Positive Feedback Promotes Oscillations in Negative Feedback Loops

Abstract

A simple three-component negative feedback loop is a recurring motif in biochemical oscillators. This motif oscillates as it has the three necessary ingredients for oscillations: a three-step delay, negative feedback, and nonlinearity in the loop. However, to oscillate, this motif under the common Goodwin formulation requires a high degree of cooperativity (a measure of nonlinearity) in the feedback that is biologically “unlikely.” Moreover, this recurring negative feedback motif is commonly observed augmented by positive feedback interactions. Here we show that these positive feedback interactions promote oscillation at lower degrees of cooperativity, and we can thus unify several common kinetic mechanisms that facilitate oscillations, such as self-activation and Michaelis-Menten degradation. The positive feedback loops are most beneficial when acting on the shortest lived component, where they function by balancing the lifetimes of the different components. The benefits of multiple positive feedback interactions are cumulative for a majority of situations considered, when benefits are measured by the reduction in the cooperativity required to oscillate. These positive feedback motifs also allow oscillations with longer periods than that determined by the lifetimes of the components alone. We can therefore conjecture that these positive feedback loops have evolved to facilitate oscillations at lower, kinetically achievable, degrees of cooperativity. Finally, we discuss the implications of our conclusions on the mammalian molecular clock, a system modeled extensively based on the three-component negative feedback loop.

Feedback Loops of the Mammalian Circadian Clock Constitute Repressilator

“…Delayed negative feedback loops constitute the basic elements of self-sustained oscillations [27, 28]. Often these negative feedback loops are interlinked with positive feedbacks ensuring robust and tunable rhythms [29–33]. Thus, we focused on which sub-networks forming feedback loops are able to generate sustained rhythms for physiologically plausible parameters.…”

“…It is known that classical Goodwin-based models need strong negative cooperativity (minimal Hill coefficient of 8—probably unrealistic biochemically) and long balanced degradation times to obtain self-sustained oscillations [28, 33, 42]. Within the repressilator, however, the delay and the required non-linearities can be distributed over the three inhibitions.…”

“…Furthermore, positive feedback loops are known to support rhythm generation [33]. A comprehensive list of loops within our gene regulatory network is given in (S4 Appendix), showing the interrelations and coherence of loops.…”

Protein sequestration versus Hill‐type repression in circadian clock models

“…1), which is known as the secant condition (see Appendix for details) [91][92][93][94]. The Michaelis-Menten type of repressor clearance also reduces the necessary Hill exponent as it can serve as an additional source of non-linearity [35,[95][96][97][98].…”

Design principles for robust oscillatory behavior

“…We do so by conducting extensive numerical simulations of every possible topological arrangement. Previous related work focused on exploring a limited number of topological variations of three-node oscillators (Tsai et al 2008;Ananthasubramaniam and Herzel 2014). Our approach, however, is not biased towards any specific architecture since we evaluate the robustness of every possible three-component network.…”