The system (PTTRS) formed by typical 2-Cys peroxiredoxins (Prx), thioredoxin (Trx), Trx reductase (TrxR), and sulfiredoxin (Srx) is central in antioxidant protection and redox signaling in the cytoplasm of eukaryotic cells. Understanding how the PTTRS integrates these functions requires tracing phenotypes to molecular properties, which is non-trivial. Here we analyze this problem based on a model that captures the PTTRS’ conserved features. We have mapped the conditions that generate each distinct response to H₂O₂ supply rates (v_sup), and estimated the parameters for thirteen human cell types and for Saccharomyces cerevisiae. The resulting composition-to-phenotype map yielded the following experimentally testable predictions. The PTTRS permits many distinct responses including ultra-sensitivity and hysteresis. However, nearly all tumor cell lines showed a similar response characterized by limited Trx-S^- depletion and a substantial but self-limited gradual accumulation of hyperoxidized Prx at high v_sup. This similarity ensues from strong correlations between the TrxR, Srx and Prx activities over cell lines, which contribute to maintain the Prx-SS reduction capacity in slight excess over the maximal steady state Prx-SS production. In turn, in erythrocytes, hepatocytes and HepG2 cells high v_sup depletes Trx-S^- and oxidizes Prx mainly to Prx-SS. In all nucleated human cells the Prx-SS reduction capacity defined a threshold separating two different regimes. At sub-threshold v_sup the cytoplasmic H₂O₂ concentration is determined by Prx, nM-range and spatially localized, whereas at supra-threshold v_sup it is determined by much less active alternative sinks and μM-range throughout the cytoplasm. The yeast shows a distinct response where the Prx Tsa1 accumulates in sulfenate form at high v_sup. This is mainly due to an exceptional stability of Tsa1's sulfenate. The implications of these findings for thiol redox regulation and cell physiology are discussed. All estimates were thoroughly documented and provided, together with analytical approximations for system properties, as a resource for quantitative redox biology.