Algorithmic algebraic model checking IV: Characterization of metabolic networks

A series of papers, all under the title of Algorithmic Algebraic Model Checking (AAMC), has sought to combine techniques from algorithmic algebra, model checking and dynamical systems to examine how a biochemical hybrid dynamical system can be made amenable to temporal analysis, even when the initial conditions and unknown parameters may only be treated as symbolic variables. This paper examines how to specialize this framework to metabolic control analysis (MCA) involving many reactions operating at many dissimilar time-scales. In the earlier AAMC papers, it has been shown that the dynamics of various biochemical semi-algebraic hybrid automata could be unraveled using powerful techniques from computational real algebraic geometry. More specifically, the resulting algebraic model checking techniques were found to be suitable for biochemical networks modeled using general mass action (GMA) based ODEs. This paper scrutinizes how the special properties of metabolic networks-a subclass of the biochemical networks previously handled-can be exploited to gain improvement in computational efficiency. The paper introduces a general framework for performing symbolic temporal reasoning over metabolic network hybrid automata that handles both GMA-based equilibrium estimation and flux balance analysis (FBA). While algebraic polynomial equations over Q[x(1),...,x(n)] can be symbolically solved using Grobner bases or Wu-Ritt characteristic sets, the FBA-based estimation can be performed symbolically by rephrasing the algebraic optimization problem as a quantifier elimination problem. Effectively, an approximate hybrid automaton that simulates the metabolic network is derived, and is thus amenable to manipulation by the algebraic model checking techniques previously described in the AAMC papers.