Studying emergent behaviours in morphogenesis using signal spatio-temporal logic

Pattern formation is an important spatio-temporal emergent behaviour in biology. Mathematical models of pattern formation in the stochastic setting are extremely challenging to execute and analyse. Here we propose a formal analysis of the emergent behaviour of stochastic reaction diffusion systems in terms of Signal Spatio-Temporal Logic, a recently proposed logic for reasoning on spatio-temporal systems. We present a formal analysis of the spatio-temporal dynamics of the Bicoid morphogen in Drosophila melanogaster, one of the most important proteins in the formation of the horizontal segmentation in the development of the fly embryo. We use a recently proposed framework for statistical model checking of stochastic systems with uncertainty on parameters to characterise the parametric dependence and robustness of the French Flag pattern, highlighting non-trivial correlations between the parameter values and the emergence of the patterning.