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ESCRT-III is a ubiquitous biological nanomachine that reshapes and cuts cell membranes in a wide range of cellular processes, from vesicle formation, HIV and Ebola release, to cell division. Our interdisciplinary team has developed the first computer model that captures the physical mechanisms of how the ESCRT-III nanomachinery might operate. We propose that the change in the ESCRT-III filament geometry produces mechanical forces to drive membrane scission. We show that this mechanism can capture all the experimentally reported cases of ESCRT-III driven membrane sculpting, including the formation of downward and upward membrane cones and tubules, and the fission of cargo-containing vesicles.