Cahn-Hilliard dynamical models for condensed biomolecular systems

Biomolecular condensates create dynamic subcellular compartments that alter systems-level properties of the networks surrounding them. One model combining soluble and condensed states is the Cahn-Hilliard equation, which specifies a diffuse interface between the two phases. Customized approaches required to solve this equation are largely inaccessible. Using two complementary numerical strategies, we built stable, self-consistent Cahn-Hilliard solvers in Python, MATLAB, and Julia. The algorithms simulated the complete time evolution of condensed droplets as they dissolved or persisted, relating critical droplet size to a coefficient for the diffuse interface in the Cahn-Hilliard equation. We applied this universal relationship to the chromosomal passenger complex, a multi-protein assembly that reportedly condenses on mitotic chromosomes. The fully constrained Cahn-Hilliard simulations yielded dewetting and coarsening behaviors that closely mirrored experiments in different cell types. The Cahn-Hilliard equation tests whether condensate dynamics behave as a phase-separated liquid, and its numerical solutions advance generalized modeling of biomolecular systems.