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Phase-transition-mediated Switching in a Multisite Phosphorylation Chain

We construct and analyze a stochastic model of a population of N protein molecules, each of which can be phosphorylated and dephosphorylated at J sites by the same kinase and phosphatase respectively. In addition, fully dephosphorylated (phosphorylated) proteins feed back to act as catalysts for all intermediate dephosphorylation (phosphorylation) reactions. The population is treated as a vector of occupancies of J + 1 sites in a one-dimensional lattice representing the phosphorylation states of each protein molecule. This model exhibits a continuous phase transition at any J ≥ 2, between a symmetric and a symmetry-breaking state. In addition, we find that the universality class of the phase transition at finite J is different than that at J → ∞.