  Steady state non-monotonic ("biphasic") dose responses are often observed in
experimental biology, which raises the control-theoretic question of
identifying which possible mechanisms might underlie such behaviors. It is well
known that the presence of an incoherent feedforward loop (IFFL) in a network
may give rise to a non-monotonic response. It has been conjectured that this
condition is also necessary, i.e. that a non-monotonic response implies the
existence of an IFFL. In this paper, we show that this conjecture is false, and
in the process prove a weaker version: that either an IFFL must exist or both a
positive loop and a negative feedback loop must exist. Towards this aim, we
give necessary and sufficient conditions for when minors of a symbolic matrix
have mixed signs. Finally, we study in full generality when a model of immune
T-cell activation could exhibit a steady state non-monotonic dose response.
