Não houve tempo para evoluçao Darwinista CHOREM Darwinistas!
There’s plenty of time for evolution
Herbert S. Wilfa,1 and Warren J. Ewensb
aDepartment of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395; and bDepartment of Biology, University of Pennsylvania,
Philadelphia, PA 19104-6018
Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved November 2, 2010 (received for review October 7, 2010)
Objections to Darwinian evolution are often based on the time
required to carry out the necessary mutations. Seemingly, exponential
numbers of mutations are needed. We show that such estimates
ignore the effects of natural selection, and that the
numbers of necessary mutations are thereby reduced to about
K log L, rather than KL, where L is the length of the genomic
“word,” and K is the number of possible “letters” that can occupy
any position in the word. The required theory makes contact with
the theory of radix-exchange sorting in theoretical computer
science, and the asymptotic analysis of certain sums that occur
there.
mutations ∣ natural selection ∣ geometric distribution
The 2009 “Year of Darwin” has seen many welcome tributes to
this great scientist, and reaffirmations of the validity of his
theory of evolution by natural selection, though this validity is
not universally accepted. One of the main objections that have
been raised holds that there has not been enough time for all
of the species complexity that we see to have evolved by random
mutations. Our purpose here is to analyze this process, and our
conclusion is that when one takes account of the role of natural
selection in a reasonable way, there has been ample time for the
evolution that we observe to have taken place.
The Calculations
Biological evolution is such a complex process that any attempt to
describe it precisely in a way similar to the description of the
dynamic processes in physics by mathematical methods is impossible.
This fact does not mean, however, that arbitrary models of
biological evolution are allowed. Any allowable model has to reflect
the main features of evolution. Our main aims, discussed
below, are to indicate why an evolutionary model often used
to “discredit” Darwin, leading to the “not enough time” claim,
is inappropriate, and to find the mathematical properties of a
more appropriate model.
Before doing so we take up some other points. Evolution as a
Darwinian-Mendelian process takes place via a succession of
gene replacement processes, whereby a new “superior” gene
arises by mutation in the population and, by natural selection,
steadily replaces the current gene. (We use here the word “gene”
rather than the more technically accurate “allele”.) It has recently
been estimated (1) that a newborn human carries some de novo
100–200 base mutations. Only about five of these can be expected,
on average, to arise in parts of the genome coding for
genes or in regulatory regions. In a population admitting a million
births in any year, we may expect something on the order of five
million such de novo mutations, or about 250 per gene in a genome
containing 20,000 genes. There is then little problem about a
supply of new mutations in any gene. However only a small proportion
of these can be expected to be favorable. We formalize
these considerations in the calculations below.
We now turn to the inappropriate evolutionary model referred
to above concerning the fixation of these genes in the population.
The incorrect argument runs along the following lines: Consider
the replacement processes needed in order to change each of the
resident genes at L loci in a more primitive genome into those
of a more favorable, or advanced, gene. Suppose that at each
such gene locus, the argument runs, the proportion of gene types
(alleles) at that gene locus that are more favored than the primitive
type is K−1. The probability that at all L loci a more favored
gene type is obtained in one round of evolutionary “trials” is K−L,
a vanishingly small amount. When trials are carried out sequentially
over time, an exponentially large number of trials (of order
KL) would be needed in order to carry out the complete transformation,
and from this some have concluded that the evolutionby-
mutation paradigm doesn’t work because of lack of time.
But this argument in effect assumes an “in series” rather than a
more correct “in parallel” evolutionary process. If a superior gene
for (say) eye function has become fixed in a population, it is not
thrown out when a superior gene for (say) liver function becomes
fixed. Evolution is an “in parallel” process, with beneficial mutations
at one gene locus being retained after they become fixed
in a population while beneficial mutations at other loci become
fixed. In fact this statement is essentially the principle of natural
selection.
The paradigm used in the incorrect argument is often formalized
as follows: Suppose that we are trying to find a specific unknown
word of L letters, each of the letters having been chosen
from an alphabet of K letters.We want to find the word by means
of a sequence of rounds of guessing letters. A single round consists
in guessing all of the letters of the word by choosing, for each
letter, a randomly chosen letter from the alphabet. If the correct
word is not found, a new sequence is guessed, and the procedure
is continued until the correct sequence is found. Under this paradigm
the mean number of rounds of guessing until the correct
sequence is found is indeed KL.
But a more appropriate model is the following: After guessing
each of the letters, we are told which (if any) of the guessed letters
are correct, and then those letters are retained. The second round
of guessing is applied only for the incorrect letters that remain
after this first round, and so forth. This procedure mimics the
“in parallel” evolutionary process. The question concerns the
statistics of the number of rounds needed to guess all of the letters
of the word successfully. Our main result is
Theorem 1. The mean number of rounds that are necessary to guess
all of the letters of an L letter word, the letters coming from an
alphabet of K letters, is
¼
logL
logð K
K−1ÞþβðLÞþOðL−1Þ ðL→∞Þ [1]
with βðLÞ being the periodic function of logL that is given by Eq. 7
below. The function βðLÞ oscillates within a range which for K ≥ 2,
is never larger than :000002 about the first two terms on the righthand
side of Eq. 7.
For example, if we are using a K ¼ 40 letter alphabet, and a
word of length 20,000 letters, then the number of possible words
is about 1034;040, but our theorem shows that a mean of only about
Author contributions: H.S.W. and W.J.E. designed research; H.S.W. and W.J.E. performed
research; and H.S.W. and W.J.E. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1To whom correspondence should be addressed. E-mail: wilf@math.upenn.edu.
22454–22456 ∣ PNAS ∣ December 28, 2010 ∣ vol. 107 ∣ no. 52
www.pnas.org/cgi/doi/10.1073/pnas.1016207107
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