Nucleation Kinetics determine the Computational Capabilities of Phase-Separation

The cytoplasm, nucleus, and lipid membrane of cells reside near numerous condensation points beyond which liquid-droplets enriched in specific biomolecules de-mix from the surrounding solution. This phase-separation is often driven by the composition of the cellular environment: de-mixing is sometimes a consequence of internal changes in biomolecular concentrations, and other times a response to external perturbations. The space of compositions is high-dimensional, as is the space of condensed phases; it is unknown how cells navigate this space with fidelity and if kinetic processes like nucleation are required to steer this navigation. We develop a sampling method that allows us to calculate nucleation pathways and rates from initially meta-stable configurations into condensed phases. We use this method to probe the computational capabilities of phase-separation. Systems prone to phase-separation map a set of compositions to a phase by de-mixing, and this can be understood as a computation that can distinguish various input compositions by their de-mixing into distinct liquid phases. Input compositions determine both the stability of output phases and nucleation rates into these phases. Some sets of inputs can be distinguished on purely equilibrium thermodynamic grounds, while other tasks --- and more compact solutions --- demand that computation is performed through the nucleation kinetics of phase-separation.