there can be no population structure, and thus no shifting of the male genetics over to heritability measured using only females.  In any case the analysis looks like this:

dur heritability random assoc.

Click on the table for a clearer view.

The sire variance component is 0.56, so the additive genetic variance for mating duration is VA =  4*0.56  =  2.24.  Since the dam variance component is negative, we can say that VD = 0.  The total variance is 16.28, which implies that the heritability = h2 = 0.14.

At this point we can artificially impose population structure.  A convenient one would be to allow only brother sister mating.  The problem is that, with the data structured the way it is, it is not possible to include dams within sires in this analysis of a subset of the data, still we can get the effective additive genetic variance.  Note that this mating structure enforces a covariance between two partners in the mating group, and should affect the heritability.

variance estimates brother sister mating

Click on the table for a clearer view.

 

Indeed it does.  In this case the variance among sires goes up to 3.87, thus the effective additive genetic variance goes up to eVA = 15.48, and the heritability goes way up to eh2 =0.77.

This is the point I am trying to make about measuring the heritability of contextual traits.  Using the same data set, if we design our experiment using random mixing of interacting partners then the heritability will miss a lot of the variance that can contribute to a response to selection. In this case nearly all of it.  In contrast if we use a breeding design that preserves those interactions we can pull in the association that the interaction structure generates.  Designing such experiments will be like standard quantitative genetics, only hard, and the resulting experiments will be like standard breeding designs only big.  (For the uninitiated that is a joke.  Breeding designs are notoriously difficult to design, and result in notoriously huge experiments.)

Griffing, Associate effects, and heritability.

Posted: March 20th, 2014 by Charles Goodnight

Last week I talked about the effects of localized mating on heritability.  If you remember we discovered the effect was small, at least for weedy species like Plantago lanceolata.  This week I would instead like to talk about the effect of interaction structure on the heritability of traits.  Much of what I will be talking about is discussed more formally in Wolf, Brodie, Cheverud, Moore, and Wade (1998. TrEE 13, 64-9).  I figure there are two conceptual approaches to measuring interaction structure and heritability, one is to preserve the actual population structure in a breeding design – I don’t think anybody has ever actually done this, and the second is to estimate the indirect genetic effects and use them to estimate variance components.  The first is probably “better” in the sense that plugging in the indirect genetic effects into a formula to estimate heritabilities will never be as accurate as actually directly measuring these effects.  Nevertheless it is the second I want to talk about because it makes the point conceptually much clearer, and because in fact, somebody has done the experiment.

That somebody is Bruce Griffing.  If you have not read Bruce Griffing’s work and you are interested in thinks like interactions among plants you should.  He published a great deal between the mid 1950s and the late 80s.  If you want to understand his thinking probably the best paper to start with is Griffing B 1977 (Selection for populations of interacting genotypes. In Proc. Int. Cong. Quant. Gen.,  Pollak, Kempthorne, Bailey (eds.), pp. 413-34 – this is the “red book” that can sometimes be a bit hard to find), but I want to talk about what I believe is his last paper (Griffing. 1989. Genetics 122, 943-56), which is an experiment using Arabidopsis.

Arabidopsis
The mouse-ear cress, Arababidopsis thaliana in its natural habitat, yes, it does have a common name and a natural habitat!  (from http://www.weedimages.org/browse/detail.cfm?imgnum=5400154)
 

It is another great weed (I used them in my thesis) for experimentation, and the nice thing is that they can be grown in sterile medium on agar.

Arabidopsis in agar

A pair of Arabidopsis growing in sterile agar.  This is similar to the experimental unit that Griffing used in his experiments (image from http://research.iheartanthony.com/tag/d2o-effects-on-life-2/)

Griffing used two strains of Arabidopsis, CHI and DI.  Because Arabidopsis is normally self-fertilizing, these plants were homozygous, so he treated them as homozygous inbred strains.  He then made then used the three possible genotypes, CHI, DI, and the F1 hybrid.  These were grown in pairs in all possible combinations in sterile agar.  He also varied the growth temperature and nutrient levels, but we will ignore that for today.

cross types

The basic experimental unit:  Pairs of plants were grown in sterile agar medium.  Each vial contained one plant assigned as the “direct” genotype and one plant assigned as the associate genotype.

