# A18 Mathematical Systems Biology

Prof. Dr. Alexander Bockmayr, Dr. Heike Siebert, Dr.-Ing. Steffen Waldherr, Dr. Yaron Goldstein, Lászlo David, Hannes Klarner

DFG Research Center MATHEON

Systems biology is a lively interdisciplinary research field that has received considerable attention in recent years. While traditional molecular biology studies the various components of a biological system (genes, RNAs, proteins,...) in isolation, systems biology aims to understand how these components interact in order to perform higher-level functions. Mathematical modeling then plays a major role not only in capturing and analyzing complex networks governing biological processes, but also as a way to verify hypotheses based on observational evidence and in designing efficient experiments to further the understanding of the system. The goal of this project is to pool our knowledge on discrete, deterministic and stochastic modeling of biological systems to pave the way towards efficient hybrid models (discrete/continuous, deterministic/stochastic) for elaborate biological networks.

**Background**

Systems biology is a lively interdisciplinary research field that has received considerable attention in recent years.

While traditional molecular biology studies the various components of a biological system (genes, RNAs, proteins,…) in isolation, systems biology aims to understand how these components interact in order to perform higher-level functions. Mathematical modeling then plays a major role not only in capturing and analyzing complex networks governing biological processes, but also as a way to verify hypotheses based on observational evidence and in designing efficient experiments to further the understanding of the system. Traditionally, differential equations are used to model time-dependent concentrations of reacting species. A number of efficient algorithms and software implementations for the solution of such systems have been developed over the years. Although these tools simplify the modeling process, the main challenge is still parameter identification. A common problem particular to biological systems as opposed to, for example, applications in physics, is the lack of reliable experimental data, e.g. kinetic parameters. Given this situation, a possible alternative to differential equation modeling is to translate the available information into a set of constraints. This in general does not allow for deterministic predictions, but permits to identify system characteristics, and a set of possible behaviors. The idea of constraint-based modeling leads to the application of discrete methods, for example steady-state analysis of metabolic networks based on stoichiometric and thermodynamic constraints, or the representation of regulatory networks as discrete dynamical systems.

A further difficulty to be considered when modeling biological systems are low copy numbers. As the number of participating molecules decreases, stochastic fluctuations come into play. In this case, modeling chemical reaction systems solely by means of the classical equations of reaction kinetics is not sufficient anymore. Rather, the chemical master equation (CME) in high dimensions has to be utilized, a discrete partial differential equation that describes the time-evolution of the probability density for the copy number of each species. The considerations above highlight some important aspects of modeling biological systems, and of course many systems call for modeling frameworks capable of combining several of those aspects. Hybrid discrete/continuous methods offer possibilities to enhance discrete dynamics with continuous processes, e.g., adding a continuous time evolution that allows for integration of more quantitative data. Hybrid continuous/stochastic methods, e.g.; integrate the chemical master equation into reaction kinetics allowing for analysis of high-dimensional models with a rather low-dimensional, yet time-dependent subspace of low copy numbers. In general, hybrid methods extend modeling power considerably, allowing for a continual model development in a comprehensive framework.

**External Cooperations: **Prof. Dr. Dr. h.c. Peter Deuflhard (ZIB), Dr. Susanna Röblitz (ZIB), Claudia Stötzel (ZIB)

**Software**

BioPARKIN (Biology related Parameter Identification in Large Kinetic networks)

FFCA (Feasibility-based Flux Coupling Analysis)

ERDA (Edge Refinement and Data Assessment)

F2C2 (Fast flux coupling calculator)

tFVA (Thermodynamically Constrained Flux Variability Analysis)

L4FC (Lattices for Flux Coupling)

CEFMISTR (Computing elementary flux modes involving a set of target reactions)

**Publications**

Reimers AC, Goldstein Y, Bockmayr A: Generic Flux Coupling Analysis. Mathematical Biosciences, 262, 28-35, Apr 2015

Goldstein Y, Bockmayr A: Double and Multiple Knockout Simulations for Genome-Scale Metabolic Network Reconstructions. Algorithms for Molecular Biology, 10:1, Jan 2015

Waldherr S, Oyarzún DA, Bockmayr A: Dynamic optimization of metabolic networks coupled with gene expression. Journal of Theoretical Biology, 365, 469-485, Jan 2015 (arXiv)

David L, Bockmayr A: Computing elementary flux modes involving a set of target reactions. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 11/6, 1099-1107, Nov/Dec 2014

Klarner H, Bockmayr A, Siebert H: Computing Symbolic Steady States of Boolean Networks. Cellular Automata, ACRI 2014, Krakow, Poland, Springer, LNCS 8751, 561-570, Sep 2014

Bockmayr A, Siebert H., Röblitz S, Schütte Ch, Deuflhard P: Advanced mathematical modeling in systems biology. MATHEON-Mathematics for Key Technologies, volume 1, pages 29-44, Peter Deuflhard, Martin Grötschel, Dietmar Hömberg, Jürg Kramer, Volker Mehrmann, Konrad Polthier, Frank Schmidt, Christof Schütte, Martin Skutela, Jürgen Sprekels (Editors), European Mathematical Society, 2014

Müller AC, Bockmayr A: Flux Modules in Metabolic Networks. Journal of Mathematical Biology, Oct 2013

Goldstein Y, Bockmayr A: A Lattice-Theoretic Framework for Metabolic Pathway Analysis. Computational Methods in Systems Biology, CMSB 2013, Klosterneuburg. Springer, LNBI 8130, 178-191, Sep 2013

David L, Bockmayr A: Constrained Flux Coupling Analysis. Workshop on Constraint based Methods for Bioinformatics, WCB'13, Uppsala, 75-83, Sep 2013

S. Röblitz, C. Stötzel, P. Deuflhard, H. M. Jones, D.-O. Azulay, P. van der Graaf, S. W. Martin: A mathematical model of the human menstrual cycle for the administration of GnRH analogues. Journal of Theoretical Biology, 321, 8–27, March 2013.

