Simplified Exactly Solvable Model for $\beta$-Amyloid Aggregation

We propose an exactly solvable simplified statistical mechanical model for the thermodynamics of $\beta$-amyloid aggregation, generalizing a well-studied model for protein folding. The monomer concentration is explicitly taken into account as well as a nontrivial dependence on the microscopic degrees of freedom of the single peptide chain, both in the $\alpha$-helix folded isolated state and in the fibrillar one. The phase diagram of the model is studied and compared to the outcome of fibril formation experiments which is qualitatively reproduced.

Figure 1

In-register parallel $\beta$ hairpins in $A\beta 40$ peptide structural model [3] (top) and schematic representation of two interacting monomers in our model (bottom). Dots correspond to amino acids, horizontal and vertical segments are ordered peptide bonds and tilted segments are unstructured ones. Dashed lines represent contacts between the two monomers. See text for details.

Figure 2

Phase diagram of the model in the plane (${ϵ}_{\alpha }$, ${ϵ}_{\beta }$) at $\rho =0.7$ and $\sigma =5$, being $\sigma$ the entropy loss associated to the aggregation. Here $B=10$ corresponding to peptides of 21 monomers. White, light-gray, and gray regions are helix, fibril, and unfolded phases, respectively. Top inset depicts the order parameters $p_{\alpha }$ and $p_{\beta }$ as a function of the TFE concentration $M$ while the bottom one shows the phase diagram in the plane ($M$, $\rho$). See text for details.