Algorithmic Reduction of Biological Networks With Multiple Time Scales

TL;DR

This paper introduces a symbolic algorithmic method for reducing biological network models with multiple time scales, using tropical geometry and invariant manifold theory, supported by computational examples.

Contribution

It provides a novel, mathematically justified reduction technique for biological networks with multiple time scales, including an algorithmic test for hyperbolicity and nested invariant manifolds.

Findings

Effective reduction of biological networks demonstrated on BioModels database

Algorithmic test for hyperbolicity based on Hurwitz criteria

Implementation compatible with existing symbolic computation tools

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 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.