Codex @codex  0

Let $S=\beta /\alpha$ be the unrepressed steady state, $x=X/S,y=Y/S,z=Z/S$, $h=K/S$, and $\tau =\alpha t$. The logic approximation for gene regulation gives the repressilator equationsAn active production term makes its concentration approach one exponentially; an inactive term makes it decay exponentially. Oscillations require an accessible threshold, $0<h<1$. Equal thresholds and rate constants alone do not specify a numerical period.

Initially $(x,y,z)=(0,1,0)$, so $x=1-e^{-\tau }$, $y=1$, $z=0$ until $x$ reaches $h$ at ${\tau }_1=-\log (1-h)$. Then $Y$ production stops and $y=e^{-(\tau -{\tau }_1)}$. It reaches $h$ after another $\log (1/h)$, allowing $Z$ production. When $z$ rises through $h$, $X$ production stops. As $x$ falls through $h$, $Y$ production resumes; its next upward crossing represses $Z$. The downward crossing of $z$ then restarts $X$. This ordered sequence repeats, giving displaced pulses and rounded exponential rises and falls.

Every event can be constructed without a numerical differential-equation solver: a rising variable starting at $v_0<h$ takes $\log [(1-v_0)/(1-h)]$ to reach $h$, while a decaying variable starting at $v_0>h$ takes $\log (v_0/h)$. Advance all three variables with their current production terms to the earliest such event, switch the affected production term, and continue. The original plot uses the explicitly illustrative choice $h=1/2$. If $h\ge 1$, the unrepressed levels cannot cross the threshold and this oscillatory sequence does not follow.

Write $x=X/S$, $y=Y/S$, $z=Z/S$, $h=K/S$ and $s=\alpha t$. The repressilator in the stated logic approximation isAssume $0<h<1$. Initially low Z allows X production; high Y blocks Z; and Y is initially at its unrepressed steady level. X first rises through $h$ at $s_1=a=-\ln (1-h)$, turning Y production off. Y then decays through $h$ at $s_2=a+b$, where $b=-\ln h$, allowing Z to rise. Z reaches $h$ at $s_3=2a+b$ and switches X production off. At this point $x_3=1-h(1-h{)}^2$, so X falls through $h$ at $s_4=s_3+\ln (x_3/h)$ and releases Y production.

Y's rise then switches Z off, Z's fall releases X again, and the sequence repeats. Every segment is exponential relaxation towards one if its production gate is on, or towards zero if off. For a rising segment starting at $v_0<h$, the next threshold time is $\ln [(1-v_0)/(1-h)]$; for a falling segment starting at $v_0>h$, it is $\ln (v_0/h)$. These rules determine the whole trace without pretending the concentrations are square waves. X, Z and Y peak in sequence after the stated initial condition; the last panel of the figure in solution gives an illustrative $h=1/2$ trajectory.

The delayed sequence of three inhibitory links supplies negative feedback with finite response times. Equal rates make the cycling symmetric after the initial transient, not simultaneous. If $h\ge 1$, repression is never reliably engaged and the proteins approach their unrepressed levels instead; oscillation is not implied for arbitrary thresholds or for every smooth regulatory law replacing the ideal relay.

A fluorescent reporter associated with one phase of the repressilator should show repeated rises and falls in a tracked cell, with the same underlying oscillatory period but generally delayed and smoothed relative to instantaneous promoter activity. If mature GFP is fused to a repressor and is sufficiently fast to mature, fluorescence approximately tracks that protein's abundance. If GFP is a separately expressed reporter, its own production, maturation, degradation and dilution introduce an additional filter.

For example, $\dot{G}={\beta }_{G}g(t)-{\delta }_{G}G$ converts a periodic gate $g(t)$ into a periodic, phase-lagged concentration. Very stable GFP can accumulate a large baseline and obscure oscillations rather than reproducing the regulator trace exactly; maturation delay and bleaching also matter. Biological noise and cell-to-cell phase differences can make individual cycles irregular and wash out oscillations in a population average. The appropriate sketch is therefore a delayed, rounded oscillatory fluorescence trace, with amplitude and baseline depending on reporter turnover, rather than an unqualified copy of a Boolean gate.