Systems Biology strives to hasten our understanding of
the fundamental principles of life by adopting a systems-level
approach for the analysis of cellular function and
behavior. One popular framework for capturing the
chemical kinetics of interacting biochemicals is Hybrid Automata.
Our goal in this thesis is to aid Systems Biology research by
improving the current understanding of hybrid automata, by developing
techniques for symbolic rather than numerical analysis of the
dynamics of biochemical networks modeled as hybrid automata, and by
honing the theory to two classes of problems: kinetic mass action based
simulation in genetic regulatory & signal transduction pathways, and
pseudo-equilibrium simulation in metabolic networks.
We first provide new constructions that prove that the "open" Hierarchical Piecewise Constant Derivative (HPCD) subclass is closer to the decidability and undecidability frontiers than was previously understood. After concluding that the HPCD-like classes are unsuitable for modeling chemical reactions, our quest for semi-decidable subclasses leads us to define the "semi-algebraic" subclass. This is the most expressive hybrid automaton subclass amenable to rigorous symbolic temporal reasoning. We begin with the bounded reachability problem, and then show how the dense-time temporal logic Timed Computation Tree Logic (TCTL) can be model-checked by exploiting techniques from real algebraic geometry, primarily real quantifier elimination. We also prove the undecidability of reachability in the Blum-Shub-Smale Turing Machine formalism. We then develop efficient approximation strategies by extending bisimulation partitioning, rectangular grid-based approximation, polytopal approximation and time discretization. We then develop a uniform algebraic framework for modeling biochemical and metabolic networks, also extending flux balance analysis. We present some preliminary results using a prototypical tool Tolque. It is a symbolic algebraic dense time model-checker for semi-algebraic hybrid automata, which uses Qepcad for quantifier elimination.
The "Algorithmic Algebraic Model Checking" techniques developed in this thesis present a theoretically-grounded mathematically-sound platform for powerful symbolic temporal reasoning over biochemical networks and other semi-algebraic hybrid automata. It is our hope that by building upon this thesis, along with the development of computationally efficient parallelizable quantifier elimination algorithms and the integration of different computer algebra tools, scientific software systems will emerge that fundamentally transform the way biochemical networks (and other hybrid automata) are analyzed.