What is “additive variance” in genetically uniform populations?

I recently got a comment from Michael Bentley at Oxford pointing out that he had a different interpretation of heritability among cells within higher organisms. His comment was:

“Please could I just clarify something you say in this piece, as it relates to something I’m working on at the moment. You say:

‘From the perspective of individuality, what this does is that it lowers the heritability at the cellular level to nearly zero.’

This confused me, since the heritability at the cell level via mitosis is nearly one, not nearly zero, isn’t it? If we take h^2 = Cov(zi,zi’)\Var(zi), where zi is parent cell phenotype, and zi’ is offspring cell phenotype (we have regressed parent phenotype against offspring phenotype and taken the gradient of the regression line to be the heritability). Assuming high fidelity, we have Cov(zi,zi’) approx = Cov(zi,zi) = Var(zi). Putting this back in we get h^2 = Var(zi)/Var(zi) = 1, and thus h = 1.”

The relevant post is here. Mr. Bentley raises a very good point. In this post I argue that because within an organism cells divide by mitosis, that there is essentially no genetic variation, and as a result, baring somatic mutations, the heritability within organisms is very near to zero. Michael argues that in fact the somatic cells have very high phenotypic fidelity when they divide. Thus, liver cells divide to make to liver cells, and skin cells divide to make to skin cells. By his reckoning the heritability should be very close to one.

So, how should this be handled. First off, I would argue that Michael is right, and I am wrong. Michael used an appropriate definition of “realized” heritability based on a phenotypic perspective, whereas, old fogey that I am, I somehow was stuck in trying to force Fisher’s model where it didn’t belong. Nevertheless, I do stand by my point that mitosis serves as a mechanism that minimizes the response to selection within organisms, I just should have been careful when I called it “heritability”

What this says is that we need to more carefully define heritability, and the additive variance. Fisher first defined additive genetic variance, and to paraphrase something that Walt Ewens, Fisher defined it, and thus we need to accept that his definition is correct. Fishers definition of the additive genetic variance is the sum of the covariances between average effects and average excesses, however, as Falconer has pointed out this definition is useless in the real world (Falconer 1985 Genet. Res. Camb. 46:337). Thus, we are stuck with making up a useful definition. Falconer provides an alternate definition of additive genetic variance statistically, for example as the variance due to regression of offspring on mid parents (I don’t have his book with me in Brazil, so I am not sure of his exact definition). However I would call this the “effective” additive genetic variance, since in real populations it will not exactly equal Fisher’s definition.  It is also relevant to mention that Falconer (in Introduction to Quantitative Genetics 1989) nicely demonstrates that the additive genetic variance is the genetic covariance between parents and offspring.

The way I have been thinking about phenotypic evolution is as a super-set of quantitative genetics. Fundamentally quantitative genetics is a phenotypic approach. The breeders equation demonstrates this:

R = h2S

Or in words, the response to selection is equal to the heritability times the selection differential. It is a phenotypic model because basically the heritability serves as the transition equation that converts the fitness weighted distribution of phenotypes in the parental generation (S) into the distribution of phenotypes in the next generation (R). What the phenotypic perspective does is to argue that this is a fundamentally correct perspective for thinking about evolution, but that a transition equation that is a single constant and (at least theoretically) includes only genetic effects is overly simplistic. Relevant to my discussion with Michael, quantitative genetics is also overly simplistic because it only applies to sexually reproducing organisms. Aside: It is hard to fault Fisher for this. His primary goals were to describe the genetics for humans and mammalian livestock, and to provide tools for animal breeders. His efforts were spectacularly successful to the point of saying that Fisher was the central figure in the new synthesis, and one could argue that he basically single handedly built the foundation for the new synthesis.

So, the bottom line is that we should stick with something similar to Falconer’s practical definition: The additive variance is the covariance between parents and offspring. Note that I did not say the “additive genetic variance”, and this is an important distinction. I suggest we should define the additive variance as the covariance between parents and offspring without regard to the cause of that covariance.

Of course in many situations that is not satisfying. In the discussion between Michael and I both of our perspectives were important. He was exactly right that there is a very high covariance between parent and offspring cells in metazoans, but I was also correct that there is essentially no genetic differences among cells in metazoans. So, what is causing the high covariance that Michael identified? I don’t know, but it is not genetic. More likely it is due to two causes. First there are epigenetic changes – silencing of some genes, and over expression of others – that give a particular cell type its phenotype, and importantly, these epigenetic changes are preserved during mitosis. Second there is a lot of cell-cell interaction that causes offspring cells to resemble parental cells due to the “developmental ecological” or “positional” situation a cell finds itself in. In development there are numerous examples of this sort of induction. It may well be that one reason the daughter cells of liver cells are also liver cells is because they are in the liver, and induced to be liver cells because of that.

I suggest the correct thing to do is to accept the general definition of additive variance, but then allow this to be broken up into components. That is the additive variance could be broken up into Additive “genetic” variance, Additive “epigenetic” variance, Additive “positional”, and so on. Thus, we should accept the single obvious definition of additive variance of the covariance between parents and offspring, but then use some form of least squares partitioning to divide it into sub components.

Of course there is a problem here. That is how do we do that division? Again, I suggest that we follow Fisher’s lead here. What is needed is an appropriate modification of parent-offspring regression and half sib design breeding experiments. For example, we might examine the additive variance in the natural setting to get the total additive variance. Second, we might look at the variance among cell lineages to get the additive genetic variance, and the variance within cell lineages to get the additive non-genetic variance. By transplanting cells to other locations we could get the additive physiological-ecological variance, and by using molecular methods to remove the epigenetic modifications get an estimate of the additive epigenetic variance.

What ever the actual experimental protocol that ends up being appropriate, what we want is:

Cov(Parent,Offspring) = Covgenetic(Parent offspring) + Covepigenetic(Parent offspring) + Covpositional(Parent offspring) + . . .

There are, of course, two major problems with this. The first is practical. If you decide to do that experiment, well good luck. At least at first blush it looks like it would be a horrific amount of work that would simply not be worth the information obtained. The second is statistical in nature. I am arguing for using a Fisherian least squares partitioning into the subcomponents of the additive variance. The good news is that, if done properly, such partitionings are orthogonal, so that the components would add up the total additive variance. The bad news is that such partitionings are context dependent, thus, the partitioning into sub components of the additive variance would change as conditions change. Nevertheless, it seems to me that this is a good way to think about simple linear transition equations from the phenotypic perspective. It is also a way to keep the excellent framework that Fisher provided, while allowing it to be conceptually expanded to other systems of reproduction, and non-genetic forms of inheritance.

One Response to “What is “additive variance” in genetically uniform populations?”

  1. Michael Bentley says:

    Hi Charles,

    I found something regarding heritability in Frank (1997) that I think deals with the artificially high covariance term. He starts with the other definition of heritability:

    B_g’z = Cov(g’,z)/Var(z),

    where g’ is the breeding value transmitted to offspring, and z is the phenotype of the parent. He showed that this could be further partitioned into

    B_g’z = B_g’g * B_gz = (Cov(g’g) / Var(g)) * (Var(g) / Var(z)).

    Thus, heritability has two components, the fidelity of transmission of breeding value B_g’g, which will be ~1 for clonal cells (like I said), and B_gz, which will be ~0 for clonal cells (like you said), since there is little genetic variance.

    Ref:
    Frank (1997) The Price Equation, Fisher’s Fundamental Theorem, Kin Selection, and Causal Analysis

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