Algorithmic Reduction of Biological Networks with Multiple Time Scales

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.

Qualitative dynamics of chemical reaction networks: an investigation using partial tropical equilibrations

“…under some conditions meaning roughly that the solutions of f (x, ȳ) = 0 are hyperbolic attracting equilibria of the equation dx dt = f (x, ȳ; k), see [3] for the rigorous statement. The first equation of ( 6) defines the quasisteady state variety and imposes constraints on x, y.…”

“…Each partial tropical equilibration solution provides a scaling of the variables. This scaling is used for identification of slow and fast variables and for automatic model reduction with algorithms introduced in [3,1]. The algorithms from [3] work when the quasi-steady state equations of fast variables satisfy hyperbolicity conditions.…”

“…The concept of tropical equilibration comes from algebraic geometry and it is a necessary condition for the existence of real Puiseux series solutions of systems of polynomial equations whose coefficients are powers or Puiseux series of some scaling parameter . The concept is naturally related to the problem of finding scalings of differential equations needed in the mathematical theory of singular perturbations for systems with multiple timescales [3].…”

Parametric Toricity of Steady State Varieties of Reaction Networks

“…3 We consider here a variant, which ignores a discrete event changing ligand concentration at time t = 2500. A nonparametric instance of this variant has been studied in [27] with respect to toricity and in [36] with respect to multiple time scale reduction. There are 8 parameters k and species concentrations x = (x 1 , .…”

ODEbase: A Repository of ODE Systems for Systems Biology

“…Recently, symbolic computation methods are playing an increasing role in systems biology and mathematical biology [1, and the references there]. Problems investigated using such methods include Hopf bifurcations, multi-stationarity, multi-scale model reduction, dynamical invariants, and structural properties of steady state varieties; for details see, e.g., [4,7,3,2,9,6]. Compared to numerical analysis and simulation, symbolic computation provides not only quantitative, but qualitative results about network dynamics, to some extent in parametric settings.…”