Bistable switches that produce all-or-none responses have been
found to regulate a number of natural cellular decision making processes,
and subsequently synthetic switches were designed to exploit their
potential. However, an increasing number of studies, particularly
in the context of cellular differentiation, highlight the existence
of a mixed state that can be explained by tristable switches. The
criterion for designing robust tristable switches still remains to
be understood from the perspective of network topology. To address
such a question, we calculated the robustness of several 2- and 3-component
network motifs, connected via only two positive feedback
loops, in generating tristable signal response curves. By calculating
the effective potential landscape and following its modifications
with the bifurcation parameter, we constructed one-parameter bifurcation
diagrams of these models in a high-throughput manner for a large combinations
of parameters. We report here that introduction of a self-activatory
positive feedback loop, directly or indirectly, into a mutual inhibition
loop leads to generating the most robust tristable response. The high-throughput
approach of our method further allowed us to determine the robustness
of four types of tristable responses that originate from the relative
locations of four bifurcation points. Using the method, we also analyzed
the role of additional mutual inhibition loops in stabilizing the
mixed state.