Text 857, 205 rader
Skriven 2004-11-18 07:39:00 av Tinyurl.Com/uh3t (1:278/230)
Ärende: Re: Metabolism Forced
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> From: an588@freenet.carleton.ca (Catherine Woodgold)
> I thought in your scenario, the "food" was plentiful, so
> all you needed was one copy of the replicator
> appearing, and then suddenly it would spread exponentially.

All you need is one copy of a catalytic-loop with once-around-loop
fecundity greater than 1 to have a chance of exponential growth. The
more the fecundity exceeds 1, the less chance of dying out before there
are enough copies to assure exponential growth. For example, if the
fecundity is appx. 1.1, there's about 2/3 chance of dying out, 1/3
chance of surviving with exponential growth. Fecundity of 1.1 doesn't
mean absolute assurance of making one copy once-around and 1/10 chance
of making a second. It means a statistical distribution expected,
whereby there might be 0 copies (extinction already) or 1 copy (back to
square one to try again) or 2 copies (now better than even odds at
least one will survive) or 3 copies (very good survival odds, but this
case is rare), hardly ever any more after a single loop, such that the
weighted average of those possibilities equals 1.1.

All you need is a situation where catalytic loops with fecundity
greater than 1 occur every so often, such that those fecundities are
spread over a wide range (because they belong to randomly-generated
molecules that have no particular reason to all have the same
fecundity) so that most of the over-1 are in fact over-1.1, so that
each time one of them occurs there's a decent chance of survival, so
that after a few such tries there's almost certainty of one or another
surviving.

> Also, I thought you were imagining it in such a way that
> only one molecule of the replicator was needed, and it
> would generate all the other molecules in the
> catalytic loop.

Yes, that's the process. But note that any individual molecule isn't
assured of completing the cycle, especially if its fecundity is only a
little more than 1.0. Some links in such a cycle have single-link
fecundity much greater than one, and some much less than one, such that
the factors nearly cancel to yield a product of single-step fecundities
that exceed 1.0 by a little bit. Let me give a hypothetical example:

Suppose about 5 times per megayear a single molecule of C1 is produced,
in a favorable location, by processes that don't involve any of the
other catalysts in the loop, and in such a favorable location the
one-step fecundity from C1 to C2 is 0.5, and from C2 to C3 is 0.7, and
from C3 to C4 is 4.0, and from C4 to C1 is 0.8. Multiplying those four
one-step fecundities we see a once-around fecundity of 1.12, right? So
this catalytic cycle qualifies as a "capable" replicator. With C1 being
formed 5 times per megayear, those C1's form a C2 about 2.5 times per
megayear, and those C2's form a C3 about 1.75 times per megayear, and
those C3's form a C4 about 7 times per megayear, and those C4's form a
C1 about 5.6 times per megayear. (The initial formation of C1 is
randomly-evenly distributed in time (called a "Poisson distribution"),
like geiger counter clicks, but the formations of C4 are clumped
together after a C3 happened to be produced from C2 from C1.) The
bottleneck is C3, the molecular species which occurs least often during
any sincle cycle. About 1.75 times per megayear one C3 gets formed by
the C1 -> C2 -> C3 process, but then C3 almost surely produces some C4,
appx. 4 on the average, usually a bit more or less than 4, hardly ever
zero and not often just 1. That batch of appx. 4 C4's will then produce
appx. 3.2 C1's, which produce appx. 1.6 C2's, which produce appx. 1.12
C3's, and we're back to the bottleneck again where there's a good
chance in fact zero C3's were made and the catalytic dies this time,
but there's also a good chance more than one C3's were made in which
case the next time around there's much less chance of dying out.

So the place to analyze the statistics of that particular catalytic
cycle is not at C1 where the cycle is entered but at C3 where the
bottleneck occurs. About 1.75 times per megayear not only does some
random chemical cascade result in one molecule of C1 being made, but
also that one molecule of C1 gets lucky and makes one molecule of C2,
and not only that but that one molecule of C2 makes one molecule of C3,
at which point there's about a 1/3 chance of exponential growth.
(There's a small chance the one molecule of C1 would make two molecules
of C2, and a moderately-small chance the one molecule of C2 would make
two molecules of C3, so the distribution of C2 and C3 aren't Poisson
but slightly clumpy, but to a first approximation we can assume
production of initial C2 and C3 are Poisson, with only one molecule of
each initially formed at any time.)

So now to directly answer your question: In the example above, with
once-around fecundity just over 1.1, with bottleneck at C3, any time a
molecule of C3 is created., it'll almost surely make several molecules
of C4, which will likely make one or more molecules of C1, which have a
good chance of making a molecule of C2, whereby there's about a 1/3
chance of one or more molecules of C3 being made but with on the
average about 1.12 molecules of C3 made once-around from the original
C3.

