In our research we focus on modeling and analyzing biological regulatory networks in cooperation with scientists from the life sciences, meeting the mathematical challenges posed by such applications. Of particular importance for us is to provide modeling and analysis methods respecting the constraints imposed by the available data. Data quality and uncertainties must be carefully taken into account to construct valid models. A major aim of model analysis in turn is to pinpoint ambiguities and generate hypotheses exploitable for experimental design. Combining different theoretical approaches for data analysis and modeling allows to bundle their distinct strengths and to process the available information in a systematic and holistic fashion. Developing such integrated methods is one of the central aims in our group.

As a starting point, we use logic-based models to represent the systems of interest to us. They are based on qualitative observations which are often more abundantly available than the quantitative data needed for constructing well-supported differential equation models. Analysis of such models can draw on a variety of mathematical methods, e.g., iteration and graph theory as well as methods from formal verification. This allows for comprehensive analysis of complex systems, often uncovering crucial structural and dynamical characteristics. It furthermore makes it possible to test large numbers of models for consistency with the available data, addressing the parameter identification problem and allowing to evaluate analysis results with respect to data uncertainty.

The computational strength of the logical formalism can also be exploited in tandem with more resolved modeling. Combining discrete, continuous and stochastic approaches yields methods tailored to optimally exploit the available data, to evaluate uncertainty and to uncover essential mechanisms of the system. Such integrated approaches are also promising for data analysis, where statistical methods can be combined with a network-based view to uncover the key players of system functionality.

We develop methods tailored to application in close cooperation with colleagues from the life sciences. In our current work, this includes oncogenic signaling networks, mammalian reproductive systems as well as bacterial infection mechanisms.

The following fields of research are of particular interest for our applications:

Model building often starts with a graph capturing the network structure. However, many different logical functions are consistent with a given network structure. Choosing a specific function can be seen as choosing parameter values for a model. Other than in the continuous case, the parameter space is finite if often very large. Formal verification methods such as model checking can be used to explore the parameter space efficiently and to identify classes of functions satisfying a given property. This allows on the one hand to check for consistency of modeling assumptions and available data, and on the other hand for insights into higher-level characteristics of model sets resulting from incomplete information.

In the group, we work on approaches based on constraint programming, model checking and graph algorithms that allow for efficient analysis of the parameter space resulting in a meaningful classification of models.

The systems of interest in biology and medical sciences become increasingly larger and more complex. Even in the coarse logical modeling formalism comprehensive analysis of the dynamics exceeding simple simulation is often not possible anymore. To provide efficient approaches to analysis, we focus mainly on two approaches that allow to reduce the complexity of the problem.

The first approach focuses on exploiting structural information of the system and augment that with a targeted analysis of dynamical characteristics linked to structural properties. For example, the importance of feedback circuits in the network structure for certain dynamical patterns is widely known. Our goal is to develop a method that provides a structured representation of the state space based on attractors and basin of attractors based on a functionality analysis of the feedback circuits of the system.

Our second approach to the complexity problem is based on modularization methods. Most such methods work on the network structure but do not take dynamical behavior into account. Clearly, that is not sufficient if we hope to extract information about the functionalities of a network from an understanding of idules. We developed an iteration theory base approach that allows us to find network modules whose analysis yields the attractors of the original system in at least a subdomain of state space. We plan to extend the idea to discover modules governing appropriately specified dynamical characteristics, not necessarily restricted to asymptotic dynamics.

Logical models are, by nature, coarse models, and may not be able to capture all aspects of interest. Also, more and more information might become available while studying the system that may not be exploitable to its full potential in an abstract discrete framework. A solution for both difficulties is a refinement process that allows to evolve a model step by step across formalism boundaries. Hybrid models enrich a discrete system with continuous time or stochastic effects and can be useful intermediaries between logical and differential equation models. In the group, we developed such hybrid formalisms both for including stochastic effects and continuous time evolution. In addition, we investigate the mathematical rules that govern loss or preservation of properties across different frameworks.

In addition, we try to exploit discrete and continuous models in tandem. Such a dual view on a system can provide a more comprehensive understanding for the dynamics of the continuous model based on state and parameter space analysis of the corresponding discrete model. For example, it might allow to answer the question how the qualitative behavior of the system changes with parameter values.