A coarse-grained, coupled oscillator toy model to describe low-frequency lattice-like vibrations of β-sheets in amyloid fibrils

We present a coupled-oscillator toy model that describes the low-frequency lattice-like vibrations observed in Raman and inelastic neutron scattering spectra of amyloid fibrils, silks, and other proteins that are rich in β-sheet structures. In this model, the strands of the β-sheets are treated as harmonically oscillating rods with uniform mass density that are coupled together by springs that represent the non-covalent bonds between the strands and sheets. The resulting equations that are derived can be used to determine the frequencies and intensities of longitudinal and transverse lattice modes, including those that derive from β-sheet accordion, buckling, shearing, and breathing motions. Remarkably, the spectra predicted by the model recapitulate key features observed in experimental low-frequency Raman spectra of amyloid fibrils and other protein assemblies with β-sheet structures between ∼5-100 cm-1. Overall, we anticipate that our model provides a relatively simple theoretical framework for understanding the origins of lattice-like modes in proteins and exploring how they spectroscopically report on supramolecular structural features such as β-sheet packing, ordering, and chirality.