  Many instances of cellular signaling and transcriptional regulation involve
switch-like molecular responses to the presence or absence of input ligands. To
understand how these responses come about and how they can be harnessed, we
develop a statistical mechanical model to characterize the types of Boolean
logic that can arise from allosteric molecules following the
Monod-Wyman-Changeux (MWC) model. Building upon previous work, we show how an
allosteric molecule regulated by two inputs can elicit AND, OR, NAND and NOR
responses, but is unable to realize XOR or XNOR gates. Next, we demonstrate the
ability of an MWC molecule to perform ratiometric sensing - a response behavior
where activity depends monotonically on the ratio of ligand concentrations. We
then extend our analysis to more general schemes of combinatorial control
involving either additional binding sites for the two ligands or an additional
third ligand and show how these additions can cause a switch in the logic
behavior of the molecule. Overall, our results demonstrate the wide variety of
control schemes that biological systems can implement using simple mechanisms.
