Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 13E / d / Solution

Factor the extrinsic term asThe first factor is environmental relative variance. The second is static susceptibility squared: differentiating the equilibrium relation gives $\delta \log m_2=H^{-1}\delta \log m_1$. The last is the time-averaging factor with effective protein response time $1/({\beta }_{2}H)$. Feedback makes response faster, which by itself weakens averaging, but simultaneously reduces the static gain. These are distinct effects and both must be retained.