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Evolution in Structured Populations

On Kinship

Posted: October 30th, 2015 by Charles Goodnight

On Facebook one of my friends posted that they were attending a conference on reconceptualizing kinship, and of course, I responded that it was tempting to put in my two cents worth on the subject. To my shock another commenter asked me to do just that. SO, given that I have been absolutely swamped since getting back from sabbatical, I figure I ought to at least pretend to have a blog (I actually have a couple of other issues that really need to be addressed. Maybe in a few weeks). In any case, here goes.

Of course, kinship means many things in different contexts. Colloquially kin can mean relatives, as in my cousin is kin. From a genetic perspective, the classic use of kinship is in inbreeding coefficients, and thus it is the probability that two individuals share common genes. Finally, kinship was coopted by Maynard-Smith to describe the “r” term of Hamilton’s rule.

Colloquially we can use kinship in a more general sense. Thus, a person might describe themselves as having kinship with others of a similar political view, or a group fighting for a cause may consider their fellow members as kin.


Sisterhood: 1) A bond between two or more girls or women not necessarily related by blood. 2) An association, society, or community of women linked by a common interest, religion, or trade. (Image from http://www.thismorleylife.com/wp-content/uploads/2013/09/25473872sisterhood.jpg, definition from Google.com, and Urban dictionary.com)

Of particular importance here from a genetic perspective is that the inbreeding coefficient definition of kinship is very consistent with the colloquial definition of kinship as relatives. This is not true of Maynard-Smith’s definition of kinship as the r term in Hamilton’s rule. In fact, Maynard-Smith’s definition comes much closer to the social extension of kinship as two individuals that share a common bond, regardless of the cause of that bond. I would suggest that Maynard-Smith made a mistake here. In common culture we make a distinction between kin as in relatives and kin as in shared interest. For example, we have no problem referring to relatives strictly as “kin”, but for those with a shared interest we are likely to put in a modifier – brother in arms, brother from another mother, BFF. I would argue that Hamilton’s r is actually a combination of these two colloquial concepts of kinship. Sharing of genes in an additive sense, and other forms of sharing of phenotypes.

The reason I say this comes from contextual analysis. First, in one of my papers (Goodnight 2013, Evolution 67:1539) I show that the mathematics of kin selection can be directly translated into the mathematics of contextual analysis. The basic process is that Hamilton’s inclusive fitness can be translated into the direct, or neighborhood fitness approach (Taylor and Frank 1996 JTB180:27 for the neighborhood fitness approach, Taylor, Wild and Gardner 2007 J. Evol. Biol. 20301 for the equivalence of inclusive fitness and direct fitness). The neighborhood fitness approach in turn is based on EXACTLY the same equation as inclusive fitness (fun fact: Contextual analysis predates neighborhood fitness by a lot. It has precedence, and I think a strong argument could be made that the neighborhood fitness approach should be re-named contextual analysis).

It is worth emphasizing that there are significant differences between kin selection and multilevel selection, and these are the basis for my reasoning behind why I don’t like kin selection. However, given the close mathematical association between contextual analysis and neighborhood fitness it is hardly surprising that it is possible to derive Hamilton’s rule using contextual analysis (Goodnight Schwartz and Stevens 1992.  American Naturalist 140:743) :

Hamiltons rule

This formulation differs from the classic kin selection version in several important regards. The first, a bit irrelevant, but something I personally can’t ignore, is that CA is treating Hamilton’s rule as a competing rates problem. That is, altruism will evolve when group selection is stronger than individual selection, and this can happen either due to differences in the intensity of selection (measured in genetic standard deviations at the two levels), or in the relative magnitudes of the variances in the group and individual trait (measured as the proportion of variance that is among groups).  In contrast, neighborhood fitness is providing the optimality solution.  This difference rather fades away when applied to contextual analysis, but it is a big difference in philosophy.

Much more important, however, is that using contextual analysis it is clear that Hamilton’s “r” is the fraction of the total variance that is among groups. This is an important point.   In an additive world where nobody interacts with anybody kinship, as measured by Wright’s FST, is exactly equal to the variance among groups. However, in a world in which interactions occur this will no longer be the case. Consider:

If there is epistasis is gene interation: For epistasis, the variance among groups will roughly be proportional to FSTN, where N is the order of the interaction. For example if there is two-locus epistasis the fraction of variance among groups will go up roughly as FST2. This is really only strictly true for additive by additive epistasis, nevertheless it makes the point that relatedness and variance among groups are not necessarily the same thing.

If there are interactions among individuals: This can be indirect genetic effects, but it can also be non-genetic effects. One of the very common features of social groups is some form of policing behavior. For example, in bees workers typically destroy worker laid eggs. Thus, even if there is genetic variation among workers in their propensity to lay eggs, it doesn’t matter since all of this variation is suppressed by the policing behavior. On an emphatically non-genetic scale, musical bands work together to enforce a steady rhythm on the band members. This rhythm produces a product (music) that is appealing, and, if the stories about rock bands are correct, has a definite effect on fitness. Similar stories can be made about everything from sports teams to military organizations. These interactions affect the heritable variation among groups, but have no effect on the heritable variation within groups. They simply are not in kin selection models, and as such they represent a complete wild card that will nearly always make altruism easier to evolve.

The point is Hamilton was thinking about a linear additive world when he was talking about the r term in his rule. This was perfectly reasonable in 1964 when he published his work. That was before the days of computers, before any math around gene interactions had been developed, and before there was any data to suggest that group selection could act on interactions among individuals. It was also over 50 years ago. I think it is probably time we considered moving on!

So, what should we do: Recognize that Maynard-Smith was incorrect to call Hamilton’s rule it kin selection. Hamilton’s r DOES NOT refer to kinship, or at least not kinship in the narrow genetic sense. Rather it refers to the ratio of the variance among groups to the total phenotypic variance. Given what we know about gene interactions and social interactions we are long past the time when we should have abandoned Hamilton’s simplistic view of the cause of the variance among groups. I leave it to the reader to decide whether kinship applies to individuals related by social interactions. Maybe we should have a “grouphood of the interacting genes”?


