Fig. 2.

Robustness properties of two gene expression circuits (auto-regulated with negative feedback versus constitutively expressed without feedback). a The model of the auto-regulated gene expression circuit. Negative feedback is achieved by a Hill-type function resulting from the multimerization of the protein P into an n-mer P _n , which in turn binds to the active gene G and represses it. Constitutive expression is modeled by an expression rate av that is independent of p. b The relative sensitivities of p ^∗, the steady-state concentration of the protein, to the model parameters in both circuits. The auto-regulated circuit is robust to parameters a and c, in contrast to the constitutively expressed circuit, which is sensitive to both parameters. The auto-regulated gene circuit is, however, sensitive to parameter n. c A graphical explanation of the differences in robustness between both circuits. The intersection of the line and the graph of h(·) in the left figure (auto-regulated circuit) gives p ^∗. Robustness in this circuit is achieved through high-gain and feedback, just as it is in the amplifier circuit. The higher the gain n the more robust the value of p ^∗ will be to variations in the parameters a, c, and b. Indeed it can be shown that \(S_{a}({\boldsymbol {\theta }}) \approx \frac {1}{n+1}\), \(S_{c}({\boldsymbol {\theta }}) \approx \frac {-1}{n+1}\), \(S_{b}({\boldsymbol {\theta }}) \approx \frac {-1}{n+1}\), and \(S_{n}({\boldsymbol {\theta }}) \approx \frac {-n\log p^{\ast }}{n+1}\). In contrast, the constitutively expressed gene circuit lacks robustness to parameters a and c, even though it shares the same protein level p ^∗ as the auto-regulated circuit. See also [20] for a general discussion of sensitivity of biochemical reactions and the effect of feedback