We describe solid state nuclear magnetic resonance (NMR) measurements on fibrils formed by the 40-residue β-amyloid peptide associated with Alzheimer's disease (Aβ ) that place constraints on the identity and symmetry of contacts between in-register, parallel β-sheets in the fibrils. We refer to these contacts as internal and external quaternary contacts, depending on whether they are within a single molecular layer or between molecular layers. The data include: (1) twodimensional 13 C-13 C NMR spectra that indicate internal quaternary contacts between sidechains of L17 and F19 and sidechains of I32, L34, and V36, as well as external quaternary contacts between sidechains of I31 and G37; (2) two-dimensional 15 N-13 C NMR spectra that indicate external quaternary contacts between the sidechain of M35 and the peptide backbone at G33; (3) measurements of magnetic dipole-dipole couplings between the sidechain carboxylate group of D23 and the sidechain amine group of K28 that indicate salt bridge interactions. Isotopic dilution experiments allow us to make distinctions between intramolecular and intermolecular contacts. Based on these data and previously-determined structural constraints from solid state NMR and electron microscopy, we construct full molecular models using restrained molecular dynamics simulations and restrained energy minimization. These models apply to Aβ 1-40 fibrils grown with gentle agitation. We also present evidence for different internal quaternary contacts in Aβ 1-40 fibrils grown without agitation.Amyloid fibrils are filamentous aggregates formed by a wide variety of peptides and proteins and distinguished from other types of protein fibrils by their appearance in electron microscope (EM) images, by their dye-binding properties, and by the presence of cross-β structural motifs within the fibrils (1,2). Amyloid fibrils are likely causative or contributing agents in diseases such as Alzheimer's disease, type 2 diabetes, Parkinson's disease, and transmissible spongiform encephalopathies (3). The formation and transmission of several prions in yeast and fungi is known to be based on the formation of amyloid fibrils by particular proteins (4). Many proteins that are not known to form amyloid fibrils in vivo can also form amyloid fibrils in vitro under conditions that destabilize their unaggregated states (5,6).Determination of the full molecular structures of amyloid fibrils requires unusual experimental approaches, due to their inherent noncrystalline, insoluble nature (2). Full structure determination requires experimental constraints at the primary, secondary, tertiary, and quaternary structural levels (7). As for monomeric peptides and proteins, primary and secondary structures refer to the amino acid sequence and to segments with standard backbone