Is a Grouphood of traveling genes a form of kinship? (http://trailers.apple.com/trailers/wb/thesisterhoodofthetravelingpants/images/sisterhood_01.jpg)

Population structure and recombination

Posted: September 24th, 2015 by Charles Goodnight

One of the joys of a genic view is the apparent constancy of things. One of the big ones is that a gene has an effect that can in some sense be considered a constant that can be written down and stored on a piece of paper in a mayonnaise jar on Funk and Wagnall’s front porch (that is a Carnac the Magnificent reference for those less than a million years old). As I have pointed out before, the way we measure the effect of an allele is to use a defined, usually homozygous, background and ask what the “mutant” and “wild type” alleles do to the phenotype (yes, I know the reality is more sophisticated than that, but really not by much!). The point is that the effect of the gene is only constant in the context of the simplified genetic background in which it is measured.


CARNAC THE MAGNIFICENT (Johnny Carson): Supercalifragilisticexpialodocious and constant gene action. ED MCMAHON (reading question): Name two phrases that have no meaning. CARNAC: May the fleas of a thousand camels infest your armpits.

I got to thinking about this and realized that there are other genetic parameters that we take as constants, that are actually functions of the context in which they are measured. The one I want to talk about today is recombination rate. Typically recombination rate is simply the map distance between two genes measured by the frequency of crossovers in a dihybrid cross.


A dihybrid cross tells us the recombination rate for a pair of loci. In this case the loci are linked with a recombination rate of 0.17 (From https://www.studyblue.com/notes/note/n/lecture-exam-2/deck/8120019)

This is all well and good, until you get to population genetics textbooks. They will then tell you about linkage disequilibrium (or gametic disequilibrium if you prefer). First a few fun asides. Linkage disequilibrium is actually the covariance between the allele state of two loci. To see this imagine we have an A locus with alleles A1 (value 1) and A2 (value 0), and a B locus with B1 (value 1) and B2 (value 0). The four haplotypes have frequencies of p11 (A1B1), p12 (A1B2), p21 (A2B1), and p22 (A2B2). Then the covariance is:

Equation 1

Second fun aside: D is the determinant a matrix of the frequencies of the gamete types in a population.

Equation 2

What that all means I am not sure. That is other than demonstrating the obvious fact that all of nature is embodied in covariances and linear algebra.

At this point your population genetics textbook will go on to tell you about the decay of linkage disequilibrium based on the recombination rate. It doesn’t take a lot of algebra to show that linkage disequilibrium decays as a function of the recombination rate. For example:

Equation 3

where the prime mark indicates the next generation and

Equation 4

At first blush this is beautiful. It demonstrates that simply by knowing the map distance between two loci we know the recombination rate, and with it the rate of decay of linkage disequilibrium. In other words, we assume that this classic measure of association is a property solely of the genome, and thus only the genome is responsible for the behavior of the genes.

Sadly, as is so often the case with the genic view, there is a hidden assumption that has been hidden for so long that it is even lost from our intuition. This hidden assumption, of course, is the assumption that the population is unstructured. The important point to realize is that the only time that recombination has any effect is in the double heterozygote. If either locus is homozygous then a crossover event produces exactly the same gametes as are produced in the absence of crossing over. The problem comes that population structure tends to reduce the frequency of heterozygotes. As a consequence, in a classically inbred population the rate of decay for linkage disequilibrium is affected by a factor of (1-f)2, where f is Wright’s the inbreeding coefficient. The easy way to think about it is (1-f) is the probability that two alleles are not correlated, usually because they are identical by descent (IBD). If the alleles in an individual are IBD that individual is by definition homozygous at that locus and crossing over will have no effect. The quantity is squared because homozygosity at either locus negates the effects of crossing over. Thus we can re-write the above equations as:

equation 5


equation 6

In other words we can think of the “effective” recombination rate as r(1-f)2.

This seemingly trivial point is actually quite important. It emphasizes that population genetic parameters are a function both of the genes and the population in which they are measured. Even something as seemingly constant as recombination rate can be changed simply by changing the population structure, and the degree of mating among relatives.

It also has some fun adaptive story telling implications. There is a battle between sex as a source of variation, and sex as breaking up well adapted genotypes. This little exercise suggests that there is a middle ground: sex with relatives. Population structure limits the field of recombination, and has the effect of reducing the recombination rate among loci. One can imagine population structure evolving as a means of preserving local adaptations and reducing the effective recombination rate. Of course this would come at a cost of decreased heterozygosity, so perhaps that would be a different battle.

Finally, I should mention that in many mammals fIS is often negative. This should have the effect of increasing the recombination rate. I leave it to you to make up adaptive stories for that one. . .


A quick review of the phenotypic perspective, pt 2

Posted: August 28th, 2015 by Charles Goodnight

Continuing on with my brief reprise of the tenets of this blog:

Individuals are the level at which we assign fitness.

First a parable. If we go back to Mayr, he thought that the species was the only natural unit of organization, and argued that the biological species concept (BSC) was the correct definition of species. Since then we have realized that species are perhaps not that monolithic. There is gene flow between species, and it is not always clear whether two different populations are the same or different species.

I argue that the same is true of individuals. (part 1, part 2, part 3)(Not so long ago it was easy to argue that individuals were “organisms”, and that they were a natural unit. The problem, as always, comes when we look too close. Just like the BSC works until we look to closely, this classic definition of individual breaks down when we look closely. Evolution is always defined in a way that precludes changes within an individual, usually called development. Thus, evolutionary biologists are adamant that the change in an individual as it grows from zygote to adult is not evolution. On the other hand we would like to study distinctly cellular level processes, such as cancer development, as evolution. Similar problems come when we study proto-multicellular organisms like volvox, or organisms that grow clonally, such as Aspen. On the one hand a clean definition of “individual” as organism works fine when we are talking about vertebrates and not looking too closely, ignoring that the vertebrate organism is a multispecies entity. However, on the other hand, such a narrow definition results in our narrowing our definition of evolution to changes at or above the organismic level. Something has to give, and I would argue that the correct thing to do is to allow flexibility on what we call an individual.


Volvox: what is the individual? (http://www.dr-ralf-wagner.de/Bilder/Volvox-aureus-DF.jpg)

Allowing a flexible definition of individual has huge advantages. We can study evolution at any level. That is it IS legitimate to study cancer as an evolutionary process. It IS legitimate to study “species selection” in the fossil record even though you do not have access to the fitness of the underlying organisms. And, when we discover that the microbiome was not something that could be simply ignored we don’t have to through out all of the evolutionary studies that have been done previously. This last point is an important one. Our understanding of biology is expanding rapidly. We need concepts and models that can incorporate these new findings naturally without having to re-do our entire conceptual system. The classic concept of individual as a colony of genetically identical cells of the same species physically separated from other colonies (I made that definition up) is simply too rigid, and cannot be adapted to our expanding understanding of living systems.

