Molecular constructs and dual recognition constructs having a sequence encoding a TCR affinity weakening motif, and DNA and RNA sequences corresponding thereto, are presented. Modified T-cells and other cells transformed with the molecular contracts express a modified TCR that imparts a reduction, in non-specific binding, -an enhancement of binding specificity and an enhancement of binding affinity for a target antigen, compared to non-transformed (wild-type, native) T-cells, are described. The modified TCRs possess an affinity enhancing motif and an affinity weakening motif. Methods of transforming cells and methods of using enriched populations of transformed cells, in the treatment of cancer and infections arid T-cell mediated pathologies are provided. The affinity weakening motif imparts a weakened interaction, of a TCR with major histocompatibility complex proteins, such as HLA proteins in humans. Soluble modified TCRs are also provided, Therapeutic preparations comprising modified T-cells, modified TCRs, and modified TCR-therapeutic agent-conjugates, are also provided.