# Levinthal's paradox

Levinthal's paradox is a thought experiment in the field of computational protein structure prediction; protein folding seeks a stable energy configuration. An algorithmic search through all possible conformations to identify the minimum energy configuration (the native state) would take an immense duration; however in reality protein folding happens very quickly, even in the case of the most complex structures, suggesting that the transitions are somehow guided into a stable state through an uneven energy landscape.

## History

In 1969, Cyrus Levinthal noted that, because of the very large number of degrees of freedom in an unfolded polypeptide chain, the molecule has an astronomical number of possible conformations. An estimate of 10$^{300}$ was made in one of his papers (often incorrectly cited as the 1968 paper). For example, a polypeptide of 100 residues will have 200 different phi and psi bond angles, two within each residue. If each of these bond angles can be in one of three stable conformations, the protein may misfold into a maximum of 3$^{200}$ different conformations (including any possible folding redundancy), not even considering the peptide linkages between each residue or the conformations of the side-chains. Therefore, if a protein were to attain its correctly folded configuration by sequentially sampling all the possible conformations, it would require a time longer than the age of the universe to arrive at its correct native conformation. This is true even if conformations are sampled at rapid (nanosecond or picosecond) rates. The "paradox" is that most small proteins fold spontaneously on a millisecond or even microsecond time scale. The solution to this paradox has been established by computational approaches to protein structure prediction.
Levinthal himself was aware that proteins fold spontaneously and on short timescales. He suggested that the paradox can be resolved if "protein folding is sped up and guided by the rapid formation of local interactions which then determine the further folding of the peptide; this suggests local amino acid sequences which form stable interactions and serve as nucleation points in the folding process". Indeed, the protein folding intermediates and the partially folded transition states were experimentally detected, which explains the fast protein folding. This is also described as protein folding directed within funnel-like energy landscapes. Some computational approaches to protein structure prediction have sought to identify and simulate the mechanism of protein folding.
Levinthal also suggested that the native structure might have a higher energy, if the lowest energy was not kinetically accessible. An analogy is a rock tumbling down a hillside that lodges in a gully rather than reaching the base.
Levinthal's paradox was cited on the first page of the Scientific Background to the 2024 Nobel Prize in Chemistry (awarded to David Baker, Demis Hassabis, and John M. Jumper for computational protein design and protein structure prediction) by way of demonstrating the sheer scale of the problem given the astronomical number of permutations.

## Suggested explanations

According to Edward Trifonov and Igor Berezovsky, proteins fold by subunits (modules) of the size of 25–30 amino acids.