Model for the Nucleation Mechanism of Protein Folding

Abstract

A nucleation-like pathway of protein folding involves the formation of a cluster containing native residues that grows by including residues from the unfolded part of the protein. This pathway is examined by using a heteropolymer as a protein model. The model heteropolymer consists of hydrophobic and hydrophilic beads with fixed bond lengths and bond angles. The total energy of the heteropolymer is determined by the pairwise repulsive/attractive interactions between nonlinked beads and by the contribution from the dihedral angles involved. The parameters of these interactions can be rigorously defined, unlike the ill-defined surface tension of a cluster of protein residues that constitutes the basis of a previous nucleation model. The main idea underlying the new model consists of averaging the dihedral potential of a selected residue over all possible configurations of all neighboring residues along the protein chain. The resulting average dihedral potential depends on the distance between the selected residue and the cluster center. Its combination with the average pairwise potential of the selected residue and with a confining potential caused by the bonds between the residues leads to an overall potential around the cluster that has a double-well shape. Residues in the inner (closer to the cluster) well are considered as belonging to the folded cluster, whereas those in the outer well are treated as belonging to the unfolded part of the protein. Transitions of residues from the inner well into the outer one and vice versa are considered as elementary emission and absorption events, respectively. The double-well character of the potential well around the cluster allows one to determine the rates of both emission and absorption of residues by the cluster using a first passage time analysis. Once these rates are found as functions of the cluster size, one can develop a self-consistent kinetic theory for the nucleation mechanism of folding of a protein. The model allows one to evaluate the size of the nucleus and the protein folding time. The latter is evaluated as the sum of the times necessary for the first nucleation event to occur and for the nucleus to grow to the maximum size (of the folded protein). Depending on the diffusion coefficients of the native residues in the range from 10(-6) to 10(-8) cm2/s, numerical calculations for a protein of 2500 residues suggest that the folding time ranges from several seconds to several hundreds of seconds.

First passage time analysis of protein folding via nucleation and of barrierless protein denaturation

“…Both the model for the nucleation mechanism of protein folding [16][17][18] and the model for barrierless thermal denaturation [34,36] considered the polypeptide chain of a protein as a heteropolymer chain which was first introduced as a simple model of a protein in the MD and MC simulations of protein folding dynamics [7,14]. The heteropolymer consists of N connected beads which can be thought of as representing the α-carbons of various amino acids.…”

“…We have assumed [16][17][18] it to be spherical so that the confining potential ϕ cp (r) = 0 for r b r cp and ϕ cp (r) = ∞ for r ≥ r cp . The radius of the confining boundary r cp is assumed to be determined [16][17][18] by the total number of residues in the protein, densities of residues in its unfolded and folded parts, and the radius of the cluster (folded part).…”

“…Recently [16][17][18], we have presented a microscopic model for the nucleation mechanism of protein folding which is believed to be one of the possible pathways for the transition of a protein from a compact "amorphous" configuration (into which an initially unfolded protein quickly transforms) to the native state. Such a transition occurs immediately following the formation of a number of tertiary (native) contacts [8,[19][20][21]; once a critical number thereof is attained, the native structure is formed without passing through any detectable intermediates.…”

“…The model involves pairwise "molecular" interactions (i.e., repulsion/attraction), configurational (dihedral angle) potentials in which protein residues are involved, and confining potential which arises because of the finite size of the protein. The protein itself is treated as a heteropolymer chain consisting of three types of beadsneutral, hydrophobic, and hydrophilic [8,16,17,20] (with all the bonds and bond angles equal and constant, see Fig. 1).…”

“…The radius of the cluster is R and the distance from the selected bead 1 to the cluster center is r. of the models is not technically very detailed, because we would like to put the emphasis on the basic underlying ideas and concepts as well as on the predictive power of the models. For more details readers are referred to the original papers [16][17][18]34,36].…”