  We introduce a framework for analyzing ordinary differential equation (ODE)
models of biological networks using statistical model checking (SMC). A key
aspect of our work is the modeling of single-cell variability by assigning a
probability distribution to intervals of initial concentration values and
kinetic rate constants. We propagate this distribution through the system
dynamics to obtain a distribution over the set of trajectories of the ODEs.
This in turn opens the door for performing statistical analysis of the ODE
system's behavior. To illustrate this we first encode quantitative data and
qualitative trends as bounded linear time temporal logic (BLTL) formulas. Based
on this we construct a parameter estimation method using an SMC-driven
evaluation procedure applied to the stochastic version of the behavior of the
ODE system. We then describe how this SMC framework can be generalized to
hybrid automata by exploiting the given distribution over the initial states
and the--much more sophisticated--system dynamics to associate a Markov chain
with the hybrid automaton. We then establish a strong relationship between the
behaviors of the hybrid automaton and its associated Markov chain.
Consequently, we sample trajectories from the hybrid automaton in a way that
mimics the sampling of the trajectories of the Markov chain. This enables us to
verify approximately that the Markov chain meets a BLTL specification with high
probability. We have applied these methods to ODE based models of Toll-like
receptor signaling and the crosstalk between autophagy and apoptosis, as well
as to systems exhibiting hybrid dynamics including the circadian clock pathway
and cardiac cell physiology. We present an overview of these applications and
summarize the main empirical results. These case studies demonstrate that the
our methods can be applied in a variety of practical settings.
