42 Robert A. Skipper, Jr.
Fisher’s “fundamental theorem of natural selection,” and is the centerpiece of his
natural selection theory.
Interestingly, inasmuch as Fisher considered his fundamental theorem the cen-
terpiece of his evolutionary theory, it happens that the theorem is also the most
obscure element of it. The theorem was thoroughly misunderstood until 1989
when Warren Ewens rediscovered George Price’s 1972 clarification and proof of it.
Fisher’s original statement of the theorem in 1930 confusingly suggests that mean
fitness can never decrease because variances cannot be negative. Price showed that
in fact the left-hand-side of the equation that describes the theorem is not the total
rate of change in fitness but rather only one component of it. That part is the
portion of the rate of increase that can be ascribed to changes in gene frequencies.
And, actually, in Fisher’s ensuing discussion of the theorem, he makes this clear.
The total rate of change in mean fitness is due to a variety of forces including gene
frequencies themselves, environmental changes, epistatic gene interaction, domi-
nance and so forth. The theorem isolates the changes in gene frequencies from the
rest, a move suggested in Fisher’s 1922 paper. The key change Price and Ewens
make in the statement of the theorem, the change that clarifies it, is to write
“additivegenetic variance” for “genetic variance.” With the theorem clarified and
proven, Price and later Ewens argue that it is not so fundamental. Given that it is
a statement about only a portion of the rate of increase in fitness, it is incomplete.
Under the rubric of the fundamental theorem, Fisher offers a geometrical proof
of his conclusion from 1922 that cumulative evolution is primarily the result of low
pressures of natural selection on mutations of small effect. Fisher’s model depicts
an organism as a point in a space of very high dimension. Each dimension is a
phenotypic trait. For Fisher, adaptation is a step-wise process to the optimum
phenotype, and each step is the substitution of an advantageous mutation having
some phenotypic effect. Consider Figure 1, which depicts Fisher’s model in two
dimensions. Two continuous phenotypes, X and Y, have a fitness optimum at O.
Fitness decreases the further X and Y moveaway from the optimum; the circle
through D and centered around O represents the phenotypes that have the same
low fitness. If a population is at D, mutations of small and large effect arising in
the population are represented, respectively, by the small and large circles centered
around D. The dashed portion of the circles corresponds to mutations that bring
the population from D to points closer to O and are therefore advantageous. The
solid portion corresponds therefore to deleterious mutations, which take the popu-
lationaway from O. In spite of the fact that the advantageous proportion is about
half of the small circle, it is much less than half of the large one. Consequently,
evolution by natural selection is more likely to proceed by small steps because
advantageous mutations of small effect will be more common than advantageous
mutations of large effect. And the higher the dimensionality of the model, the
greater the bias against mutations of large effect. (Discussion follows [Burch and
Chao, 1999].)