Figure 1.
Instances of three-component negative feedback loops with positive feedback in biochemical oscillators and the generic motifs studied in this work.
(A) Some biochemical oscillators utilizing the three-component negative feedback loop motif. Positive feedback loops (in blue) are often found along with the core negative feedback loop (in orange) (B) Oscillatory motifs involving the three-component negative feedback loop (top) with all possible auxiliary positive feedback loops (in blue). The motifs encountered in biochemical oscillators are clearly recognizable. The kinetic model formulation for each motif is also listed with the term representing the positive feedback in blue. The three components of the negative feedback loop are activator , intermediate
and repressor
. The parameter
tunes the strength of the positive feedback, with a value of
representing no positive feedback. (C) Waveform of the Goodwin oscillator with equal degradation in all steps with cooperativity increased from 3 (left) to 10 (right).
Figure 2.
Minimum cooperativity required for oscillations for the core motif versus the positive feedback motifs.
The motifs with positive feedback are grouped according to the type of mechanism: self-activation (SA), Michaelis-Menten (MM) degradation or cross-activation (CA). The component (activator, intermediate or repressor) on which the positive feedback is acting is also indicated by color. Each point represents one choice of parameters for the motif and the comparison is made with common parameters having identical values. In the scatter plots, points below the y = x line represent cases where the positive feedback motif oscillates at a smaller cooperativity than the core motif. The data in the three scatter plots is summarized in the bar graph, i.e., the percentage of parameter sets where positive feedback motifs oscillate with smaller cooperativity than the core motif.
Figure 3.
Reduction in the required cooperativity among positive feedbacks within each class.
For each class, the additional reduction achieved in the minimum cooperativity by placing the positive feedback on the fastest step (largest degradation rate or smallest half-life) in the three-component motif, relative to positive feedback elsewhere, is shown as a histogram for the same data in Figure 2. For the class CA, this translates to a positive feedback between the two fastest steps in the core motif. For each choice of parameter values, the motif that happens to have the positive feedback on the fastest step (or between the fastest steps) is compared against the remaining two motifs within each class and labeled with name of the former. We compare different motifs keeping common parameters at identical values using the color scheme used in Figure 2.
Figure 4.
Period of oscillations for the core motif versus the positive feedback motifs.
The ratio of the period of the positive feedback motif and period of the core motif for different choices of parameter values categorized by class: SA, MM and CA.
Figure 5.
Cumulative effect of two positive feedback loops on the core motif.
The percentage of random parameter choices for which the cooperativity was improved by adding a second positive feedback to the nine positive feedback motifs in Figure 1. The original positive feedback motifs (shown on the y-axis) were modified by adding a second positive feedback different from the first, i.e., each of the other eight.
Figure 6.
Components of the mammalian circadian oscillator.
Interactions within the clock network have been reduced in order to make simple motifs from Figure 1 easy to identify. We use the same color for positive and negative interactions as in Figure 1, i.e., orange for the negative feedback loop and blue for the positive feedbacks.