Pseudo-one-dimensional nucleation in dilute polymer solutions

Abstract

Pathogenic protein fibrils have been shown in vitro to have nucleation-dependent kinetics despite the fact that one-dimensional structures do not have the size-dependent surface energy responsible for the lag time in classical theory. We present a theory showing that the conformational entropy of the peptide chains creates a free-energy barrier that is analogous to the translational entropy barrier in higher dimensions. We find that the dynamics of polymer rearrangement make it very unlikely for nucleation to succeed along the lowest free-energy trajectory, meaning that most of the nucleation flux avoids the free-energy saddle point. We use these results to construct a three-dimensional model for amyloid nucleation that accounts for conformational entropy, backbone H bonds, and side-chain interactions to compute nucleation rates as a function of concentration.

FIG. 1

Cartoon of a nucleating trimer illustrating the two-dimensional reaction coordinate used in the theory. A configuration of the trimer is mapped to the x–y plane where the x coordinate is the number of H-bonds between the middle and lower molecules and y is the number of H-bonds between the top and middle molecules. The formation and breakage of H-bonds results in a random walk in this 2D space. The unequal free energies of H-bond formation mean that the random walk is biased by drift velocities that push the system toward the x = y diagonal.

FIG. 2

(a) Probability of a trimer containing x and y intermolecular bonds to proceed to the fully bound state before either terminal molecule unbinds. (b) Grand free energy (in units of k_BT) of a trimer as a function of the number of intermolecular bonds. The increase in free energy when x or y increase from 0 to 1 arises from the loss of translational entropy upon molecular binding. This jump, given by k_BT ln C₁, is arbitrarily set to 2k_BT for convenient visualization. There is a pronounced trough along the x = y diagonal (green line), however, due to the low probability of nucleation when x and y are both small (panel a) most nucleation trajectories (blue arrows) avoid this path. (c) The nucleation flux as a function of the number of bonds in the starting dimer (see Eq. 3). This is proportional to the product of the nucleation probability (panel a) along the y = 1 line and the dimer Boltzmann weight (y = 0 line in panel b). The competition between these terms gives a non-monotonic function that has a peak significantly removed from the lowest free energy pathway (x = 1). This is shown schematically by the blue arrows in the middle panel. L = 10, f_strong = −0.5, f_weak = 0.3 unless noted.

FIG. 3

Schematic of the assembly process described by the 3D model for (a) peptides that form a single β-sheet and (b) peptides that form two β-sheets separated by a hairpin. Each block represents a β strand consisting of x amino acids. Disordered peptide segments have been omitted for clarity. The two β-sheets of the nascent fibril are color coded so red/blue (light grey/dark grey) interfaces represent sidechain steric zipper interactions while red/red and blue/blue interfaces represent backbone H-bond interactions.

FIG. 4

Nucleation rate of hairpin and single strand molecules. Molecules that form hairpin structures nucleate much faster than molecules that only contribute one strand because of the reduced translational entropy penalty to form the nucleus. The curvature at high concentration is due to the saturation of the Zeldovich factor.