The down side is also analogous to issues with species concepts. It is now understood that, in order to avoid confusion, when you talk about species and speciation you need to explicitly state what you mean by species. As with species concepts, our understanding of what is and is not evolution changes dramatically depending on the level at which we define fitness. Everything that is at or above the level of the individual can be studied as evolution. Everything below the level of the individual has to be studied as some form of development. This means that when Gardner argues that cancer cannot be studied as evolution (Gardner’s paper; My critique of Gardner; Gardner’s critique of me) and I argue that it can we are both right. In his view the individual is rigidly defined as the organism, and as such cancer must be treated as changes occurring within the organism. He can call it development, disease, what ever, but he cannot call it evolution. I, on the other hand, by embracing a more flexible definition of the individual, CAN call it evolution. What I am doing is defining the cell to be the individual, then the faster cell division of cancer cells is selection. This is opposed by “group” selection in that, with a few notable exceptions, the spread of a cancer is ultimately stopped by the death of the organism.

cancer and selection

Cancer as an evolutionary process (from http://www.nature.com/nature/journal/v481/n7381/full/nature10762.html)

Multilevel selection:

Viewed from the perspective of a flexible definition of the individual, depending on the level at which you assign fitness potentially ALL selection is multilevel selection. Classic “individual selection” is selection on organisms, which is a group of cells, and actually on a community of cells. It is group selection if we assign fitness at the level of the cell, it is individual selection if we assign fitness at the level of the organism. Given that genic view “individual” selection is actually group selection, and is nearly ubiquitous, is it really surprising that “group selection” works at other levels as well?

This raises a second point. The phenotypic approach follows the lead of quantitative genetics, and makes a sharp distinction between selection and the response to selection. Selection is the ecological process by which some entities leave more offspring than other entities. The response to selection is the evolutionary consequences of that. Organismal selection is the differential survival and reproduction of individuals. The evolutionary consequence is a difference in the distribution of offspring that is due to that differential survival and reproduction of the parents. The point is, regardless of the consequences, the level of selection is the level at which selection acts. Thus, if both individual selection and group selection cause exactly the same change in gene frequency, they are still different things, because they are different ecological processes. If I dye a shirt red it may be the same color as if I wove it out of red thread, but nobody would pretend that dying fabric and weaving fabric are the same thing, even if the outcome was the same. Of course, as with dying versus weaving fabric, selection at different levels has qualitatively different consequences. The result is that we do our understanding of evolution a huge disservice when we decide to dismiss levels of selection a priori. This is especially true when we realize that whether or not selection is acting at a particular level is an empirical question, and that we have the tools we need to answer that question.


Dyed or woven, you’re still dead. (from http://anartistcalledred.deviantart.com/art/Curse-of-the-RedShirt-173225300)


A quick review of the phenotypic perspective, pt 1

Posted: August 11th, 2015 by Charles Goodnight

Some of the recent comments I have received have made me realize that maybe I should re-emphasize some of the very early points I made on this blog. The point of this blog is to blatantly promote a phenotypic view of evolution, and do try to dislodge the dominant paradigm of the gene as the center of evolution. In the discussion that follows it is convenient to use Dawkins as a straw man. My own feeling, based on no evidence, is that most evolutionary biologists accept Fisher as a brilliant founder of modern genetics, and accept his as a very genic view of evolution. Interestingly, Dawkins perspective, working through the lens of Williams, is the logical outcome of taking Fisher’s work to its extremes. So, just as I feel most evolutionary biologists accept Fisher, I feel that that they are deeply uncomfortable with Dawkins, but most of these biologists would have trouble articulating exactly why Dawkins is wrong. Somewhere between Fisher’s deeply mathematical prose and Dawkins polemics something has gone awry. My feelings are that where Dawkins goes astray is very fundamental, and goes all the way back to Fisher. Basically Fisher imagined a genetical world that was a reasonable abstraction for a world in which we had no idea what a gene was, we were at the very beginnings of our understanding of inheritance, and we lacked the computational machinery to do anything more than relatively simple analytical models.

Another reading of Fisher, however, is that quantitative genetics is fundamentally a phenotypic model. The average offspring is the mean of the parents, but the loss of variation due to averaging is recovered in the form of within family variation. We can interpret Fisher’s book is an example of the phenotypic view of evolution that is illustrated using a simple Mendelian model of genetics, but which can be expanded as necessary and as computational power allows. Viewed in this way the phenotypic perspective I am advocating may be more of a descendent of Fisher’s legacy than the more classical genic view.

Matt Foley

Just a bit of self promotion, and maybe a bit of motivational speaking (http://gallery4share.com/c/chris-farley-snl-matt-foley.html)

So here are some of the relevant points:

1) Phenotypes create new phenotypes: At first blush this is just a change in perspective. At the risk of setting up a Dawkinsonian straw man, the classic genic view is that genes make copies of themselves, and use phenotypes as a mechanism to protect themselves, and help them survive to the next generation. This is why Dawkins refers to DNA as “immortal coils”. In the phenotypic perspective parent phenotypes create offspring phenotypes using “transition equations”. These transition equations are accepted to be impossibly complex, and so we accept at the outset that the best we can do are approximations. The simplest approximation to a transition equation is probably the heritability of quantitative genetics, or the simple Mendelian math of a Punnett square, however, in many situations it will be useful to add complications ranging from maternal inheritance and indirect genetic effects, to epigenetic effects, and all the way up to cultural effects.

What this change of perspective buys us is that genes are no longer the center of evolution. There are no such things as vehicles and replicators. These are the construct of a fevered mind that deeply misunderstands evolution. Instead, genes are relegated to being a prominent, but certainly not the only, contributor to the transition equation. This leaves the transition equation as an open ended construct that can incorporate new scientific findings. Rather than having to totally reconstruct our understanding of evolution every time we come up with a new mode of inheritance, we simply need to recognize that the transition equation was more complex than we had originally thought, and we need to modify that equation appropriately.

2) Some aspects of our understanding of evolution change with a shift to a phenotypic perspective, but our basic understanding remains remarkably similar. There have been many definitions of evolution, some of which have relied on a genic view. For example, a classic definition is that evolution is change in gene frequency. Re-framing our understanding to a phenotypic perspective demands a careful rethinking of what we mean by evolution. My own definition is evolution is the change in the distribution of phenotypes in a population due to the gain or loss of individuals. This definition is consistent with phenotypically oriented classic definitions, but ends up being more specific in many ways.

