Towards universality of delayed and quickened bifurcations in biological signalling

延迟或“临界减速”分岔是当控制参数动态跨越分岔时普遍发生的非平衡过程，因此与大量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，并且可以表现为明显的滞后，模糊了平衡分岔的关键特性。在最近使用动力系统模拟生物信号的工作中，观察到相反的效果；在系统中标记高激活区和低激活区之间有效移动边界的分岔加快，可以抵消动态延迟效应。与延迟的情况一样，这种加速效应掩盖了分岔的确切性质。在研究生物信号中普遍存在的开关反应时，理解这些影响及其相互作用尤为重要，因为这些突然的“全有或全无”的生物反应对应于系统中的分叉。生物信号传导实践中遇到的分岔通常可以简化为完整高维系统中的低维流形上的动力学，例如通过干草叉或跨临界分岔。因此，大量涉及生物信号传导复杂环境的时空系统（例如，基因调控、细胞-细胞通信、模式形成、流行病学等）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分类和解释生化信号系统中超灵敏反应的功能。我将通过分析它们在时空波动存在下的非平衡分岔结构，并描述加速和延迟之间的动态相互作用来做到这一点。我将通过将系统多尺度分析与数值模拟相结合的协同方法，系统地推导出常见分岔类型的适当非平衡范式。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。这将包括自催化群体感应和基因表达，以及在异质环境和生长域中激活剂-抑制剂诱导的模式。特别感兴趣的将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鲁棒性产生有趣的影响。

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.