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Slow-Fast Systems in Cellular Biology

Periodic Reporting for period 1 - SFSysCellBio (Slow-Fast Systems in Cellular Biology)

Reporting period: 2016-08-30 to 2018-08-29

Biologists have identified a wealth of molecular components and regulatory mechanisms underlying the
control of the fundamental cellular processes such as cell differentiation, cell cycle, and cell death. A
multitude of signalling pathways controlling the response of cells to environmental variability or stimuli is
critical for these processes. Mathematical modelling has emerged as an important tool to handle the overwhelmingly structural
complexity of cellular processes and to gain better understanding of their functioning and dynamics.

This project was devoted to the mathematical analysis of ODE models arising in cell biology. Pertinent mathematical questions of
biological interest are: existence and stability of equilibria, periodic oscillations, switching phenomena, and bifurcations, i.e. changes
of these dynamical properties as key parameter vary. The approach in this project relied strongly on novel dynamical systems
methods for systems with multiple time scale dynamics, known as geometric singular perturbation theory (GSPT). These methods
have been successfully used in many areas of mathematical physiology, e.g. mathematical neuroscience and calcium signaling.
However, slow-fast analysis and GSPT of mathematical models of the cell cycle and for signaling pathways is much less established.
We have focused on several models based on Michaelis-Menten type kinetics. Interestingly, the models under investigation do not have the standard form of slow-fast systems and were not covered by the existing theory. It turns out that such systems have a specific hidden multi-scale structures caused by the occurrence of very large or very small parameters in important types of nonlinearities commonly found in such models, e.g. Michaelis-Menten-type terms and Hill functions.
In this project an extended version of GSPT was adopted to singular perturbation problems of general type arising in these contexts, in order to analyze the induced dynamics, in particular switching phenomena and oscillatory behavior. The research generated novel mathematical results and led to considerable advances in the field of GSPT.
The methods were applied to specific signalling problems in collaboration with our project partner from systems biology.
A visualization tool, which supports our geometric approach, has been developed in collaboration with partners from computer science working in visualization.
Impact on mathematics and modelling in cell biology.

The project advanced multidisciplinary research at the boundary of mathematics and modelling in cell biology.
New geometric techniques and dynamical systems methods for the analysis of medium sized models with complicated slow-fast dynamics have been developed which
considerably extend the scope and applicability of GSPT. Systematic mathematical analysis of biological models supports simulation and interpretation of
results. This leads to better understanding of the cellular processes and their dynamics. The techniques and tools used and developed in the project are applicable in the analysis of other important signaling and control pathways.
visualization by manylands