Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 13E (Mathematical Biology)

Let $m_i=⟨x_i⟩$. The mean equations give $m_1={\lambda }_1/{\beta }_1$ and $m_2={\lambda }_2{m}_1/{\beta }_2$. For relative deviations, the drift Jacobian and normalized reaction-noise covariance areThe normalized stationary fluctuation-dissipation relation for a reaction network is $A\eta +\eta A^T+D=0$ with this negative-drift convention. Its entries successively giveThe birth and death channels act on one species at a time, so the diffusion matrix is diagonal even though the stationary covariance is not.

If RNA concentration does not fluctuate, protein synthesis is an immigration-death process, so its intrinsic normalized variance is $\boxed{1/m_2}$. If protein birth/death noise is suppressed and protein responds deterministically to the fluctuating RNA input, the remaining normalized variance isWriting ${\tau }_i=1/{\beta }_i$, its factor is ${\tau }_1/({\tau }_1+{\tau }_2)$. Slow environmental changes are followed almost fully; rapid environmental fluctuations are averaged over the longer protein response time. Thus extrinsic noise is the magnitude of relative RNA fluctuations multiplied by a dynamical averaging factor.

At the deterministic feedback equilibrium use ${\lambda }_2{m}_{1}f(m_2)={\beta }_2{m}_2$. The indicated elasticity isThe normalized drift becomes $A=\left(\begin{array}{cc}-{\beta }_1& 0\\ {\beta }_2& -{\beta }_{2}H\end{array}\right)$; equal mean birth and death rates still give $D=\mathrm{diag}(2{\beta }_1/m_1,2{\beta }_2/m_2)$. Solving the same covariance equation yields the feedback attenuation of gene-expression noise:For $f(x_2)=k/x_2$, $h=1,H=2$, soThe inverse-feedback expression is interpreted near a positive equilibrium; a microscopic birth rate singular at zero needs a separate regularization there.

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.

For the constitutive system, every propensity is affine in the copy numbers. The exact first- and second-moment equations therefore close: expectations of drift are determined by the means, and covariance drift by the covariance. The stationary fluctuation-dissipation calculation is consequently exact, even though the full joint stationary distribution need not be Gaussian.

For feedback, expectations such as $E[x_{1}f(x_2)]$ involve nonlinear moments and correlations. Replacing them by a linear expansion about the mean is a linear noise approximation, so the covariance formula is approximate. It works well around a stable single equilibrium when relative fluctuations are small and the rates vary smoothly over their typical range, generally at sufficiently large molecule numbers. Low counts, strong nonlinearities, multiple attractors and proximity to instability can invalidate it.

With response rates and elasticity effectively fixed, changing ${\lambda }_2$ changes $m_2$ and hence intrinsic noise but not the environmental term. Observation (i) therefore indicates extrinsic noise dominates, making the mean-dependent intrinsic contribution negligible. For a general feedback law whose elasticity changes substantially with the mean, compensation between terms is another possibility; independence from ${\lambda }_2$ alone would then not logically identify the dominant term.

Observation (ii) measures the local relaxation rate ${\beta }_{2}H$, which is larger under negative feedback. Observation (iii) is not explained by faster adjustment alone. At fixed susceptibility, increasing the response rate increases ${\beta }_{2}H/({\beta }_1+{\beta }_{2}H)$, reducing time averaging and increasing transmitted environmental noise. Feedback lowers variance primarily by reducing susceptibility and the intrinsic term. Thus rapid adjustment and low noise share a feedback mechanism but are not a direct causal equivalence.