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.
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 · log N · log log N ). De, Kurur, Saha and Saptharishi (DKSS) presented an asymptotically faster algorithm with a better time bound of N · log N · 2 O(log * N ) . For this paper, a simplified DKSS multiplication was implemented. Assuming a sensible upper limit on the input size, some required constants could be precomputed. This allowed to simplify the algorithm to save some complexity and run-time. Still, run-time is about 30 times larger than SSA, while memory requirements are about 2.3 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.
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 using a recent result by Cardin and Teixeira. The existence of invariant manifolds is subject to hyperbolicity conditions, which we test algorithmically using 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.
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.
Summary
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, corresponding 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 suitable input data derived from established existing biomodels, covering in particular the BioModels database.
Availability and implementation
ODEbase is available free of charge at https://odebase.org.
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