Griffing, Associate effects, and heritability.

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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