The Wnt signalling pathway plays an important role in development, disease, and normal tissue function. Mathematical models for Wnt signalling have predominantly focused on quantitatively predicting changes in steady-state $\beta$-catenin concentrations (the main downstream protein regulated by canonical Wnt signalling). One of the genes targeted for expression by Wnt/$\beta$-catenin signalling is the negative Wnt regulator Axin2. Recently, a number of authors have indicated a potential theoretical role of Axin2 feedback to induce oscillatory behaviour in the pathway and this has been observed in a number of detailed mathematical models. Due to the complexity of these models, the investigations to date have been limited to numerical experiments and parameter sensitivity analyses. In this manuscript, we study the fundamental structure of the dynamical system underlying the Wnt signalling mechanism with Axin2 feedback to gain some insight into why and when oscillations occur in models with this structure. We semi-rigorously analyse three simple models and, for these models, gain deep understanding of the characteristic set of conditions that are necessary and sufficient for oscillations to be induced. We discuss the possible biological consequences of these findings for Wnt signalling pathway oscillations. They include; to promote oscillations (1) Keeping all other parameters constant, the Wnt signal strength should neither be too high or too low but within a single finite window of values, (2) Wnt receptor complexes should fully deactivate Axin rather than temporarily deconstruct it from other scaffold proteins, (3) In the absence of stochastic effects or more complicated mechanisms, a critical delay in Axin2 feedback in the system is necessary, (4) Deactivation of Axin by the Wnt receptor complex needs to be critically efficient relative to $\beta$-catenin removal by Axin, and (5) conditions necessary are less strict if Axin2 feedback occurs after a fixed time rather than a Poisson-distributed time with the same average.

Wnt信号通路在发育，疾病和正常组织功能中起重要作用。Wnt信号传递的数学模型主要集中于定量预测稳态变化$\beta$-catenin浓度（主要的下游蛋白质受经典Wnt信号调节）。Wnt /靶向表达的基因之一$\beta$-catenin信号是负的Wnt调节剂Axin2。最近，许多作者指出了Axin2反馈在诱导该途径中的振荡行为方面的潜在理论作用，并且已经在许多详细的数学模型中观察到了这一点。由于这些模型的复杂性，迄今为止的研究仅限于数值实验和参数敏感性分析。在本文中，我们研究了具有Axin2反馈的Wnt信号机制基础的动力学系统的基本结构，以了解为什么和何时在具有这种结构的模型中发生振荡。我们对三个简单的模型进行半严格的分析，对于这些模型，我们可以深入理解引起振荡所需的条件集。我们讨论了这些发现对于Wnt信号通路振荡的可能的生物学后果。它们包括; 为了促进振荡（1）保持所有其他参数不变，Wnt信号强度不应太高或太低，而应在一个有限的数值窗口内；（2）Wnt受体复合物应完全使Axin失活，而不是暂时将其与其他毒素解构支架蛋白，（3）在没有随机作用或更复杂的机制的情况下，系统中Axin2反馈的关键延迟是必要的，（4）Wnt受体复合物使Axin失活相对于$\beta$如果Axin2反馈发生在固定时间之后而不是具有相同平均值的Poisson分布时间之后，则Axin去除-catenin的时间和（5）所需条件不太严格。