We do this by keeping stringent accounts of the nonlinearities in the core three-component motif and identifying cooperativity as the key metric to compare different positive feedbacks. We conjecture that this facilitation of oscillations explains the frequent occurrence of positive feedbacks within these negative feedback systems. We consider the three-component negative feedback loop motif represented by the Goodwin oscillator with various auxiliary loops listed in Figure 1B. This motif is simple enough to allow theoretical analyses of its oscillatory properties. Nevertheless, properties of the motifs for different choices of kinetic parameters require numerical evaluations of analytically-derived conditions. We call the three components in the motif, the activator, intermediate and feedback repressor. The positive feedbacks can be grouped into three classes based on the underlying mechanism: self-activation, Michaelis-Menten degradation and cross-activation. Here, we consider the simplest possible mathematical representations of the motifs to emphasize the generality of our result. However, more complex interactions in systems with Jacobians having patterns of nonzero elements described in the Voxtalisib Analogue Analysis section will also obey our observations. We empirically test the properties of 2000 kinetic rate parameter choices for each motif in Figure 1B. A similar Monte-Carlo approach was used to study the Sodium Butyrate conditions for oscillations in arbitrary metabolic networks in. In order to obtain general conclusions, we compare different motifs holding common kinetic parameters at identical values. Since we use non-dimensionalized models, all metrics presented in this Results section are without units. We first study the minimum degree of cooperativity required to produce oscillations in all motifs in Figure 1B. The degree of cooperativity of a motif is measured by the Hill coefficient in the negative feedback regulation. For each choice of parameters, the positive feedback motifs are compared against the core motif without positive feedback. Generally, addition of any positive feedback loop reduces the required cooperativity to produce oscillations.