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We report on theoretical studies of electronic transport in the archetypical molecular hybrid formed by DNA wrapped around single-walled carbon nanotubes (CNTs). Using a Green's function formalism in a π-orbital tight-binding representation, we investigate the role that spin-orbit interactions play on the CNT in the case of the helicoidal electric field induced by the polar nature of the adsorbed DNA molecule. We find that spin polarization of the current can take place in the absence of magnetic fields, depending strongly on the direction of the wrapping and length of the helicoidal field. These findings open new routes for using CNTs in spintronic devices.