Mathematical Analysis of Biological Oscillator Models and Machine Learning Conservation Laws

In the first part of this thesis, we develop for method to machine learning conservation laws of dynamical systems. We present a novel approach to machine learning conservation laws using kernel methods, more precisely kernel ridge regression. Our basic tool is the inhomogeneous polynomial kernel, meaning that we search mainly for polynomial conservation laws. We have shown how conservation laws of arbitrary dynamical systems can be discovered with an indeterminate form of a kernel ridge regression. Our kernel methods always provide explicit closed-form expressions for the discovered conservation laws so that no subsequent symbolic regression is necessary. We furthermore believe that such an approach is not only computationally much more efficient than neural networks, but it also requires much less training data. This point becomes important if one is not working with synthetic numerical data but with experimental data. As long as numerical trajectory data are used, it is usually fairly cheap to produce large amounts of data. However, in principle, such approaches can also be applied to experimental data, which may be much harder to get in large quantities. Of course, this raises the question of how robust the used methods are against noisy data. In our method, kernel ridge regression includes a regularization parameter to mitigate the effects of noise, thus enhancing the reliability of the discovered conservation laws. We also discovered multiple (functionally independent) conservation laws from a single data set. Furthermore, we have developed an approach for making the discovered conservation laws sparse, which significantly improves the usefulness of the results. The second part focuses on mathematical analysis of biological oscillator models. We developed a mathematical model to describe the cell cycle dynamics in Drosophila melanogaster during early embryogenesis. In the early stages of their development, Drosophila melanogaster embryos possess very fast and well-coordinated cell cycles. In the cell cycle, CDK is regulated by the binding of CDK to cyclin to form an active complex, as well as by phosphorylation via CDC25 and dephosphorylation via Wee1. A biochemically sound model of CDK activation and inactivation was developed, consisting of 5 differential equations and 16 parameters. The model was then reduced to a simplified form consisting of 3 differential equations and 8 parameters by exploiting a conservation law and the dynamics on different time scales via a quasi-steady-state approximation, as well as by reducing the number of parameters through scaling symmetries. The reduced system demonstrates qualitatively similar behavior to that observed in experiments, namely periodic behavior with an increasing period length. The model highlights how small changes in the level of cyclin and APC have a significant effect on cell cycle progression by identifying key thresholds for CDK activation and inactivation driven by these components. We reduced cyclin accumulation (or synthesis) slightly in each cycle to represent the lengthening of the cell cycle after cycle 10. For better alignment with experimental results, the model is further refined by including time-dependent cyclin synthesis.