arXiv subjects

Christoph Lüders

Publications and source records attributed to Christoph Lüders.

6 recordsLinked to original sources

arXiv2023

A Computational Approach to Polynomial Conservation Laws

For polynomial ODE models, we introduce and discuss the concepts of exact and approximate conservation laws, which are the first integrals of the full and truncated sets of ODEs. For fast-slow systems, truncated ODEs describe the fast dynamics. We define compatibility classes as subsets of the state space, obtained by equating the conservation laws to constants. A set of conservation laws is complete when the corresponding compatibility classes contain a finite number of steady states. Complete sets of conservation laws can be used for model order reduction and for studying the multistationarity of the model. We provide algorithmic methods for computing linear, monomial, and polynomial conservation laws of polynomial ODE models and for testing their completeness. The resulting conservation laws and their completeness are either independent or dependent on the parameters. In the latter case, we provide parametric case distinctions. In particular, we propose a new method to compute polynomial conservation laws by comprehensive Gröbner systems and syzygies. Keywords: First integrals, chemical reaction networks, polynomial conservation laws, syzygies, comprehensive Gröbner systems.

math.DS↗

arXiv2023

Reduction of Chemical Reaction Networks with Approximate Conservation Laws

Model reduction of fast-slow chemical reaction networks based on the quasi-steady state approximation fails when the fast subsystem has first integrals. We call these first integrals approximate conservation laws. In order to define fast subsystems and identify approximate conservation laws, we use ideas from tropical geometry. We prove that any approximate conservation law evolves slower than all the species involved in it and therefore represents a supplementary slow variable in an extended system. By elimination of some variables of the extended system, we obtain networks without approximate conservation laws, which can be reduced by standard singular perturbation methods. The field of applications of approximate conservation laws covers the quasi-equilibrium approximation, well known in biochemistry. We discuss reductions of slow-fast as well as multiple timescale systems. Networks with multiple timescales have hierarchical relaxation. At a given timescale, our multiple timescale reduction method defines three subsystems composed of (i) slaved fast variables satisfying algebraic equations, (ii) slow driving variables satisfying reduced ordinary differential equations, and (iii) quenched much slower variables that are constant. The algebraic equations satisfied by fast variables define chains of nested normally hyberbolic invariant manifolds. In such chains, faster manifolds are of higher dimension and contain the slower manifolds. Our reduction methods are introduced algorithmically for networks with monomial reaction rates and linear, monomial or polynomial approximate conservation laws. We propose symbolic algorithms to reshape and rescale the networks such that geometric singular perturbation theory can be applied to them, test the applicability of the theory, and finally reduce the networks. As a proof of concept, we apply this method to a model of the TGF-b signaling pathway.

math.DS↗

arXiv2022

ODEbase: A Repository of ODE Systems for Systems Biology

Recently, symbolic computation and computer algebra systems have been successfully applied in systems biology, especially in chemical reaction network theory. One advantage of symbolic computation is its potential for qualitative answers to biological questions. Qualitative methods analyze dynamical input systems as formal objects, in contrast to investigating only part of the state space, as is the case with numerical simulation. However, symbolic computation tools and libraries have a different set of requirements for their input data than their numerical counterparts. A common format used in mathematical modeling of biological processes is SBML. We illustrate that the use of SBML data in symbolic computation requires significant pre-processing, incorporating external biological and mathematical expertise. ODEbase provides high quality symbolic computation input data derived from established existing biomodels, covering in particular the BioModels database.

q-bio.MN↗

arXiv2021

Algorithmic Reduction of Biological Networks With Multiple Time Scales

We present a symbolic algorithmic approach that allows to compute invariant manifolds and corresponding reduced systems for differential equations modeling biological networks which comprise chemical reaction networks for cellular biochemistry, and compartmental models for pharmacology, epidemiology and ecology. Multiple time scales of a given network are obtained by scaling, based on tropical geometry. Our reduction is mathematically justified within a singular perturbation setting. The existence of invariant manifolds is subject to hyperbolicity conditions, for which we propose an algorithmic test based on Hurwitz criteria. We finally obtain a sequence of nested invariant manifolds and respective reduced systems on those manifolds. Our theoretical results are generally accompanied by rigorous algorithmic descriptions suitable for direct implementation based on existing off-the-shelf software systems, specifically symbolic computation libraries and Satisfiability Modulo Theories solvers. We present computational examples taken from the well-known BioModels database using our own prototypical implementations.

q-bio.MN↗

arXiv2020

Computing Tropical Prevarieties with Satisfiability Modulo Theories (SMT) Solvers

A novel way to use SMT (Satisfiability Modulo Theories) solvers to compute the tropical prevariety (resp. equilibrium) of a polynomial system is presented. The new method is benchmarked against a naive approach that uses purely polyhedral methods. It turns out that the SMT approach is faster than the polyhedral approach for models that would otherwise take more than one minute to compute, in many cases by a factor of 60 or more, and in the worst case is only slower by a factor of two. Furthermore, the new approach is an anytime algorithm, thus offering a way to compute parts of the solution when the polyhedral approach is infeasible.

cs.LO↗

arXiv2015

Fast Multiplication of Large Integers: Implementation and Analysis of the DKSS Algorithm

The Schönhage-Strassen algorithm (SSA) is the de-facto standard for multiplication of large integers. For $N$-bit numbers it has a time bound of $O(N \cdot \log N \cdot \log \log N)$. De, Kurur, Saha and Saptharishi (DKSS) presented an asymptotically faster algorithm with a better time bound of $N \cdot \log N \cdot 2^{O(\log^* N)}$. In this diploma thesis, results of an implementation of DKSS multiplication are presented: run-time is about 30 times larger than SSA, while memory requirements are about 3.75 times higher than SSA. A possible crossover point is estimated to be out of reach even if we utilized the whole universe for computer memory.

cs.MS↗