Math Forum Discussions

David Wilson

Posts: 3
Registered: 12/12/04

Re: Several questions on evolution, and mutation (rate)
Posted: Sep 8, 1996 9:44 PM

Plain Text

Note followups.

In article <Pine.A32.3.91.960815144016.39360B-100000@flute.aix.calpoly.edu>
<asinger@flute.aix.calpoly.edu> (Andrew Singer) wrote:

> .. I just pick up the book "Darwin on Trial" by Phillip Johnson
>
> ... Let me take a quick excerpt from the book from which I have a few
> questions to throw out:
> "..the mathematician D.S. Ulam argued that it was highly
> improbable that the eye could have evolved by the accumulation of small
> mutations, because the number of mutations would have to be so large and
> the time available was not nearly long enough for them to appear..."
>

This quotation seems to be typical of the general quality of scholarship
Johnson displays in his book. In the first place, he gets the name wrong.
Stanislaw Ulam (or S.M. Ulam) was a well-known and highly respected
mathematician. He was also the only person with that surname to have
presented a paper at the meeting which Johnson was referring to. In the
second place, Johnson's claim, "...Ulam argued that it was highly improbable
that the eye could have evolved by the accumulation of small mutations...",
is simply false; Ulam did no such thing.

Near the beginning of his paper, Ulam explicitly stated that the mathematical
models he was going to present were "certainly ... not correct in a realistic
sense", to use his own words. His main aim was to present some models which
might serve as a starting point for the development of better ones, and to
challenge the biologists present to find ways of determining the values of
various parameters that would be needed for any such models to be useful.
His remarks indicated that it would be a long time, in his opinion, before
these models would ever become sufficiently accurate to be of any practical
use.

> Now I have a ton of questions stemming from this one sentence. I hope
> someone out there can help.
>
> 1- How does someone determine how long it would take for a series of
> mutations to occur (all being presumably favorable mutations)?
>
In the paper which Johnson was referring to, Ulam outlined some ideas for
models aimed at answering this question for 3 forms of reproduction:
asexual, sexual with random mating, and sexual with recognition of
favourable phenotypes and preferential mating between them. Of the last
two, he presented very few specific details. Even for his first model,
he did not give enough details for anyone reading the paper to reconstruct
it exactly. However, he did give enough to provide a pretty fair idea of
how it worked. Some of its main features were:

1. Fixed population size, N
2. Fixed time interval t between generations.
3. A number F of successive favourable mutations is needed for the organism
to acquire a certain novel characteristic.
4. With respect to this series of favourable mutations, each individual has
a fixed probability 1-p of being born with exactly the same "genetic
complement" as its parent and a probability p of being born with the
favourable mutation which is next in the series from the one which its
parent carried.
5. An organism with a favourable mutation produces an (average?) number
of offspring which is a fixed multiple 1 + g of the (average?) number
produced by one without it.

By way of illustration, Ulam took N = 10^11, t = 1 day, F = 10^6
p = 10^(-10) and g = 10^(-6). Without providing details of his calculation,
he stated that with these parameters the model would give 10^(13) as the
number of generations required for most of the population to acquire the
novel characteristic in question. At 1 generation per day, this would amount
to some 27 billion years, so it is presumably this calculation which Johnson
is relying upon to justify his claim about Ulam's argument. However, Ulam
himself drew _no_ factual conclusions whatsoever from his calculations,
either about the probability of the evolution of the eye, or about any other
aspect of evolution.

>
> 2- How accurate should one assume these calculations are?
>

As I indicated above, Ulam explicitly admitted that his models were
unrealistic. He also noted that the values of most of the parameters he
used in his calculations were unknown by up to several of orders
of magnitude, and could well have been off by that amount.

--------------------------------------------------------------------
David Wilson
david@defcen.gov.au