  We consider the general problem of sensitive and specific discrimination
between biochemical species. An important instance is immune discrimination
between self and not-self, where it is also observed experimentally that
ligands just below discrimination threshold negatively impact response, a
phenomenon called antagonism. We characterize mathematically the generic
properties of such discrimination, first relating it to biochemical adaptation.
Then, based on basic biochemical rules, we establish that, surprisingly,
antagonism is a generic consequence of any strictly specific discrimination
made independently from ligand concentration. Thus antagonism constitutes a
"phenotypic spandrel": a phenotype existing as a necessary by-product of
another phenotype. We exhibit a simple analytic model of discrimination
displaying antagonism, where antagonism strength is linear in distance from
detection threshold. This contrasts with traditional proofreading based models
where antagonism vanishes far from threshold and thus displays an inverted
hierarchy of antagonism compared to simpler models. The phenotypic spandrel
studied here is expected to structure many decision pathways such as immune
detection mediated by TCRs and FC$\epsilon$RIs, as well as endocrine
signalling/disruption.
