= Genetic toggle switch stability margin
{title2=$m=1-L$}
A \b[<genetic toggle switch stability margin>] quantifies how far a modeled operating state lies from losing local attraction. There is no single universal margin for every <genetic toggle switch>. A useful dimensionless choice compares the local <mutual repression> gain with the product of the two removal rates; it extends the <mutual repression stability criterion> to unequal rates.
Consider an original deterministic model
$$
\dot x=f(y)-\delta_x x,\qquad
\dot y=g(x)-\delta_y y,
$$
where $\delta_x,\delta_y>0$ and $f',g'<0$ at the chosen <equilibrium>. Its <Jacobian matrix> is
$$
J=\begin{pmatrix}-\delta_x&f'\\g'&-\delta_y\end{pmatrix},
\qquad
\det J=\delta_x\delta_y-f'g'.
$$
The trace is negative. Define the local gain and the chosen normalized margin by
$$
L=\frac{f'g'}{\delta_x\delta_y},\qquad \boxed{m=1-L}.
$$
For this two-variable model, $m>0$ gives negative real <eigenvalues> and local asymptotic stability; $m<0$ gives one positive <eigenvalue>. At $m=0$, a zero <eigenvalue> prevents deciding attraction from the linear terms alone. The negative trace by itself is insufficient.
When both removal rates equal $\delta$, the <eigenvalues> are $-\delta\pm\delta\sqrt L$. The slow return rate is therefore
$$
k_{\mathrm{return}}=\delta(1-\sqrt L),\qquad
\tau=\frac1{k_{\mathrm{return}}}.
$$
For a stipulated $\delta=1\,\mathrm{h^{-1}}$, $L=0.25$ gives margin $0.75$ and <relaxation time> 2 hours, whereas $L=0.81$ gives margin $0.19$ and <relaxation time> 10 hours. Approaching the local threshold produces slow recovery even while the state remains attracting. The calculation concerns small disturbances after the input has returned to its operating value.
The original toggle-switch experiment measured switching and single-cell expression distributions, not this particular normalized margin.<Construction of a genetic toggle switch in Escherichia coli by Timothy S. Gardner and others>[Figures 4–6, persistence, expression distributions and switching times]{ref} An experimental local test would apply small disturbances around one state and estimate the recovery dynamics with an appropriate reporter model. Reporter maturation or measurement averaging can introduce additional timescales. A favorable local margin does not measure the size of a <basin of attraction>, prove global <bistability>, or determine the rate of noise-driven switching.
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