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Treadmilling protein filaments perform essential cellular functions by growing from one end while shrinking from the other, driven by nucleotide hydrolysis. Bacterial cell division relies on the primitive tubulin homolog FtsZ, a target for antibiotic discovery that assembles into single treadmilling filaments that hydrolyse GTP at an active site formed upon subunit association. We determined high-resolution filament structures of FtsZ from the pathogen Staphylococcus aureus in complex with different nucleotide analogs and cations, including mimetics of the ground and transition states of catalysis. Together with mutational and biochemical analyses, our structures reveal interactions made by the GTP γ-phosphate and Mg2+ at the subunit interface, a K+ ion stabilizing loop T7 for co-catalysis, new roles of key residues at the active site and a nearby crosstalk area, and rearrangements of a dynamic water shell bridging adjacent subunits upon GTP hydrolysis. We propose a mechanistic model that integrates nucleotide hydrolysis signaling with assembly-associated conformational changes and filament treadmilling. Equivalent assembly mechanisms may apply to more complex tubulin and actin cytomotive filaments that share analogous features with FtsZ. Bacterial cell division critically relies on the tubulin homolog FtsZ, which assembles into filaments that treadmill, fuelled by GTP hydrolysis. This structural and biochemical study of FtsZ from Staphylocuccus aureus reveals the mechanism of GTP hydrolysis and its connection with filament dynamics.