Mathematical Approaches to Biological Complexity in Systems Biomedicine

Living organisms represents the maybe most complex systems in the universe. This complexity is rooted in the necessity of life to be robust and adaptable. During evolution, life has therefore developed diverse regulatory strategies that are implemented by interactions of a plethora of entities and driven by the indispensable non-equilibrium character of living matter. The resulting intrinsic complexity of biological systems has been a major obstacle to deeply understand the underlying principles of life. Systems biology and biomedicine address this challenge by interdisciplinary approaches where mathematical modeling represents a key element to reveal and dissect the sources of complexity from large and big data sets.

In this spirit, the presented thesis applies bottom-up and top-down systems biomedicine approaches to investigate biological complexity at different levels and from different angels for metabolism and cell differentiation.

First, a bottom-up approach is targeting the mathematical properties of the stoichiometric matrix, which is the essential mathematical object in biochemical reaction networks and thus of metabolism. Applying graph and hypergraph theory, we present the key mathematical properties of the stoichiometric matrix and exploit biochemical properties of such networks to obtain a moiety-based decomposition of the stoichiometric matrix and consequently of biochemical reaction networks. These insights lay the foundation for a more descriptive characterization of metabolism.

Second, a novel top-down approach is presented to identify cell differentiation properties from single cell transcriptomic data by a combination of binarization, information theory and neural networks in terms of self organizing maps. This distribution-based analysis of cell fate is applied to blood cell differentiation and to the differentiation of induced pluripotent stem cells (iPSCs) into dopaminergic neurons in the context of Parkinson's disease (PD). This methodology allows for an alternative and efficient characterization of differentially expressed genes and the robust identification of critical points in cell differentiation. Comparing iPSCs with a PD-associated mutation within the LRKK2 gene to a healthy control cell line shows a faster maturation process in the disease context. By adapting concepts form non-equilibrium statistical physics, an entropy-based methodology in form of the Kullback-Leibler divergence is introduced to quantify the non-equilibrium character of cell fate, which reveals complementary essential biological processes of differentiation.

Finally, a potential integrative approach in form of Bayesian networks is introduced that will eventually allow for efficient and robust mechanistic inference from big data. In particular, the optimization approach is based on Markov chain Monte Carlo methods for sampling from distributions with non-smooth potential functions and uses Langevin stochastic equations for an advanced optimization strategy. The potential of the introduced approach is demonstrated by a first application to a logistic regression function.

Overall, the thesis applies complementary mathematical techniques to develop new tools for the characterization of biological complexity and the identification of underlying principles that are appearing in living systems.