Figure 1. General model of multisite phosphorylation with substrate S having n sites, kinase E and phosphatase F.

The phospho-forms are denoted S_u, where u is a bit string indicating the absence/presence (0/1) of phosphate. Kinase reactions for n=3 with one phosphorylation per reaction (a, distributivity²⁹) and multiple phosphorylations per reaction (b, processivity²⁹). Phosphatases act similarly in removing phosphates. c, Each enzyme (X=E or F) uses a standard biochemical mechanism³⁰ but may form multiple products, with associated parameters ($a_u^X,b_u^X,c_{u,v}^X$) for mass-action kinetics. ATP is assumed to be held constant and synthesis and degradation are ignored. d, The 3×2ⁿ differential equations, where $u\stackrel{X}{\to }v$ signifies that X converts S_u to S_v. The same symbol is used for a chemical species and for its concentration. Note that X(S_u) here indicates the product of X and S_u.