Positive Feedback Promotes Oscillations in Negative Feedback Loops

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.

三组分负反馈环是生物化学振荡器中反复出现的经典基序。该基序可产生振荡，因为其具备了振荡所需的三项核心要素：三步延迟机制、负反馈调控以及环路内的非线性特性。然而，在经典的古德温模型（Goodwin formulation）框架下，该基序要实现振荡，需要反馈环节具备极高的协同性（非线性程度的量化指标），而这种高协同性在生物学层面上并不具备现实可行性。此外，这类反复出现的负反馈基序通常会通过正反馈相互作用得到增强与补充。本研究表明，此类正反馈相互作用可在更低的协同性水平下促成振荡，由此可统一多种已知的促进振荡的常见动力学机制，例如自激活与米氏降解。当正反馈环路作用于寿命最短的组分时，其增益效果最为显著：此时正反馈通过平衡不同组分的分子寿命来发挥调控功能。在以“实现振荡所需协同性的降幅”作为收益衡量标准的多数考察场景中，多重正反馈相互作用的增益效应具有累积性。这类正反馈基序还能够产生相较于仅由组分寿命决定的振荡周期更长的振荡信号。据此我们可推测，这类正反馈环路在演化过程中被选择，以实现在动力学上可实现的更低协同性水平下的振荡过程。最后，我们讨论了本研究结论对哺乳动物分子钟的启示——该系统正是基于三组分负反馈环构建并被广泛研究的经典模型之一。