Cohesin is a fundamental protein complex that holds sister chromatids together. Separase protease cleaves a cohesin subunit Rad21/SCC1, causing the release of cohesin from DNA to allow chromosome segregation. To understand the functional organization of cohesin, we employed next-generation whole-genome sequencing and identified numerous extragenic suppressors that overcome either inactive separase/Cut1 or defective cohesin in the fission yeast  Schizosaccharomyces pombe  Unexpectedly, Cut1 is dispensable if suppressor mutations cause disorders of interfaces among essential cohesin subunits Psm1/SMC1, Psm3/SMC3, Rad21/SCC1, and Mis4/SCC2, the crystal structures of which suggest physical and functional impairment at the interfaces of Psm1/3 hinge, Psm1 head-Rad21, or Psm3 coiled coil-Rad21. Molecular-dynamics analysis indicates that the intermolecular β-sheets in the cohesin hinge of  cut1  suppressor mutants remain intact, but a large mobility change occurs at the coiled coil bound to the hinge. In contrast, suppressors of  rad21-K1  occur in either the head ATPase domains or the Psm3 coiled coil that interacts with Rad21. Suppressors of  mis4-G1326E  reside in the head of Psm3/1 or the intragenic domain of Mis4. These may restore the binding of cohesin to DNA. Evidence is provided that the head and hinge of SMC subunits are proximal, and that they coordinate to form arched coils that can hold or release DNA by altering the angles made by the arched coiled coils. By combining molecular modeling with suppressor sequence analysis, we propose a cohesin structure designated the "hold-and-release" model, which may be considered as an alternative to the prevailing "ring" model.