The probability of an average protein molecule made up of 500 amino acids being arranged in the correct quantity and sequence in addition to the probability of all of the amino acids it contains being only left-handed and being combined with only peptide bonds is "1" over 10^950 (1 followed by 950 zeros).
The Molecular Impasse of Evolution
Talkorigins is SO handy in answering this type of probability related questions.
Here is a experiment you can do yourself: take a coin, flip it four times, write down the results, and then do it again. How many times would you think you had to repeat this procedure (trial) before you get 4 heads in a row?
Now the probability of 4 heads in a row is is (1/2)^4 or 1 chance in 16: do we have to do 16 trials to get 4 heads (HHHH)? No, in successive experiments I got 11, 10, 6, 16, 1, 5, and 3 trials before HHHH turned up. The figure 1 in 16 (or 1 in a million or 1 in 10^40) gives the likelihood of an event in a given trial, but doesn't say where it will occur in a series. You can flip HHHH on your very first trial (I did). Even at 1 chance in 4.29 x 10^40, a self-replicator could have turned up surprisingly early. But there is more.
1 chance in 4.29 x 10^40 is still orgulously, gobsmackingly unlikely; it's hard to cope with this number. Even with the argument above (you could get it on your very first trial) most people would say "surely it would still take more time than the Earth existed to make this replicator by random methods". Not really; in the above examples we were examining sequential trials, as if there was only one protein/DNA/proto-replicator being assembled per trial. In fact there would be billions of simultaneous trials as the billions of building block molecules interacted in the oceans, or on the thousands of kilometers of shorelines that could provide catalytic surfaces or templates
Let's go back to our example with the coins. Say it takes a minute to toss the coins 4 times; to generate HHHH would take on average 8 minutes. Now get 16 friends, each with a coin, to all flip the coin simultaneously 4 times; the average time to generate HHHH is now 1 minute. Now try to flip 6 heads in a row; this has a probability of (1/2)^6 or 1 in 64. This would take half an hour on average, but go out and recruit 64 people, and you can flip it in a minute. If you want to flip a sequence with a chance of 1 in a billion, just recruit the population of China to flip coins for you, you will have that sequence in no time flat.
So, if on our prebiotic earth we have a billion peptides growing simultaneously, that reduces the time taken to generate our replicator significantly.
Okay, you are looking at that number again, 1 chance in 4.29 x 10^40, that's a big number, and although a billion starting molecules is a lot of molecules, could we ever get enough molecules to randomly assemble our first replicator in under half a billion years?
Yes, one kilogram of the amino acid arginine has 2.85 x 10^24 molecules in it (that's well over a billion billion); a tonne of arginine has 2.85 x 10^27 molecules. If you took a semi-trailer load of each amino acid and dumped it into a medium size lake, you would have enough molecules to generate our particular replicator in a few tens of years, given that you can make 55 amino acid long proteins in 1 to 2 weeks.
So how does this shape up with the prebiotic Earth? On the early Earth it is likely that the ocean had a volume of 1 x 10^24 litres. Given an amino acid concentration of 1 x 10^-6 M (a moderately dilute soup, see Chyba and Sagan 1992 ), then there are roughly 1 x 10^50 potential starting chains, so that a fair number of efficent peptide ligases (about 1 x 10^31) could be produced in a under a year, let alone a million years. The synthesis of primitive self-replicators could happen relatively rapidly, even given a probability of 1 chance in 4.29 x 10^40 (and remember, our replicator could be synthesized on the very first trial).
Assume that it takes a week to generate a sequence. Then the Ghadiri ligase could be generated in one week, and any cytochrome C sequence could be generated in a bit over a million years (along with about half of all possible 101 peptide sequences, a large proportion of which will be functional proteins of some sort).
Although I have used the Ghadiri ligase as an example, as I mentioned above the same calculations can be performed for the SunY self replicator, or the Ekland RNA polymerase. I leave this as an exercise for the reader, but the general conclusion (you can make scads of the things in a short time) is the same for these oligonucleotides.
Arnab: By the way, EVOLUTION is not the discipline that deals with the emergence of life(self-replicating molecule) from non-living matters. That discipline is called ABIOGENESIS.
[Edited on 4-8-2003 by Arnab]