**Project Heads**

*Heike Siebert, Christian Haase*

**Project Members**

Robert Schwieger (FU) (until 05/2020), Hannes Klarner (FU)

**Project Duration**

01.01.2019 – 31.12.2021

**Located at**

FU Berlin

The project aims at establishing a new application field for algebraic methods in systems biology, connecting to Boolean modeling which has been shown to be a fruitful framework for investigating molecular interaction networks. A model is given as a Boolean function, where the coordinate functions specify processing rules of the components. The dynamics of this function can be represented by a non-deterministic state transition system that can be efficiently analyzed using logic-based verification methods such as model checking, Answer Set Programming (ASP) or SAT approaches. However, results often are of the type of yes/no answers to property queries and do not offer insight into the structural properties of the model that give rise to the behavior.

Exploiting more of the problem structure becomes possible in an algebraic setting. Boolean functions can be identified with polynomials over the field with two elements. In this setting we want to tap into the potential of Gröbner bases and related algebraic approaches, which have already been used for, e.g., reverse engineering problems. Here, we apply them to a new set of questions and integrate them with the formal verification methods developed for related tasks. The motivating biological question is the analysis of cell fate decision processes, such as cell differentiation in embryogenesis or cell alterations in diseases such as cancer. Cell fates can be identified with classes of model attractors, called phenotypes, of systems processing complex input signals. They are defined via projection on marker components. In application, the interest now lies in finding all minimal sets of such marker components and to identify and understand the mechanisms that govern the branching into different phenotypes

This problem is related to computing classifiers for the different phenotypes whose characteristics, e.g., which components need to be considered for classification, can then be linked back to the model. We developed logic-based approaches to this and related problems, but they do not exploit any of the algebraic structure inherent in the setting. Here, we address this by developing a complementary approach utilizing Gröbner bases. Aside from the theoretical work we are also aiming at efficient implementations. Using data structures tailored to Boolean systems, Gröbner bases for Boolean models can be computed for networks with several hundred variables, opening up an efficient and comprehensive analysis approach capable of dealing with the complex molecular network models considered in application.

**Project Webpages**

**Selected Publications
**

- R. Schwieger and H. Siebert. Structure and behavior in Boolean monotonic model pools. Biosystems, 214:104610, 2022.
- L. Cifuentes-Fontanals, E. Tonello, and H. Siebert. Control in Boolean networks with model checking, 2021. Submitted to Frontiers in Applied Mathematics and Statistics. arXiv
- H. Klarner, E. Tonello, L. Fontanals, F. Janody, C. Chaouiya and H. Siebert. Detection of markers for discrete phenotypes. Proceedings of CSBio 2021, ACM, 64-68, 2021.

- R. Schwieger, M. Bender, H. Siebert and C. Haase. Classifier construction in Boolean networks using algebraic methods. Proceedings of Computational Methods in Systems Biology (CMSB), LNCS 12314, 159-175, 2020.
- H. Klarner, F. Heinitz, S. Nee, H. Siebert: Basins of Attraction, Commitment Sets and Phenotypes of Boolean Networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2019.
- M. Nowicka, H. Siebert: Designing Distributed Cell Classifier Circuits Using a Genetic Algorithm. Computational Methods in Systems Biology. CMSB 2019. Lecture Notes in Computer Science, 11773, 96-119, 2019.
- K. Becker, H. Klarner, M. Nowicka, H. Siebert: Designing miRNA-based Synthetic Cell Classifier Circuits Using Answer Set Programming. Frontiers in Bioengineering and Biotechnology, 6:70, 2018.

**Selected Pictures
**

The figure shows the interaction graph of a model by Calzone and colleagues capturing a signaling network governing cell fate decisions, namely survival and different types of cell death. These are represented by sets of steady states. In our project we determine minimal sets of model components, also called biomarkers, the measurement of which is sufficient to distinguish between the different sets without explicit calculation of the steady states.

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