Molecular modeling is a promising method for assessing protein structures that is capable of presenting an energetically beneficial protein conformation with atomic precision. This method is of great importance for studying molecular interactions and confirming the pathogenic significance of the changes in protein structures caused by particular mutations. In this study, we used molecular modeling to assess mutations in the SOD1 gene in patients with amyotrophic lateral sclerosis (ALS), a severe neurodegenerative disorder characterized by the loss of spinal and cerebral motor neurons. The product of SOD1 is a cytosolic dimeric enzyme Cu/Zn superoxide dismutase (SOD1) responsible for the detoxification of cellular superoxide radicals. We showed that all eight revealed coding-point mutations of the gene led to moderate or significant changes in SOD1 protein energy. The mutation His49Arg increased protein energy, and the reconstruction of the respective model indicated the spatial destabilization of the molecule and abnormal interactions with the metal ion inside the active center. Conversely, the other seven mutations (Gly17Ala, Leu85Val, Asn87Ser, Asp91Ala, Ser106Leu, Glu134Gly, and Leu145Phe) led to a decrease in protein energy and an increase in the spatial stability of SOD1, which is usually accompanied by an increased tendency for the inert mutant molecule to misfold and demonstrate cellular aggregation. Therefore, the results of the in silico analysis of the SOD1 gene mutations confirms that ALS belongs to the class of the so-called conformational diseases of the central nervous system, a characteristic feature of which is the formation of cytotoxic, insoluble protein inclusions in neurons.