Tumor cells switch phenotypes — epithelial to mesenchymal, drug-sensitive to drug-resistant — through stochastic transitions that depend on their neighbors. Each cell type has different adhesive properties: some stick to their own kind, others stick to the opposite kind, and the local composition of the tumor microenvironment influences the switching rate. The interplay between adhesion (spatial organization) and switching (phenotypic dynamics) determines whether the tumor maintains a mixed population or segregates into distinct phenotypic compartments.
Chae and Choi (arXiv:2603.12802) formalize this as a nonlocal adhesion model for two interacting cell types with random phenotypic switching. The microscopic description — interacting particles that diffuse, attract or repel each other, and randomly switch type — has a mean-field limit described by a McKean-Vlasov equation. The question: does the mean-field description capture the particle behavior uniformly in time?
In the weak-interaction regime, yes. The propagation of chaos — the principle that particles become approximately independent when there are many of them — holds uniformly in time, not just over finite intervals. The McKean-Vlasov equation contracts exponentially in the Wasserstein distance, meaning the macroscopic description is not only accurate but self-correcting: perturbations die out.
In the strong-interaction regime, the uniform behavior breaks down through bifurcation. The stationary distribution loses stability, and the system transitions to a state with multiple stable equilibria — phenotypic segregation. The tumor can now maintain distinct compartments rather than a well-mixed population. The bifurcation is the mathematical signature of the biological observation: above a critical interaction strength, phenotypic heterogeneity becomes spatially organized rather than randomly distributed.