Mathematical Modeling

One of the signature observations and enduring mysteries of PCP signaling is how clones of some, but not all, PCP signaling components perturb the polarity of neighboring wild-type cells on a characteristic side of the clone (referred to as domineering non-autonomy). This phenomenon is perhaps most important because it provides a rich source of clues that could aid in understanding the function of the underlying PCP signaling mechanism. Models to explain this phenomenon have often invoked diffusible factors, referred to as factor X or Z because they have not yet been identified, or ill-defined "vectors" of directional information. We have proposed instead, that the local competition between proximal and distal components, combined with the global directional cue, is sufficient to explain domineering non-autonomy. However, other researchers have looked at the same data, and instead concluded that our proposed signaling network cannot explain one or more experimental observations. This collective inability to deduce, given a particular signaling network hypothesis, definitive links between molecular genetic interventions and tissue patterning effects has severely hampered progress in understanding PCP signaling. Indeed, it has become impossible to rely on intuition to design experiments that most observers would agree serve to test and understand the relationship between the underlying signaling network and the resulting tissue level patterns.

To rigorously examine the sufficiency of our proposed PCP signaling network to explain the constellation of characteristic PCP phenotypes, we are collaborating with Claire Tomlin’s group in the Department of Aeronautics and Astronautics at Stanford to mathematically model our proposed network. Claire, a control theorist, and some of her students have provided expertise in mathematical modeling. Together, we have developed a reaction diffusion model of PCP signaling that is based on our biological model, and shown that our proposed mechanism can recapitulate all of the most characteristic PCP phenotypes. This observation reinforces our belief that our model of the Fz competition machine is a reasonably correct picture of the underlying biology. Furthermore, by studying the properties of this model, and by verifying predictions of the model with biological experiments, we gained important insight into the features controlling autonomy. The model, and experimental validation of model predictions, reveal why some components or alleles display domineering non-autonomy, while others are nearly autonomous. We are using this model to address other questions in PCP signaling.

In addition, we are developing more powerful modeling techniques that will help address other questions in PCP signaling. Of paramount interest is the development of hybrid methods that will facilitate modeling interactions between the global and core PCP signaling systems. These methods will be broadly applicable to problems in developmental patterning.