Sewall Wright, Speciation and Migration

Many thanks to several members of the Zufall lab for bringing me up to date on mutation accumulation experiments .  Her lab is actively involved in MA experiments (http://www.genetics.org/content/early/2013/07/29/genetics.113.153536.abstract), and what was satisfying to me, quite interested in dissecting out dominance and epistatic effects of new mutants.  Hopefully we will get more data on this in the future.  In any case, lets move on to the last, and perhaps, from a theoretical perspective, least interesting, of the four forces of evolution, migration.

To point out just how mundane it is, consider the infinite alleles model of mutation drift balance:

Migration equation 2

All we need to change this into migration drift balance is change the font for the mu from symbol to times. . .

Migration equation 1

More seriously, it is fun to consider that old chestnut “one migrant every other generation destroys genetic variation”.  I have no idea where this form comes from, but ultimately it is derived from Wright (1931. Genetics 16: 93-159) in which he says “How little interchange would appear necessary to hold a large population together may be seen from the consideration that m = 1/2N means an interchange of only one individual every other generation, regardless of the size of the subgroup.”  Interestingly he goes on to say “However, this estimate must be much qualified by the consideration that the effective N of the formula is in general much smaller than the actual size of the population or even than the breeding stock, and by the further consideration that q, of the formula refers to the gene frequency of actual migrants and that a further factor must be included if q, is to refer to the species as a whole. Taking both of these into account, it would appear that an interchange of the order of thousands of individuals per generation between neighboring subgroups of a widely distributed species might well be insufficient to prevent a considerable random drifting apart in their genetic compositions.”  By the way, if you were hoping I would say something scandalous about Sewall Wright, other than his penchant for erasing black boards with guinea pigs, well it ain’t gunna happen.  I had the great privilege of meeting him several times, the first time when I was a starting graduate student.  He is one of my heroes.

So, what was he really saying about migration.  He is absolutely correct that migration will tie a metapopulation together genetically.  Looking at his original figure you can see that what happens in a two allele system is that when m = 1/2N is that every gene frequency is equally likely.

Wright Figure

(From Wright 1931. Genetics 16: 93-159)

Note that this shows just how miss-stated that old platitude really is.  Far from destroying genetic variance, all gene frequencies are equally likely (at a gene frequency of 0.5), or in other words, anything is possible.  More important, it is a statement about variance in gene frequency, not variance in phenotype, and of course, there is no epistasis.  This last is hardly surprising as this was published 50 some odd years before the first theoretical treatment of the effect of epistasis on phenotypic variance (Goodnight 1983. Ph.D. thesis University of Chicago – I have to establish my bragging rights at some point).

If others can establish silly rules of thumb about migration and variation, I can do the same thing.  In my chapter on metapopulation quantitative genetics in Hanski and Gaggiotti’s book (Goodnight 2004. In: Ecology, Genetics and Evolution of Metapopulations. Hanski and Goggiotti. eds.) I make the perhaps rash assumption that we can solve for the standard approximation:

Migration equation 3

and using this to calculate the equilibrium additive genetic variance due to various forms of gene interactions we can get the following graph:

VA by migrants graph

Additive genetic variance within demes as a function the number of migrants per deme at equilibrium.  Total genetic variance in the outbred population (F=0) is standardized at 1.  (from Goodnight 2004. In: Ecology, Genetics and Evolution of Metapopulations. Hanski and Goggiotti. eds.)

There are a couple of things to notice about this graph.  First, as you might expect, for simple additive systems the greater the migration rate the greater the additive genetic variance.  Thus, in a purely additive world the additive genetic variance is a simple tradeoff with population differentiation.  The greater the migration, the greater the VA, and the less the population differentiation.  With other forms of genetic effects the tradeoff is not quite so simple.  Isolation leads to fixation, and with it the conversion of the gene interaction into additive genetic variance.  Too little migration and you get simple fixation, and no genetic variance, additive or otherwise.  To much migration and the populations are effectively panmictic, and there is no conversion taking place. This tradeoff is readily seen in the figure above.  What can be seen is that for simple dominance the greatest conversion takes place around one migrant every other generation (M = Nm = ½).  For the other forms of genetic interaction it can be seen that the additive genetic variance is maximized at around one migrant every 4 generations (M = Nm = ¼).

This raises an important point, and I am realizing I am out of space to really discuss it properly.  It will probably take at least two weeks, but I will discuss how there are two types of population differentiation:  differentiation for population means and differentiation for genetic effects.  It turns out that an increase in additive genetic variance with increasing inbreeding coefficient is clear evidence for a shift in what genes are doing to the phenotype in different populations.  Thus, the increase in additive genetic variance we see as a function of migration rate means that the subpopulations in the metapopulation are differentiating.  This means that we can utter our very own migration platitude:

One migrant every four generations is optimal for speciation to occur.

Do I believe it?  Well I certainly think it is more believable than the platitude I uttered at the beginning of this essay.

As a final note:  The semester has started.  I will try to keep up my weekly post, but please be patient if I start to fall short as things heat up.

2 Responses to “Sewall Wright, Speciation and Migration”

  1. Jamie;

    Great to hear from you. I knew that blogging had some sort of value!

    To your questions:

    1) No, nobody has done any experimental work. It seems like a bit of a dangerous research route since all things considered the effects might be pretty subtle. What I am getting at is that with one migrant every 4 generations you get the best balance of loss of variation due to fixation of genes, and the increase in additive variance due to the conversion of epistasis into additive variance. This “new” additive variance will involve shifting of the average effects that has the potential to send populations down different evolutionary trajectories (hopefully the topic of this weeks post).

    2) Sewall Wright denies that he erased blackboards with guinea pigs, but I am pretty sure I was once told by his last graduate student, Janice Spofford, that she once saw him erase a board with a guinea pig. So, third hand, your call whether or not it is true. He definitely brought guinea pigs into class, and would line them up on the lectern while he talked about their genetics, so erasing boards with them does not seem out of the question.

    If I was lecturing with guinea pigs I would for sure do it. The squealing alone would be worth it.

  2. Jamie says:

    1. Has lab-based work been done to examine optimal migration rate for speciation? Or has most research in this area been theoretical? How many generations might it take for speciation to occur?

    2. Is it true about the guinea pigs? I suppose a little chalk never hurt anyone.

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