Figure 1

(Color online) (a) The dock-and-convert free energy landscape picture of the elongation process proposed in [9, 10, 11, 12, 13]. $S_0$ denotes the initial state, i.e., a free monomer, and $S_5$ denotes the final state with monomer being part of the fibril. (b) The free-energy landscape picture advocated in this work. The schematic underneath the landscape picture depicts one particular conformation in the $S_{2\leftrightarrow 3}$ state trapped due to misalignment. Note that there are three $D$ bonds formed with the fibril and the transparent cloud encloses the dangling end with two unbound $A$ beads (cf. Fig. 2 for more details on the schematic).Reuse & Permissions

Figure 2

(Color online) Schematic pictures depicting the model adopted in this work. (a) Each amino acid is simplistically represented by two beads, the gray beads represent the peptide backbone of an amino acid, the red (dark gray) bead for a hydrophobic side-chain, and the green (light gray) bead for a hydrophilic side chain. We stress that the representation is only meant to be qualitative. (b) The five-amino-acid peptide employed in this work with alternating hydrophilic-hydrophobic side chains. The alternating pattern has been shown to promote amyloid fibril formation [36, 37, 38, 39, 40, 41]. (c) A segment of fibril consisting of four peptides in two layers of cross-beta sheets. (d) A cartoon depicting the four beta strands corresponding to (c). The green (light gray) and red (dark gray) faces of the panel depicts the hydrophilic and hydrophobic) sides, respectively.Reuse & Permissions

Figure 3

(Color online) The bonded interactions for the model. (a) Besides the white bonds between the beads, the orange (gray) bonds are employed to fix the angle between the side chains and the peptide backbone. Note that the side-chain at the end of the peptide, $B_5$, is bonded to $A_4$ in order to maintain the angle $B_5-A_5-A_4$. (b) The six bonding interactions in the $D$ bond between beads $A_k$ and $A_h$. If one of the $A$ beads involved is at the end of the peptide, e.g., $A_h=A_5$, then the cross-peptide $A\text{−}A$ bonds will be between are $A_k-A_5$, $A_{k+1}-A_5$, and $A_k-A_4$ instead.Reuse & Permissions

Figure 4

(Color online) A snapshot of one simulation run in progress. The fibril is put in the middle of the simulation box and the monomer is diffusing toward it. The fibrillar axis is along the $z$ axis, the cross-beta sheets are along the $x$-axis, and the coordinates of the $A_3$ bead for the peptides are $\left(-0.25,0.6,k/2-1\right)$ and $\left(0.25,0.6,k/2-1.25\right)$, $k=0,\dots 4$. The corners of the simulation box are at $x,y,z=\pm 3$.Reuse & Permissions

Figure 5

A segment of the data from the simulations. (a) The original time series consists of the times when the number of $D$ bond is changed. This is then transformed into a time series on the transitions of the set of states $\left\{S\right\}$ (b). The procedure of transformation is described in the text.Reuse & Permissions

Figure 6

The numbers of transitions within the set of states $\left\{S\right\}$. Note the $ij$ entry denotes the number of transitions from state $S_{i-1}$ to $S_{i-1\leftrightarrow j}$.Reuse & Permissions

Figure 7

The transition rates for the set of states $\left\{S\right\}$ constructed from the time series as described in the text. The units are in ${\text{ps}}^{-1}$.Reuse & Permissions

Figure 8

A schematic diagram depicting the scenario where the elongation process is viewed as a diffusion process over a rugged energy landscape in 1D. The linear dimension denotes the “reaction coordinate” and the position $x_0$ indicates the location when the monomer is first bound to the fibril’s end.Reuse & Permissions