  We propose a novel theoretical biomolecular design to implement any Boolean
circuit using the mechanism of DNA strand displacement. The design is scalable:
all species of DNA strands can in principle be mixed and prepared in a single
test tube, rather than requiring separate purification of each species, which
is a barrier to large-scale synthesis. The design is time-responsive: the
concentration of output species changes in response to the concentration of
input species, so that time-varying inputs may be continuously processed. The
design is digital: Boolean values of wires in the circuit are represented as
high or low concentrations of certain species, and we show how to construct a
single-input, single-output signal restoration gate that amplifies the
difference between high and low, which can be distributed to each wire in the
circuit to overcome signal degradation. This means we can achieve a digital
abstraction of the analog values of concentrations. Finally, the design is
energy-efficient: if input species are specified ideally (meaning absolutely 0
concentration of unwanted species), then output species converge to their ideal
concentrations at steady-state, and the system at steady-state is in (dynamic)
equilibrium, meaning that no energy is consumed by irreversible reactions until
the input again changes.
  Drawbacks of our design include the following. If input is provided
non-ideally (small positive concentration of unwanted species), then energy
must be continually expended to maintain correct output concentrations even at
steady-state. In addition, our fuel species - those species that are
permanently consumed in irreversible reactions - are not "generic"; each gate
in the circuit is powered by its own specific type of fuel species. Hence
different circuits must be powered by different types of fuel. Finally, we
require input to be given according to the dual-rail convention, so that an
input of 0 is specified not only by the absence of a certain species, but by
the presence of another. That is, we do not construct a "true NOT gate" that
sets its output to high concentration if and only if its input's concentration
is low. It remains an open problem to design scalable, time-responsive,
digital, energy-efficient molecular circuits that additionally solve one of
these problems, or to prove that some subset of their resolutions are mutually
incompatible.
