BioND — Dynamics of Biological Networks

Extension of generalized modeling and application to problems from cell biology

Martin Zumsande
PhD thesis in Physics, Dresden Technical University, 2011.



Mathematical modeling is an important tool in improving the understanding of complex biological processes. However, mathematical models are often faced with challenges that arise due to the limited knowledge of the underlying biological processes and the high number of parameters for which exact values are unknown. The method of generalized modeling is an alternative modeling approach that aims to address these challenges by extracting information about stability and bifurcations of classes of models while making only minimal assumptions on the specific functional forms of the model. This is achieved by a direct parameterization of the Jacobian in the steady state, introducing a set of generalized parameters which have a biological interpretation.

In this thesis, the method of generalized modeling is extended and applied to different problems from cell biology. In the first part, we extend the method to include also the higher derivatives at the steady state. This allows an analysis of the normal form of bifurcations and thereby a more specific description of the nearby dynamics. In models of gene-regulatory networks, it is shown that the extended method can be applied to better characterize oscillatory systems and to detect bistable dynamics.

In the second part, we investigate mathematical models of bone remodeling, a process that renews the human skeleton constantly. We investigate the connection between structural properties of mathematical models and the stability of steady states in different models. We find that the dynamical system operates from a stable steady state that is situated in the vicinity of bifurcations where stability can be lost, potentially leading to diseases of bone.

In the third part of this thesis, models of the MAPK signal transduction pathway are analyzed. Since mathematical models for this system include a high number of parameters, statistical methods are employed to analyze stability and bifurcations. Thereby, the parameters with a strong influence on the stability of steady states are identified. By an analysis of the bifurcation structure of the MAPK cascade, it is found that a combination of multiple layers in a cascade-like way allows for additional types of dynamic behavior such as oscillations and chaos.

In summary, this thesis shows that generalized modeling is a fruitful alternative modeling approach for various types of systems in cell biology.