Conformational entropy limits the transition from nucleation to elongation in amyloid aggregation

Abstract

The formation of β-sheet-rich amyloid fibrils in Alzheimer's disease and other neurodegenerative disorders is limited by a slow nucleation event. To understand the initial formation of β-sheets from disordered peptides, we used all-atom simulations to parameterize a lattice model that treats each amino acid as a binary variable with β- and non-β-sheet states. We show that translational and conformational entropy give the nascent β-sheet an anisotropic surface tension that can be used to describe the nucleus with 2D classical nucleation theory. Since translational entropy depends on concentration, the aspect ratio of the critical β-sheet changes with protein concentration. Our model explains the transition from the nucleation phase to elongation as the point where the β-sheet core becomes large enough to overcome the conformational entropy cost to straighten the terminal molecule. At this point the β-strands in the nucleus spontaneously elongate, which results in a larger binding surface to capture new molecules. These results suggest that nucleation is relatively insensitive to sequence differences in coaggregation experiments because the nucleus only involves a small portion of the peptide.

Figure 1

MD snapshots from the sampling of the conversion between β- and non-β-sheet states. (a) Kinetic parameters for postnucleation (“strong”) bonds are sampled from the terminal molecules on an established cluster (side chains are not shown for clarity). (b) Prenucleation (“weak”) bonds are sampled using a dimer that is harmonically restrained at the central amino acids. (c) Ribbon representation of the β-sheet bilayer simulation showing the stacking of the two sheets. The six core molecules are restrained in β conformation. The top molecule in the white sheet and the bottom molecule in the magenta sheet are free to sample β- and non-β-sheet states. This side view shows the staggered alignment of β-strands in which the white strands are positioned between the magenta strands in the opposite sheet. This staggering ensures that each molecule addition results in an equivalent set of H-bond and side-chain interactions. (d) Atomistic view of the bilayer showing the interdigitation of side chains between the two β-sheets. To see this figure in color, go online.

Figure 2

Schematic of the mapping between the lattice and all-atom models shown in three representations: (a) lattice, (b) ball-and-stick, and (c) atomistic (side chains are hidden for clarity). The lattice has a width given by the number of amino acids per molecule and a height given by the current number of molecules in the cluster. Lattice sites (represented by vertical lines) can be occupied, representing an H-bond between connected amino acids, or empty, indicating that at least one of the adjacent amino acids is in the random coil state. The alternating direction of H-bonds (seen in the atomistic view (c) means that only every-other site can have a bond (green circles in (a)). To see this figure in color, go online.

Figure 3

The time distributions of the unbound (a) and bound state (b) of H-bond pairs in the hexamer. Blue points indicate the H-bond pair immediately adjacent to the restrained bonds (red dashes in (c)), whereas orange points indicate the second pair from the restraints, as indicated by the highlighted squares in (c). Solid lines show exponential fits. (d) Average H-bond pair transition times associated with three different force fields. The open and closed markers represent the times of unbound and bound state of the H-bond pair, respectively. Red indicates strong bonds and green indicates weak bonds. Error bars indicate one standard deviation. To see this figure in color, go online.

Figure 4

Probability of successful nucleation trajectories with parameters from the AMBER14SB force field as a function of the number of molecules and the total number of H-bonds in the cluster. Increasing concentration shifts the transition surface (50%, red dotted line) to smaller clusters and reduces the need for β structure due to increased monomer deposition rate. Red dots indicate the free energy saddle points calculated from Eqs. 7 and 8. To see this figure in color, go online.

Figure 5

(a) Solution concentration determines the shape of the β-sheet core in critical nuclei as seen by the number of H-bonds in transition clusters (defined by the 50% committor). Line shows fit to Eq. 6. Low concentration nuclei (inset left) have extensive β structure, whereas higher concentration nuclei (inset right) have shorter β-strands. (b) The number of H-bonds per β-strand in the transition cluster decreases with concentration. Line shows Eq. 8. Inset: the ratio $N^{‡}/{\ell }^{‡}$ is nearly independent of concentration, consistent with the prediction of Eq. 7 that the number of molecules in the critical cluster does not vary with concentration. Error bars indicate one standard deviation. To see this figure in color, go online.

Figure 6

Contour plot of cluster free energy, Eq. 9, as a function of the number of molecules, M, and the number of intermolecular H-bonds, N. Increasing the concentration lowers the overall free energy, decreases the slope of the lowest free energy path (dotted lines) and shifts the free energy saddle point (red dots) to lower values of N. To see this figure in color, go online.