  In order to transmit biochemical signals, biological regulatory systems
dissipate energy with concomitant entropy production. Additionally, signaling
often takes place in challenging environmental conditions. In a simple model
regulatory circuit given by an input and a delayed output, we explore the
trade-offs between information transmission and the system's energetic
efficiency. We determine the maximally informative network, given a fixed
amount of entropy production and delayed response, exploring both the case with
and without feedback. We find that feedback allows the circuit to overcome
energy constraints and transmit close to the maximum available information even
in the dissipationless limit. Negative feedback loops, characteristic of shock
responses, are optimal at high dissipation. Close to equilibrium positive
feedback loops, known for their stability, become more informative. Asking how
the signaling network should be constructed to best function in the worst
possible environment, rather than an optimally tuned one or in steady state, we
discover that at large dissipation the same universal motif is optimal in all
of these conditions.