Classically there have been four forces of evolution  that have been identified: mutation, migration, selection and drift. These have mostly been defined in genetic terms. Thus, drift is often called “genetic drift”, mutation is discussed in terms of change in DNA structure.   However, these terms can be defined and discussed in phenotypic terms without reference to the specifics of the underlying mechanisms of inheritance. Clearly migration and selection do not need reference to genes, and our understanding of them really does not need to change at all. From a phenotypic perspective “mutation” need not be genetic change. It can be any change that randomly alters the phenotype of an individual, and that does not correlate with fitness. There is the interesting caveat here, however, that based on our definition of evolution, such random changes do not become “evolution” until they are passed on to offspring. Similarly, drift can be viewed as a change in phenotype frequencies due to the random gain or loss of individuals. With the phenotypic perspective, however, a fifth force must be recognized. This force is easily ignored in the genic world, but cannot be ignored in the phenotypic perspective. This is force is secular environmental change. A lasting change in the environment, such as global warming, can change the distribution of phenotypes directly, and in at least some cases it will be an intergenerational event. For example, global warming is changing sex ratios in some reptiles. If we assume an individuals sex is fixed at hatching, then indeed this change in the distribution of males and females is an evolutionary change by our definition.

I seem to have run into my self imposed thousand word limit, so I will continue this review next week.


Down the rabbit hole: More on multispecies organisms

Posted: July 23rd, 2015 by Charles Goodnight

I just tripped and fell down another rabbit hole. I was going to skip this week, but I would love input on this issue, so here it is. Earlier I argued that the organism was a multispecies entity. This makes perfect sense if we consider mitochondria to be symbiotic bacteria in a host cell, and we talk about the microbiome. Now here is the question: If you catch the flu, or get a bacterial infection (to keep it cellular), is that disease part of you as an organism?


Dang another rabbit hole.

There are two important points to remember. First, in the phenotypic view I am advocating considering the phenotype to be a vector through time, with every trait (a measured aspect of the phenotype) having a time element. Thus, it is not my weight, but my weight when I am 19001055824 seconds old (that is approximately how old I am while writing this). This means that even very temporary things such as whether you are inhaling or exhaling is technically a valid trait. Thus, if you have a fever of 104 degrees on a Saturday morning, that is the value of the trait “body temperature”  at that particular moment. The question is, do we make a distinction because that temperature is “caused” by a flu virus? The truth is I am beginning to believe we cannot make that distinction.

Taking a clearer example. Consider a person who chooses to dye their hair purple. This color comes out of a bottle, and it is no sense genetic or otherwise heritable (well, maybe in some odd cultural sense). That said, it is part of the phenotype. If you were to categorize people by the trait “hair color”. this person would go into the “purple”. Thus, it is a valid trait, and a valid part of their phenotype. How do we deal with this? I would argue that the best way would to consider the bottle of hair color to be a non-heritable or environmental influence on the phenotype. By analogy, I think it is perfectly reasonable to suggest your 104-degree fever is also part of your phenotype.

purple hair

This woman has a purple hair. It is certainly part of her phenotype, but probably not heritable.   (from http://darkuro.tumblr.com/)

So, your fever is part of your phenotype, but is the virus part of you as an organism? Certainly, we would not consider the bottle of hair color to be part of an organism. It is an external aspect of the environment that changes your hair color. Cold air temperatures may cause you to put on a coat (the coat wearing trait?), but it is certainly not part of your body. However, the virus differs here. It is IN your body, and in fact it is in your cells.

Consider our microbiome. There certainly are aspects of the microbiome that are acquired from our parents, either at birth, or because we live next to them as infants, and many of these we will pass on to our children.   Thus, they are heritable from the phenotypic perspective. However, others are picked up late in life, perhaps when we temporarily change our diet, and then lost again, perhaps when we revert to our old diet, and are not heritable. I think a strong argument can be made that this microbiome should be considered part of the multispecies organism: Selection acts on the whole organism; outside of perhaps prokaryotes, single species organisms don’t exist; as far as I know, animals cannot survive without their symbionts. From an experimental perspective, it is difficult or impossible to separate symbionts that are heritable from those that are non-heritable, and perhaps more important both can have significant effects on our phenotypes in ways that can affect our fitness. Thus, I think it can be argued that all aspects of the microbiome, whether heritable or not, should be considered part of the organism. Nor does it makes sense to me to argue that there is a minimum residence time before a symbiont or disease should be considered part of the organism. Such a waiting time is necessarily arbitrary, and as a result there will always be situations that are ambiguous.

Now comes the question: Should we make the distinction between the bacteria that we picked up on vacation that makes it easier to digest shrimp from another bacteria that gives us diarrhea? I cannot think of a criterion that does not require special pleading that incorporates the former, but not the latter into the organism.

One final caveat is that it is important to remember that the most appropriate unit to assign fitness depends on the trait being investigated. Thus, the colony might be the appropriate unit if we are examining colony defense, the organism if we are examining foraging behavior, and the cell if we are examining cancer. Perhaps the organism is best thought of as being equally fluid. A flu infection is an assault on our bodies, thus if our trait is immune response, maybe the organism is everything but the flu virus, whereas if we are looking at body temperature the organism is everything including the flu virus. This is a bit of a conundrum for me, and I am happy to get any feedback that anybody else may have.


Some thoughts on aging and the phenotype

Posted: July 16th, 2015 by Charles Goodnight

I have been gone a while. Something of a creative meltdown after the Evolution meetings. Perhaps one to many Caparinha, at what might have been the best party ever at an Evolution meeting. Leave it to the Brazilians to throw a party with enough food and liquor and the wackiest live music ever. In any case, I am back in Vermont, eating kale and other healthy things that us aging hippies do, and thinking about the pain of having to ply my profession as a teacher. Time to get back to work.


Admittedly a terrible picture, but the party was one of the best.   If you missed the Evolution meetings this year you made a mistake.

Jake Moorad started a discussion with me about how aging affects individuality. My first thought was that I have no idea. As usual, such an answer means that it is a really interesting question. After giving it more thought I have come to realize that it has no effect at all. After all, an individual is the level at which we assign fitness, which is potentially quite arbitrary. In most cases the “individual” will be an organism and its associated symbionts. Thus, despite the fact that an individual changes as it ages, I think it should have little influence on what we call an individual. What it does change, however, is it complicates what we think of as the phenotype. The fact that the phenotype changes over time is not a trivial issue, and it is one that needs to be given some attention.

