More from less, increased variance from bottlenecks

I have stated before that additive genetic variance is the relevant component of variance when modeling the response to selection in relation to a quantitative trait. In other words:

Response = (additive genetic variance)/(total phenotypic variance) X Selection

Consider height, which is about 80% heritable in the narrow sense in modern developed nations. What do I mean 80% heritable in the narrow sense? I mean that 4/5 of the variation in height, which is distributed in a normal fashion, is controlled by additive variation in the genotype. In other words, if I substitute allele 2 for allele 1 it is going to have an effect of deviation z upon the phenotype. What are the other components of variation? Obviously environmental variation. In regards to height we're probably talking about nutrition, but since in modern societies we have something of a nutritionally saturated environment this doesn't really vary too much (if you eat more food you just get fatter, not taller, beyond a certain point). There are also other possibilities, such as dominance effects (non-additivity within a locus), epistasis (non-independence between loci) and gene-environment interaction (non-linear dynamics, e.g., norm of reaction).

In any case, with respect to height you can see that the breeder's equation, R = h2S, will result in some dividends, as most of the variation is heritable in the narrow sense. As you reduce the heritable variation, you reduce the ability of selection to make an impact upon a quantitative trait. So, one assumes that a population bottleneck will result in a reduced ability of a population to respond to selection because of the implied reduction in genetic variation as a whole. Bottlenecks tend to result in founder effects because of the increased power of sampling variance. Like a low resolution photocopy a founder population invariably exhibits a loss of genetic information as rare allelic variants go extinct, while the proportional relationship between more common variants tends to be shifted.

But biology is the science of exceptions: in some cases populations which go through bottlenecks may actually be more responsive to selection upon quantitative traits because of increased additive genetic variation. How does this happen? Well, imagine that dominance or epistatic variation is converted to additive variation! Here's a toy illustration:

You have a locus with two alleles, 1 & 2. The three genotypes map onto phenotypic values like so:

11 = a (positive deviation from the mean when substituted)
12 = d (this measures the extent of dominance)
22 = -a (negative deviation from the mean when substituted)

The additive & dominance genetic variations can be modeled by the expression:

additive variance = 2pq(a + d(q - p))2
dominance variance = (2pqd)2

(the formalism should be reasonably familiar)

Assume complete dominance, so d = a. Also, for simplicity, a = 1. This means that:

additive variance = 8pq3 (substitute & do the algebra)

Take original allele proportions to be p = 0.75 & q = 0.25, and plug & chug:

additive variance = 8(0.75)(0.25)3 = 0.094

Assume that the population passes through a bottleneck which deviates the allele frequencies through sampling error to p = 0.70 and q = 0.30, and plug & chug:

additive variance = 8(0.7)(0.3)3 = 0.1512

Bingo! Additive genetic variance is now greater than before. There are more complex models which take epistatic genetic variance and convert it to additive genetic variance. It is important to recall that much of quantitative genetical theory, and R.A. Fisher's original work, tends to assume an average genetic background. The dynamics of populations though often result in a shift of that background, and in regards to quantitative genetics the act of selection itself tends to shift the allele frequencies in a manner which gives rise to new combinations and radically different genetic architectures many deviations away from the original phenotypic mean. Averages by their nature tell us what we can expect, but it is always important to note that quite often in evolution it is the dispersions away from expectation which hold the keys to the kingdom.

Note: Based on the treatment in Evolutionary Genetics: Concepts and Case Studies.

Related: Breeding the breeder's equation.

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Fascinating stuff, but as someone outside your field, I'm not quite keeping up with the implication of those equations.

Even if variables like p and q are standard, including a full variable key for articles like this would help a great deal.

By Spaulding (not verified) on 07 May 2007 #permalink

spaulding, sure, thanks for keeping me honest ;-) p & q are standard representations for a diallelic system, e.g., allele 1 = p and allele 2 = q. so if p = 0.75, then 75% of the alleles within the gene pool are p, and 25% are q. diallelic systems are neat because they make the algebra easier, q is by definition 1 - p.

in any case, think of it this way. let's say you have two alleles, 1 & 2. ok, an additive scenario might be like this in a diploid organism:

11 = 2 units
12 = 1 unit
22 = 0 units

11, 12 and 22 are the homozygote and heterozygote genotypes. you see here the phenotype is dependent simply on the flavor of allele in the slot, it's a pretty straightforward linear relationship. now, assume that there is complete dominance. now you have:

11 = 1 unit
12 = 1 unit
22 = 0 units

in a random mating populations with all things equal the relations between homozygotes & het. can be represented like so (HWE):

p2 + 2pq + q2

in a situation of complete dominance the frequency of q, which is "recessive" in its phenotypic expression, has a big effect on the outcome. as q gets smaller the effect of dominance becomes overwhelming because q is mostly resident within heterozygote individuals. at the boundary condition as q approaches 0 there is no phenotypic variantion produced by the genetic variance. the illustration above shows that bottlenecks can tilt the balance so that q's frequency is more balanced with p, and so genetic variation is more easily translated into phenotypic variation as q2 increases in its ratio to 2pq.

Regarding bottlenecks, isn't it the case that a given mutation has a better chance of surviving random drift longer in a small population than if it arose in a larger population, swamped by the other alleles? Kind of the flip side of bottlenecks leading to quicker extinction of established genotypes--must be that the new candidates stand a better chance of taking over?

By Joe Knapp (not verified) on 14 May 2007 #permalink

Would the value for the variable "a" be in standard deviation units?

By HellKaiserRyo (not verified) on 30 May 2007 #permalink

Would the value for the variable "a" be in standard deviation units?

not in this case, which is exceedingly simple. but yes, that would be convenient for a host of reasons.

Joe Knapp said:

"isn't it the case that a given mutation has a better chance of surviving random drift longer in a small population than if it arose in a larger population, swamped by the other alleles?"

Not of surviving random drift. What matters for that is the absolute (not proportional) number of alleles in the population.

If it is of selective advantage it has a better chance of not being "swamped" in a small population, especially if the homozygote is selected for. But that doesn't apply to neutral genes and random loss.

There is a bigger chance of becoming *fixed* in a smaller population of course, because that chance is the same as the proportion of the gene in the population and a single rare mutation will be a larger proportion of a small population.

And in finite time the chance is higher than that - every neutral allelle has an equal chance of getting to fixation but in practice if you ar a member of a species of billions of breeders that's hardly ever going to happen. So species that have had large populations for a long time will tend to have huge numbers of rare neutral or slighly deleterious alleles, with apparently random near-fixation of old mutations coming down the pipeline from millions of generations ago (That this is undoubtabtly the case is just one more reason for believeing that most selective advantages or disadvantages aren't additive)