These plants were raised, then harvested, washed, dried and weighed, and gave the following results (at 28o, ½ nutrient level):
Means for Griffing

From this it is a small matter to plug these into a two-way ANOVA, and because of the nice balanced structure do a priori contrasts, yielding the following results:

ANOVA Griffing

Basically, what this tells us is that there are highly significant direct and associate effects, and most of those are due to the F1 hybrid that seems to be showing heterosis.

This analysis shows us that there is a genetic effect both on the individual and on its neighbors, however, it is not an estimate variance components the trait.   This is because the ANOVA was done as a balanced unweighted design.  Variance components are a property of both the genotype and the population it is found in.  Essentially, if we do it as a weighted regression of the direct effects this will give us the additive genetic variance for direct effects, and similarly, the weighted linear regression of the associate effects will give us the additive genetic variance for associate effects.  HOWEVER, what we are interested in is the effect of the associate effects on the genetics of the trait under consideration.  This will be given by the additive genetic variance for the associate effects multiplied by the correlation between the (direct) parent and offspring for the associate effects of the interacting individual.  In other words, if pairs are assigned randomly each generation the correlation is zero, and there is no heritability due to the associate effects.  On the other hand, if the interacting pair of the offspring is identical to the interacting pair of the parent, then the correlation is one, and the additive genetic variance of associate effects are fully translated into additive genetic variance for the trait.

direct and associate effects

The trait is measured only in the “direct plant”; however, it is influenced both by the direct effect of the plant on itself, and the associate effect of its partner.   If it helps, (it hurts my soul to suggest this) think of it as the genes in the associate plant affecting the trait in the direct plant.  These associate effects become heritable to the extent that there is a correlation between the associate effects in the parent pair and the offspring pair.

Thus, redoing the analysis using weighted regressions.  Because the effects are almost entirely seen in the heterozygote (i.e., this system mostly has heterosis) the best examples are seen when the gene frequencies are far away from 0.5.  In fact at a gene frequency of 0.5 there is no additive genetic variance for either direct or associate effects.  Therefore, I chose to use a gene frequency of 0.1 as an example.  In that case we get the following results:

Variance estmates Griffing

Thus, at a gene frequency of 0.1 there is additive variance for the trait due to the direct effects of an individual on itself, and there is potentially additive genetic variance due to the interacting partner.  Whether or not this associate effect additive genetic variance is heritable or not depends on the between generation correlation between the interacting partners.  Thus, if the interacting partner is randomly chosen from the population every generation then the correlation will be zero and the associate effect variance will be zero.  In contrast if the interacting individual always has the same genotype every generation then the associate effect variance will be the full value of 0.002.  It turns out that in this simple system unless we use clones of the parents this correlation will tend to be very low, so in this system the contribution of associate effects to the additive genetic variance for an individuals traits will be small. (sadly, it took me a lot of work to figure that out!).

Nevertheless, this raises an important point.  In this study no contextual traits were measured.  Nevertheless, it shows that that depending on the interaction structure the genetics of interacting individuals can contribute to the expression of traits in other individuals.  Under most circumstances the correlation in interaction between generations will be small enough that the associate effects can be considered to be environmental variance, but under certain circumstances that need not be the case.  This will be especially true for things like maternal traits, which because they are heritable means that, for example, a mammal that has a mother with rich milk might be likely to also have rich milk for her babies.  It also emphasizes the point that in traits associated with yield the genetic covariance between direct and associate effects tend to be negative (In this example the correlation is very nearly -1).  With individual selection this negative correlation can lead to an overall negative response to individual selection, something that I saw in my thesis (Goodnight 1985 Evol. 39:545).

As a final note, I dedicate this post to Sunny, who was as fine a cat as I have ever known.  Recently she was diagnosed with lymphoma, and sadly today I had to take her for her last trip to the vet.  She will be missed.

Sunny sleeping

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