Genetical Theory, chapter 2

It's been nearly a month since I last posted on The Genetical Theory of Natural Selection. I've been holding off because I didn't know how to approach chapter 2, in many ways it is the most important and ambitious chapter (though not technically the most taxing). I think I will likely post twice on this chapter, and in this entry I'll avoid talking about the difference between "average effect" and "average excess" and what not. Rather, I'll focus on two issues:

1) The Fundamental Theorem of Natural Selection.

2) Fisher's view on the nature of adaptation

Both of these are rather simple concepts. In regards to the Fundamental Theorm, Fisher states that "The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time." Formally it is represented as:

Σ(αdp) = Wdt

α is the "average effect" (roughly, the quantitative contribution of a locus, or gene, to the value of a trait), and W is fitness. Both of these are changing with respect to the frequency, p, of the allele (diallelic model on one locus) and t, time. As time progresses fitness increases on a single locus in proportion to the variance of fitness. To put in another way, this is often take to imply that the rate of change in evolution is proportional to the heritable variation (additive genetic variance as a proportion of total variance) extant within the population. This relates to another equation:

R = h²S

where R is the response to S, selection, proportional to h², heritable variation, within the population . This goes to the intuitive understanding we have that evolution operates predominantly upon intrapopulational variation, and without this variation generated by mutation evolution would eventually exhaust itself. The implication of Fisher's theorem is that over time evolution will remove its necessary precondition (variation), but the reality is that simple traits attain a mutation-selection balance where exhaustion and replenishment of variation match each other, and the "mutation" rate of polygenes and the combinations possible on a quantitative trait are both sufficient to maintain variation for an interminable number of generations. Many long term agricultural breeding programs (e.g., flax seed) have managed to shift the median quantitative value of a trait many standard deviations from the original value as rare alleles are driven to higher frequencies so as to meet and generate exceptional phenotypes (i.e., if the initial frequency of two alleles is 0.0001, their chances of combining are low, but as they increase in frequency their chances of combining increase).

There are of course major issues with Fisher's conception. Ultimately Fisher wanted something that was like an "an ideal gas law" of evolution, i.e., PV = nRT. Many (most?) people do not believe this will ever occur because the significance of theorem as a precise deducible formalism beyond one locus is questioned. I will leave that for later, and others.

A second major point in this second chapter relates to the phenotypic changes induced by selection. Using a geometrical analogy Fisher argues that large mutations, large changes in trait, are far more likely to reduce fitness than small changes. Consider a fitness optimum, represented by a point in space. Now, consider a sphere which represents a fixed distance from this optimum. If a point of current fitness lay embedded within the surface of the sphere a small radius of deviation around that point is as likely to increase as decrease fitness, while a larger radius will tend to be "outside" the sphere, and so further away from the fitness optimum, more than it will be inside the sphere. Ultimately, when the "jump" becomes large enough the full arc of possible points distance "x" from the current fitness optimum will be outside the sphere and so "overshoot" any increase in fitness as a matter of definition (one can imagine it as overshooting a peak mountain climbing, as you near it your leaps need to decrease or you'll go back down the hill).

Fisher represents this model with the formula:

1/2 (1 - r/d),

where r is the radius of the point deviated from the current optimum, while d represents the diameter of the sphere. As r, the radius of the deviation approaches 0 you approach a 1/2 chance of increasing fitness. As r approaches d the chance nears 0, until all possibilities now reduce fitness. Fisher acknowledges that 3-dimensions is artficial, and concludes that a trait will explore a much higher dimensionality in regards to fitness. And, in relation to this dimensionality the constraints alluded to above become much more prohibitive, as Fisher shows that a there is a "standard magnitude of change," d/(√n), where n is the dimensionality. As n increases the standard magnitude drops, and as the ratio between this and the magnitude of genuine change determines the probability of increased fitness, that probability decreases for a given change in a trait as dimensionality increases. Short form: complex traits have to change at a slow rate.

Heady stuff, the formalism is quite baby-like (though the derivation not always), but verbally translating this isn't too easy. The take home from this is two-fold:

1) Variation is necessary
2) Change is gradual

This is the tradition which led to Richard Dawkins, and the seed for the one that Stephen Jay Gould attacked as "Ultra-Darwinian." Love it or hate it, you should know it.