  The tryptophan (trp) operon in E. coli codes for the proteins responsible for
the synthesis of the amino acid tryptophan from chorismic acid, and has been
one of the most well-studied gene networks since its discovery in the 1960s.
The tryptophanase (tna) operon codes for proteins needed to transport and
metabolize it. Both of these have been modeled individually with delay
differential equations under the assumption of mass-action kinetics. Recent
work has provided strong evidence for bistable behavior of the tna operon. The
authors of (Orozco, 2019) identified a medium range of tryptophan in which the
system has two stable steady-states, and they reproduced these experimentally.
In this paper, we will show how a Boolean model can capture this bistability.
We will also develop and analyze a Boolean model of the trp operon. Finally, we
will combine these two to create a single Boolean model of the transport,
synthesis, and metabolism of tryptophan. In this amalgamated model, the
bistability disappears, presumably reflecting the ability of the trp operon to
produce tryptophan and drive the system toward homeostasis. All of these models
have longer attractors that we call "artifacts of synchrony", which disappear
in the asynchronous automata. This curiously matches the behavior of a recent
Boolean model of the arabinose operon in E. coli, and we discuss some
open-ended questions that arise along these lines.