Now let's change the example, by doubling the first and last
single-step fecundities, so the overall once-around fecundity is 4
times the original, i.e. 4.48, specifically now have
C1 --(1.0)-> C2 --(0.7)-> C3 --(4.0)-> C4 --(1.6)-> C1
The bottleneck is still at C3. About 5 every megayear a C1 is produced,
and about the same rate but clumped after C1's there are C2's produced,
and about 3.5 times per megayear (clumped again) there's a C3 produced,
but look what happens each time a C3 is produced: Appx. 4 C4's are
produced, which produce appx. 6.4 C1's, which produce appx. 6.4 C2's,
which produce appx. 4.48 C3's, and nowhere in that loop is there any
good chance of the actual number being zero, so almost surely this
single C3 has produced several C3's the first time around the loop,
which produce a whole bunch of C3's the second time around, which will
produce many C3's the third time around. So it's very likely that the
very first time cascade -> C1 -> C2 -> C3 occurs, that one molecule of
C3 will surely survive to obtain exponential growth.

So now to directly answer your question for a catalytic cycle of
fecundity this nicely large: Whichever is the bottleneck molecule, as
soon as just one molecule of it is produced, almost surely it'll
survive on the very first attempt.

> you seem to be saying that the same
> molecules that occur in the replicator would occasionally
> occur, but would be unimportant, before the replicator
> came into existence.

Nothing important happens until the first successful replicator occurs,
i.e. among the capable replicators which occur from time to time, the
very first time such a capable replicator survives to achieve several
copies of the bottleneck molecule instead of dying out. Each time a
capable replicator dies out at the single-molecule level, nothing
important has really happened (no significant change to the ecosystem
has occurred, and this particular species of molecule will leave no
trace of how close it got to surviving), and as time goes on many other
randomly created molecules will get their chance, and most likely it'll
be some other molecular species rather than this one again which avoid
bad luck and succeed at exponential growth.

To directly answer your question: Various capable replicators with
fecundity just a little bit more than 1.0 might occur and quickly died
out, and that's all unimportant, until one random capable replicator
avoids bad luck (or has very high fecundity like second example above
so luck is not so important for it) and becomes successful. Also, often
molecules other than the bottleneck molecule may appear but then die
out before that point i.e. not produce the bottleneck molecule. In fact
with widely varying single-step fecundities, because some catalysts are
very very good with fecundities in the millions given just the
particular reactants we want to use and with fecundities in the
thousands even with a random mix of reactants most of which aren't the
ones that produce the desired product, but with many other catalysts
very weak, like they might have only one chance out of ten of
catalyzing even one desired reaction before they are broken apart, I
would expect the bottle neck in any capable catalytic loop to be more
severe than the first example above. For example:
C1 --(0.07)-> C2 --(0.03)-> C3 --(9742)-> C4 --(0.055)-> C1
with an once-around fecundity of 1.125., with bottleneck at C3. Many
many times C1 would be created but no C2, and of those few times C2 is
created usually C3 is not created. Only those very very rare times when
not only is C1 created but it makes C2 which makes C3, *then* suddenly
thousands of C4 would surely be created making hundreds of C1 making
tens of C2 having good chance of making one or more C3 to continue from
the bottleneck again. If the bottleneck occurs at the very first
catalyst in the loop, the one which is spontaneously made via some
other path, then it's unlikely any of the other catalysts in the loop
would be made except if the loop is going to be successful. But in the
more common case, such as that third example in this paragraph, the
entry point is not at the bottleneck, so the loop gets entered many
times before it makes it to the bottleneck even once.

So to directly answer your question, yes I expect many times
non-bottleneck catalysts in any given cycle will be created at random
before the bottleneck catalyst is ever created. Most capable catalytic
loops would be like that, only a few would be entered precisely at the
bottleneck.

> By "closing the catalytic loop" you mean that the food for the last
> link in the chain began to be available.

No. I'm assuming food is in equilibrium, being created and destroyed at
the same rate, sitting there for tens of millions of years with slowly
fluctuating quantities due to varying amounts of UV or asteroid/comet
crashes or volcanic activity, but basically near-constant during any
period of time when a capable catalytic cycle would by chance occur
once. By "closing the catalytic loop" I mean the chance occurrance of
the bottleneck in the catalytic loop, which is the compound probability
of (1) the entry point occurring at random, and (2) after that,
successful progression from the entry point to the bottleneck. For
example, in that third example, with C1 occurring about 5 times per
megayear, C2 gets formed from it only about 0.35 times per megayear,
and C3 gets formed from C2 only about once every hundred million years,
and about 2/3 of those rare times C3 doesn't survive, while in 1/3 of
those rare times C3 survives to exponential growth. Once the bottleneck
is reached, regardless of whether the replicator survives, I guess we
could say that since a single C3 makes thousands of C4 which make
hundreds of C1, at that point the cycle has been closed. So closing the
loop in that sense gives decent chance of survival even if once-around
fecundity is only 1.1, and almost sure survival if fecundity is much
bigger such as 4.48 in second example.

In summary: Mere entry into the catalytic loop at some point other than
the bottleneck doesn't give good chance of survival at all, but the
rare times the loop is not only entered but reaches the bottleneck,
that's when the loop is sure to make it all the way around to many
copies of the entry-point catalyst (even one copy of that would qualify
for the loop to be "closed" in the sense I defined above), and good
chance of making it to the bottleneck a second time, and when there's a
decent chance of survival to exponential growth.
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