How to view the phenotype as a vector through time is a topic I have discussed before, and one that is a general issue for the phenotypic view of evolution. My solution is to treat the phenotype of an individual as a vector through time that begins at formation of the individual, and ends at their dissolution. There are a couple of interesting things in that sentence. Note that I am specifying the “individual”. It seems to me that, although perhaps not essential, it makes sense to assign phenotypes at the same level at which we assign fitness. I could be argued out of that, but for the moment it seems right (which is not terribly convincing to me, let alone anybody else). Second, I speak of the “formation of the individual” and “dissolution” of the individual. If we assign fitness at the level of the organism, this will be at conception or birth, depending on your perspective, and ends when the organism dies. But remember two things: First, if we assign fitness at a level other than the organism, “birth” and death may not be an appropriate terms. However formation and dissolution will always be appropriate, since our definition of evolution involves the gain and loss of evolution, individuals must of necessity have a beginning and an ending if they are to be considered to evolve.

If we take a classic genic perspective of the individual as a single species organism it is easy to ignore the time dependent aspects of the phenotype. Most importantly under a genic perspective genes are the only important heritable effects on the phenotype. These all enter at the time of formation (birth) and are unchanging through time (this last ignores somatic mutations of course).   This is not true from a phenotypic perspective. Culture is a prime example. The language you speak whether you are most comfortable with a fork and knife or with chopsticks are all heritable aspects of phenotype that are added after birth. Similarly, your cultural parents will not necessarily be the same as your genetic parents. Children learn their earliest language from their parents, but the vast majority of their language comes from peers and children that are slightly older than they are. This acquisition of heritable elements is not limited to culture, of course. We acquire much of our microbiome from our parents, and other individuals with which we live, and in many social insects, such as termites, trophallaxis is essential for their survival.

Screen Shot 2015-04-20 at 12.25.47 PM

The phenotype as a vector through time. Effects entering from the top are possibly heritable inputs, effects leaving below the lines are products of the phenotype. Note that the phenotype ends at death. This is why products such as beaver dams and human produced things like books are part of the phenotype, and not part of the distended (or is it extended, I always get it wrong) phenotype.

However, it is not just the acquisition of heritable elements that affect the need to consider the time element in the phenotype. Traits also change over time.   At a simple level every trait will have a time element. One way that this is handled is to simply to measure traits at a time when they are stabilized. For example, most vertebrates have targeted growth. Thus, there is a period between when adulthood is reached, and before senescent decline that traits are stable enough that we can effectively ignore the time element. In reality, of course, we should always include the age component, thus, it should be considered weight at age X, not adult body weight. However, speculating about how things ought to be done is different than doing things, and well, I for one will not be angry at people who simply measure adult body weight.

This does raise one additional interesting point. That is we can measure the time element of a trait to whatever precision we choose. Thus, in principle we could measure a trait such as whether an animal is inhaling or exhaling, or whether their heart is in systole or diastole. From an evolutionary perspective it would be silly to measure such highly time dependent transient traits, nevertheless it emphasizes the point that traits are aspects of the phenotype we choose to measure, and as such can be measured to whatever precision is appropriate.

More interesting, however, is the expression of traits with non-genetic inheritance. In some instances traits might not be expressed at all until the causal elements are acquired. For example, the trait of speaking a language cannot be expressed until the language is learned. Language is not acquired at birth, and if it is not the mother tongue, it may be acquired quite late in life. Further, if this person goes on to teach others their newly acquired language we can say that they have a heritable trait (the language understanding) that they acquired late in life.

Finally, some traits we may be interested in might be rate of change over time. Such traits might be the slope of the decline in fertility with age since puberty. To be honest I am not sure of the language to use to describe such traits, since such traits are explicitly incorporating the time element, and do not fit well with my phenotype as vector metaphor. If anybody has any ideas I would be pleased to hear them.

The point of this is that the evolution of aging remains an important issue. However, I am inclined to think it will have at most a small effect on our understanding of individuality. Where I think it will have its big impact will be on how we think about the phenotype, and the necessity to think of traits as being age dependent.



Why there is no Genic Selection

Posted: June 17th, 2015 by Charles Goodnight

This is the week before the Evolution meetings, so the big question of the day is what can I post that I believe to be true, and will rile enough people up to get a good discussion going. I decided Sam Scheiner was a good target – we were graduate students together, he is a good guy, and a great scientist at NSF. BUT, one day on an online discussion, I don’t remember where, he took umbrage at my statement that there is no such thing as genic selection. So, that is today’s thesis: There is no selection on genes. I will actually soften that a little, and bring up a special case that is indeed genic selection.

Sam Scheiner

Sam Schiener – Currently at NSF, long ago a fellow graduate student with me (Sadly, all my incriminating photos are in Vermont). We spent long hours hanging out and working in the Barnes Greenhouses, which have since been torn down. (left: https://www.researchgate.net/profile/Samuel_Scheiner right: http://ian.umces.edu/blog/2011/02/19/more-randy-alberte-memories/ )

First, an important philosophical point. As I have probably said more times than necessary, quantitative geneticists divide evolution by natural selection into selection and the response to selection.   Selection is an ecological process that has no reference to whether or not a trait is heritable. As an extreme example, consider a situation in which one person is painting random numbers on the backs of turtles, and a second investigator, seeing the numbers decides to select for those turtles with the largest numbers on their backs. This is selection, even though it obviously has no genetic basis. More realistic traits can have a heritability anywhere from 0 to 1, so there can be no logical cutoff where we say it is no longer selection.   This is of more than philosophical importance. It is of practical importance. The ecological process of selection really is only studied in nature.   That is, we, as evolutionary biologists, are mainly interested in the selective forces acting in natural natural populations in natural settings. Further, the appropriate way of studying selection in nature is to use the regression approaches of Lande, Arnold and Wade (Lande & Arnold. 1983. Evolution 37: 1210-1226; Arnold & Wade. 1984. Evolution 38: 709-718). In contrast, the response to selection is a function of genetics. From a practical perspective heritabilities are measured using breeding designs and statistically comparing relatives.  In these designs, the mating structure is forced, and comparisons are most easily done in the laboratory (e.g., Falconer and Mac Kay 1996, introduction to quantitative genetics). Finally, there will be numerous situations where we will be interested the measurement of selection acting on interesting organisms or in interesting situations, but we have no knowledge of the underlying heritability of the traits.  The distinction between selection and the response to selection tells us that such studies are interesting even without simultaneously doing genetic studies.

