Cellular Potts Model — Cell Sorting & Engulfment

Differential adhesion hypothesis (Steinberg): cells sort by adhesion energy, like immiscible fluids — the cell type with stronger self-adhesion ends up in the center

J₁₁ (type-A self-adhesion) 2.0

J₂₂ (type-B self-adhesion) 8.0

J₁₂ (cross-adhesion) 12.0

Temperature T 4.0

CPM Hamiltonian: H = Σ J(σᵢ,σⱼ) over neighbor pairs. Contact energy J(σ,σ')=0 if same cell, J_τ(σ)τ(σ') between cell types. Metropolis acceptance: ΔH<0 always accepted; ΔH>0 accepted with probability e^(−ΔH/T). Cells minimize interfacial energy → Steinberg's differential adhesion hypothesis: sorting is a thermodynamic minimum. If J₁₂ > (J₁₁+J₂₂)/2: types separate (like oil/water). Engulfment (one type surrounds the other) occurs when J₂₂ ≫ J₁₁ — type-B maximizes self-contact.