Jamshidi S, Siebert H, Bockmayr A: Preservation of Dynamic Properties in Qualitative Modeling Frameworks for Gene Regulatory Networks. Biosystems, 112/2, 171-179, 2013

Lorenz, Therese and Siebert, Heike and Bockmayr, Alexander: Analysis and characterization of asynchronous state transition graphs using extremal states. Bulletin of Mathematical Biology, 75/6, 920-938, 2013

Bockmayr A, Siebert H: Bio-Logics: Logical Analysis of Bioregulatory Networks. Programming Logics. Essays in Memory of Harald Ganzinger. Springer, LNCS 7797, 19-34, 2013

Müller AC, Bockmayr A: Fast Thermodynamically Constrained Flux Variability Analysis. Bioinformatics, 29/7, 903-909, 2013

Marashi SA, David L and Bockmayr A: On flux coupling analysis of metabolic subsystems. J. Theoretical Biology, 302, 62-69, 2012.

M. Rügen, A. Bockmayr, J. Legrand, and G. Cogne. Network reduction in metabolic pathway analysis: Elucidation of the key pathways involved in the photoautotrophic growth of the green alga Chlamydomonas reinhardtii. Metabolic Engineering, 14, 458-467, 2012

Hannes Klarner, Adam Streck, David Safránek, Juraj Kolcák, Heike Siebert: Parameter Identification and Model Ranking of Thomas Networks. CMSB 2012: 207-226

H. M. T. Boer, C. Stötzel, S. Röblitz, and H. Woelders. A differential equation model to investigate the dynamics of the bovine estrous cycle. In Advances in Systems Biology, volume 736 of Advances in Experimental Medicine and Biology, pages 59-606. Springer, 2012.

S. Jamshidi, H. Siebert, and A. Bockmayr. Comparing discrete and piecewise affine differential equation models of gene regulatory networks. In Proc. 9th Int. Conf. Information Processing in Cells and Tissues, IPCAT 2012, Cambridge, UK, Springer, LNCS 7223, 17-24, 2012

H .M. T. Boer, S. Röblitz, C. Stötzel, B. Kemp, R. F. Veerkamp, and H. Woelders. Mechanisms regulating follicle wave patterns in the bovine estrous cycle investigated with a mathematical model. J. Dairy Sci., 94(12):5987-6000, 2011.

H .M. T. Boer, C. Stötzel, S. Röblitz, P. Deuhard, R. F. Veerkamp, and H. Woelders. A simple mathematical model of the bovine estrous cycle: follicle development and endocrine interactions. J. Theoret. Biol., 278(1):20-31, 2011.

G. Cogne, M. Rügen, A. Bockmayr, M. Titica, C. G. Dussap, J. F. Cornet, and J. Legrand. A model-based method for investigating bioenergetic processes in autotrophically growing eukaryotic microalgae: Application to the green algae Chlamydomonas reinhardtii. Biotechnology Progress, 27/3:631-40, 2011.

L. David, S. A. Marashi, A. Larhlimi, B. Mieth, and A. Bockmayr. FFCA: a feasibility-based method for flux coupling analysis of metabolic networks. BMC Bioinformatics, 12:236, 2011.

M. Hegland and J. Garcke. On the numerical solution of the chemical master equation with sums of rank one tensors. In Proc. 15th Biennial Computational Techniques and Applications Conference, CTAC-2010, volume 52 of ANZIAM J., pages C628-C643, 2011

H. Klarner, H. Siebert, and A. Bockmayr. Parameter inference for asynchronous logical networks using discrete times series. Computational Methods in Systems Biology, CMSB 2011, Paris, 2011.

S.A. Marashi and A. Bockmayr. Flux coupling analysis of metabolic networks is sensitive to missing reactions. BioSystems, 103:57-66, 2011.

A. Palinkas and A. Bockmayr. Petri nets for integrated models of metabolic and gene regulatory 10 networks. Workshop on Constraint based Methods for Bioinformatics, WCB'11, Perugia, 2011.

A. Rezola, L.F. de Figueiredo, M. Brock, J. Pey, A. Podhorski, C. Wittmann, S. Schuster, A. Bockmayr, and F. J. Planes. Exploring metabolic pathways in genome-scale networks via generating flux modes. Bioinformatics, 27/4:534-540, 2011.

H. Siebert. Analysis of discrete bioregulatory networks using symbolic steady states. Bull. Math. Biol., 73:873-898, 2011.

S. Twardziok, H. Siebert, and A. Heyl. Stochasticity in reactions: a probabilistic boolean modeling approach. Computational Methods in Systems Biology, CMSB 2010, Trento, Italy, pages 76-85. ACM, 2010.