Low hydrogen peroxide concentrations are essential for eukaryotic cell physiology, but high concentrations trigger an antioxidant response. The redox-dependent activation of specific transcription factors is a critical feature of this response. Deleting these transcription factors increases the sensitivity of cells to hydrogen peroxide stress, but their constitutive activation is also harmful. Curiously, many redox transcription factors require multiple oxidation events for full activation. E. coli OxyR needs four events, mammalian Nrf2-Keap1 requires three, Yap1 in baker’s yeast needs three to four, and Pap1 in S. pombe requires at least two oxidation events. We investigated the purpose of these multiple oxidation events using computational modelling of a basic system and the fission yeast Pap1 system. Our results demonstrated that multiple oxidation steps increased the system’s ability to attenuate signal activation at low hydrogen peroxide concentrations and amplify it at higher peroxide concentrations. This high-pass filtering property, in part, explains how eukaryotic cells can tolerate low hydrogen peroxide levels without triggering an adaptive response.