On temporal logic constraint solving for analyzing numerical data time series

Temporal logics and model-checking have proved successful in expressing biological properties of complex biochemical systems, and automatically verify their satisfaction, in both qualitative and quantitative models. In this article, we go beyond model-checking and present a constraint solving algorithm for quantifier-free first-order temporal logic formulae, with constraints over the reals. This algorithm computes the domain of the real valued variables occurring in a formula that makes it true in a model. We illustrate this approach for the automatic generation of a temporal logic specification from biological data time series. We provide a set of biologically relevant patterns of formulae, and apply them to numerical data time series of models of the cell cycle control and MAPK signal transduction. We show in these examples that this approach infers automatically semi-qualitative, semi-quantitative information about concentration thresholds, amplitude of oscillations, stability properties, checkpoints and influences between species.