.Model reduction of fast-slow chemical reaction networks based on the quasi-steady state approximation fails when fast subsystem has first integrals. We call these integrals approximate conservation laws. In order to define subsystems and identify laws, we use ideas from tropical geometry. prove that any law evolves more slowly than all species involved in it therefore represents a supplementary slow variable an extended system. By elimination some variables system, obtain without which can...

Abstract Summary Recently, symbolic computation and computer algebra systems have been successfully applied in biology, especially chemical reaction network theory. One advantage of is its potential for qualitative answers to biological questions. Qualitative methods analyze dynamical input as formal objects, contrast investigating only part the state space, case with numerical simulation. However, corresponding tools libraries a different set requirements their data than counterparts. A...

The Schönhage-Strassen algorithm (SSA) is the de-facto standard for multiplication of large integers. For N-bit numbers it has a time bound O(N log N N). De, Kurur, Saha and Saptharishi (DKSS) presented an asymptotically faster with better N• 2O(log* this paper, simplified DKSS was implemented. Assuming sensible upper limit on input size, some required constants could be precomputed. This allowed to simplify save complexity run-time. Still, run-time about 30 times larger than SSA, while...

Abstract 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 cellular biochemistry, compartmental models pharmacology, epidemiology ecology. Multiple time scales of given network are obtained by scaling, based on tropical geometry. Our reduction is mathematically justified within singular perturbation setting. The existence subject...

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

.For polynomial ODE models, we introduce and discuss the concepts of exact approximate conservation laws, which are first integrals full truncated sets ODEs. For fast-slow systems, ODEs describe fast dynamics. We define compatibility classes as subsets state space, obtained by equating laws to constants. A set is complete when corresponding contain a finite number steady states. Complete can be used for model order reduction studying multistationarity model. provide algorithmic methods...

Abstract The concept of attractor dynamic biochemical networks has been used to explain cell types and alterations in health disease. We have recently proposed an extension the notion take into account metastable regimes, defined as long lived dynamical states network. These regimes correspond slow dynamics on low dimensional invariant manifolds networks. Methods based tropical geometry allow compute represent them polyhedra space logarithms species concentrations. are looking for sensitive...

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

Recently, symbolic computation and computer algebra systems have been successfully applied in biology, especially chemical reaction network theory. One advantage of is its potential for qualitative answers to biological questions. Qualitative methods analyze dynamical input as formal objects, contrast investigating only part the state space, case with numerical simulation. However, tools libraries a different set requirements their data than counterparts. A common format used mathematical...

Model reduction of fast-slow chemical reaction networks based on the quasi-steady state approximation fails when fast subsystem has first integrals. We call these integrals approximate conservation laws. In order to define subsystems and identify laws, we use ideas from tropical geometry. prove that any law evolves slower than all species involved in it therefore represents a supplementary slow variable an extended system. By elimination some variables system, obtain without which can be...

For polynomial ODE models, we introduce and discuss the concepts of exact approximate conservation laws, which are first integrals full truncated sets ODEs. fast-slow systems, ODEs describe fast dynamics. We define compatibility classes as subsets state space, obtained by equating laws to constants. A set is complete when corresponding contain a finite number steady states. Complete can be used for model order reduction studying multistationarity model. provide algorithmic methods computing...