Background: So far, little is known about the molecular mechanisms of amyotrophic lateral sclerosis onset and progression caused by SOD1 mutations. One of the hypotheses is based on SOD1 misfolding resulting from mutations and subsequent deposition of its cytotoxic aggregates. This hypothesis is complicated by the fact that known SOD1 mutations of similar clinical effect could be distributed over the whole protein structure. Results: In this work, a measure of hydrogen bond stability in conformational states was studied with elastic network analysis of 35 SOD1 mutants. Twenty-eight hydrogen bonds were detected in nine of 35 mutants with their stability being significantly different from that with the wild-type. These hydrogen bonds were formed by the amino acid residues known from the literature to be located in contact between SOD1 aggregates. Additionally, residues disposed between copper binding sites of both protein subunits were found from the models to form a stiff core, which can be involved in mechanical impulse transduction between these active centres. Conclusions: The modelling highlights that both stability of the copper binding site and stability of the dimer can play an important role in ALS progression.