Symbolic Methods in Simulation

Abstract

Ongoing Projects

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SYMBIONT is an interdisciplinary project ranging from mathematics via computer science to systems biology and systems medicine. The project has a clear focus on fundamental research on mathematical methods, and prototypes in software, which is in turn benchmarked against models from computational biology databases. Computational models in systems biology are built from molecular interaction networks and rate parameters resulting in large systems of differential equations. These networks are foundational for systems medicine. The currently prevailing numerical approaches shall be complemented with our novel algorithmic symbolic methods, which will address fundamental problems in this area. One important problem is that statistical estimation of model parameters is computationally expensive and many parameters are not identifiable from experimental data. In addition, there is typically a considerable uncertainty about the exact form of the mathematical model itself. The parametric uncertainty (with wide potential variations of parameters by several orders of magnitudes) leads to severe limitations of numerical approaches even for rather small and low dimensional models. Furthermore, extant model inference and analysis methods suffer from the curse of dimensionality that sets an upper limit of about ten variables to the tractable models. For those reasons, the formal deduction of principle properties of large and very large models has a very high relevance. The main goal of SYMBIONT is to combine symbolic methods with model reduction methods for the analysis of biological networks. We propose new methods for symbolic analysis, which overcome the above mentioned obstacles and therefore can be applied to large networks. In order to cope more effectively with the parameter uncertainty problem we impose an entirely new paradigm replacing thinking about single instances with thinking about orders of magnitude. Our computational methods are diverse and involve various branches of mathematics such as tropical geometry, real algebraic geometry, theories of singular perturbations, invariant manifolds and symmetries of differential systems. The foundations and validity of our methods will be carefully secured by mathematical investigation. Corresponding computer algebra problems are NP-hard, but experiments point at their feasibility for biological networks. We have already shown that complexity parameters such as tree-width or number of distinct metastable regimes grow only slowly with size for models available in existing biological databases. We will exploit this observation to solve challenging problems in network analysis including determination of parameter regions for the existence and stability of attractors, model reduction, and characterization of qualitative dynamics of nonlinear networks. The methods developed in this project will be benchmarked against existing biological models and also against more challenging models, closer to the needs of systems and precision medicine that will be generated using biological pathways databases.

Completed Projects

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We will analyse bifurcations and singularities of algebraic systems of ordinary differential equations with particular emphasis on questions concerning the existence of oscillations.

Publications

 
Hassan Errami, Markus Eiswirth, Dima Grigoriev, Werner M. Seiler, Thomas Sturm, and Andreas Weber
In: Journal of Computational Physics (2015), 291(279-302)
 
 
Dima Grigoriev and Andreas Weber
In proceedings of Computer Algebra in Scientific Computing - 14th International Workshop (CASC 2012), Maribor, Slovenia, Springer, Sept. 2012
 
 
N. N. Vassiliev (Editors)
Hassan Errami, Thomas Sturm, and Andreas Weber
In proceedings of Polynomial Computer Algebra, Saint Petersburg, pages 25-28, The Euler International Mathematical Institute, Apr. 2011
 
 
Vladimir P. Gerdt, Wolfram Koepf, E. W. Mayr, and E. V. Vorozhtsov (Editors)
Andreas Weber, Thomas Sturm, W. M. Seiler, and Essam O. Abdel-Rahman
In proceedings of Computer Algebra in Scientific Computing - 12th International Workshop (CASC 2010), Tsakhkadzor, Armenia, pages 267-279, Springer, Sept. 2010
 
 
Thomas Sturm, Andreas Weber, Essam O. Abdel-Rahman, and M. El Kahoui
In: Mathematics in Computer Science (Mar. 2009), 2:3
 
 
Katsuhisa Horimoto, Georg Regensburger, Markus Rosenkranz, and Hiroshi Yoshida (Editors)
Thomas Sturm and Andreas Weber
In proceedings of Algebraic Biology - Third International Conference (AB 2008), Springer-Verlag, 2008
 
 
In proceedings of Computer Algebra in Scientific Computing (CASC '06), pages 279-283, 2006
 
 
C. W. Brown, M. El Kahoui, Dominik Novotni, and Andreas Weber
In: Journal of Symbolic Computation (2006), 41:11(1157-1173)
 
 
V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov (Editors)
In proceedings of Computer Algebra in Scientific Computing (CASC '05), pages 387-398, Springer-Verlag, Sept. 2005
 
 
A. Dolzmann, A. Seidl, and Thomas Sturm (Editors)
C. W. Brown, M. El Kahoui, Dominik Novotni, and Andreas Weber
In proceedings of Algorithmic Algebra and Logic, pages 59-63, Apr. 2005
 
 
V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov (Editors)
C. W. Brown, M. El Kahoui, Dominik Novotni, and Andreas Weber
In proceedings of Computer Algebra in Scientific Computing (CASC '04), pages 51-60, July 2004
 
 
V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov (Editors)
W. M. Seiler and Andreas Weber
In proceedings of Computer Algebra in Scientific Computing (CASC'03), Sept. 2003
 
 
V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov (Editors)
M. El Kahoui and Andreas Weber
In proceedings of Computer Algebra in Scientific Computing (CASC~2002), pages 71-83, Sept. 2002
 
 
Andreas Weber, M. Weber, and P. Milligan
In: Mathematical Biosciences (2001)
 
 
M. El Kahoui and Andreas Weber
In: Journal of Symbolic Computation (Aug. 2000), 30:2(161-179)
 
 
In: SIGSAM Bulletin (Sept. 1996), 30:117(11-20)
 
 
J. Fleischer, J. Grabmeier, F. W. Hehl, and W. Küchlin (Editors)
C. Chauvin, M. Müller, and Andreas Weber
In proceedings of Computer Algebra in Science and Engineering, Zentrum für Interdisziplinäre Forschung, pages 287-296, World Scientific, Aug. 1994