On generating secondary helical structures from polypeptide-solvent interactions
Proteins are known to carry out their biological functions only when they are folded into specific three-dimensional shapes or conformations. How does the protein know what shape it must assume? We analytically investigate this problem by noting that a newly synthesized protein starts as an unfolded linear chain of amino acids. When the chain diffuses through a crowded cell and folds into a specific conformation, each amino acid in the chain interacts differently with the aqueous medium since it may be hydrophobic or hydrophilic, positively or negatively charged, among other properties.
We approach this protein folding problem by viewing various protein conformations as arrays of diffusion paths. The solution of the Fokker-Planck equation for this diffusion process is considered where the conditional probability density, P(r1,L; r0,0), is expressed as a path integral which sums over all histories, or all possible diffusion paths with endpoints at r0 and r1. The l corresponds to the length of a monomer or repeating unit in the chain, where a biopolymer consisting of N monomers has a total length, L = Nl.
Since amino acid-solvent interaction could vary along the length of a protein, we allow the drift coefficient to be length dependent. In particular, we look at secondary structures of multiple helical-linear segments in the chain using a winding probability function obtained for Besselian drift coefficients. The probability function, P(r1,L; r0,0), can be evaluated for several choices of the length-dependent drift coefficient. Once the probability function has been evaluated, one could proceed to calculate the probabilities for various winding conformations of a biopolymer, W(n) = Pn / P(L), where n is a winding number. Applications are explored for systems where the formation of helices between linear segments can be mapped to the conformations of actual proteins.