Thus, for both philosophical practical reasons it is reasonable to separate selection from the response to selection. So, what is the point of this? Genic selectionists are arguing that we can act AS IF selection were acting on genes. As I have argued in the past doing this is fraught with dangers. But more importantly, we have to ask the question do we want to play “as if” games if we are scientists. It would be great for the molecular biologist to be able to assign fitnesses to individual alleles, but they are interested in the genes, not in how selection is working. Such reductionism is very suspect, and, if it does work, probably working for the wrong reasons, so even if your idea of understanding evolution is ignoring the actual process, and just focusing on the change of gene frequencies you are likely to be disappointed.

If on the other hand, we are interested in how selection is acting, then “as if” doesn’t cut it. We want to know where selection is acting and such reductionism tells us nothing about the ecology of how selection actually works. For this we need to study selection, not its consequences.  When we talk about selection it is best to always identify an “among” and a “within”. The “among” is what level of organization selection is acting on, and “within” is what the range, or scope, of selection is. Thus we can have selection among groups within a metapopulation, selection among organisms within a population, selection among cells within an organism. The point is, to say that even if selection at different levels causes the same change in gene frequency (as I keep saying, they don’t) is to say they have the same consequence, not to say that they are the same thing.

So what about genic selection? Well, first off we need an among and a within. The among is pretty obvious, it is alleles. But what is the among within? Except in special cases I will get to, the among within cannot be the organism. Loci and thus alleles are always grouped into genomes and genomes are properties of cells (ignoring viruses here). Further, mitosis stops within genome selection. If you are heterozygous today, you will be heterozygous tomorrow. This is easily handled by contextual analysis. Simply put, because of mitosis, there is no variation in fitness among genes within the genome. Even if we assign fitness at the level of the allele, there can be no variation in fitness among the alleles within the genome, and indeed the lowest level at which it is possible for there to be variation in fitness is at the level of the cell. Thus the lowest level at which it is even theoretically possible for selection to act is at the level of the cell.


A Genic selection. Which is the best Gene? All are good, but the correct answer is Gene Kelly. He is well known as a jumping Gene. Top row (left to right) Gene Wilder, Gene Hackman, Gene Simmons. Bottom row (left to right) Gene Kelly, Gene Autry, Gene Tierney.

Of course there is an exception to this, and that is transposable elements. Now you can have selection among transposons within the genome. That is because each insertion site can be thought of as a bit of habitat for the transposon to insert into, and for most transposons a transposition event is a form of reproduction. Thus, selection among transposons within the genome is a special case example of genic selection. Lest you want to revel in having finally justified the term genic selection be aware that having high transposition rates is usually not good for the organismal phenotype, so there is selection at the level of the organism against jumping genes. Not only is this an example of genic selection, it is also an example of multilevel selection in which the gene is the individual, and the organism is the group. And, McClintock help us, genes that “choose” not to jump are being altruists (uggh.  Hate that term).



Why reductionism DOES work: Individuals to genes

Posted: June 4th, 2015 by Charles Goodnight

In the last couple of posts I have suggested that reductionism is for chumps. Two weeks ago I argued that gene interactions made average effects wonder around all over the place, and last week I argued that indirect genetic effects mucked up the works if there was population structure. This would seem to imply that quantitative genetics doesn’t work. Tell that to anybody who works in the agricultural breeding industry and they will laugh at you. Possibly more than any other field you can take quantitative genetics to the bank. You want lean pork, more marbling in your steak, more lysine in your corn? Quantitative genetics will do it for you. Furthermore, heritability estimates are technically only valid for the generation in which they are measured, however, the reality is that the common rule of thumb is that they are generally usable for 10 or more generations, and often appear to be pretty close after 100 generations. So, if reductionism doesn’t work why does quantitative genetics work so well? I will argue that as may be true of many complex systems, it works for the wrong reasons.

1. Within populations genetic effects will tend to be additive for statistical reasons.

In other words, selection and drift make gene interactions go away. Yes, genetic drift and selection can cause the additive genetic variance to increase, but it happens by statistically depleting the epistatic variance. After only a few generations of small population size the population can be treated as if there was no epistasis. In other words, as long as you stay within populations reductionism often provides a fairly accurate picture of the world. But you need to be careful. Another population may also act additively, but it will be a different additivity with alleles having different effects on the phenotype.

VA by generation

Twenty five generations of brother sister mating starting with equal amounts of AXA epistasis and additive genetic variance. The small population size increases the additive genetic variance, but also decreases the epistatic genetic variance, which is the difference between the green and red lines. After only a few generations of small population size there is very little epistatic variance available.

2. In a well connected metapopulation you may not see much differentiation for local average effects.

The way to detect epistasis is to look among populations either by examining the variance in local average effects (Goodnight 2000. Heredity 84:587), or the variance in local breeding values (Goodnight 1995, Evolution 49:502). This runs into two problems. First, nobody ever listens to me, so this experiment has been exactly twice (plus one in progress unpublished experiment) (De Brito, et al. 2005. Evolution 59: 2333, Drury & Wade 2011. JEB 24:168), and second, the variance in local breeding values is a function of the migration rate among subpopulations within the metapopulation.

I have not published this work, so I am violating my personal rule to not put unpublished results on my blog, but I think this is relevant, and it is part of a much larger model on speciation. Lets just say there is more than one paper coming out of this model, I am coming off of sabbatical, and well, it might be a year before this part gets written up. In any case If we look at a single metapopulation with an infinite number of demes. By the way, infinite demes is an assumption that is very suspect. For example, the approximation that equation works pretty well if there are infinite demes, but falls apart in finite metapopulations (yea, that is another paper that will out of this model. . . ).

With that in mind if we look at the variance in local breeding values as a function of Nm it becomes apparent that in order to get a significant variance in local breeding values migration rates need to be relatively low.


The effect of migration on the variance in local breeding values. Gene interaction is much more detectable among populations than within populations.   The appropriate measure being the variance in local breeding values or the variance in local average effects. Shown here is the variance in local breeding values as a function of Nm, or the number of migrants entering a deme per generation.   Note that migration rates must be below one migrant per generation before variance in local breeding values will be statistically detectable. Green dots (upper left corner) is zero migration, red dots are migration rates of 0.005. Scatter is due to different deme sizes. VAA = 1, VA = 0, generation 30,000.

