Fisher’s fundamental theorem and Gaussian adaptation
Gregor Kjellström
Retired specialist in simulated evolution
Abstract
It will be shown – by a recapitulation of the proof of Fishers theorem - that the fitness of genes depends on the fitness and selection of individuals ruling the enrichment of genes. Besides, a drawback of Fisher’s theorem is that it does not tell us the increase in mean fitness from the offspring in one generation to the offspring in the next (which would be expected), but only from offspring to parents in the same generation. Gaussian adaptation may serve as a complement being a good second order approximation on the individual level. In this case the gradient of mean fitness and phenotypic disorder are pointing in the same direction.
Introduction
In modern terminology Fisher’s theorem has been stated as:
“The rate of increase in the mean fitness of any organism at any time ascribable to natural selection acting through changes in gene frequencies is exactly equal to its genic variance in fitness at that time”. (A.W.F. Edwards, 1994, see also Wikipedia).
The mean fitness is determined as a mean over the set of genes (alleles) in a large population assuming that a gene may have a fitness of its own being a unit of selection. The variance is a variance in fitness values of different genes.
As earlier shown, the theorem is formally correct. Even though individuals, having individual fitness values, replace genes, in which case the selection of individuals rules the enrichment of genes – the theorem is formally correct. It is also possible to make good simulations using numbers as genotypes (phenotypes) and digits as alleles.
An obstacle, however is the definition of fitness as given in many text books, see for instance Maynard Smith (see reference page 38): ”Fitness is a property, not of an individual, but of a class of individuals – for example homozygous for allele A at a particular locus. Thus the phrase ’expected number of offspring’ means the average number, not the number produced by some one individual. If the first human infant with a gene for levitation were struck by lightning in its pram, this would not prove the new genotype to have low fitness, but only that the particular child was unlucky.”
Even if the definition is useful in breeding programs, it can hardly be of any use as a basis of a theory of an evolution selecting individuals. It seems as if this definition denies the fitness of the individual. Nevertheless, the individual fitness is needed, because otherwise the “expected average number of offspring” from a certain class of individuals can’t be determined. In addition, if it is possible to define the fitness of an allele as an average of individual fitness values, then, it must be possible to define the mean fitness of a whole population from such values. So it can’t be forbidden to use the fitness of the individual.
In this paper we use the fitness defined by Hartl as the probability s(x) that the individual having the n characteristic parameters xT = (x1, x2, ..., xn) – where xT is the transpose of x - will survive, i. e. become selected as a parent of new individuals in the progeny. This measure is perhaps not very useful in breeding programs, but from a philosophical point of view the impact of the lightning may also be considered.
Recapitulation of the proof
Following Maynard Smith (see reference page 117), we may let the frequency of genotypes
g = (g1, g2, …, gn) before selection be p = (p1, p2, …, pn) and their fitness
w = (w1, w2, …, wn).
Then the mean fitness becomes (summation is over the set of indices i)
W = å piwi ; (1)
After selection has operated, the frequency of gi becomes
pi* = piwi/W (2)
and hence the mean fitness of the selected parents is
W* = å pi* wi
Hence the selection differential on fitness is
S = W* - W = å piwi(wi – W)/W;
We want now to show that S is equal to Vw, the variance of fitness before selection. Thus
Vw = å pi(wi – W)2
= å piwi(wi – W),
and since, in a density regulated population we have W = 1, we have Fisher’s theorem
S = Vw.
Exactly the same proof will hold even though w represent the fitness of individuals in a large population. In this case W is replaced - in order to distinguish between the genotypic and phenotypic level - by the probability P, defined as a mean of s(x), which may be < 1 even though the population is density regulated. Therefore the following variants of Fisher’s theorem are preferred:
S = Vw/W or S = Vs(x)/P
The theory of Gaussian adaptation
P is defined as:
P(m) = ò s(x) N(m – x) dx
In the special case when N is a Gaussian (normal) p. d. f.
N(m – x) = (2p)-n/2(det M)-1/2 exp{ -(m – x)TM-1(m – x) } with mean m (= [mi]) and moment matrix M (= [mij]), the gradient of P with respect to m may easily be determined. The variance or variability of interest here is represented by M.
The theorem of GA
1. The gradient of the mean fitness of a normal p. d. f. with respect to m is equal to
gradmP(m) = P M-1 ( m* – m).
The maximizing necessary condition for mean fitness is m* = m (at selective equilibrium).
2. The gradient of phenotypic disorder (entropy, average information, diversity) with respect to m – assuming P constant - points in the same direction as gradmP(m).
Unfortunately, nothing has earlier been published about the gradient because focus has been on the last sentence in part 1, which was sufficient when NA was used as simulated evolution for parametric optimization.
Proof: Since differentiation is here allowed to the right of the integral sign we get
¶P(m)/¶mj = ò s(x) { ¶ N(m – x) /¶mj } dx
= - ò s(x) N(m – x) { ujT M-1(mj – xj) + (mj – xj)T M-1 uj } / 2 dx,
where the components of the vector uj are = 0, except for the component number j
which is = 1. Thus we have
gradm P(m) = - ò s(x) N(m – x) M-1 (m – x) dx
= - P M-1 { m - ò x s(x) N(m – x) dx / P }
= P M-1 ( m* – m )
where we have introduced the mean of phenotypes of the set of selected parents
m* = [mj*] = ò x s(x) N(m–x) dx / ò s(x) N(m–x) dx =
= ò x s(x) N(m–x) dx / P
which proves part 1 of the theorem.
A proof of part 2 of the theorem may be found in appendix.
The increase in mean fitness due to GA
A correspondence to Fisher’s increase in mean fitness may now be derived from NA. In this case the increase is defined from the offspring in one generation to the offspring in the next (it is assumed that M is fairly constant from one generation to the next). We have
DP = (¶P/¶m1) Dm1 + (¶P/¶m2) Dm2 + … + (¶P/¶mn) Dmn
If the Gaussian is moved from m to m* we get the approximation
DP = P (m* - m)T M-1 (m* - m);
A simulation using digits as alleles
Suppose, for sake of simplicity, that each individual organism is determined as a point in a 2-dimensionnal space by two phenotypes (x, y) each represented by a 3-digit number where each digit is an allele. Thus we have six locus each having ten alleles
0, 1, …, 9 or w = (w1, w2, …, w60).
The figure shows two different cases (upper and lower) of individual selection, where the green points with fitness = 1 - between the two lines - will be selected, while the red points with fitness = 0 outside will not. The fraction of green feasible points is the same in both cases, but the variance in the horizontal direction is larger in the lower case. m is black, m* is red.
