Symbolic Methods for Biological Networks
Bioinformatics and Theoretical Biology
Data Management, Data-Intensive Systems, Computer Science Methods in Business Informatics
Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
Theoretical Computer Science
Final Report Abstract
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 benchmarked against models from computational biology databases. In systems biology, computational models are built from molecular interaction networks and rate parameters resulting in large systems of differential equations. These networks are foundational for systems medicine. 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, responsible to the introduction of parameters with wide potential variations of by several orders of magnitudes. This in turn causes severe limitations of numerical approaches even for rather small and low dimensional models. Furthermore, existant model inference and analysis methods suffer from the curse of dimensionality that sets an upper limit of about ten variables to the tractable models. Hence the formal deduction of principal qualitative properties of large and very large models is highly relevant. Symbolic methods. 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 symbolic computational methods are diverse and involve various branches of mathematics such as automated reasoning, tropical geometry, real algebraic geometry, theories of singular perturbations, invariant manifolds and symmetries of differential systems. The validity of our methods is established on within rigorous mathematical frameworks. Although, we are frequently dealing with NP-hard problems, which would be categorized as “infeasible” in theoretical computer science, we achieve feasibility in the special case of biological networks. For instance, we observe that natural complexity parameters such as tree-width or number of distinct metastable regimes grow only slowly with size for models in biological databases. This allows to combine symbolic methods with model reduction methods for the analysis of biological networks in order 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. Major results of the project. We have obtained a geometric understanding the steady states of reaction kinetics, even in the presence of parametric reaction rates. We furthermore have developed an algorithmic framework for reducing reaction kinetics to multiple time scales, with a solid foundation in Analysis.
Publications
- “Graphical Requirements for Multistationarity in Reaction Networks and their Verification in BioModels”. In: Journal of Theoretical Biology 459 (Dec. 2018), pp. 79–89
A. Baudier, F. Fages, and S. Soliman
(See online at https://doi.org/10.1016/j.jtbi.2018.09.024) - “Coordinate-independent singular perturbation reduction for systems with three time scales”. In: Mathematical Biosciences and Engineering 16.5 (2019), pp. 5062–5091
N. Kruff and S. Walcher
(See online at https://doi.org/10.3934/mbe.2019255) - “Identifying the parametric occurrence of multiple steady states for some biological networks”. In: Journal of Symbolic Computation 98 (May 2020), pp. 84–119
R. Bradford, J. H. Davenport, M. England, H. Errami, V. Gerdt, D. Grigoriev, C. Hoyt, M. Košta, O. Radulescu, T. Sturm, and A. Weber
(See online at https://doi.org/10.1016/j.jsc.2019.07.008) - “On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations”. In: Mathematics in Computer Science 14.2 (June 2020), pp. 281–293
E. Braun, W. Seiler, and M. Seiß
(See online at https://doi.org/10.1007/s11786-019-00423-6) - “Singular perturbations and scaling”. In: Discrete and Continuous Dynamical Systems - Series B 25.1 (2020), pp. 1–29
C. Lax and S. Walcher
(See online at https://doi.org/10.3934/dcdsb.2019170) - “Tikhonov–Fenichel Reduction for Parameterized Critical Manifolds with Applications to Chemical Reaction Networks”. In: Journal of Nonlinear Science 30.4 (Aug. 2020), pp. 1355–1380
