Towards universality of delayed and quickened bifurcations in biological signalling

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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.