Protein synthesis by RNA can be represented by the stochastic system

$$\begin{array}{lll} x_{1} \stackrel{\lambda_{1}}{\longrightarrow} x_{1}+1 & \text { and } & x_{1} \stackrel{\beta_{1} x_{1}}{\longrightarrow} x_{1}-1 \\ x_{2} \stackrel{\lambda_{2} x_{1}}{\longrightarrow} x_{2}+1 & \text { and } & x_{2} \stackrel{\beta_{2} x_{2}}{\longrightarrow} x_{2}-1 \end{array}$$

in which $x_{1}$ is an environmental variable corresponding to the number of RNA molecules per cell and $x_{2}$ is a system variable, with birth rate proportional to $x_{1}$, corresponding to the number of protein molecules.

(a) Use the normalized stationary Fluctuation-Dissipation Theorem (FDT) to calculate the (exact) normalized stationary variances $\eta_{11}=\sigma_{1}^{2} /<x_{1}>^{2}$ and $\eta_{22}=$ $\sigma_{2}^{2} /<x_{2}>^{2}$ in terms of the averages $<x_{1}>$ and $<x_{2}>$.

(b) Separate $\eta_{22}$ into an intrinsic and an extrinsic term by considering the limits when $x_{1}$ does not fluctuate (intrinsic), and when $x_{2}$ responds deterministically to changes in $x_{1}$ (extrinsic). Explain how the extrinsic term represents the magnitude of environmental fluctuations and time-averaging.

(c) Assume now that the birth rate of $x_{2}$ is changed from the "constitutive" mechanism $\lambda_{2} x_{1}$ in (1) to a "negative feedback" mechanism $\lambda_{2} x_{1} f\left(x_{2}\right)$, where $f$ is a monotonically decreasing function of $x_{2}$. Use the stationary FDT to approximate $\eta_{22}$ in terms of $h=\left|\partial \ln f / \partial \ln x_{2}\right|$. Apply your answer to the case $f\left(x_{2}\right)=k / x_{2}$.

[Hint: To reduce the algebra introduce the elasticity $H_{22}=\partial \ln \left(R_{2}^{-} / R_{2}^{+}\right) / \partial \ln x_{2}$, where $R_{2}^{-}$and $R_{2}^{+}$are the death and birth rates of $x_{2}$ respectively.]

(d) Explain the extrinsic term for the negative feedback system in terms of environmental fluctuations, time-averaging, and static susceptibility.

(e) Explain why the FDT is exact for the constitutive system but approximate for the feedback system. When, generally speaking, does the FDT approximation work well?

(f) Consider the following three experimental observations: (i) Large changes in $\lambda_{2}$ have no effect on $\eta_{22}$; (ii) When $x_{2}$ is perturbed by $1 \%$ from its stationary average, perturbations are corrected more rapidly in the feedback system than in the constitutive system; (iii) The feedback system displays lower values $\eta_{22}$ than the constitutive system.

What does (i) imply about the relative importance of the noise terms? Can (ii) be directly explained by (iii), i.e., does rapid adjustment reduce noise? Justify your answers.