Multisite phosphorylation is an important mechanism for fine‐tuned regulation of protein function. Mathematical models developed over recent years have contributed to elucidation of the functional consequences of a variety of molecular mechanisms involved in processing of the phosphorylation sites. Here we review the results of such models, together with salient experimental findings on multisite protein phosphorylation. We discuss how molecular mechanisms that can be distinguished with respect to the order and processivity of phosphorylation, as well as other factors, regulate changes in the sensitivity and kinetics of the response, the synchronization of molecular events, signalling specificity, and other functional implications.

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.

No abstract

Experimental studies have demonstrated that Ca(2+)-regulated proteins are sensitive to the frequency of Ca(2+) oscillations, and several mathematical models for specific proteins have provided insight into the mechanisms involved. Because of the large number of Ca(2+)-regulated proteins in signal transduction, metabolism and gene expression, it is desirable to establish in general terms which molecular properties shape the response to oscillatory Ca(2+) signals. Here we address this question by analyzing in detail a model of a prototypical Ca(2+)-decoding module, consisting of a target protein whose activity is controlled by a Ca(2+)-activated kinase and the counteracting phosphatase. We show that this module can decode the frequency of Ca(2+) oscillations, at constant average Ca(2+) signal, provided that the Ca(2+) spikes are narrow and the oscillation frequency is sufficiently low--of the order of the phosphatase rate constant or below. Moreover, Ca(2+) oscillations activate the target more efficiently than a constant signal when Ca(2+) is bound cooperatively and with low affinity. Thus, the rate constants and the Ca(2+) affinities of the target-modifying enzymes can be tuned in such a way that the module responds optimally to Ca(2+) spikes of a certain amplitude and frequency. Frequency sensitivity is further enhanced when the limited duration of the external stimulus driving Ca(2+) signaling is accounted for. Thus, our study identifies molecular parameters that may be involved in establishing the specificity of cellular responses downstream of Ca(2+) oscillations.