Past exam of the biological physics course of the University of Oxford / 2018 / 4 / e / Solution

The quadratic-Hill mutual repression switch is explicit: for $n=2$, steady state requires $Y(1+Z^2)=\gamma$ and $Z(1+Y^2)=\gamma$. Subtraction givesThus either $Y=Z=x$, giving the unique positive cubic root $x^3+x-\gamma =0$, or $YZ=1$. In the latter case, $Y+1/Y=\gamma$, henceThe non-symmetric pair is real, positive and distinct exactly when $\gamma >2$. For $0<\gamma <2$ only the symmetric solution exists. Eliminating one variable gives the source's hinted factorization,with no singular denominator for real concentrations. The cubic is strictly increasing since $3Y^2+1>0$, so it contributes only the symmetric state.

One can verify the switch interpretation without a lengthy stability analysis. At symmetry the Jacobian matrix eigenvalues are $-1\pm 2x^2/(1+x^2)$: symmetry is stable for $x<1$ and becomes a saddle for $x>1$, corresponding to $\gamma <2$ and $\gamma >2$. At either asymmetric equilibrium the eigenvalues are $-1\pm 2/\gamma$, both negative when $\gamma >2$. ThereforeAt $\gamma =2$, all three branches meet at $(1,1)$ with a zero linear eigenvalue, the switching threshold rather than two distinct robust memory states. The lower panels of the original figure in solution show representative nullclines on either side.