xia@cc.helsinki.fi (10/16/90)
Average of Fitness, Evolution of Sex and Others
===============================================
In case that someone forgets, let me remind you that the average
of fitness of a gene over generations is represented by geometric
mean, not by arithmetic mean. Allow me to use an example to
illustrate the reason behind this.
If a population of a gene makes 100 copies of itself in one generation,
but 0 copies in the next generation and become extinct, then the
average fitness of the gene is 0 (=square root of 100*0), not
50 (=(100+0)/2).
Suppose a locus with two alleles, A and a. Allele A always makes X
copies of itself per generation, but allele a makes X+s copies in
50% of generations and X-s in the other 50% of generations. After
2N generations, the number of copies of allele A will be
2N
N(A) = X ,
while the average fitness of allele a during the same period is
N N N
N(a) = (X+s) *(X-s) = ((X+s)*(X-s))
2 2 N
=(X - s ) .
2 2 2
Apparently, N(A) > N(a) because X > X - s. Therefore, allele A
will eliminate allele a over the long run.
(BTW, the above simple formulation is the foundation of the so-called
bet-hedgeing in life-history theory.)
One theorem we can draw from above is that an allele with less
variable fitness over generations will eliminate an allele with
more variable fitness over generations, although both have the
same arithmatic mean fitness over generations.
A corollary of the theorem is that any gene that reduces fitness
variability of its carier will be favoured by natural selection.
The gene for sexual reproduction is such a gene, it reduces the
fitness variation of its carrier 1.414 (=square root of 2) times.
The gene for promiscuity is also such a gene when an animal is not
sure of fitness potential of its mates. This may even be applied
to human societies.
(to be continued)
(Please let me know if my writing is interesting so that I won't
keep posting things that you do not read.)
blot@URZ.UNIBAS.CH (Michel Blot) (10/16/90)
keep on writing. This will be the best answer to the debate that occurred a feww
days ago on the usefulness of this list.
Michel Blot, biozentrum Basel.MOYLEK@SSCVAX.CIS.MCMASTER.CA ("Ken Moyle.... aka Jose.... aka That CIS guy...") (10/17/90)
> > > Average of Fitness, Evolution of Sex and Others > =============================================== > >. >. >. >. >(to be continued) > >(Please let me know if my writing is interesting so that I won't >keep posting things that you do not read.) > This is very interesting... I've never seen this particular mathematical reasoning before (but then, I'm a biochemist, not an evolutionary geneticist). Can you give any referneces which delve into these evolutionary theories? ....Ken Moyle McMaster University Hamilton, Ontario
joe@GENETICS.WASHINGTON.EDU (Joe Felsenstein) (10/17/90)
In comment on Xia's posting on: > Average of Fitness, Evolution of Sex and Others > =============================================== > In the example of two alleles with fitnesses (X+s):X in one generation and (X-s):X in the next, the assumption is implicit that both alleles have the same arithmetic mean fitness. But if they don't, then it is not obvious which one will win out without computing the geometric means. For example, if one generation the alleles have fitnesses X(1+2s) : X and in the other X/(1+s) : X, then the MORE variable one wins since (1+2s)/(1+s) > 1. > (BTW, the above simple formulation is the foundation of the so-called > bet-hedgeing in life-history theory.) > A corollary of the theorem is that any gene that reduces fitness > variability of its carier will be favoured by natural selection. So it is not just a matter of bet-hedging: if there is a cost of bet-hedging then it can be selected against. > The gene for sexual reproduction is such a gene, it reduces the > fitness variation of its carrier 1.414 (=square root of 2) times. If the square root of two is based (as I suspect) on the fact that there is a two-fold cost of sexual reproduction, then this won't work as the reduction of mean is too great to make the reduction of variance worhtwhile. > (Please let me know if my writing is interesting so that I won't > keep posting things that you do not read.) > Do keep it up. ---- Joe Felsenstein, Dept. of Genetics, Univ. of Washington, Seattle, WA 98195 Internet/ARPANet: joe@genetics.washington.edu (IP No. 128.208.128.1) Bitnet/EARN: felsenst@uwalocke UUCP: ... uw-beaver!evolution.genetics!joe
toms@fcs260c2.ncifcrf.gov (Tom Schneider) (10/17/90)
In article <3347.2719f222@cc.helsinki.fi> xia@cc.helsinki.fi writes: > > Average of Fitness, Evolution of Sex and Others > =============================================== > >In case that someone forgets, let me remind you that the average >of fitness of a gene over generations is represented by geometric >mean, not by arithmetic mean. Allow me to use an example to >illustrate the reason behind this. I have two problems with this thesis. The first is that the concept of fitness is ill-defined (flame away!!!), however much it may be discussed. How do you measure fitness? What are the units? I think that it is an arbitrary measure that assumes that the organism does not affect the environment it is in. But every organism strongly affects its environment! "We" made oxygen! We made oil! We put the oil back into the atmosphere! The oxygen precipitated iron and uranium from the oceans (correct me if I'm wrong on this) so we were the cause of the uranium deposits that allow us to spread radioactive particles all over... The idea of 'fitness' ought to be dropped and replaced with better measures. >If a population of a gene makes 100 copies of itself in one generation, >but 0 copies in the next generation and become extinct, then the >average fitness of the gene is 0 (=square root of 100*0), not >50 (=(100+0)/2). My second objection is that if fitness is the time geometric mean, then highly successful animals like the dinosaur and (yes) humans AND ALMOST ALL SPECIES have zero fitness, since most species die off eventually! Tom Schneider National Cancer Institute Laboratory of Mathematical Biology Frederick, Maryland 21702-1201 toms@ncifcrf.gov
aalto@cc.helsinki.fi (10/18/90)
In article <1910@fcs280s.ncifcrf.gov>, toms@fcs260c2.ncifcrf.gov (Tom Schneider) writes: > My second objection is that if fitness is the time geometric mean, then > highly successful animals like the dinosaur and (yes) humans AND ALMOST > ALL SPECIES have zero fitness, since most species die off eventually! > It is not meaningful to speak about the fitness of a species. Only about the fitness of an individual, genotype or a gene and only within a population. A brilliant analysis of the problems of the concept of fitness is given by Richard Dawkins in the chapter 'An Agony in Five Fits' in the book 'The Extended Phenotype'. Erkki Aalto University of Helsinki Finland