  We investigate weakly coupled quarter-filled ladders with model parameters
relevant for NaV_2O_5 using density-matrix renormalization group calculations
on an extended Hubbard model coupled to the lattice. NaV2O5 exhibits
super-antiferroelectric charge order with a zigzag pattern on each ladder. We
show that this order, with a periodicity of four ladders, causes a spin
dimerization along the ladder and a corresponding spin gap of the same
magnitude as that observed experimentally. The spin gap is destroyed again at
large charge order due to a restructuring of the spins. An analysis of an
effective spin model predicts a re-creation of the gap by inter-ladder singlets
when the charge order increases further.
