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BIMSA Computational Math Seminar
Switching and multiple time-scales in gene regulatory networks
Switching and multiple time-scales in gene regulatory networks
演讲者
Kristian Kristiansen
时间
2025年04月10日 15:00 至 16:00
地点
Online
线上
Zoom 787 662 9899
(BIMSA)
摘要
ODE-based models for gene regulatory networks (GRNs) can often be formulated as smooth singular perturbation problems with multiple small parameters, some of which are related to time-scale separation, whereas others are related to 'switching', i.e. proximity to a non-smooth singular limit. This motivates the study of reduced models which are obtained after (i) a quasi-steady state reduction (QSSR) which utilises the time-scale separation, and (ii) piecewise-smooth approximations which reduce the nonlinearity of the model by viewing the highly nonlinear sigmoidal terms as singular perturbations of step functions. We investigate the interplay between the reduction methods (i)-(ii), in the context of a 4-dimensional GRN which has been used as a low-dimensional representative of an important class of (generally high-dimensional) GRN models in the literature. In particular, we identify a region in the small parameter plane for which this problem can be formulated as a smooth singularly perturbed system on a blown-up space, uniformly in the switching parameter. In this way, we can apply Fenichel's coordinate-free theorems and obtain a rigorous reduction to a 2-dimensional protein-only system, that is a perturbation of the system obtained via QSSR. We also show that the reduced system features a Hopf bifurcation which does not appear in the QSSR system, due to the influence of higher order terms. Overall, our findings suggest that the relative size of the small parameters is important for the validity of QSS reductions and the determination of qualitative dynamics in GRN models more generally. Moreover, although the focus is on the 4-dimensional GRN, our approach is applicable to higher dimensions.