To see the interplay between migration rate and deme size a three dimensional graph may help:

3d Graph

3d graph of the variance in local breeding values as a function of migration rate (M) and deme size (N). Unfortunately, JMP does not render surfaces exactly correctly. The graph should reach up to a value of 2 for zero migration.

There are a few caveats. First this is a drift model. There is no selection. If selection were to be added (good luck with that) I would speculate that selection against migrants offspring (e.g., hybrids) would mean much higher levels of population differentiation. Second, this model uses island model migration with infinite number of demes. Isolation by distance would dramatically increase the population differentiation and allow detectable gene interactions at much higher migration rates.

So this puts us in a relatively interesting situation. The models of drift and selection within demes tells that epistasis will be difficult to detect, and models of migration among demes tells us that migration rates above about 1 migrant per generation will also make gene interactions difficult to detect. Thus migration has the effect of tying the population together, and as a result preserving a lot of alleles.   The larger the metapopulations and the more the migration the more overall number alleles that will be preserved. However, such situations are ripe to explode if migration is ever restricted, or two metapopulations are separated. The variation is there, and thus no measurable epistasis, but once the populations are separated those interactions will pop out of hiding and show up again between species where migration rates are lower or non-existent. This would argue that again, the additive model is working for the wrong reasons. Just because gene interaction is statistically hard to detect doesn’t mean it isn’t there. It may simply mean that the conditions are such that it is hidden.


Not a whole lot of ice from my view from the ships deck! (http://blogdasa.com/2012/12/27/5-documentarios-que-me-tornaram-uma-pessoa-mais-bacana/)

One last thought. This also argues that it is reasonable to speculate that Dobzhansky and Muller are wrong. You don’t need two, or even any mutations for speciation to occur, just a barrier to gene flow that can be anything from isolation by distance to a road to disruptive selection.Barriers to reproduction will naturally arise.

Next time:  I will NOT talk about why you cannot reduce group selection to individual selection.  I say this for one simple reason:  I got nothin.  As far as I can tell indirect genetic effects are so powerful that any attempt to reduce group selection to individual selection is destined to end in tears.





Why reductionism doesn’t work, Part 2: Groups to individuals

Posted: May 26th, 2015 by Charles Goodnight

Williams (1966) famously wrote “In explaining adaptation, one should assume the adequacy of the simplest form of natural selection, that of alternative alleles in Mendelian populations, unless the evidence clearly shows that this theory does not suffice.” This principle of parsimony makes two interesting points. The first phrase “In explaining adaptation” makes the point that Williams was interested in examining patterns, and then using those patterns to infer how selection acted in the past. This is very different than modern MLS approaches in which the process of selection is examined. This is why parsimons (A bit of artistic license with the spelling) are so unimportant in modern MLS theory: such rules are not necessary if you are studying the process rather than inferring the process from standing patterns. More importantly, this principle implies that group selection can in many cases be reduced to individual selection, or even genic selection. The only thing that stands in the way of doing this is the ecology. Unless the trait is “altruism”, and thus impossible to evolve at a lower level, there is no reason not to act as if it was one of these lower levels of selection.


The principle of persimmony: persimmons come from a persimmonious tree (https://www.flickr.com/photos/giagir/5185254421).

But is this really true?   Last week I discussed why individual selection can’t be reduced to genic selection. It turns out that the situation is worse trying to reduce group to selection on the underlying individuals. So with that long-winded introduction out of the way, the main reason that group selection cannot be reduced to individual selection is indirect genetic effects (IGEs). Indirect genetic effects occur when genes in one individual affect the phenotype of another individual.

This is an effect that has been seen time and time again. The most aggressive chickens lay the most eggs, but also suppress the egg laying of their cage mates (Muir 1996, Poultry Science 75:447), crop plants aggressively interact such that the highest producing plants most strongly suppress their neighbors (Griffing 1977 in: Proceedings of the International Congress on Quantitative Genetics, August 16-21, 1976.) and many more examples. The important thing is that interactions that are internal to the unit of selection can contribute to the response to selection, whereas if they are external to the unit they cannot. Thus group selection can act on IGEs, but individual selection cannot.

To see this it is easiest to use the Price equation. The Price equation divides the covariance between a trait and relative fitness into within and between group components. It is easy and convenient to use this partitioning to make the point I want to make, but it is important to emphasize that the Price partitioning should never be equated with group and individual selection (Are you listening West and Gardner?).

Imagine we have a metapopulation in which individuals interact within groups but not between groups. The individuals interact in some manner that affects all individuals in the group in the same way. That is, perhaps they release waste products into their environment and everybody gets equally poisoned, or on a more positive note, perhaps they release some chemical public good. Further imagine that we have a trait, z, that is influenced by direct genetic effects (DGE), indirect genetic effects (IGE) and environmental effects. Thus, the trait value of the ith individual in the jth deme is:

Zij = DGEij + IGE.j + eij

Further imagine that the fitness of the ijth individual relative to the metapopulation mean fitness is wij, and the correlation between environmental effects and fitness is zero (just to get them out of the way).

To bring this back to my posts on Gardner, if I was following his model, at this point what I would want to do is partition the “total breeding value” so I could compare it with his partitioning of Fisherian breeding values. “Breeding value” is defined by Fisher (1930, Falconer and MacKay 1996) to be the average value of an individual’s offspring measured as a deviation from the population mean. This breeding value assumes that there is no population structure and that offspring interact randomly with other individuals in the population. Because they ignore population structure Fisherian breeding values cannot be partitioned. Bijma and Wade (2008. JEB 21: 1175-1188) solved this by defining “Total Breeding Value” to be the average value of an individual’s offspring measured in their native social environment as a deviation from the metapopulation mean. Unlike Fisherian breeding values, total breeding values can be partitioned. If you prefer to partition total breeding values replace “z” with total breeding value in the equation below, and replace DGE’s and IGE’s with their additive genetic equivalent.

If we put all this together, using the Price equation to partition the covariance between total breeding value and relative fitness we get an algebraic explosion!

