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Imagine that there was only a single bond between each amino acid in a protein of 101 amino acid residues. Imagine that there were only three possible configurations around each of those bonds. This means that the protein could adopt 3100, or 5 × 1047 different conformations.
If the protein is able to sample 1013 different bond configurations per second then it would take 1027 years to sample all possible conformations of the protein (Zwanzig et al. 1992). This is quite a long time. Far longer, in fact, than the age of the universe.
Small proteins usually fold spontaneously within seconds and even the largest proteins fold within minutes. The difference between the theoretical calculation and the observed result is known as Levinthal's Paradox.
It isn't really a paradox. Levinthal knew full well that proteins did not fold by sampling all possible conformations. He knew that protein folding involved local cooperative interactions such as formation of α helices and that the formation of such secondary structure elements proceeded in parallel and not sequentially as his thought experiment proposed.
We now know that protein folding is largely driven by hydrophobic collapse as the regions of secondary structure come together to exclude water. This process is a global process involving simultaneous rearrangements of hundreds of bonds at the same time. That's why proteins fold so rapidly. Cyrus Levinthal knew this.
The point of Levinthal's paradox is to demonstrate that when a mathematical calculation shows that some routine process is impossible, then it's the calculation that's wrong, or the assumptions behind the calculation. This point is lost on most Intelligent Design Creationists. They are tremendously fond of complex calculations proving that some biological process is impossible. To them, this is not proof that their calculations are flawed—it's proof that a miracle occurred.
Levinthal, C. (1969) How to Fold Graciously. Mossbauer Spectroscopy in Biological Systems: Proceedings of a meeting held at Allerton House, Monticello, Illinois. J.T.P. DeBrunner and E. Munck eds., University of Illinois Press Pages 22-24 [complete text]
Zwanzig, R., Szabo, A. and Bagchi, B. (1992) Levinthal's paradox. PNAS 89:20-22. [PNAS]