Consider the simplified version of the chemotaxis circuit in Fig. 24.10.

(a) Derive the conditions for the process gain K that ensure that the receptor activity is always reset to zero and even for the case of a persistent ligand signal.
(b) Show that the closed-loop transfer function from the ligand to the receptor activity is equivalent to a first-order transfer function with numerator dynamics.
(c) Comment on the biological relevance of the result in part (b), particularly for a ligand signal that is fluctuating.

yo K -1 -x Figure 24.10 Integral control feedback circuit representation of chemotaxis (Adapted from Yi, T.M., Y. Huang, M. I. Simon, and J. Doyle, Robust Perfect Adaptation in Bacterial Chemotaxis through Integral Feedback Control, Copyright 2000 National Academy of Sciences, U.S.A.").