Helical modulations of the electric potential were observed in several devices during application of resonant magnetic perturbations (RMPs). To address the implication of the helical modulation on RMP-induced transport, we derive a system of 1D equations for zonal flows (ZFs) and helical potential in the presence of RMPs. As ZFs are turbulence-driven, turbulence plays a major role in this plasma self-organization towards a quasi-equilibrium with 3D helical potential. The model reveals how RMPs modify an initially given a saturated-state of coexisting turbulence and ZFs. It is shown that RMPs trigger a transport bifurcation by allowing energy-transfer out of turbulence-driven ZFs into ZF-driven helical potential.