Kinetic models of phosphorylation cycles: A systematic approach using the rapid-equilibrium approximation for protein–protein interactions

Activation–inactivation cycles of signalling proteins and transcription factors catalysed by kinases and phosphatases are a core component of cellular signal transduction. We present a systematic kinetic analysis of a phosphorylation cycle that starts from the description of elementary protein–protein interaction and catalytic steps. A rapid-equilibrium approximation for protein interactions is used to reduce the set of parameters. The resulting description consists of a kinetic equation for the phosphorylation of the target and a set of conservation conditions for kinase and phosphatase. Generally no explicit rate laws exist for the two enzymes; linear or Michaelis–Menten rate equations can be obtained in special cases. Key parameters that determine the stimulus–response curve and the response time of the cycle are the concentrations of kinase and phosphatase relative to the target protein and the affinities of the two enzymes for the different phosphorylation states of the target. Characterizing the response curve by the global response coefficient, we obtain a phase diagram that shows the existence of three kinds of behaviours: graded, ultrasensitive, and a previously undescribed biphasic response. Two kinds of competition effect turn out to shape the behaviour: (1) the degree of product inhibition of each enzyme, and (2) the competition between kinase and phosphatase to bind the target protein, as determined by their relative target affinities. The approach outlined here may be useful also for analysing more complex systems, including multiple phosphorylation and kinase cascades.