Simple model characterization and experimental results of the two-switch negative-feedback oscillator (Design I). The phase diagram of dynamic behaviors shows two domains with respect to changes in α and β (unitless parameters proportional to [T12^{tot}] and [T21^{tot}]) versus *n* and *m* (apparent Hill exponents of the activation and inhibition functions). (a, b) Phase portraits using unitless dynamic variables proportional to [rA1] and [rI2], *x* and *y*. The nullclines for *x* (violet lines) and *y* (black lines) are superimposed with temporal trajectories (cyan lines). There are fixed points where *x* and *y* nullclines intersect; stable (gray circle) and unstable (red circle) fixed points are marked. The system exhibits (a) limit cycle oscillation or (b) damped oscillation to a monostable steady state. Parameters are [T12^{tot}]=[T21^{tot}]=[A1^{tot}]=100 nM, *K*_{A}=*K*_{I}=1 μM, *n=m*=5, τ=500 s, γ=1, *k*_{d}=0.002/s, and respectively *k*_{p}=0.04/s or *k*_{p}=0.025/s. In experiments, normalized fluorescence time courses measured the percent OFF-state switch Sw12 (TAMRA-labeled T12, green) and the percent OFF-state switch Sw21 (Texas Red-labeled T21, red). Two examples are shown: reaction #8 with stable oscillations and reaction #13 with strongly damped oscillations. Experimental parameters were, respectively, ([T21^{tot}], [A1^{tot}], [dI1^{tot}],[T12^{tot}], [A2^{tot}], [RNAP^{tot}], [RNaseH^{tot}])=(250, 250, 700, 120, 350, 92, 8.3) nM and (250, 250, 1000, 80, 500, 125, 15.0) nM. Reaction #13 used higher [dI1^{tot}], higher [A2^{tot}], and lower [T12^{tot}], compared with reaction #8. Thus, reaction #13 is mapped to a higher *n* and a lower *α* than reaction #8, and lay within the monostable domain rather than the oscillating domain. Experimental results () were mapped to the phase diagram with respect to a reference oscillation, as discussed in section 1.1, and shown as ‘stable' (circles, damping coefficient <0.15/h), ‘damped' (dots, damping coefficient between 0.15/h and 0.5/h), and ‘strongly damped or too slow to measure' (crosses, damping coefficient >0.5/h) oscillations.

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