  A central goal of synthetic biology is the design of molecular controllers
that can manipulate the dynamics of intracellular networks in a stable and
accurate manner. To address the fact that detailed knowledge about
intracellular networks is unavailable, integral-feedback controllers (IFCs)
have been put forward for controlling molecular abundances. These controllers
can maintain accuracy in spite of the uncertainties in the controlled networks.
However, this desirable feature is achieved only if stability is also
maintained. In this paper, we show that molecular IFCs can suffer from a
hazardous instability called negative-equilibrium catastrophe (NEC), whereby
all nonnegative equilibria vanish under the action of the controllers, and some
of the molecular abundances blow up. We show that unimolecular IFCs do not
exist due to a NEC. We then derive a family of bimolecular IFCs that are
safeguarded against NECs when uncertain unimolecular networks, with any number
of molecular species, are controlled. However, when IFCs are applied on
uncertain bimolecular (and hence most intracellular) networks, we show that
preventing NECs generally becomes an intractable problem as the number of
interacting molecular species increases.
