The geometry of multisite phosphorylation.

Reversible protein phosphorylation on multiple sites is a key regulatory mechanism in most cellular processes. We consider here a kinase-phosphatase-substrate system with two sites, under mass-action kinetics, with no restrictions on the order of phosphorylation or dephosphorylation. We show that the concentrations of the four phosphoforms at steady state satisfy an algebraic formula-an invariant-that is independent of the other chemical species, such as free enzymes or enzyme-substrate complexes, and holds irrespective of the starting conditions and the total amounts of enzymes and substrate. Such invariants allow stringent quantitative predictions to be made without requiring any knowledge of site-specific parameter values. We introduce what we believe are novel methods from algebraic geometry-Gröbner bases, rational curves-to calculate invariants. These methods are particularly significant because they make it possible to treat parameters symbolically without having to specify their numerical values, and thereby allow us to sidestep the parameter problem. We anticipate that this approach will have much wider applications in biological modeling.