Multi-Scale Modeling in Morphogenesis: A Critical Analysis of the Cellular Potts Model

Cellular Potts models (CPMs) are used as a modeling framework to elucidate mechanisms of biological development. They allow a spatial resolution below the cellular scale and are applied particularly when problems are studied where multiple spatial and temporal scales are involved. Despite the increasing usage of CPMs in theoretical biology, this model class has received little attention from mathematical theory. To narrow this gap, the CPMs are subjected to a theoretical study here. It is asked to which extent the updating rules establish an appropriate dynamical model of intercellular interactions and what the principal behavior at different time scales characterizes. It is shown that the longtime behavior of a CPM is degenerate in the sense that the cells consecutively die out, independent of the specific interdependence structure that characterizes the model. While CPMs are naturally defined on finite, spatially bounded lattices, possible extensions to spatially unbounded systems are explored to assess to which extent spatio-temporal limit procedures can be applied to describe the emergent behavior at the tissue scale. To elucidate the mechanistic structure of CPMs, the model class is integrated into a general multiscale framework. It is shown that the central role of the surface fluctuations, which subsume several cellular and intercellular factors, entails substantial limitations for a CPM's exploitation both as a mechanistic and as a phenomenological model.