Versatile regulation of multisite protein phosphorylation by the order of phosphate processing and protein-protein interactions

Multisite protein phosphorylation is a common regulatory mechanism in cell signaling, and dramatically increases the possibilities for protein-protein interactions, conformational regulation, and phosphorylation pathways. However, there is at present no comprehensive picture of how these factors shape the response of a protein's phosphorylation state to changes in kinase and phosphatase activities. Here we provide a mathematical theory for the regulation of multisite protein phosphorylation based on the mechanistic description of elementary binding and catalytic steps. Explicit solutions for the steady-state response curves and characteristic (de)phosphorylation times have been obtained in special cases. The order of phosphate processing and the characteristics of protein-protein interactions turn out to be of overriding importance for both sensitivity and speed of response. Random phosphate processing gives rise to shallow response curves, favoring intermediate phosphorylation states of the target, and rapid kinetics. Sequential processing is characterized by steeper response curves and slower kinetics. We show systematically how qualitative differences in target phosphorylation - including graded, switch-like and bistable responses - are determined by the relative concentrations of enzyme and target as well as the enzyme-target affinities. In addition to collective effects of several phosphorylation sites, our analysis predicts that distinct phosphorylation patterns can be finely tuned by a single kinase. Taken together, this study suggests a versatile regulation of protein activation by the combined effect of structural, kinetic and thermodynamic aspects of multisite phosphorylation.