Figure 2

Analysis of the relaxation oscillator design centered on the set of parameter values automatically determined for the oscillatory phenotype. Results from the automated strategy without specifying values for the parameters (left panels) are compared with results from a previous study¹⁰ based on experimentally measured and estimated values for the parameters (right panels). (a,b) System design space with the effective rate constant for inactivation of the two regulators on the x and y axes. (a) Enumeration of the qualitatively distinct phenotypes identified by color. (b) The number of eigenvalues with positive real part represented as a heat map on the z axis: blue for 0 eigenvalues with positive real part (mono-stability); red for an overlap of Cases consisting of one with 1 and two with 0 eigenvalues having positive real part (bi-stability); yellow for two complex conjugate eigenvalues with positive real part (unstable focus); orange for an overlap of Cases consisting of one with 0, one with 1 and one with 2 eigenvalues having positive real part. The overlaps represented by orange regions correspond to three fixed points: a stable node, an unstable node and an unstable focus; boundaries between orange and yellow regions have the potential for Saddle-Node into Limit Cycle (SNIC) bifurcations that produce transitions between stable steady-state behavior and large-amplitude oscillations.¹⁰ (c) Temporal behavior of repressor concentration X₄ determined by simulation of the full system with parameter values from the automatic strategy (● in left panels) and with experimentally determined values from the previous study (★ in right panels). Note that the values of the parameters on the x and y axes of both panels are near the center of the region of potential oscillation (yellow+orange). The values in the left panel are automatically determined, whereas those in the right panel are manually selected to be near the center of the region of potential oscillation.