## Section: Research Program

### Logical Paradigm for Systems Biology

Our group was among the first ones in 2002 to apply **model-checking** methods to systems biology
in order to reason on large molecular interaction networks, such as Kohn's map of the mammalian cell cycle (800 reactions over 500 molecules) (N. Chabrier-Rivier, M. Chiaverini, V. Danos, F. Fages, V. Schächter. Modeling and querying biochemical interaction networks. Theoretical Computer Science, 325(1):25–44, 2004.).
The logical paradigm for systems biology that we have subsequently developed for quantitative models can be summarized by the following identifications :

biological model = transition system $K$

dynamical behavior specification = temporal logic formula $\phi $

model validation = model-checking $\phantom{\rule{4pt}{0ex}}K,\phantom{\rule{4pt}{0ex}}s\phantom{\rule{4pt}{0ex}}\vDash ?\phantom{\rule{4pt}{0ex}}\phi $

model reduction = sub-model-checking, ${K}^{\text{'}}\subset K$ s.t. ${K}^{\text{'}}?,\phantom{\rule{4pt}{0ex}}s\phantom{\rule{4pt}{0ex}}\vDash \phantom{\rule{4pt}{0ex}}\phi $

model prediction = formula enumeration, $\phi $ s.t. $K,\phantom{\rule{4pt}{0ex}}s\phantom{\rule{4pt}{0ex}}\vDash \phantom{\rule{4pt}{0ex}}\phi ?$

static experiment design = symbolic model-checking, state $s$ s.t. $K,\phantom{\rule{4pt}{0ex}}s?\phantom{\rule{4pt}{0ex}}\vDash \phantom{\rule{4pt}{0ex}}\phi $

model synthesis = constraint solving $\phantom{\rule{4pt}{0ex}}K?,\phantom{\rule{4pt}{0ex}}s\phantom{\rule{4pt}{0ex}}\vDash \phantom{\rule{4pt}{0ex}}\phi $

dynamic experiment design = constraint solving $\phantom{\rule{4pt}{0ex}}K?,\phantom{\rule{4pt}{0ex}}s?\phantom{\rule{4pt}{0ex}}\vDash \phantom{\rule{4pt}{0ex}}\phi $

In particular, the definition of a continuous satisfaction degree for **first-order temporal logic** formulae with constraints over the reals,
was the key to generalize this approach to quantitative models,
opening up the field of model-checking to model optimization (On a continuous degree of satisfaction of temporal logic formulae with applications to systems biology
A. Rizk, G. Batt, F. Fages, S. Soliman
International Conference on Computational Methods in Systems Biology, 251-268) This line of research continues with the development of temporal logic patterns with efficient constraint solvers
and their generalization to handle stochastic effects.