Towards universality of delayed and quickened bifurcations in biological signalling

延迟分叉是指当控制参数动态地跨越分叉时普遍发生的非平衡过程，因此与大量的物理和生物系统相关。这种效应延迟了系统突然变化的开始，并可能表现为明显的滞后，掩盖了平衡分叉的关键属性。在最近使用动力学系统对生物信号进行建模的工作中，观察到了相反的效果：标记通过系统的高激活区和低激活区之间的有效移动边界的分叉加快，这可以抵消动态延迟效应。与延迟情况一样，这种加速效应掩盖了分叉的确切性质。在研究生物信号中普遍存在的开关式反应时，了解这些效应及其相互作用尤为重要，因为这些突然的、要么全有要么全无的生物反应对应于系统中的分叉。在实际中，生物信号中遇到的分叉通常可以归结为整个高维系统中较低维流形上的动力学，例如通过干草叉子或跨临界分叉。因此，大量涉及生物信号传递中复杂环境(如基因调控、细胞间通信、模式形成、流行病学等)的时空系统可以通过研究低维分叉的加速及其与延迟的相互作用来表征。我将建立一个理论框架来普遍地分类和解释生化信号系统中超敏感反应的功能。我将通过分析它们在时空涨落存在的情况下的非平衡分叉结构，并描述加速和延迟之间的动态相互作用来实现这一点。我将通过系统的多尺度分析和数值模拟相结合的方法，系统地推导出常见分叉类型的适当的非平衡范式。这将使我能够描述系统中可能的非平衡行为的具体类型。然后，我将把我得出的一般结果应用于特定的特征生物信号系统。这将包括自动催化群体感应和基因表达，以及在不同环境和不断增长的领域中激活-抑制诱导的模式。特别有趣的是了解哪些物理系统在非平衡行为类型之间切换，因为这将对它们的空间或时间健壮性产生有趣的影响。

Delayed or 'critically slowed' bifurcations are nonequilibrium processes that occur universally when a control parameter dynamically crosses a bifurcation, and are therefore relevant to a vast number of physical and biological systems. This effect delays the onset of the sudden change to the system and can manifest as an apparent hysteresis, obscuring key properties of the equilibrium bifurcation. In recent work on modelling biological signalling using dynamical systems, the opposite effect has been observed; a quickening of the bifurcations that mark effective moving boundaries between high- and low-activation regions through the system, which can counteract the dynamic delaying effect. As with the delayed case, this quickening effect obscures the precise nature of the bifurcation. Understanding these effects and their interaction is particularly important when studying the switch-like responses that are ubiquitous in biological signalling, since these sudden 'all or nothing' biological responses correspond to bifurcations in the system.Bifurcations encountered in practice in biological signalling can typically be reduced to dynamics on a lower-dimensional manifold within the full higher-dimensional system, such as through pitchfork or transcritical bifurcations. Hence, a vast number of spatio-temporal systems involving complex environments in biological signalling (e.g., gene regulation, cell-cell communication, pattern formation, epidemiology, and many more) could be characterised by investigating quickening and its interaction with delay for low-dimensional bifurcations.I will build a theoretical framework to universally classify and interpret the function of ultrasensitive responses in biochemical signalling systems. I will do this by analysing their nonequilibrium bifurcation structure in the presence of spatio-temporal fluctuations, and characterising the dynamic interaction between quickening and delay. I will systematically derive the appropriate nonequilibrium normal forms for common bifurcations type through a synergistic approach combining systematic multiscale analysis with numerical simulations. This will allow me characterise the specific types of possible nonequilibrium behaviour in the system. I will then apply the general results I derive to specific characteristic biological signalling systems. This will include autocatalytic quorum sensing and gene expression, and activator-inhibitor induced patterning in heterogeneous environments and growing domains. Of particular interest will be understanding which physical systems switch between nonequilibrium behaviour types, since this will have interesting implications for their spatial or temporal robustness.