Dynamical Mean Field Theory for Confluent Tissues and Continuous Constraint Satisfaction Problems

Confluent tissues are biological complex systems of interacting and self-propelling cells. Experiments show that such systems undergo phase transitions that play an important role in tissue formation and in the spread of metastatic cancer. Furthermore, the transition shares many properties similar to the jamming transition observed in particulate matter. Inspired by these observations, a model has been proposed where the Voronoi description of confluent tissues is mapped to a random Continuous Constraint Satisfaction Problem (CCSP): by solving the Hamiltonian with replica method, the model predicts the same rigidity transition of Vertex/Voronoi models for confluent tissues. In this paper, we re-propose the same model to study the dynamical properties of confluent tissues, under zero-temperature Langevin dynamics and in the mean field limit. With the help of the Dynamical Mean Field Theory (DMFT) description for statistical mechanics, we derive the dynamical equations and we propose an efficient algorithm for their integration. By comparing the results with numerical simulations, we confirm the correctness of the theory and the existence of the rigidity transition observed both experimentally and theoretically. In addition and in the context of optimisation science, we show that Gradient Descent (GD) is blind to Replica Symmetry Breaking (RSB), when it occurs at zero temperature.