In distributive phosphorylation catalytic constants enable non-trivial dynamics

TL;DR

This paper demonstrates that catalytic constants in distributive double phosphorylation, whether sequential or cyclic, enable complex dynamics such as multistationarity and oscillations, with implications for intracellular signaling.

Contribution

It shows that the same inequality involving catalytic constants governs both multistationarity and oscillations in different phosphorylation motifs, revealing a unified dynamic mechanism.

Findings

Cyclic distributive phosphorylation can exhibit oscillations via Hopf bifurcations.

The same catalytic constant inequality underpins multistationarity and oscillations.

A procedure is provided to generate parameters for sustained oscillations.

Abstract

Ordered distributive double phosphorylation is a recurrent motif in intracellular signaling and control. It is either sequential (where the site phosphorylated last is dephosphorylated first) or cyclic (where the site phosphorylated first is dephosphorylated first). Sequential distributive double phosphorylation has been extensively studied and an inequality involving only the catalytic constants of kinase and phosphatase is known to be sufficient for multistationarity. As multistationarity is necessary for bistability it has been argued that these constants enable bistability. Here we show for cyclic distributive double phosphorylation that if its catalytic constants satisfy the very same inequality, then Hopf bifurcations and hence sustained oscillations can occur. Hence we argue that in distributive double phosphorylation (sequential or distributive) the catalytic constants enable non-trivial dynamics. In fact, if the rate constant values in a network of cyclic distributive double phosphorylation are such that Hopf bifurcations and sustained oscillations can occur, then a network of sequential distributive double phosphorylation with the same rate constant values will show multistationarity -- albeit for different values of the total concentrations. For cyclic distributive double phosphorylation we further describe a procedure to generate rate constant values where Hopf bifurcations and hence sustained oscillations can occur. This may, for example, allow for an efficient sampling of oscillatory regions in parameter space. Our analysis is greatly simplified by the fact that it is possible to reduce the network of cyclic distributive double phosphorylation to what we call a network with a single extreme ray. We summarize key properties of these networks.