Equation 1

Or much more simply:

equation 2

So, in words, this simply tells us that the within demes covariance between phenotype and relative fitness (red in the equation) includes ONLY direct genetic effects, whereas the between demes covariance between phenotype and relative fitness (blue in the equation) includes both direct and indirect genetic effects. This is shown graphically in the following figure:

Screen Shot 2015-03-25 at 2.04.15 PM

The sources of variation for a trait and the group mean of the trait. For clarity I have left the total variance proportions the same for the group mean trait, even though in most situations the direct genetic effects and the environmental effects would be reduced due to averaging. Although the genetic components underlying the trait are unchanged by taking the average, the heritable component does change. For the individual trait only the direct effects are heritable, whereas for the group mean trait both the direct and indirect genetic effects are heritable.

What this is saying is that from an evolutionary perspective a trait and the group mean of a trait are actually different traits. Because group selection can act on both direct and indirect effects it can produce genetic changes that are qualitatively different than selection acting on the individual level. As I have pointed out numerous times this is not a minor theoretical issue that experimentalists can ignore. Indirect genetic effects have shown up as major factors in the response to group selection in every situation where it has been possible to infer there presence, including both experiments specifically designed to detect them (e.g., Goodnight 1990 Evolution 44:1625), or where it was obvious even though the experiment did not have explicit treatments to detect them (e.g., Muir 1996).

Next week, as promised for this week, but not delivered:  Why reductionism does work.



Why reductionism doesn’t work; Part 1, Individuals to genes

Posted: May 18th, 2015 by Charles Goodnight

One thing that often used to happen, perhaps not so much any more, is that people will say that we don’t need to worry about levels of selection because all selection can be reduced to selection acting directly on genes. George Williams perhaps put this view best, first with his principle of parsimony, which argues that reductionism is the right perspective:

“In explaining adaptation, one should assume the adequacy of the simplest form of natural selection, that of alternative alleles in Mendelian populations, unless the evidence clearly shows that this theory does not suffice”

and in the same book, and more explicitly, which says that reductionism is works:

“No matter how functionally dependent a gene may be, and no matter how complicated its interactions with other genes and environmental factors, it must always be true that a given gene substitution will have an arithmetic mean effect on fitness in any population.”

All I can say to this is GAHHHH!

Brave 2 frustrated

Merida expresses her opinion on genetic reductionism (taken from http://giphy.com)

I think a lot of people know that you cannot think of selection as acting on genes, but a lot of people also can’t articulate why it doesn’t work. So, if anybody asks you, the simple answer is that reductionism doesn’t work because of interactions. At the individual level this will primarily be gene interactions of dominance and epistasis.

In a fully additive system there would be no problem, and this IS the problem.  Our intuition about genetics was developed using simple additive models.  In an additive system, knowing at what level selection was acting would be nice information, but the fitness of the phenotype can always be algebraically reduced to fitness effects on individual loci.   In other words, in additive systems, how the genes are packaged really doesn’t affect the effect of genes on the phenotype. To see this consider a phenotype affected by a single locus additive trait:

Genotype A1A1 A1A2 A2A2
Frequency p2 2pq q2
Fitness 1 1-Z/2 1-Z

(I use Z to emphasize that we are not talking about fitness. Selection will be affected by the packaging for the simple reason that some of the selection is on heterozygotes). We can calculate the average effect of the A1 allele on the phenotype we would discover that it is:

Original genotype genotype after substitution probability change
A1A1 A1A1 p2 0
A1A2 A1A2 ½ 2pq 0
A1A1 ½ 2pq Z/2
A2A2 A1A2 q2 Z/2

So, the average effect of the A1 allele is:

Screen Shot 2015-05-16 at 12.37.15 PM

Now consider a haploid system

Genotype A1 A2
Frequency p q
Fitness 1 1-Z/2

The average effect with the same phenotypic effects (adjusted for ploidy). Now the local average effect of the A1 allele is:

Original genotype genotype after substitution probability change
A1 A1 p 0
A2 A1 q Z/2

So, the average effect of the A1 allele is: you guessed it:

Screen Shot 2015-05-16 at 12.37.24 PM

The effect of the allele on the phenotype is not affected by the packaging.

Now lets do the same thing with a dominant system:

Genotype A1A1 A1A2 A2A2
Frequency p2 2pq q2
Fitness 1 1 1-Z

Now the average effect of the A1 allele on the phenotype becomes:

Original genotype genotype after substitution probability change
A1A1 A1A1 p2 0
A1A2 A1A2 ½ 2pq 0
A1A1 ½ 2pq 0
A2A2 A1A2 q2 Z

So, the average effect of the A1 allele is:

Screen Shot 2015-05-16 at 12.37.34 PM

turning to the haploid system

Genotype A1 A2
Frequency p q
Fitness 1 1-Z/2

Now the local average effect of the A1 allele is:

Original genotype genotype after substitution probability change
A1 A1 p 0
A2 A1 q Z/2

The average effect in the haploid system is now different than in the diploid system,.

Screen Shot 2015-05-16 at 12.37.24 PM

In other words, if we add the simplest possible form of nonadditivity the packaging does matter. Trust me it gets worse. I am way to lazy to put up tables for average effects in epistatic systems, but I have talked about this before. It turns out that the variance in local average effects is a measure of how the average effects of alleles are to genetic background. I have talked about these before, but it bears re-posting the relevant figure:

Drift and epistasis LAE graph

The important point is that the variance in local average effects is zero in additive systems, but non-zero when there are any sort of interactions. This means that the reducability of fitness effects on to genes is a reasonable exercise in additive system, but simply is not meaningful in epistatically interacting systems. To see how bad this can be, consider long-term directional selection in a system with AXA epistasis. Depending on the starting gene frequencies the average effect of an allele can actually reverse signs.  For what it is worth, the dashed lines are the local average effects for an additive system, and the solid lines are the local average effects for AXA epistasis.  This shows the contrast between additive systems and epistatic systems.  For the additive system, if you were to evaluate the fitness effects in generation zero they would provide a pretty good estimate of the fitness at the end (in this deterministic system an exact estimate).  On the other hand, for the epistatic system, estimates of allelic effects made in generation zero rapidly become useless, and by the time fixation is reached they are exactly wrong.

figure 12 AXA LAE

In one sense, Williams is absolutely correct. At any given instant it is certainly possible, in principle, to do a least squares regression analysis and assign fitness effects to individual loci. However in an epistatically interacting system those fitness assignments are ONLY good for the moment, or perhaps the generation, in which the assignment is done. Those effects will change as gene frequencies change, and not just gene frequencies at the locus under study, but gene frequencies at any other loci as well. So, my point is not that the assignment cannot be done, but rather that the assignment carries no information that is useful beyond the moment.

Next time I talk about why reductionism does work!

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