Familial amyotrophic lateral sclerosis caused by mutations in copper-zinc superoxide dismutase (SOD1) is characterized by the presence of SOD1-rich inclusions in spinal cords. It has been shown that a reduced intra-subunit disulfide bridge apo-SOD1 can rapidly initiate fibrillation forming an inter-subunits disulfide under mild, physiologically accessible conditions. Once initiated, elongation can proceed via recruitment of either apo or partially metallated disulfide-intact SOD1 and the presence of copper, but not zinc, ions inhibit fibrillation. We propose a structural model, refined through molecular dynamics simulations, that, taking into account these experimental findings, provides a molecular explanation for the initiation and the elongation of SOD1 fibrils in physiological conditions. The model indicates the occurrence of a new dimeric unit, prone to interact one with the other due to the presence of a wide hydrophobic surface and specific electrostatic interactions. The model has dimensions consistent with the SOD1 fibril size observed through electron microscopy and provides a structural basis for the understanding of SOD1 fibrillation.