E. Feliu, N. Kruff, and S. Walcher
(See online at https://doi.org/10.1007/s00332-020-09610-3) - “A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equations”. In: Mathematics in Computer Science 15.2 (June 2021), pp. 333–352
W. M. Seiler, M. Seiß, and T. Sturm
(See online at https://doi.org/10.1007/s11786-020-00485-x) - “A Short Contribution to the Theory of Regular Chains”. In: Mathematics in Computer Science 15.2 (2021), pp. 177–188
F. Boulier, F. Lemaire, M. Moreno Maza, and A. Poteaux
(See online at https://doi.org/10.1007/s11786-020-00477-x) - “Algorithmic Reduction of Biological Networks with Multiple Time Scales”. In: Mathematics in Computer Science 15.3 (Sept. 2021), pp. 499–534
N. Kruff, C. Lüders, O. Radulescu, T. Sturm, and S. Walcher
(See online at https://doi.org/10.1007/s11786-021-00515-2) - “Efficiently and Effectively Recognizing Toricity of Steady State Varieties”. In: Mathematics in Computer Science 15.2 (June 2021), pp. 199–232
D. Grigoriev, A. Iosif, H. Rahkooy, T. Sturm, and A. Weber
(See online at https://doi.org/10.1007/s11786-020-00479-9) - “Is the diatom sex clock a clock?” In: Journal of the Royal Society Interface 18.179 (June 2021), p. 20210146
T. Fuhrmann-Lieker, N. Kubetschek, J. Ziebarth, R. Klassen, and W. Seiler
(See online at https://doi.org/10.1098/rsif.2021.0146) - “No Chaos in Dixon’s System”. In: International journal of bifurcation and chaos in applied sciences and engineering 31.03 (Mar. 2021), p. 2150044
W. Seiler and M. Seiß
(See online at https://doi.org/10.1142/S0218127421500449) - “On Initials and the Fundamental Theorem of Tropical Partial Differential Geometry”. In: Journal of Symbolic Computation (2021)
S. Falkensteiner, C. E. Garay-Lopez, M. Haiech, M. P. Noordman, F. Boulier, and Z. Toghani
(See online at https://doi.org/10.1016/j.jsc.2022.08.005) - “On the quasi-steady-state approximation in an open Michaelis–Menten reaction mechanism”. In: AIMS Mathematics 6.7 (2021), pp. 6781–6814
J. Eilertsen, M. Roussel, S. Schnell, and S. Walcher
(See online at https://doi.org/10.3934/math.2021398) - “Quantitative imaging of transcription in living Drosophila embryos reveals the impact of core promoter motifs on promoter state dynamics”. In: Nature Communications 12.1 (Dec. 2021)
V. L. Pimmett, M. Dejean, C. Fernandez, A. Trullo, E. Bertrand, O. Radulescu, and M. Lagha
(See online at https://doi.org/10.1038/s41467-021-24461-6) - “Singular initial value problems for scalar quasi-linear ordinary differential equations”. In: Journal of Differential Equations 281 (Apr. 2021), pp. 258–288
W. Seiler and M. Seiß
(See online at https://doi.org/10.1016/j.jde.2021.02.010) - “Singularities of algebraic differential equations”. In: Advances in Applied Mathematics 131 (Oct. 2021), p. 102266
M. Lange-Hegermann, D. Robertz, W. Seiler, and M. Seiß
(See online at https://doi.org/10.1016/j.aam.2021.102266) - “ODEbase: A Repository of ODE Systems for Systems Biology”. In: Bioinformatics Advances (2022)
C. Lüders, T. Sturm, and O. Radulescu
(See online at https://doi.org/10.1093/bioadv/vbac027) - “Quasi-Steady-State and Singular Perturbation Reduction for Reaction Networks with Noninteracting Species”. In: SIAM Journal on Applied Dynamical Systems 21.2 (June 2022), pp. 782–816
E. Feliu, C. Lax, S. Walcher, and C. Wiuf
(See online at https://doi.org/10.1137/20M1364503) - “The control of transcriptional memory by stable mitotic bookmarking”. In: Nature Communications 13.1 (Dec. 2022), p. 1176
M. Bellec, J. Dufourt, G. Hunt, H. Lenden-Hasse, A. Trullo, A. Zine El Aabidine, M. Lamarque, M. Gaskill, H. Faure-Gautron, M. Mannervik, M. Harrison, J.-C. Andrau, C. Favard, O. Radulescu, and M. Lagha
(See online at https://doi.org/10.1038/s41467-022-28855-y)
